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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 620961, 15442]*) (*NotebookOutlinePosition[ 621693, 15467]*) (* CellTagsIndexPosition[ 621649, 15463]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Half-String Coordinate Modes Derivation", "Subtitle"], "\nBen Sauerwine\n\nVersion 6: June 22, 2004\nThe newest version of this \ derivation can be downloaded from http://www.snakebyte.biz/string/day1.html\n\ \n", StyleBox["References Used", "Subsection"], "\n\nHalf-String Oscillator Approach to String Theory\nJ. Bordes, Chan \ Hong-Mo, Lukas Nellen, Tsou Sheung Tsun\n\nEric W. Weisstein. \"Fourier \ Series.\" \nFrom ", StyleBox["MathWorld", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], "--A Wolfram Web Resource. ", StyleBox["http://mathworld.wolfram.com/FourierSeries.html", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " \n\nOperator Formulation of Interacting String Field Theory (I)\nDavid J. \ Gross, Antal Jevicki\n\n", StyleBox["Definitions and Boundaries", "Subsection"], "\n\nGiven a String defined from 0 to \[Pi] by the formula\n\n[1a]\nx(\ \[Sigma]) =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x\_0\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\)]], "cos(n \[Sigma])\n[1b]\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " = ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "\[ImaginaryI] (", Cell[BoxData[ \(TraditionalForm\`a\_0\)]], "- ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(a\_0\), "TraditionalForm"], "+"], TraditionalForm]]], ")\n", Cell[BoxData[ \(TraditionalForm\`p\_0\)]], " = (", Cell[BoxData[ \(TraditionalForm\`a\_\(\(0\)\(\ \)\) + \ \(a\_0\^+\)\)]], ")\n[1c]\n", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], "= ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ \(TraditionalForm\`\[ImaginaryI]\ \@\(2\/n\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`a\_n\)]], "- ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(a\_n\), "TraditionalForm"], "+"], TraditionalForm]]], ")\n", Cell[BoxData[ \(TraditionalForm\`p\_n\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\@\(n\/2\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`a\_\(\(n\)\(\ \)\) + \ \(a\_n\^+\)\)]], ")\n\n\nI wish to discuss it in terms of its left and right side, which I \ will call ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], " and", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Chi]\_R\)\)\)]], ", respectively. The left and right sides will be discussed in terms of \ displacement from a central reference point, and so I choose the central \ reference point to be ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], " define\n\n[2a]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], " = x(\[Sigma]) - x(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ")\n[2b]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_R\)(\[Sigma])\)]], " = x(\[Pi] - \[Sigma]) - x(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ")\n\nOver the interval \[Sigma] \[Element] [0, ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], "]\n\nNote that in these definitions, the meaning of \[Sigma] is distance \ inwards from the outer limit of the string and that I choose to include the \ border point, ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ", in both substrings.\n\nBy examination, I see from [1] that x'(0) = x'(\ \[Pi]) = 0. Therefore, from definitions [2a] and [2b], ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "'(0) = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "'(0) = 0. Also, since the two half-strings share the common midpoint, ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ") = 0 (this is obvious from the definitions).\n\n", StyleBox["Fourier Series\n\n", "Subsection"], "Keeping in mind that the goal is to find an expression for ", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_R\)(\[Sigma])\)]], " in terms of Fourier series and then connecting these expressions to that \ of the original x(\[Sigma]), I want solutions like\n\n[3a]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], "= ", Cell[BoxData[ \(TraditionalForm\`A\_\(L\ 0\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] A\_\(L\ n\)\)]], "Cos(n \[Sigma]) + ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] B\_\(L\ n\)\)]], "Sin(n \[Sigma])\n[3b]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_R\)(\[Sigma])\)]], "= ", Cell[BoxData[ \(TraditionalForm\`A\_\(R\ 0\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] A\_\(R\ n\)\)]], "Cos(n \[Sigma]) + ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] B\_\(R\ n\)\)]], "Sin(n \[Sigma])\n\nHowever, noting the boundary conditions, I see that \ x'(0) = x'(\[Pi]) = 0 and so the B coefficients are all zero. I now have\n\n\ [4a]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], "= ", Cell[BoxData[ \(TraditionalForm\`A\_\(L\ 0\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] A\_\(L\ n\)\)]], "Cos(n \[Sigma]) \n[4b]\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "(\[Sigma]) = ", Cell[BoxData[ \(TraditionalForm\`A\_\(R\ 0\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] A\_\(R\ n\)\)]], "Cos(n \[Sigma]) \n\nFurther, the periodicity of the Cosine function plus \ the constraint that ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ") = 0 implies that these half-wave functions must be odd about ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\/2\) . \ \ Therefore\)\(,\)\)\)]], "the even indices A are all zero and these half-waves can be rewritten\n\n\ [5a]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( 1\/2\) A\_\(L\ n\)\)]], "(1 - ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^n\)]], ")Cos(n \[Sigma]) \n[5b]\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "(\[Sigma]) = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( 1\/2\) A\_\(R\ n\)\)]], "(1 - ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^n\)]], ")Cos(n \[Sigma]) \n\nWith the new terms being added to eliminate even \ values of n without having to redefine the indices on A or modify the \ Cosines.\nBy the standard procedure for finding the coefficients of a Fourier \ series, then, (Let A subsequently be denoted by \[Chi])\n\n[6a]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`1\/\[Pi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-\[Pi]\)\%\[Pi]\(\( \[Chi]\_L\)(\ \[Sigma])\)\ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], "\nBut letting even-indices equal zero per [5]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-\[Pi]\)\%\[Pi]\(\( \[Chi]\_L\)(\ \[Sigma])\)\ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], "\nHowever, this requires ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\_L\)(\[Sigma])\)\(\ \)\)\)]], "to be extended. I will choose an odd extension about ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], " and an even extension about 0, so that\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(\[Sigma]) = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(\[Pi] - \[Sigma]) and ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(-\[Sigma]) = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], "(\[Sigma])\nAnd now\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-\[Pi]\)\%0\(\( \[Chi]\_L\)(\[Sigma])\ \)\ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Pi]\(\( \[Chi]\_L\)(\[Sigma])\)\ \ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], "-", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(-\[Pi]\)\(\(\[Chi]\_L\)(\[Sigma])\)\ \ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Pi]\(\( \[Chi]\_L\)(\[Sigma])\)\ \ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Pi]\(\( \[Chi]\_L\)(\(-\[Sigma]\))\ \)\ \(Cos(\(-n\)\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Pi]\(\( \[Chi]\_L\)(\[Sigma])\)\ \ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\(2 \[Pi]\)\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], "2", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Pi]\(\( \[Chi]\_L\)(\[Sigma])\)\ \ \(Cos(n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Sigma])\ \)\ \(Cos( n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\ + \ \[Integral]\_\(\ \[Pi]\/2\)\%\[Pi]\(\( \[Chi]\_L\)(\[Sigma])\)\ \(Cos( n\ \[Sigma])\)\ \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Sigma])\ \)\ \(Cos( n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\ - \ \[Integral]\_0\%\ \(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Pi]\ - \[Sigma]\ )\)\ \(Cos( n\ \((\[Pi]\ - \[Sigma])\))\)\ \ \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(1\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", "("}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Sigma])\ \)\ \(Cos( n\ \[Sigma])\)\ \[DifferentialD]\[Sigma]\ - \ \[Integral]\_0\%\ \(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Sigma]\ )\)\ \(Cos( n\ \((\[Pi]\ - \[Sigma])\))\)\ \ \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\(\(\[Chi]\_L\)(\[Sigma])\ \)\ \(Cos(n\ \[Sigma])\)\ \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)\)]], "\n[6b]\nAnd following similar arguments,\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\(\(\[Chi]\_R\)(\[Sigma])\ \)\ \(Cos(n\ \[Sigma])\)\ \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)\)]], "\n\nSubstituting [2] into [6], \n\n[7a]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ RowBox[{"(", RowBox[{\(x(\[Sigma])\), "-", RowBox[{"x", "(", FormBox[\(\[Pi]\/2\), "TraditionalForm"], ")"}]}], ")"}], " ", \(Cos(n\ \[Sigma])\), " ", \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}], TraditionalForm]]], "\n[7b]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ RowBox[{"(", RowBox[{\(x(\[Pi] - \[Sigma])\), "-", RowBox[{"x", "(", FormBox[\(\[Pi]\/2\), "TraditionalForm"], ")"}]}], ")"}], " ", \(Cos(n\ \[Sigma])\), " ", \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}], TraditionalForm]]], "\n\nAnd [1a] into [7],\n\n[8a]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ RowBox[{"(", " ", RowBox[{\(x\_0\), "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity] x\_m\), "TraditionalForm"], \(Cos(m\ \[Sigma])\)}], "-", RowBox[{"(", " ", RowBox[{\(x\_0\), "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity] x\_m\), "TraditionalForm"], \(Cos(m\ \[Pi]\/2)\)}]}], ")"}]}], " ", ")"}], \(Cos(n\ \[Sigma])\), " ", \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}], TraditionalForm]]], "\nRemember: n is odd\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"2", " ", FormBox["", "TraditionalForm"]}], "\[Pi]"], RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " ", \(\[Sum]\+\(m = 1\)\%\[Infinity]\)}], TraditionalForm]]], " (", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(Cos(m\ \[Sigma])\), \(Cos( n\ \[Sigma])\)}]}], "-", RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(Cos(m\ \[Pi]\/2)\), \(Cos( n\ \[Sigma])\), \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}], TraditionalForm]]], ")\n\n(the ", Cell[BoxData[ FormBox[ FormBox[\(\@2\), "TraditionalForm"], TraditionalForm]]], " was incorporated into the definition of \[Chi])\n\nNow, looking at the \ integral, \n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(Cos(m\ \[Sigma])\), \(Cos( n\ \[Sigma])\)}]}], TraditionalForm]]], " \[DifferentialD]\[Sigma] \n\nis ", Cell[BoxData[ \(\(n\ \[Pi] + Sin \((n\ \[Pi])\)\)\/\(4\ n\)\)]], "if m = n, 0 if m is odd and m \[NotEqual] n, but is in general\n\n", Cell[BoxData[ \(\(m\ Cos \((\(n\ \[Pi]\)\/2)\)\ Sin \((\(m\ \[Pi]\)\/2)\) - n\ Cos \ \((\(m\ \[Pi]\)\/2)\)\ Sin \((\(n\ \[Pi]\)\/2)\)\)\/\(m\^2 - n\^2\)\)]], "\nAnd the integral \n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\)]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FormBox[\(x\_m\), "TraditionalForm"]}], \(Cos(m\ \[Pi]\/2)\), \(Cos( n\ \[Sigma])\), \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}], TraditionalForm]]], "\nis zero if m is odd and ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\(\[Pi]\/2\)\)]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], "(", \(-1\), ")"}], \(m\/2\)]}], " ", \(Cos(n\ \[Sigma])\), \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}], TraditionalForm]]], " if m is even, or\n", Cell[BoxData[ \(\(-\(\(\((1\ + \ \((\(-1\))\)\^m)\) \((\(-1\))\)\^\(m\/2\)\ Sin \ \((\(n\ \[Pi]\)\/2)\)\ x\_m\)\/\(2 n\)\)\)\)]], "\nSo now \n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], \((\ \ \ \[Sum]\+\(m = 1\)\%\[Infinity] x\_m\)}], TraditionalForm]]], Cell[BoxData[ \(\(m\ Cos \((\(n\ \[Pi]\)\/2)\)\ Sin \((\(m\ \[Pi]\)\/2)\) - n\ Cos \ \((\(m\ \[Pi]\)\/2)\)\ Sin \((\(n\ \[Pi]\)\/2)\)\)\/\(m\^2 - n\^2\)\)]], " ", Cell[BoxData[ \(\(-\(\[Sum]\+\(m = 1\)\%\[Infinity]\(\((1\ + \ \((\(-1\))\)\^m)\) \ \((\(-1\))\)\^\(m\/2\)\ Sin \((\(n\ \[Pi]\)\/2)\)\ x\_m\)\/\(2 n\)\)\)\)]], ")\n\n\nSimplifying the sums individually,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity] \ x\_m\)\)\)]], Cell[BoxData[ \(\(m\ Cos \((\(n\ \[Pi]\)\/2)\)\ Sin \((\(m\ \[Pi]\)\/2)\) - n\ Cos \ \((\(m\ \[Pi]\)\/2)\)\ Sin \((\(n\ \[Pi]\)\/2)\)\)\/\(m\^2 - n\^2\)\)]], "\n\nBecomes\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_m\)]], Cell[BoxData[ \(\(\(-n\)\ Cos \((\(m\ \[Pi]\)\/2)\)\ \((\(-1\))\)\^\(\(n - 1\)\/2\)\)\ \/\(m\^2 - n\^2\)\)]], "\n\nSince n is odd, except when m = n, when it becomes ", Cell[BoxData[ \(\(\((n\ \[Pi] + Sin \((n\ \[Pi])\))\)\ x\_n\)\/\(4\ n\)\)]], "\n\nFor ", Cell[BoxData[ \(\(\(\ \)\(\[Pi]\ x\_n\)\)\/\(\(4\)\(\ \)\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(\[Sum]\+\(m = 1, \ m\ \[NotEqual] \ n\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_m\)]], Cell[BoxData[ \(\(\(-n\)\ \((1\ + \ \((\(-1\))\)\^m)\)\ \((\(-1\))\)\^\(\(\(m\)\(\ \ \)\)\/2\)\ \((\(-1\))\)\^\(\(n - 1\)\/2\)\)\/\(2 \((m\^2 - n\^2)\)\)\)]], "\n\nOr ", Cell[BoxData[ \(\(\(\ \)\(\[Pi]\ x\_n\)\)\/\(\(4\)\(\ \)\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\(\(-n\)\ \ \((\(-1\))\)\^m\ \((\(-1\))\)\^\(\(n - 1\)\/2\)\)\/\(4 m\ \^2 - n\^2\)\)]], "since odd m yield 0 except when n = m\n\nAnd simplifying the other sum, \n\ \n", Cell[BoxData[ \(\[Sum]\+\(m = 1\)\%\[Infinity]\(\((1\ + \ \((\(-1\))\)\^m)\) \((\(-1\ \))\)\^\(m\/2\)\ Sin \((\(n\ \[Pi]\)\/2)\)\ x\_m\)\/\(2 n\)\)]], "\nyields\n", Cell[BoxData[ \(\[Sum]\+\(m = 1\)\%\[Infinity]\(\((1\ + \ \((\(-1\))\)\^m)\) \((\(-1\ \))\)\^\(m\/2\)\ \((\(-1\))\)\^\(\^\(\(n\ - \ 1\)\/2\)\)\ x\_m\)\/\(2 \ n\)\)]], "\nor\n", Cell[BoxData[ \(\[Sum]\+\(m = 1\)\%\[Infinity]\(\((\(-1\))\)\^m\ \((\(-1\))\)\^\(\^\(\ \(n\ - \ 1\)\/2\)\)\ x\_\(2 m\)\)\/n\)]], "\n\nRecombining, \n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}]}], TraditionalForm]]], "(", Cell[BoxData[ \(\(\(\ \)\(\[Pi]\ x\_n\)\)\/\(\(4\)\(\ \)\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\(\(-n\)\ \ \((\(-1\))\)\^m\ \((\(-1\))\)\^\(\(n - 1\)\/2\)\)\/\(4 m\ \^2 - n\^2\)\)]], "-", Cell[BoxData[ \(\[Sum]\+\(m = 1\)\%\[Infinity]\(\((\(-1\))\)\^m\ \((\(-1\))\)\^\(\^\(\ \(n\ - \ 1\)\/2\)\)\ x\_\(2 m\)\)\/n\)]], ")\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{\(\(\(2\)\(\ \)\)\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}]}], TraditionalForm]]], "(", Cell[BoxData[ \(\(\(\ \)\(\[Pi]\ x\_n\)\)\/\(\(4\)\(\ \)\)\)]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(m\)\(\ \)\)\ \((\(-1\))\)\^\(\(n - 1\)\/2\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`n\/\(4 m\^2 - n\^2\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/n\)]], ")))\n\nOr,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "= 0 \nif n is even\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(4\)\(\ \)\)\/\[Pi]\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(n\ \ + \ 2 m\ - \ 1\)\/2\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`n\/\(4 m\^2 - n\^2\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/n\)]], ")\nif n is odd\n\n\n[8b]\nBy a similar argument, noting that Cos(\[Pi] - \ \[Sigma]) = - Cos(\[Sigma]), \n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\[Pi]\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], " "}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ RowBox[{"(", " ", RowBox[{\(x\_0\), "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity] x\_m\), "TraditionalForm"], \(Cos( m\ \((\[Pi]\ - \[Sigma])\))\)}], "-", RowBox[{"(", " ", RowBox[{\(x\_0\), "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity] x\_m\), "TraditionalForm"], \(Cos(m\ \[Pi]\/2)\)}]}], ")"}]}], ")"}], " ", \(Cos(n\ \[Sigma])\), " ", \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}], TraditionalForm]]], "\n\nSimplifies to \n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "= 0 \nif n is even\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(-\ x\_n\), "TraditionalForm"]}], TraditionalForm]]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(4\)\(\ \)\)\/\[Pi]\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(n\ + \ 2 m\ - 1\)\/2\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`n\/\(4 m\^2 - n\^2\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/n\)]], ")\nif n is odd\n\n\n[9]\n\nLet ", Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^\(\(n\ + \ m\ + \ \ 1\)\/2\)\/\[Pi]\)]], "(", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ + \ m\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ - \ m\)\)]], ")\n\nSo, for odd n,\n\n", Cell[BoxData[ \(TraditionalForm\`B\_\(2 n\ - \ 1, \ 2 m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^\(\(2 n\ \ - \ 1\ + \ 2 m\ + \ \ 1\)\/2\)\/\[Pi]\)]], "(", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\ - \ 1\ + \ 2 m\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\ - \ 1\ - \ 2 m\)\)]], ")\n\nLet ", Cell[BoxData[ \(TraditionalForm\`B\_\(2 n\ - \ 1, \ 2 m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^\(n\ + \ m\)\/\[Pi]\)]], "(", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\ - \ 1\ + \ 2 m\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/\(2 n\ - \ 1\ - \ 2 m\)\)]], ")\n\n[10a]\n\nAnd continuing for odd n,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(4\)\(\ \)\)\/\[Pi]\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(n\ + \ 2 m\ - 1\)\/2\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`n\/\(4 m\^2 - n\^2\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`1\/n\)]], ")\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(4\)\(\ \)\)\/\[Pi]\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(n\ + \ 2 m\ - 1\)\/2\)\)]], Cell[BoxData[ \(TraditionalForm\`\(4 m\^2\)\/\(n\ \((4 m\^2\ - \ n\^2)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(2\)\(\ \)\)\/\[Pi]\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(\((\(-1\))\)\^\(\(n\ + \ 2 m\ + 1\)\/2\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 m\)\/n\)]], Cell[BoxData[ \(TraditionalForm\`\(2 \((2 m)\)\)\/\(\((2 m)\)\^2\ - \ n\^2\)\)]], "\n\nNote that ", Cell[BoxData[ \(TraditionalForm\`\(2 \((2 m)\)\)\/\(\((2 m)\)\^2\ - \ n\^2\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ + \ 2 m\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ - \ 2 m\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(\(L\)\(\ \)\(n\)\(\ \)\)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(2\ \ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 m\)\/\(\(n\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 m\)\)]], " (n odd)\n\n[10b]\n\nSimilarly,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(R\ n\)\)\(\ \)\)\)]], "= -", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(x\_n\), "TraditionalForm"]}], TraditionalForm]]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(2\ \ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 m\)\/\(\(n\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 m\)\)]], " (n odd)\n\nThe final goal is to invert this in order to find the value of \ x in terms of ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_L\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], ". Conveniently,\n\n[11]\n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(\[Chi]\_\(L\ n\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ n\)\), "TraditionalForm"]}], TraditionalForm]]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(4\ \[Sum]\+\(m = \ 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 m\)\/n\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 m\)\)]], "(n odd)\n\n", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{" ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ n\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ n\)\), "TraditionalForm"]}], ")"}]}], \(\(4\)\(\ \)\)], TraditionalForm]]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 m\)\/n\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 m\)\)]], "(n odd)\n\nwill help me find even-indexed values of x.\n\nInterestingly, \ this is analogous to a matrix operation with\n\n[\[Chi]] = [B] [x]\n\nWhere [\ \[Chi]] is a column vector with the n-th item equal to ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{" ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ n\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ n\)\), "TraditionalForm"]}], ")"}]}], "4"], TraditionalForm]]], "\n[x] is a column vector with the k-th item equal to ", Cell[BoxData[ \(TraditionalForm\`x\_\(2 k\)\)]], "\nand [B] being defined with n increasing down columns and k increasing \ across rows as ", Cell[BoxData[ \(TraditionalForm\`\(2\ k\)\/n\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 k\)\)]], "\n\nClearly, knowledge of ", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(\(-1\)\(\ \)\)\)]], "would allow a solution for x in terms of \[Chi]. \n\nGuess: ", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(\(-1\)\(\ \)\)\)]], "is similar to ", Cell[BoxData[ \(TraditionalForm\`\(\([B]\)\^+\)\)]], "\n\n[12a]\nAs a side note, \n\n", Cell[BoxData[ \(TraditionalForm\`B\_\(a, \ b\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(-b\)\/a\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(b, \ a\)\)]], "\n\nVerify: Across the diagonal (n = m) the result is constant\nVerify by \ evaluating ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\(2\ n\)\/k\), FormBox[\(B\_\(2 k, \ n\)\), "TraditionalForm"]}], TraditionalForm]]], " (n is odd)\n[12b]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ n\)\)]], "\n-", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ 2 k\)\)]], " (using [12a])\n-", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), SuperscriptBox[ RowBox[{"(", RowBox[{\(\((\(-1\))\)\^\(\(n\ + \ 2 k\ + \ \ 1\)\/2\)\/\[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(n\ - 2 k\)\), "TraditionalForm"]}], ")"}]}], ")"}], "2"]}], TraditionalForm]]], "\n-", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{\(1\/\[Pi]\^2\), SuperscriptBox[ RowBox[{"(", RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(n\ - 2 k\)\), "TraditionalForm"]}], ")"}], "2"]}]}], TraditionalForm]]], "\n-", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(k = 1\)\%\[Infinity]\( 4\/\[Pi]\^2\) \((\((2 k)\)\^2\/\((\((2\ k)\)\^2 - \ n\^2)\)\^2)\)\)]], "\n-", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(TraditionalForm\`lim\_\(\[Phi] \[Rule] 1\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(k = 1\)\%\[Infinity]\((\((2 \ k)\)\^2\/\((\(\((2\ k)\)\^2\) \[Phi] - n\^2)\)\^2)\)\)]], "\n-", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(TraditionalForm\`lim\_\(\[Phi] \[Rule] 1\)\)]], "[-", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\[Phi]\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(k = 1\)\%\[Infinity]\((1\/\(\(\((2\ \ k)\)\^2\) \[Phi] - n\^2\))\)\)]], "]\n-", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(TraditionalForm\`lim\_\(\[Phi] \[Rule] 1\)\)]], "[-", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\[Phi]\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`1\/\[Phi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(k = 1\)\%\[Infinity]\((1\/\(\((2\ k)\)\^2 \ - \((n\/\@\[Phi])\)\^2\))\)\)]], "]\n\nBy contour integration,\n-", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(TraditionalForm\`lim\_\(\[Phi] \[Rule] 1\)\)]], "[", Cell[BoxData[ \(TraditionalForm\`\(-\[PartialD]\_\[Phi]\ \(1\/\[Phi]\)\) \(2\ \[Phi] \ - n\ \[Pi]\ \@\[Phi]\ \(Cot(\(n\ \[Pi]\)\/\(2\ \@\[Phi]\))\)\)\/\(4\ \ n\^2\)\)]], "]\n\n", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(TraditionalForm\`lim\_\(\[Phi] \[Rule] 1\)\)]], "[", Cell[BoxData[ \(\(\[Pi]\ Csc \((\(n\ \[Pi]\)\/\(2\ \@\[Phi]\))\)\^2\ \((\(-n\)\ \[Pi] \ + \@\[Phi]\ Sin \((\(n\ \[Pi]\)\/\@\[Phi])\))\)\)\/\(16\ n\ \[Phi]\^2\)\)]], "]\n\n", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(\(\[Pi]\ Csc \((\(n\ \[Pi]\)\/2)\)\^2\ \((\(-n\)\ \[Pi] + Sin \((n\ \ \[Pi])\))\)\)\/\(16\ n\)\)]], "\n\nSince n is odd, \n", Cell[BoxData[ \(TraditionalForm\`4\/\[Pi]\^2\)]], Cell[BoxData[ \(\(\[Pi]\ \ \((\(-n\)\ \[Pi])\)\)\/\(16\ n\)\)]], "\n\n-", Cell[BoxData[ \(TraditionalForm\`1\/4\)]], "\n\nIn other words, the inverse of ", Cell[BoxData[ FormBox[ RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], " appears to be -4 ", Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ n\)\)]], ", henceforth ", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(-1\)\)]], "\n\nNow it is important to verify that the off-diagonal cells for the \ product of [B] and my tentative ", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(-1\)\)]], " is 0. \n\nVerify: Non-diagonal cells (n \[NotEqual] m) the result is \ zero. Constants multiplied across the sum will be ignored.\nVerify by \ evaluating ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ \ m\)\)]], " (n is odd)\n\nLet ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_c\)]], "(a) = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(b = c\)\%\[Infinity]\(-1\)\/\(a\ + \ \ \ b\)\)]], "\n\nNote that the digamma function ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_c\)]], "(a) has the property that ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_c\)]], "(", Cell[BoxData[ \(TraditionalForm\`\(-n\)\/2\)]], ") = -", Cell[BoxData[ \(TraditionalForm\`\[Psi]\_c\)]], "(", Cell[BoxData[ \(TraditionalForm\`n\/2\)]], ") where n is an integer.\n[12c]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}]}], TraditionalForm]]], "(-4) ", Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ \ m\)\)]], " (n is odd)\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", RowBox[{"k", " ", FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ \ m\)\)]], "(constant factors ignored)\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(m, \ \ 2 k\)\)]], "(using [12a], constant factors ignored)\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{\(\((\(-1\))\)\^\(\(n\ + \ 2 k\ + \ 1\)\/2\)\/\[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(n\ - \ 2 k\)\), "TraditionalForm"]}], ")"}], \(\((\(-1\))\)\^\(\(m\ + \ 2 k\ + \ 1\)\/2\)\/\[Pi]\ \), RowBox[{"(", RowBox[{ FormBox[\(1\/\(m\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(m\ - \ 2 k\)\), "TraditionalForm"]}], ")"}]}], "TraditionalForm"]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(n\ - \ 2 k\)\), "TraditionalForm"]}], ")"}], RowBox[{"(", RowBox[{ FormBox[\(1\/\(m\ + \ 2 k\)\), "TraditionalForm"], "-", FormBox[\(1\/\(m\ - \ 2 k\)\), "TraditionalForm"]}], ")"}]}], "TraditionalForm"]}], TraditionalForm]]], "(constant factors ignored)\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{ RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], FormBox[\(1\/\(m\ + \ 2 k\)\), "TraditionalForm"]}], "-"}], "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{ RowBox[{ FormBox[\(1\/\(n\ - \ 2 k\)\), "TraditionalForm"], FormBox[\(1\/\(m\ + \ 2 k\)\), "TraditionalForm"]}], "-"}], "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{ RowBox[{ FormBox[\(1\/\(n\ + \ 2 k\)\), "TraditionalForm"], FormBox[\(1\/\(m\ - \ 2 k\)\), "TraditionalForm"]}], "+"}], "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[ RowBox[{ FormBox[\(1\/\(n\ - \ 2 k\)\), "TraditionalForm"], FormBox[\(1\/\(m\ - \ 2 k\)\), "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\ + \ 2 k\) - 1\/\(m\ + \ 2 k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\ - \ 2 k\) + 1\/\(m\ + \ 2 k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\ + \ 2 k\) + 1\/\(m\ - \ 2 k\))\) + 1\/\(m\ - \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\((1\/\(m\ - \ 2 k\) - 1\/\(n\ - \ 2 k\))\), "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "(", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ + \ k\) - 1\/\(m\/2\ + \ k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ - \ k\) + 1\/\(m\/2\ + \ k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ + \ k\) + 1\/\(m\/2\ - \ k\))\) + 1\/\(m\ - \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\((1\/\(m\/2\ - \ k\) - 1\/\(n\/2\ - \ k\))\), "TraditionalForm"]}], TraditionalForm]]], ")\n", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ + \ k\) - 1\/\(m\/2\ + \ k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ - \ k\) + 1\/\(m\/2\ + \ k\))\) - 1\/\(m\ + \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\(\((1\/\(n\/2\ + \ k\) + 1\/\(m\/2\ - \ k\))\) + 1\/\(m\ - \ n\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), " ", FormBox[\((1\/\(m\/2\ - \ k\) - 1\/\(n\/2\ - \ k\))\), "TraditionalForm"]}], TraditionalForm]]], " (constant factors ignored)\n", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], Cell[BoxData[ \(TraditionalForm\`\((\(\[Psi]\_1\)(n\/2)\ - \ \(\[Psi]\_1\)( m\/2))\) - 1\/\(m\ + \ n\)\)]], Cell[BoxData[ FormBox[ RowBox[{\((\(-\ \(\(\[Psi]\_1\)(\(-n\)\/2)\)\)\ + \ \(\[Psi]\_1\)( m\/2))\), " ", FormBox[\(-\(1\/\(m\ + \ n\)\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\((\(\[Psi]\_1\)( n\/2)\ - \ \(\[Psi]\_1\)(\(-m\)\/2))\)\ + 1\/\(m\ - \ n\)\)]], Cell[BoxData[ \(TraditionalForm\`\((\ \(\[Psi]\_1\)(\(-n\)\/2)\ - \ \ \(\[Psi]\_1\)(\(-m\)\/2))\)\)]], "\n", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)(n\/2)\ - \ \(\[Psi]\_1\)(m\/2)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(+\ \(\(\[Psi]\_1\)(\(-n\)\/2)\)\)\ - \ \ \(\[Psi]\_1\)(\(-m\)\/2)\)]], ") - ", Cell[BoxData[ \(TraditionalForm\`1\/\(m\ + \ n\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\(-\ \(\(\[Psi]\_1\)(\(-n\)\/2)\)\)\ + \ \ \(\[Psi]\_1\)(m\/2)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)( n\/2)\ + \ \(\[Psi]\_1\)(\(-m\)\/2)\)]], ")\n", Cell[BoxData[ \(1\/\(m\ - \ n\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)(n\/2)\ - \ \(\[Psi]\_1\)(m\/2)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(-\ \(\(\[Psi]\_1\)(n\/2)\)\)\ + \ \(\[Psi]\_1\)( m\/2)\)]], ") - ", Cell[BoxData[ \(TraditionalForm\`1\/\(m\ + \ n\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\[Psi]\_1\)(n\/2)\ + \ \(\[Psi]\_1\)( m\/2)\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)(n\/2)\ - \ \(\[Psi]\_1\)(m\/2)\)]], ")\n0\n\nAnd so it has been shown that the inverse of the infinite matrix \ given by ", Cell[BoxData[ FormBox[ RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], "is ", Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ n\)\)]], ", but should be multiplied by -4 on inversion as inversion creates a \ factor of ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\/4\)]], ". In other words, -4", Cell[BoxData[ FormBox[ RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`B\_\(2 k, \ n\)\)]], "is the infinite identity matrix.\n\n[\[Chi]] = [B] [x]\n", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(-1\)\)]], "[\[Chi]] = [x]\n\nFor \n\n", Cell[BoxData[ \(TraditionalForm\`x\_\(2 m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"-", RowBox[{\(B\_\(2 m, \ 2 n - 1\)\), "(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ n - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 n - 1\)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " =", Cell[BoxData[ \(TraditionalForm\`\ \ \)]], "(for all positive integers m)\n\nThe knowledge of the values of \ odd-indexed values of \[Chi] will help to find odd-indexed values of x, from \ which I can find using the relation\n\n[13]\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(L\ n\)\)\(\ \)\)\)]], "-", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Chi]\_\(R\ n\)\), "TraditionalForm"]}], TraditionalForm]]], " =", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(2 x\_n\), "TraditionalForm"]}], TraditionalForm]]], "(n odd)\n\n", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], " = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(\(\[Chi]\_\(L\ n\)\)\(\ \)\), "TraditionalForm"], "-", FormBox[ RowBox[{" ", FormBox[\(\[Chi]\_\(R\ n\)\), "TraditionalForm"]}], "TraditionalForm"]}], "2"], TraditionalForm]]], " (n odd)\n\nFinally, I need knowledge of ", Cell[BoxData[ \(TraditionalForm\`x\_0, \ which\ can\ be\ found\ from\ [ 1 a], \ \(\(by\)\(\ \)\(substituting\)\(\ \)\(the\)\(\ \)\(other\)\ \(\ \)\(coefficients\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`x\_n\)]], " and knowing x(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ").\n\n[14]\n\nx(\[Sigma]) =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x\_0\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\)]], "cos(n \[Sigma])\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\)]], "cos(n ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ")\n(in this case, n must be even)\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_\(2 n\)\)]], Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^n\)]], "\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), " ", RowBox[{\(\((\(-1\))\)\^n\), " ", RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\(B\_\(2 n, \ 2 m - 1\)\), "(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), " ", RowBox[{\(\((\(-1\))\)\^n\), " ", RowBox[{\(B\_\(2 n, \ 2 m - 1\)\), "(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`\@2\/\[Pi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n\nThe next goal is to find the conjugate momemntum P\n\nLet ", Cell[BoxData[ \(TraditionalForm\`P\_\(L\ n\) = \ \(-\ \[ImaginaryI]\)\ \ \(\(\[PartialD]\_\(\[Chi]\_\(L\ n\)\)\)\(\ \)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`P\_\(R\ n\) = \ \(-\ \[ImaginaryI]\)\ \ \[PartialD]\_\(\[Chi]\_\(R\ n\)\)\)]], " and P = - \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\(x(\[Pi]\/2)\)\)\(\ \)\)\)], "DisplayFormula"], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(p\)\(\ \)\)\_n = \ \(-\ \[ImaginaryI]\)\ \ \[PartialD]\_\(x\_\(\(\ \)\(n\)\)\)\)]], "\n\n[15a]\n\nI wish to evaluate\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(L\ n\) = \ \(-\ \[ImaginaryI]\)\ \ \[PartialD]\_\(\[Chi]\_\(L\ n\)\)\)]], "(x) ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_x\)\(\ \)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`P\_\(L\ n\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(-\ \[ImaginaryI]\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = \ 0\)\%\[Infinity]\[PartialD]\_\(\[Chi]\_\(L\ n\)\)\([x\_m]\)\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_m\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\ \[PartialD]\_\(\[Chi]\_\(L\ \ n\)\)\)]], Cell[BoxData[ FormBox[ RowBox[{"[", " ", RowBox[{\(x(\[Pi]\/2)\), "-", RowBox[{ FormBox[\(\@2\/\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}], "]"}], "TraditionalForm"]}]}]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\(x\_0\)\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\ \[PartialD]\_\(\[Chi]\_\(L\ \ n\)\)\)]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{ RowBox[{"[", RowBox[{\(\[Sum]\+\(q = 1\)\%\[Infinity]\), RowBox[{"-", RowBox[{\(B\_\(2 m, \ 2 q - 1\)\), "(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2 q - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 q - 1\)\), "TraditionalForm"]}], ")"}]}]}], "]"}], \(\[PartialD]\_\(x\_\(2 m\)\)\)}]}], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\ \[PartialD]\_\(\[Chi]\_\(L\ \ n\)\)\)]], " [", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), FractionBox[ RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2 m\ - \ 1\)\)\(\ \)\), "TraditionalForm"], "-", FormBox[ RowBox[{" ", FormBox[\(\[Chi]\_\(R\ 2 m\ - \ 1\)\), "TraditionalForm"]}], "TraditionalForm"]}], "2"]}], TraditionalForm]]], "]", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_\(2 m - 1\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " = 0 (n even, no dependence on ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(L\ n\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(-\ \[ImaginaryI]\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ RowBox[{"[", " ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[\(\(\ \)\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\)\(\ \ \)\(]\)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\(x\_0\)\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(+\ \[ImaginaryI]\)\(\ \)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\([B\_\(2 m, \ n\)]\) \ \[PartialD]\_\(x\_\(2 m\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(-\ \[ImaginaryI]\)\(\ \)\)\)]], " [", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity] 1\/2\)]], "]", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_n\)\)]], "(n odd)\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[ RowBox[{" ", RowBox[{\(\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\) p\_\(\_0\)\), "-", FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\([B\_\(2 m, \ n\)]\), \(p\_\(\_\(2 m\)\)\), " ", FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(p\_\(\_n\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]}], "TraditionalForm"]}]}], TraditionalForm]]], " (n odd)\n\nAnd similarly,\n\n[15b]\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(R\ n\)\)\)]], " = 0 (n even, no dependence on ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(R\ n\)\)]], ")\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(R\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[\(\(\ \)\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\)\(\ \)\), "TraditionalForm"]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\(\(p\_\(\_0\)\)\(\ \)\)\)]], " -", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\([B\_\(2 m, \ n\)]\) p\_\(\_\(2 m\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(-\)\(\ \)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ \(TraditionalForm\`p\_\(\_n\)\)]], "(n odd)\n\n[15c]\n\nAlso note\n\nP = - \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\(x(\[Pi]\/2)\)\)\(\ \)\)\)], "DisplayFormula"], "[", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 0\)\%\[Infinity] x\_m\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_m\)\)]], "]\n\nP = - \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_\(x(\[Pi]\/2)\)\)\(\ \)\)\)], "DisplayFormula"], "[", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], "]", Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_0\)\)]], "\n\nP =", Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ \(TraditionalForm\`p\_\(\_0\)\)]], "\n\nNow I need the full-string momenta as a function of these half-string \ momenta.\n\n[16]\n\nNote that \n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(R\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`p\_\(\_n\)\)]], " (n odd)\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(R\ n\)\)\)]], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"-", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], FormBox[\(\(\ \)\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\)\(\ \)\), "TraditionalForm"], FormBox[\(\(p\_\(\_0\)\)\(\ \)\), "TraditionalForm"]}], "-", FormBox[\(2 \(\[Sum]\+\(m = 1\)\%\[Infinity]\( B\_\(2 m, \ n\)\) p\_\(\_\(2 m\)\)\)\), "TraditionalForm"], " "}]}], TraditionalForm]]], " (n odd)\n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " + ", Cell[BoxData[ FormBox[ RowBox[{\(P\_\(\(\ \)\(R\ n\)\)\), " ", "+", " ", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], FormBox[\(\(\ \)\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\)\(\ \)\), "TraditionalForm"], "P"}]}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(-2\) \(\[Sum]\+\(m = 1\)\%\[Infinity]\( B\_\(2 m, \ n\)\) p\_\(\_\(2 m\)\)\)\), "TraditionalForm"], " "}], TraditionalForm]]], " (n odd)\n\n-", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(P\_\(\(\ \)\(L\ n\)\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(P\_\(\(\ \)\(R\ n\)\)\), " ", "+", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], FormBox[\(\(\ \)\(\(\((\(-1\))\)\^\(\(n + 1\)\/2\)\/n\) \(P\ \)\(\ \)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], "2"], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\( B\_\(2 m, \ n\)\) p\_\(\_\(2 m\)\)\), "TraditionalForm"], " "}], TraditionalForm]]], " (n odd)\n\nis analogous to a matrix operation similar to the one \ encountered earlier.\n\n[P] = [B] [p]\n\nwhere [P] is a column vector with \ entry k defined by -", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(P\_\(\(\ \)\(L\ 2 k\ - \ 1\)\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(P\_\(\(\ \)\(R\ 2 k\ - \ 1\)\)\), " ", "+", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], FormBox[\(\(\ \)\(\(\((\(-1\))\)\^k\/\(2 k\ - \ 1\)\) \(P\)\(\ \)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], "2"], TraditionalForm]]], "\n[B] is an infinite matrix with j defined across rows and k defined down \ columns as ", Cell[BoxData[ \(TraditionalForm\`B\_\(2 j, \ 2 k\ - \ 1\)\)]], "\nand [p] is a column vector with entry k defined by ", Cell[BoxData[ \(TraditionalForm\`p\_\(\_\(2 k\)\)\)]], "\n\nknowledge of ", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(-1\), \ then, \ would\ allow\ me\ to\ invert\ this\ \(\(relation\)\(.\)\)\)]], "\n\nFrom my previous inversion, which showed that ", Cell[BoxData[ FormBox[ RowBox[{\(\(2\ k\)\/n\), FormBox[\(B\_\(n, \ 2 k\)\), "TraditionalForm"]}], TraditionalForm]]], "is the inverse of -4 ", Cell[BoxData[ \(TraditionalForm\`\(\(B\_\(2 k, \ n\)\)\(,\)\)\)]], "I could expect that -4", Cell[BoxData[ FormBox[ RowBox[{\(\(2\ j\)\/\(2 k\ - \ 1\)\), FormBox[\(B\_\(2 k\ - \ 1, \ 2 j\)\), "TraditionalForm"]}], TraditionalForm]]], " is the inverse of [B] so that\n\n", Cell[BoxData[ \(TraditionalForm\`\([B]\)\^\(-1\)\)]], " [P] = [p] \n\nfor\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{"-", RowBox[{"[", FractionBox[ RowBox[{ FormBox[\(P\_\(\(\ \)\(L\ 2 k\ - \ 1\)\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(P\_\(\(\ \)\(R\ 2 k\ - \ 1\)\)\), " ", "+", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], FormBox[\(\(\ \)\(\(\((\(-1\))\)\^k\/\(2 k\ - \ 1\)\) \(P\)\(\ \)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], "2"]}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\(2 j\)\/\(2 k - \ 1\)\), RowBox[{"(", RowBox[{"-", FormBox[\(\(\(4\)\()\)\)\ B\_\(2 k\ - \ 1, \ 2 j\)\), "TraditionalForm"]}]}]}], TraditionalForm]]], " ]= ", Cell[BoxData[ \(TraditionalForm\`p\_\(2\ j\)\)]], " (j \[GreaterEqual] 1)\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{"-", RowBox[{"[", RowBox[{"(", RowBox[{ FormBox[\(P\_\(\(\ \)\(L\ 2 k\ - \ 1\)\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(P\_\(\(\ \)\(R\ 2 k\ - \ 1\)\)\), " ", "+", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], FormBox[\(\(\ \)\(\(\((\(-1\))\)\^k\/\(2 k\ - \ 1\)\) \(P\)\(\ \)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}]}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(j\/\(2 k - \ 1\)\), FormBox[\(\(\ \)\(\((\(-4\))\)\ B\_\(2 k\ - \ 1, \ 2 j\)\)\), "TraditionalForm"]}], TraditionalForm]]], " ]= ", Cell[BoxData[ \(TraditionalForm\`p\_\(2\ j\)\)]], " (j \[GreaterEqual] 1)\n\nOr evaluating the sum for the P term,\n\n", Cell[BoxData[ \(TraditionalForm\`p\_\(2\ j\)\)]], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(k = 1\)\%\[Infinity]\), RowBox[{"[", RowBox[{"2", " ", RowBox[{"(", RowBox[{ FormBox[\(P\_\(\(\ \)\(L\ 2 k\ - \ 1\)\)\), "TraditionalForm"], " ", "+", " ", FormBox[\(P\_\(\(\ \)\(R\ 2 k\ - \ 1\)\)\), "TraditionalForm"]}]}]}]}]}], TraditionalForm]]], ")", Cell[BoxData[ FormBox[ RowBox[{\(\(2 j\)\/\(2 k - \ 1\)\), FormBox[\(B\_\(2 k\ - \ 1, \ 2 j\)\), "TraditionalForm"]}], TraditionalForm]]], " ] + ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\(\((\(-1\))\)\(\ \)\)\^j\)]], " P (j \[GreaterEqual] 1)\n\nand \n\n", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(L\ n\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`P\_\(\(\ \)\(R\ n\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`p\_\(\_n\)\)]], " (n odd)\n\nNext, I wish to show that the usual commutators hold:\n\n[", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(Q\ n\)\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ m\)\)]], "] = i ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(Q\ S\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(n\ m\)\)]], "\n\n[17a]\n\n[", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(Q\ n\)\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ m\)\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ n\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ m\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`P\_\(S\ m\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ n\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ n\)\)]], Cell[BoxData[ \(TraditionalForm\`\((\(-\ \[ImaginaryI]\)\ \[PartialD]\_\(\[Chi]\_\(S\ \ m\)\))\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\((\(-\ \[ImaginaryI]\)\ \[PartialD]\_\(\[Chi]\_\(S\ \ m\)\))\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ n\)\)]], "\n0 + ", Cell[BoxData[ \(TraditionalForm\`\[ImaginaryI]\ \[PartialD]\_\(\[Chi]\_\(S\ \ n\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ m\)\)]], "\nWhich, of course, yields \[ImaginaryI] if S=Q and n=m, and 0 otherwise. \ \n\nand \n\n[17b]\n\nEvaluate to verify that \n\n[", Cell[BoxData[ \(TraditionalForm\`\(\(x(\[Pi]\/2)\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`P\)]], "] =\[ImaginaryI]\n\n[", Cell[BoxData[ \(TraditionalForm\`\(\(x(\[Pi]\/2)\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`P\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], Cell[BoxData[ \(TraditionalForm\`P\)]], " - ", Cell[BoxData[ \(TraditionalForm\`P\)]], Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], "(", Cell[BoxData[ FormBox[ RowBox[{\(-\[ImaginaryI]\), FormBox[\(\[PartialD]\_\(x(\[Pi]\/2)\)\), "TraditionalForm"]}], TraditionalForm]]], ") - ", Cell[BoxData[ FormBox[ RowBox[{\(-\[ImaginaryI]\), FormBox[\(\[PartialD]\_\(x(\[Pi]\/2)\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], "\n0 + \[ImaginaryI]\n\[ImaginaryI]\n\nDefine the half-string creation and \ annihilation operators in the usual way\n\n[18a]\n", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n\nand\n\n[18b]\n\n", Cell[BoxData[ \(TraditionalForm\`\(b\_\(Q\ n\)\^+\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n\nand\n\n[18c]\n\n", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_\(Q\ n\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`n\^\(1\/2\)\)]], Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ \(n\ + \ 1\)\/2\)\)]], "\n\nand\n\n[18d]\n\n", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{ FormBox[\(\[Beta]\_\(Q\ n\)\), "TraditionalForm"], " "}], "+"], TraditionalForm]]], "= ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_\(Q\ - n\)\)]], "\n\nVerify that the commutator \n\n[", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`b\_\(S\ m\)\)]], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(Q\ S\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(n\ m\)\)]], "\n\n[", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(S\ m\)\^+\)\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], Cell[BoxData[ \(TraditionalForm\`\(b\_\(S\ m\)\^+\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(S\ m\)\^+\)\)]], Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 m\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ m\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S2\ m\ - \ 1\)\)]], ") - ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 m\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ m\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ 2 m\ - \ 1\)\)]], ")", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 m\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "((", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")", Cell[BoxData[ \(TraditionalForm\`\ \)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ m\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ 2 m\ - \ 1\)\)]], ") - (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ m\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(S\ 2 m\ - \ 1\)\)]], ")(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], "))\n\nThen, if Q \[NotEqual] S or n \[NotEqual] m,\n\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 m\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "((", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], ") - (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(S\ \ 2 m\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "))\n0\n\nBut if Q = S and n = m,\n\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ FormBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}], TraditionalForm]]], "((", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ") + \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")) - (", Cell[BoxData[ FormBox[ RowBox[{\(\[Chi]\_\(Q\ \ 2 n\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\[Chi]\_\(Q\ 2 n\ - \ 1\)\), "TraditionalForm"], "+", RowBox[{"\[ImaginaryI]", FormBox[\(2\/\(2\ n\ - \ 1\)\), "TraditionalForm"], FormBox[\(P\_\(Q\ 2 n\ - \ 1\)\), "TraditionalForm"]}]}], ")"}]}], TraditionalForm]]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(P\_\(\(Q\ 2 n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "(", RowBox[{ FormBox[\(\[Chi]\_\(Q\ 2 n\ - \ 1\)\), "TraditionalForm"], "+", RowBox[{"\[ImaginaryI]", FormBox[\(2\/\(2\ n\ - \ 1\)\), "TraditionalForm"], FormBox[\(P\_\(Q\ 2 n\ - \ 1\)\), "TraditionalForm"]}]}], ")"}], TraditionalForm]]], ")\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}], "("}], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], ") - (", Cell[BoxData[ \(TraditionalForm\`\(\(\[Chi]\_\(Q\ \ 2 n\ - \ 1\)\^2\)\(\ \)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], "))\n", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], Cell[BoxData[ FormBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`4\/\(2\ n\ - \ 1\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`1\)]], "\n\nThe next goal is to rewrite the half-string creation and annihilation \ operators, b, from [18] in terms of the full-string creation and annihilation \ operators, a, from [1b]\n\n[19a]\n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(L\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(L\ 2 n\ - \ 1\)\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(x\_\(2 n\ - \ 1\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(2\ \ \[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[\(x\_\(2 m\)\), "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), "TraditionalForm"]}]}]}], TraditionalForm]]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[ RowBox[{" ", RowBox[{\(\(\((\(-1\))\)\^\(\(2 n\ - \ 1\ + 1\)\/2\)\/\ \(2 n\ \ - \ 1\)\) p\_\(\_0\)\), "-", FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \ \(p\_\(\_\(2 m\)\)\), " ", FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(p\_\(\_\(2 n\ \ - \ 1\)\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]}], "TraditionalForm"]}]}], "TraditionalForm"], ")"}], TraditionalForm]]], ") \n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(x\_\(2 n\ - \ 1\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(2\ \ \[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[\(x\_\(2 m\)\), "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), "TraditionalForm"]}]}]}], TraditionalForm]]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[ RowBox[{" ", RowBox[{\(\(\((\(-1\))\)\^n\/\(2 n\ \ - \ 1\)\) p\_\(\_0\)\), "-", FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \ \(p\_\(\_\(2 m\)\)\), " ", FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(p\_\(\_\(2 n\ \ - \ 1\)\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]}], "TraditionalForm"]}]}], "TraditionalForm"], ")"}], TraditionalForm]]], ") \n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(-\ \[ImaginaryI]\)\/2\)]], Cell[BoxData[ FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(x\_\(2 n\ - \ 1\)\), "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{ FormBox[\(-\ \[ImaginaryI]\), "TraditionalForm"], FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], "TraditionalForm"], " ", \(\[Sum]\+\(m = 1\)\%\[Infinity]\)}]}], "TraditionalForm"], FormBox[\(x\_\(2 m\)\), "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), "TraditionalForm"]}], TraditionalForm]]], "+ ", Cell[BoxData[ FormBox[ FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"]}], FormBox[ RowBox[{" ", RowBox[{\(\(\((\(-1\))\)\^n\/\(2 n\ \ - \ 1\)\) p\_\(\_0\)\), "-", FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \ \(p\_\(\_\(2 m\)\)\), " ", FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(p\_\(\_\(2 n\ \ - \ 1\)\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]}], "TraditionalForm"]}]}], "TraditionalForm"], ")"}], TraditionalForm]]], ") \n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], TraditionalForm]]], " -", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\/2\)]], Cell[BoxData[ FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(x\_\(2 n\ - \ 1\)\), "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{ FormBox[\(-\ \[ImaginaryI]\), "TraditionalForm"], FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], "TraditionalForm"], " ", \(\[Sum]\+\(m = 1\)\%\[Infinity]\)}]}], "TraditionalForm"], FormBox[\(x\_\(2 m\)\), "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[ RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), FormBox["", "TraditionalForm"], FormBox[ RowBox[{ RowBox[{"-", FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]]}], FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \ \(p\_\(\_\(2 m\)\)\), " ", FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]], FormBox[\(p\_\(\_\(2 n\ \ - \ 1\)\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{\(b\_\(L\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "-", RowBox[{ FormBox[\(\(\(\ \)\(\[ImaginaryI]\)\)\/2\), "TraditionalForm"], FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], "TraditionalForm"], \(1\/2\), FormBox[\(\[ImaginaryI]\ \@\(2\/\(2 n - 1\)\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(2 n - 1\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 n - 1\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}]}], "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{ FormBox[\(-\ \[ImaginaryI]\), "TraditionalForm"], FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], "TraditionalForm"], " ", \(\[Sum]\+\(m = 1\)\%\[Infinity](\)}]}], "TraditionalForm"], FormBox[ RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(\[ImaginaryI]\ \@\(2\/\(2 m\)\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}], "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[ RowBox[{\(\(B\_\(2 n\ \ - \ 1, \ 2 m\)\)\()\)\), FormBox["", "TraditionalForm"], FormBox[ RowBox[{ RowBox[{"-", FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]]}], FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{ RowBox[{"(", RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \(\@\(\ \(2 m\)\/2\)\), RowBox[{"(", FormBox[\(a\_\(\(2\) \(m\)\(\ \)\) + \ \ \(a\_\(2 m\)\^+\)\), "TraditionalForm"], ")"}]}], " ", ")"}], FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"], FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]], FormBox[ RowBox[{\(\@\(\(2 n\ - \ 1\)\/2\)\), RowBox[{"(", FormBox[\(a\_\(\(2 n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \ \)\) + \ \(a\_\(2 n\ - \ 1\)\^+\)\), "TraditionalForm"], ")"}]}], "TraditionalForm"]}]}], "TraditionalForm"], " "}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{\(b\_\(L\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(\@2\/4\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(2 n - 1\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 n - 1\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}]}], "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{ FormBox[\(-\ \[ImaginaryI]\), "TraditionalForm"], FormBox[ SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]], "TraditionalForm"], " ", \(\[Sum]\+\(m = 1\)\%\[Infinity](\)}]}], "TraditionalForm"], FormBox[ RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(\[ImaginaryI]\ \@\(2\/\(2 m\)\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}], "TraditionalForm"], FormBox[\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\), "TraditionalForm"], FormBox[ RowBox[{\(\(B\_\(2 n\ \ - \ 1, \ 2 m\)\)\()\)\), FormBox["", "TraditionalForm"], FormBox[ RowBox[{ RowBox[{"-", FractionBox["1", SqrtBox[ FormBox[\(2 n\ - \ 1\), "TraditionalForm"]]]}], FormBox[ RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{ RowBox[{"(", RowBox[{\([B\_\(2 m, \ 2 n\ \ - \ 1\)]\), \(\@\(\ \(2 m\)\/2\)\), RowBox[{"(", FormBox[\(a\_\(\(2\) \(m\)\(\ \)\) + \ \ \(a\_\(2 m\)\^+\)\), "TraditionalForm"], ")"}]}], " ", ")"}], FormBox[\(\(+\)\(\ \)\), "TraditionalForm"], " ", \(\@2\/4\), FormBox[ RowBox[{"(", FormBox[\(a\_\(\(2 n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \ \)\) + \ \(a\_\(2 n\ - \ 1\)\^+\)\), "TraditionalForm"], ")"}], "TraditionalForm"]}]}], "TraditionalForm"], " "}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(L\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(1\/\@2\)\), "TraditionalForm"], FormBox[ RowBox[{\(a\_\(2 n - 1\)\), FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \[Sum]\+\(m = 1\)\%\ \[Infinity]\)\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}], "TraditionalForm"], FormBox[\(\@\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\), "TraditionalForm"], FormBox[ RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), FormBox["", "TraditionalForm"], FormBox[ RowBox[{"-", FormBox[ RowBox[{" ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), \(\@\(\(2 m\)\/\(2 n\ - \ 1\)\)\), RowBox[{"(", FormBox[\(a\_\(\(2\) \(m\)\(\ \)\) + \ \(a\ \_\(2 m\)\^+\)\), "TraditionalForm"], ")"}], " "}]}], "TraditionalForm"]}], "TraditionalForm"]}], "TraditionalForm"]}], "TraditionalForm"]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(L\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(1\/\@2\)\), "TraditionalForm"], FormBox[ RowBox[{\(a\_\(2 n - 1\)\), FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "TraditionalForm"], "-", " ", \(a\_\(2 m\)\^+\)}], ")"}]}], "TraditionalForm"], FormBox[ RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), FormBox["", "TraditionalForm"], FormBox[ RowBox[{"+", FormBox[ RowBox[{ RowBox[{" ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), "(", FormBox[\(\(-a\_\(\(2\) \(m\)\(\ \)\)\) - \ \ \(a\_\(2 m\)\^+\)\), "TraditionalForm"], ")"}], " ", ")"}], " "}], "TraditionalForm"]}], "TraditionalForm"]}], "TraditionalForm"]}], "TraditionalForm"]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(L\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "+", RowBox[{ FormBox[\(\(\ \)\(1\/\@2\)\), "TraditionalForm"], FormBox[ RowBox[{ RowBox[{\(a\_\(2 n - 1\)\), FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{ RowBox[{ "(", \(B\_\(2 n\ \ - \ 1, \ 2 m\)\ - \ B\_\(2 m, \ 2 n\ \ - \ 1\)\), " ", FormBox[")", "TraditionalForm"]}], \(a\_\(2 m\)\)}], "TraditionalForm"]}], "TraditionalForm"]}], "-", " ", RowBox[{"(", RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), " ", "+", " ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), FormBox[ RowBox[{ RowBox[{")", SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}]}]}], "TraditionalForm"]}]}]}], TraditionalForm]]], "\n\nAnd similarly\n\n[19b]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(R\ n\)\), "=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], "-", RowBox[{ FormBox[\(\(\ \)\(1\/\@2\)\), "TraditionalForm"], FormBox[ RowBox[{ RowBox[{\(a\_\(2 n - 1\)\), FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{ RowBox[{ "(", \(B\_\(2 n\ \ - \ 1, \ 2 m\)\ - \ B\_\(2 m, \ 2 n\ \ - \ 1\)\), " ", FormBox[")", "TraditionalForm"]}], \(a\_\(2 m\)\)}], "TraditionalForm"]}], "TraditionalForm"]}], "-", " ", RowBox[{"(", RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), " ", "+", " ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), FormBox[ RowBox[{ RowBox[{")", SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}]}]}], "TraditionalForm"]}]}]}], TraditionalForm]]], "\n\nwith ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(b\_\(Q\ n\)\), "TraditionalForm"], "+"], TraditionalForm]]], " obtainable by using the ", Cell[BoxData[ \(TraditionalForm\`\(\[Placeholder]\^+\)\)]], " operation on each a present.\n\nNow I would like to invert this \ relationship.\n\n[20a]\n\nClearly, \n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(L\ n\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`b\_\(R\ n\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(2\/\@2\) a\_\(2 n - 1\)\)]], "\n", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "-", " ", FormBox[\(b\_\(R\ n\)\), "TraditionalForm"]}], \(\@2\)], TraditionalForm]]], "= ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 n - 1\)\)]], " \n\nfinds odd-indexed a. To find the corresponding ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 n - 1\)\), "TraditionalForm"], "+"], TraditionalForm]]], " operator, simply use the ", Cell[BoxData[ \(TraditionalForm\`\(\[Placeholder]\^+\)\)]], " operator on each side.\n\n", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\)\(\ \)\), "TraditionalForm"]}], "2"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{ RowBox[{ "(", \(B\_\(2 n\ \ - \ 1, \ 2 m\)\ - \ B\_\(2 m, \ 2 n\ \ - \ 1\)\), " ", FormBox[")", "TraditionalForm"]}], \(a\_\(2 m\)\)}], "TraditionalForm"]}], "TraditionalForm"]}], "-", " ", RowBox[{"(", RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), " ", "+", " ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), FormBox[ RowBox[{ RowBox[{")", SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}]}]}], "TraditionalForm"]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "2"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \)\(\(+\ \(1\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{ RowBox[{ "(", \(B\_\(2 n\ \ - \ 1, \ 2 m\)\ - \ B\_\(2 m, \ 2 n\ \ - \ 1\)\), " ", FormBox[")", "TraditionalForm"]}], \(a\_\(2 m\)\^+\)}], "TraditionalForm"]}], "TraditionalForm"]}], "-", " ", RowBox[{"(", RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), " ", "+", " ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), FormBox[ RowBox[{ RowBox[{")", FormBox[\(a\_\(2 m\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}]}]}], "TraditionalForm"]}]}], TraditionalForm]]], "\n\n[21a]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\@2\), RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], "4"], " ", "-", " ", RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[ FormBox[\(\(B\_\(2 m, \ 2 n\ \ - \ 1\)\)\((\)\(\(\@\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\) \ \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "+", FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], "TraditionalForm"], TraditionalForm]]], "\n\n[21b]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{"(", FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], "4"], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[ FormBox[\(B\_\(2 n\ \ - \ 1, \ 2 m\)\ \((\(\@\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\) \ \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\nI'll have to invert both 21a and 21b first, then from the results I'll \ be able to find ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], " and ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], TraditionalForm]]], ". As I've seen earlier, the inverse of the infinite matrix given by ", Cell[BoxData[ \(TraditionalForm\`B\_\(a, \ b\)\)]], "is ", Cell[BoxData[ FormBox[ RowBox[{\(a\/b\), FormBox[\(B\_\(b, \ a\)\), "TraditionalForm"]}], TraditionalForm]]], " and requires a factor of -4 to be multiplied after inversion\n\n[21c]: \ Inverse of [21a]\n\n", Cell[BoxData[ \(TraditionalForm\`B\_\(a, \ b\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(-b\)\/a\)]], Cell[BoxData[ \(TraditionalForm\`B\_\(b, \ a\)\)]], "\n\n[b] = [B][a]\n", Cell[BoxData[ \(TraditionalForm\`\(\([B]\)\^\(-1\)\)[b]\ = \ \([a]\)\)]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\@2\), RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], "4"], " ", "-", " ", RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[ FormBox[\(\(B\_\(2 m, \ 2 n\ \ - \ 1\)\)\((\)\(\(\@\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\) \ \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "+", FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], "TraditionalForm"], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\@2\), RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], "4"], " ", "-", " ", RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], \(1\/\@\(2\ n\ - \ 1\)\), FormBox[ FormBox[\(\(B\_\(2 m, \ 2 n\ \ - \ 1\)\)\((\)\(\(\@\(2 m\)\) \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "+", FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], "4"], " ", "-", " ", RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\(2\ m\)\/\(2\ n\ - \ 1\)\), \(B\_\(2\ n\ - \ 1, \ 2\ m\)\), FormBox[ FormBox[\(\(B\_\(2 m, \ 2 n\ \ - \ 1\)\)\((\)\(\(\@\(\(2 n\ - \ 1\)\/\(2\ m\)\)\) \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}]}]}], "+", FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], "4"], " ", "-", " ", RowBox[{ FractionBox[\(\(\@2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ FormBox[" ", "TraditionalForm"], RowBox[{ RowBox[{\(\(-1\)\/4\), FormBox[ FormBox[\(\(\@\(\(2 n\ - \ 1\)\/\(2\ m\)\)\) \((a\_\(2 m\)\)\), "TraditionalForm"], "TraditionalForm"]}], "+", FormBox[ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"], " ", "-", " ", RowBox[{ FractionBox[\(4 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ FormBox[" ", "TraditionalForm"], RowBox[{ FormBox[ FormBox[\(\(\@\(\(2 n\ - \ 1\)\/\(2\ m\)\)\) \((a\_\(2 m\)\)\), "TraditionalForm"], "TraditionalForm"], "+", FormBox[ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"], " ", "-", " ", RowBox[{ FractionBox[\(4 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ FormBox[" ", "TraditionalForm"], RowBox[{ FormBox[ FormBox[\(\(\@\(\(2 n\ - \ 1\)\/\(2\ m\)\)\) \((a\_\(2 m\)\)\), "TraditionalForm"], "TraditionalForm"], "+", FormBox[ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"], " ", "-", " ", RowBox[{ FractionBox[\(4 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}]}], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ RowBox[{ FormBox[" ", "TraditionalForm"], RowBox[{ FormBox[ FormBox[\((a\_\(2 m\)\), "TraditionalForm"], "TraditionalForm"], "+", FormBox[ RowBox[{ RowBox[{ FormBox[\(a\_\(2 m\)\^+\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], TraditionalForm]]], "\n\n[21d]: Inverse of [21b]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{"(", FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], "4"], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[ FormBox[\(B\_\(2 n\ \ - \ 1, \ 2 m\)\ \((\(\@\(\(2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\) \ \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\ B\_\(2 m, \ 2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{"(", FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], "4"], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \ \)\(\[Sum]\+\(m = 1\)\%\[Infinity]\)\), "TraditionalForm"], FormBox[ FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\(\( 2\ m\)\/\(2\ n\ - \ 1\)\) \(B\_\(2 m, \ 2\ n\ - \ 1\)\) B\_\(2 n\ \ - \ 1, \ 2 m\)\ \((\(\@\(\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\/\ \(\(2\) \(m\)\(\ \)\)\)\) \((a\_\(2 m\)\)\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], ")"}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\ B\_\(2 m, \ 2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{"(", FractionBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], "4"], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \ \ \)\(\(-1\)\/4\)\), "TraditionalForm"], FormBox[ FormBox[\(\(\@\(\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \ \)\)\/\(\(2\) \(m\)\(\ \)\)\)\) \((a\_\(2 m\)\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n- ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\ B\_\(2 m, \ 2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\@2\), RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[" ", "TraditionalForm"], FormBox[ FormBox[\(\(\@\(\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \ \)\)\/\(\(2\) \(m\)\(\ \)\)\)\) \((a\_\(2 m\)\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n- ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\ B\_\(2 m, \ 2\ n\ - \ 1\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\@2\), \(\@\(\(\(2\) \(m\)\(\ \)\)\/\(\(2 n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\), RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", FormBox[ RowBox[{ RowBox[{ FormBox[" ", "TraditionalForm"], FormBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "TraditionalForm"]}], "-", " ", FormBox[ RowBox[{ SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n\n[22e]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@2\), \(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"], " ", "-", " ", RowBox[{ FractionBox[\(4 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}]}]}], TraditionalForm]]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\)\)]], Cell[BoxData[ FormBox[ RowBox[{\(\@2\), \(\@\(\(\(2\) \(m\)\(\ \)\)\/\(\(2 n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\), RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}]}], TraditionalForm]]], " = 2 ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(1\/\@2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", RowBox[{ RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"], " ", "-", " ", RowBox[{ FractionBox[\(4 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], "P"}]}], ")"}], "-", RowBox[{ FormBox[\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], ")"}]}]}]}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{ RowBox[{"-", " ", FractionBox[\(4 \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]]}], "P", " ", \(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), \(\@\(\(2\ m\)\/\(2\ \ n\ - \ 1\)\)\)}]}], "+", RowBox[{\(1\/\@2\), \(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", RowBox[{ RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}], "-", RowBox[{ FormBox[\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], ")"}]}]}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "\n\nOr evaluating the sum for the P term,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\((\(-1\))\)\^m\/\@\(2\ m\)\) P\), " ", "+", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", RowBox[{ RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "+", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}], "-", RowBox[{ FormBox[\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(b\_\(R\ n\)\), " ", "-", " ", FormBox[\(b\_\(L\ n\)\^+\), "TraditionalForm"], " ", "-", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"], " "}], "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}], ")"}]}]}]}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\((\(-1\))\)\^m\/\@\(2\ m\)\) P\), " ", "+", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "-", FormBox[\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(\(b\_\(R\ n\)\)\()\)\), " ", "+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "+", FormBox[\(\(\ \)\(B\_\(2 m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \((\ \(b\_\(L\ n\)\^+\)\)}], "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}], ")"}]}]}]}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "\n\n[22f]\n\nThe expression for ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], " is obtainable by switching ", Cell[BoxData[ \(TraditionalForm\`\(a\_\(2 m\)\^+\)\)]], " to ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], " in the formula above and by switching each of the ", Cell[BoxData[ \(TraditionalForm\`b\)]], " terms to the corresponding ", Cell[BoxData[ \(TraditionalForm\`\(b\^+\)\)]], "\n\n[23a]\n\nLetting \n", Cell[BoxData[ \(TraditionalForm\`b\_\(\((-)\) n\)\ = \ \(1\/\@2\) \((b\_\(L\ n\)\ - \ \ b\_\(R\ n\))\)\)]], "\nand \n", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\) n\)\ = \ \(1\/\@2\) \((b\_\(L\ n\)\ + \ \ b\_\(R\ n\))\)\)]], "\n\n[23b]\n\nNow writing the generic fom for the full-string vacuum, ", StyleBox["0", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\)\ n\)\) \(\ \[Phi]\_\(n\ m\)\) b\_\(\((+)\)\ m\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices)\n\nWhere the bold ", StyleBox["0 ", FontWeight->"Bold"], StyleBox["takes the place of the traditional ket notation.\n\nNote that in \ the generic form, combinations ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(n + -\)\) \(\ \[Phi]\_\(n\ m\)\) b\_\(m\ + -\)\)\)]], " with a minus subscript were ignored since any minus subscript appearing \ in the exponential would annihilate the vacuum for each term in the expansion \ except the first, leaving a 1 factor for that term since ", Cell[BoxData[ FormBox[ RowBox[{\(a\_n\), " ", StyleBox["0", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}], TraditionalForm]]], "= 0 and ", Cell[BoxData[ \(TraditionalForm\`\(1\/\@2\) \((b\_\(L\ n\)\ - \ b\_\(R\ n\))\)\ = \ a\_\(2 n - 1\)\)]], "\n\nFurthermore, I know that \n\n", Cell[BoxData[ \(TraditionalForm\`\(\(b\_\(\((+)\) \(n\)\(\ \)\)\)\(=\)\(\ \)\)\)]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FormBox[ RowBox[{ FractionBox[\(2 \(\@ 2\) \((\(-1\))\)\^\(n - 1\)\), RowBox[{"\[Pi]", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(2 n\ - \ 1\), "TraditionalForm"], ")"}], \(3\/2\)]}]], \(p\_\(\_0\)\)}], "TraditionalForm"], FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \ \)\(\(+\ \(2\/\@2\)\)\ \(\[Sum]\+\(m = \ 1\)\%\[Infinity]\@\(\( 2 m\)\/\(\(2 n\)\(\ \ \)\(-\)\(\ \)\(1\)\(\ \)\)\)\)\ \)\), "TraditionalForm"], FormBox[ RowBox[{"(", " ", FormBox[ RowBox[{ RowBox[{ "(", \(B\_\(2 n\ \ - \ 1, \ 2 m\)\ - \ B\_\(2 m, \ 2 n\ \ - \ 1\)\), " ", FormBox[")", "TraditionalForm"]}], \(a\_\(2 m\)\)}], "TraditionalForm"]}], "TraditionalForm"]}], "-", " ", RowBox[{"(", RowBox[{\(B\_\(2 n\ \ - \ 1, \ 2 m\)\), " ", "+", " ", RowBox[{\(B\_\(2 m, \ 2 n\ \ - \ 1\)\), FormBox[ RowBox[{ RowBox[{")", SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"], FormBox["", "TraditionalForm"]}], ")"}], "TraditionalForm"]}]}]}]}], "TraditionalForm"]}]}], TraditionalForm]]], "\n\nActing with ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], ", where k is even, in order to isolate ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_\(n\ m\)\)]], "\n\n[24a]\n\n", StyleBox["0", FontWeight->"Bold"], " =", Cell[BoxData[ FormBox[ StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (summing over repeated indices)\n", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " ", StyleBox["0", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(a\_k\), " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(a\_k\), \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\ \)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n\nNow using the identity ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\(w\^T\)[A]\^\(-1\)\) w\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ A)\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i = 1\)\%N\( \ \[ExponentialE]\^\(\(-\ x\^T\) A\ x\)\) \(\[ExponentialE]\^\(\(w\^T\) x\)\) \[DifferentialD]x\_i\)\)]], "\nwhere w is a vector, T represents a transpose operation, N represents \ the dimension of A, and x is a vector with the i-th element indexed ", Cell[BoxData[ \(TraditionalForm\`x\_i\)]], "where i \[Element] Integers \[Element] [1, N]\n\nThe multiplication under \ the ", Cell[BoxData[ \(TraditionalForm\`\[Product]\)]], "affects only terms with an index i.\n\nI take for granted that this \ extends to infinite-dimensional matrices and that the determinant of \[Phi] \ exists and is nonzero.\n\n[24b]\n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(a\_k\), " ", \(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\ \)\), \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}]}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " \n\nAnd now I seek to commute ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " through [24b] to act on ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], ", leaving me a new gaussian and an expression for [\[Phi]].\n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\ \)\), \(a\_k\), " ", \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}]}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " \n\nExpressing ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " in terms of b, \n\n[24c]\n\n", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\(\((\(-1\))\)\^\(k\/2\)\/\@k\) P\), " ", "+", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(\(b\_\(R\ n\)\)\()\)\), " ", "+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \((\ \(b\_\(L\ n\)\^+\)\)}], "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}], ")"}]}]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\(\((\(-1\))\)\^\(k\/2\)\/\@k\) P\), " ", "+", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox[ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], "+"], " ", "+", " ", FormBox[ RowBox[{\(\(b\_\(R\ n\)\^+\)\()\)\), " ", "+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \((\ b\_\(L\ n\)\)}], "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}], ")"}]}]}]}], TraditionalForm]]], " (k even)\n\nI seek to simplify this definition somewhat: since I am \ operating on a vacuum state, the P term can be ignored\n\n", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", RowBox[{"(", RowBox[{ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], " ", "+", " ", FormBox[ RowBox[{\(\(b\_\(R\ n\)\)\()\)\), " ", "+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \((\ \(b\_\(L\ n\)\^+\)\)}], "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\^+\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/\@2\), \(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", RowBox[{"(", RowBox[{ SuperscriptBox[ FormBox[\(b\_\(L\ n\)\), "TraditionalForm"], "+"], " ", "+", " ", FormBox[ RowBox[{\(\(b\_\(R\ n\)\^+\)\()\)\), " ", "+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \((\ b\_\(L\ n\)\)}], "TraditionalForm"], " ", "+", " ", FormBox[\(\(b\_\(R\ n\)\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n\n\n", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], FormBox[ RowBox[{"+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \(b\_\(\((+)\)\ n\)\^+\)}], "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(k\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "-", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{"+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(\(2\ n\ - \ 1\)\(,\)\(\ \)\(k\)\(\ \)\)\), "+", FormBox[\(\(\ \)\(B\_\(k, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \(b\_\(\((+)\)\ n\)\)}], "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n[24d]\n\nDefining\n\n", Cell[BoxData[ \(TraditionalForm\`M\_\(\((1)\)\ n, \ m\)\ = \ \(1\/\[Pi]\) \@\(\(2\ n\ \)\/\(2\ m\ - \ 1\)\)\ \((\(-1\))\)\^\(n\ + \ m\)\/\(n\ - \ \((\ m\ - \ 1\ \/2)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`M\_\(\((2)\)\ n, \ m\)\ = \ \(1\/\[Pi]\) \@\(\(2\ n\ \)\/\(2\ m\ - \ 1\)\)\ \((\(-1\))\)\^\(n\ + \ m\)\/\(n\ + \ \((\ m\ - \ 1\ \/2)\)\)\)]], "\n\nAnd recalling that\n\n", Cell[BoxData[ \(TraditionalForm\`B\_\(n, \ m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^\(\(n\ + \ m\ + \ \ 1\)\/2\)\/\[Pi]\)]], "(", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ + \ m\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`1\/\(n\ - \ m\)\)]], "),\n\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\_\(2 m\)\^+\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "-", FormBox[\(\(\ \)\(B\_\(2\ m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{"+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "+", FormBox[\(\(\ \)\(B\_\(2\ m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \(b\_\(\((+)\)\ n\)\)}], "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\_\(2 m\)\^+\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(\((\(-1\))\)\^\(\(\ \)\(n\ + \ \ \ m\)\)\/\[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(2 n\ - \ 1\ + \ 2\ m\)\), "TraditionalForm"], "-", FormBox[\(1\/\(2 n\ - \ 1\ - \ 2\ m\)\), "TraditionalForm"]}], ")"}]}], "-", RowBox[{\(\((\(-1\))\)\^\(m\ + \ n\)\/\[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(2\ m\ + \ \((2\ n\ - \ 1)\)\)\), "TraditionalForm"], "-", FormBox[\(1\/\(2\ m\ - \ \((2\ n\ - \ 1)\)\)\), "TraditionalForm"]}], ")"}]}]}], ")"}], " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{"+", " ", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "+", FormBox[\(\(\ \)\(B\_\(2\ m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \(b\_\(\((+)\)\ n\)\)}], "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\_\(2 m\)\^+\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(M\_\(\((1)\)\ m, \ n\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ \@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(B\_\(2\ n\ - \ 1, \ \(\(2\)\(\ \)\(m\)\(\ \)\)\)\), "+", FormBox[\(\(\ \)\(B\_\(2\ m, \ 2\ n\ - \ 1\)\)\), "TraditionalForm"]}], ")"}], \(b\_\(\((+)\)\ n\)\)}], "TraditionalForm"]}]}], "TraditionalForm"]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " =", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox[ FormBox[\(a\_\(2 m\)\), "TraditionalForm"], "+"]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(M\_\(\((1)\)\ m, \ n\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ \@\(\(2\ m\)\/\(2\ n\ - \ 1\)\)\), FormBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(\((\(-1\))\)\^\(\(\ \)\(n\ + \ \ m\)\)\/\ \[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(2 n\ - \ 1\ + \ 2\ m\)\), "TraditionalForm"], "-", FormBox[\(1\/\(2 n\ - \ 1\ - \ 2\ m\)\), "TraditionalForm"]}], ")"}]}], "+", RowBox[{\(\((\(-1\))\)\^\(m\ + \ n\)\/\[Pi]\), RowBox[{"(", RowBox[{ FormBox[\(1\/\(2\ m\ + \ \((2\ n\ - \ 1)\)\)\), "TraditionalForm"], "-", FormBox[\(1\/\(2\ m\ - \ \((2\ n\ - \ 1)\)\)\), "TraditionalForm"]}], ")"}]}]}], ")"}], \(b\_\(\((+)\)\ n\)\)}], "TraditionalForm"]}]}], "TraditionalForm"]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`\(a\_k\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(a\_\(2 m\)\^+\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(M\_\(\((1)\)\ m, \ n\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ M\_\(\((2)\)\ m, \ n\)\), FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"]}]}], "TraditionalForm"]}]}], TraditionalForm]]], " (k even)\n\n", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " = ", Cell[BoxData[ \(TraditionalForm\`a\_\(2 m\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(M\_\(\((1)\)\ m, \ n\)\), " ", FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ M\_\(\((2)\)\ m, \ n\)\), FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}]}], "TraditionalForm"]}]}], TraditionalForm]]], " (k even)\n\n[24e]\n\nSubstituting,\n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\ \)\), \(a\_k\), " ", \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}]}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (k even)\n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ \ x\)\), "(", RowBox[{\(\[Sum]\+\(A = 1\)\%\[Infinity]\), RowBox[{\(M\_\(\((1)\)\ k\/2, \ A\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ A\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ M\_\(\((2)\)\ k\/2, \ A\)\), FormBox[\(b\_\(\((+)\)\ A\)\), "TraditionalForm"]}]}], "TraditionalForm"]}]}], ")"}], " ", \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], TraditionalForm]]], " (k even)\n\nWhich I then wish to commute. The ", Cell[BoxData[ \(TraditionalForm\`b\_\((+)\)\)]], "part is the hard part, since [", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(b\_\(\((+)\)\(\ \)\)\), "TraditionalForm"], "+"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(b\_\(\((+)\)\(\ \)\)\), "TraditionalForm"], "+"], TraditionalForm]]], "] = 0\n\nTo make things as simple as possible, I note that the only things \ here inhibiting commutativity are the b values. Therefore, I'll commute test \ values and substitute back into [24e].\n\n[24f]\n ", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\(\ \)\(m\)\(\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(b\_\((+)\)\^+\) \ \[Phi]\ \(b\_\((+)\)\^+\)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " \n ", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\(\ \)\(m\)\(\ \)\)\), " ", \(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\ \)\), " ", \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], TraditionalForm]]], " \n ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \(x\^T\)[\[Phi]]\^\(-1\)\)\ \ x\)\), \(b\_\(\((+)\)\(\ \)\(m\)\(\ \)\)\), " ", \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}]}], TraditionalForm]]], " \n \n ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[PartialD]\_\(\[Rho]\_m\)\ \ \(\(\[ExponentialE]\)\(\ \)\)\^\(\(b\_\((+)\)\) \[Rho]\)\), \(\[ExponentialE]\ \^\(\(b\_\((+)\)\^+\) x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \n Where \[Rho] is defined as a vector indexed like b.\n\nNow commuting \ the exponentials, \n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[PartialD]\_\(\[Rho]\_m\)\ \[ExponentialE]\^\(\(b\ \_\((+)\)\^+\) x\)\), " ", \(\(\(\[ExponentialE]\)\(\ \)\)\^\(\(b\_\((+)\)\) \ \[Rho]\)\), \(\[ExponentialE]\^\([\(b\_\((+)\)\) \[Rho], \ \ \ \ \(b\_\((+)\)\^+\) x]\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[PartialD]\_\(\[Rho]\_m\)\ \[ExponentialE]\^\(\(b\ \_\((+)\)\^+\) x\)\), " ", \(\(\(\[ExponentialE]\)\(\ \)\)\^\(\(b\_\((+)\)\) \ \[Rho]\)\), \(\[ExponentialE]\^\(\[Rho]\ x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \nNow allowing the ", Cell[BoxData[ \(TraditionalForm\`\(\(\[ExponentialE]\)\(\ \)\)\^\(\(b\_\((+)\)\) \ \[Rho]\)\)]], "to act on the vacuum states, ", Cell[BoxData[ \(TraditionalForm\`b\_\((+)\)\)]], " annihilates the vacuum in each term of \[ExponentialE]'s expansion except \ for the first term, 1.\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[PartialD]\_\(\[Rho]\_m\)\[ExponentialE]\^\(\(b\_\ \((+)\)\^+\) x\)\), " ", \(\[ExponentialE]\^\(\[Rho]\ x\)\), \(\ \[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_m\)\(\(\[Pi]\^\(N\/2\)\)(det\ \ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\) + \ \ \[Rho])\)\ x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \n\nRe-combining my gaussian,\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_m\)\(\(\[Pi]\^\(N\/2\)\)(det\ \ \[Phi])\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ \ \(x\^T\)[\[Phi]]\^\(-1\)\)\ x\)\), " ", \(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\) + \ \ \[Rho])\)\ x\)\), \(\[DifferentialD]x\_i\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " \n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_m\)\[ExponentialE]\^\(\(-\(1\/2\)\ \) \((\(b\_\(\((+)\) A\)\^+\) + \ \[Rho]\_A)\)\ \(\(\[Phi]\_\(A, \ \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\) + \ \[Rho]\_B)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " (summing over repeated indices)\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[PartialD]\_\(\[Rho]\_m\)\)\)\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\(-\(1\/2\)\) \((\(b\_\(\((+)\) A\)\^+\) + \ \ \[Rho]\_A)\)\ \(\(\[Phi]\_\(A, \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\) + \ \[Rho]\_B)\))\), \ \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\(\((+)\) A\)\^+\) + \ \ \[Rho]\_A)\)\ \(\(\[Phi]\_\(A, \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\) + \ \[Rho]\_B)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}]}], TraditionalForm]]], " (summing over repeated indices)\n", Cell[BoxData[ \(TraditionalForm\`\(-\(1\/2\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((\[Phi]\_\(m, \ B\)\ \((\(\(b\_\((+)\)\)\_B\^+\) + \ \ \[Rho]\_B)\) + \(\[Phi]\_\(A, \ m\)\)(\(b\_\(\((+)\) A\)\^+\) + \ \[Rho]\_A))\), \(\ \[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\(\((+)\) A\)\^+\) + \ \[Rho]\_A)\)\ \ \(\(\[Phi]\_\(A, \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\) + \ \[Rho]\_B)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], \( | \_\(\(\ \)\(\[Rho]\ = \ 0\)\)\)}], TraditionalForm]]], " (summing over repeated indices)\n", Cell[BoxData[ FormBox[ RowBox[{\(-\(1\/2\)\), \((\[Phi]\_\(B, \ m\)\^T\ \ \((\(\(b\_\((+)\)\)\_B\^+\))\) + \ \(\[Phi]\_\(A, \ m\)\)(\(b\_\(\((+)\) A\)\^+\)))\), \ \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\(\((+)\) A\)\^+\))\)\ \ \(\(\[Phi]\_\(A, \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\))\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices)\n\nRe-Indexing, \n\n", Cell[BoxData[ FormBox[ RowBox[{\(-\(1\/2\)\), \(\((\[Phi]\^T + \ \[Phi])\)\_\(m, \ C\)\), " ", \((\(\(b\_\((+)\)\)\_C\^+\))\), \(\[ExponentialE]\^\(\(-\(1\/2\ \)\) \((\(b\_\(\((+)\) A\)\^+\))\)\ \(\(\[Phi]\_\(A, \ B\)\)(\(\(b\_\((+)\)\)\_B\^+\))\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices)\n\nAnd assuming that ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\^T\)]], "= \[Phi],\n \ncommuting ", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\(\ \)\(m\)\(\ \)\)\)]], " through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \ \(b\_\((+)\)\^+\) \[Phi]\ \(b\_\((+)\)\^+\)\)\)\)]], "yields\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[ RowBox[{\(b\_\(\((+)\)\(\ \)\(m\)\(\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \ \(b\_\((+)\)\^+\) \[Phi]\ \(b\_\((+)\)\^+\)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], " ", "=", RowBox[{\(-\ \[Phi]\_\(m, \ C\)\), " ", \((\(\(b\_\((+)\)\)\_C\^+\))\), \ \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\))\)\ \ \(\[Phi](\(b\_\((+)\)\^+\))\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}], TraditionalForm]]], " (summing over repeated indices)\n \n[24g]\nNow substituting into the \ original equation, \n\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(a\_k\), \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\ \)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n0 = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(M\_\(\((1)\)\ k\/2, \ A\)\), " ", FormBox[\(b\_\(\((+)\)\ A\)\), "TraditionalForm"], " ", FormBox[ RowBox[{" ", RowBox[{\(+\ M\_\(\((2)\)\ k\/2, \ A\)\), FormBox[\(b\_\(\((+)\)\ A\)\^+\), "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n0 = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(M\_\(\((2)\)\ k\/2, \ A\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ A\)\), "TraditionalForm"], "+"], FormBox[ RowBox[{" ", RowBox[{\(-\ M\_\(\((1)\)\ k\/2, \ A\)\), FormBox[\(\(\ \)\(\[Phi]\_\(A, \ B\)\ \((\(\(b\_\((+)\)\)\ \_B\^+\))\)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \nRe-indexing\n0 = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(M\_\(\((2)\)\ k\/2, \ B\)\), " ", SuperscriptBox[ FormBox[\(b\_\(\((+)\)\ B\)\), "TraditionalForm"], "+"], " ", FormBox[ RowBox[{" ", RowBox[{\(-\ M\_\(\((1)\)\ k\/2, \ A\)\), FormBox[\(\(\ \)\(\[Phi]\_\(A, \ B\)\ \((\(\(b\_\((+)\)\)\ \_B\^+\))\)\)\), "TraditionalForm"]}]}], "TraditionalForm"]}], ")"}], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n0 = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(M\_\(\((2)\)\ k\/2, \ B\)\), " ", FormBox[ RowBox[{\(-\ M\_\(\((1)\)\ k\/2, \ A\)\), FormBox[\(\(\ \)\(\[Phi]\_\(A, \ B\)\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], ")"}], \(\(b\_\((+)\)\)\_B\^+\), " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) m\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \nDividing out the troublesome parts,\n0 = ", Cell[BoxData[ FormBox[ RowBox[{\(M\_\(\((2)\)\ k\/2, \ B\)\), " ", FormBox[ RowBox[{\(-\ M\_\(\((1)\)\ k\/2, \ A\)\), FormBox[\(\(\ \)\(\[Phi]\_\(A, \ B\)\)\(\ \)\), "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n ", Cell[BoxData[ FormBox[ RowBox[{\(M\_\(\((2)\)\ k\/2, \ B\)\), " ", FormBox[ RowBox[{"=", " ", SubscriptBox[ FormBox[\(\(\ \)\((\ \(M\_\((1)\)\) \[Phi]\ )\)\), "TraditionalForm"], \(k\/2, \ B\)]}], "TraditionalForm"]}], TraditionalForm]]], " (summing over repeated indices: ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is even) \n ", Cell[BoxData[ FormBox[ RowBox[{\(M\_\(\((2)\)\(\ \)\)\), " ", FormBox[ RowBox[{"=", " ", FormBox[\(\(\ \)\(\(M\_\((1)\)\) \(\[Phi]\)\(\ \)\)\), "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(\(\ \)\(\[Phi]\)\(\ \)\), "TraditionalForm"], "=", \(M\_\((1)\)\^\(-1\)\ M\_\((2)\)\)}]}], TraditionalForm]]], "\n \n Now all I need to do is evaluate \[Phi]. Unfortunately, the \ matrices M are not easy to invert. \n \n Noticing a few identities involving \ M, \n \n ", Cell[BoxData[ \(TraditionalForm\`M\_\(\((1)\)\ n, \ m\)\ = \ \(1\/\[Pi]\) \@\(\(2\ n\ \)\/\(2\ m\ - \ 1\)\)\ \((\(-1\))\)\^\(n\ + \ m\)\/\(n\ - \ \((\ m\ - \ 1\ \/2)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`M\_\(\((2)\)\ n, \ m\)\ = \ \(1\/\[Pi]\) \@\(\(2\ n\ \)\/\(2\ m\ - \ 1\)\)\ \((\(-1\))\)\^\(n\ + \ m\)\/\(n\ + \ \((\ m\ - \ 1\ \/2)\)\)\)]], "\n\n [24h]\n \n", Cell[BoxData[ \(TraditionalForm\`\(\(\(M\_\(\((1)\)\(\ \)\)\) M\_\(\((1)\)\(\ \)\)\^T - \ \(M\_\(\((2)\)\(\ \)\)\) M\_\(\((2)\)\(\ \)\)\^T\)\(\ \)\(=\)\(\ \)\([I]\)\(\ \)\)\)]], "\n\n[24i] (trivial)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(M\_\(\((1)\)\(\ \)\)\) M\_\(\((2)\)\(\ \)\)\^T - \ \(M\_\(\((2)\)\(\ \)\)\) M\_\(\((1)\)\(\ \)\)\^T\)\(\ \)\(=\)\(\ \)\([0]\)\(\ \)\)\)]], "\n\nAlso note that\n\n", Cell[BoxData[ \(TraditionalForm\`\((A\^\(-1\))\)\^T\ = \ \((A\^T)\)\^\(-1\)\)]], "\n\nand \n\n", Cell[BoxData[ \(TraditionalForm\`\((A\ B)\)\^T\ = \ \(\(B\^T\)\(\ \)\(A\^T\)\(\ \)\)\ \)]], "\n\nand\n\n", Cell[BoxData[ \(TraditionalForm\`\((A\ B)\)\^\(-1\)\ = \ B\^\(-1\)\ A\^\(-1\)\)]], "\n\nHowever, the inverse is still elusive, therefore I will choose to \ verify that\n\n", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_\(n, \ m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\@\(2\ n\ - \ 1\)\) \(\@\(2\ m\ - \ 1\)\) \(1\/\(2\ \((n\ + \ m\ - \ 1)\)\)\) \((\(n\ - \ 1\)\&\(\(-1\)\/2\))\) \((\(m\ - \ \ 1\)\&\(\(-1\)\/2\))\)\)]], "\nBy showing that\n\n ", Cell[BoxData[ FormBox[ RowBox[{\(M\_\(\((2)\)\(\ \)\)\), " ", FormBox[ RowBox[{"=", " ", FormBox[\(\(\ \)\(\(M\_\((1)\)\) \(\[Phi]\)\(\ \)\)\), "TraditionalForm"]}], "TraditionalForm"]}], TraditionalForm]]], "\n \n [24h]\n \n Evaluating the expression ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\(M\_\((2)\)\) \[Phi]\)\(\ \ \)\(,\)\)\)\)]], "\n \n ", Cell[BoxData[ \(TraditionalForm\`M\_\(\((2)\)\ a, \ c\)\ = \ \[Sum]\+\(b = 1\)\%\ \[Infinity] M\_\(\((1)\)\ a, \ b\)\ \[Phi]\_\(b, \ c\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\[Sum]\+\(b = 1\)\%\[Infinity]\( 1\/\[Pi]\) \@\(\(2\ a\)\/\(2\ b\ - \ 1\)\)\ \ \((\(-1\))\)\^\(a\ + \ b\)\/\(a\ - \ \((\ b\ - \ 1\/2)\)\)\ \(\@\(2\ b\ - \ \ 1\)\) \(\@\(2\ c\ - \ 1\)\) \(1\/\(2\ \((b\ + \ c\ - \ 1)\)\)\) \((\(b\ - \ 1\)\&\(\(-1\)\/2\))\) \((\(c\ - \ \ 1\)\&\(\(-1\)\/2\))\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \((\(c\ - \ 1\)\&\(\(-1\)\/2\))\) \ \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(\[Sum]\+\(b = 1\)\%\[Infinity]\ \ \((\(-1\))\)\^b\/\(a\ - \ \((\ b\ - \ 1\/2)\)\)\ \(1\/\((b\ + \ c\ - \ 1)\)\) \((\(b\ - \ 1\)\&\(\(-1\)\/2\))\)\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \((\(c\ - \ 1\)\&\(\(-1\)\/2\))\) \ \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(\[Sum]\+\(b = 0\)\%\[Infinity]\ \ \((\(-1\))\)\^b\/\(a\ - \ \((\ b\ - \ 1\/2)\)\)\ \(1\/\((b\ + \ c\ )\)\) \((b\&\(\(-1\)\/2\))\)\)\)\)\)]], " (The b=0 term is zero and does not contribute to the sum)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \((\(c\ - \ 1\)\&\(\(-1\)\/2\))\) \ \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(\[Sum]\+\(b = 0\)\%\[Infinity]\ \(\((\(-1\ \))\)\^b\) \((1\/\(b\ - \ \((a\ + \ 1\/2)\)\) - \ 1\/\(b\ + \ c\))\)\ 1\/\(c\ + \ a\ - \ 1\/2\)\ \ \((b\&\(\(-1\)\/2\))\)\)\)\)\)]], " (Identity substitution)\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\(-1\)\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \((\(c\ - \ 1\)\&\(\(-1\)\/2\))\) \ \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(1\/\(c\ + \ a\ - \ 1\/2\)\) \(\[Sum]\+\(b = 0\)\%\[Infinity]\ \ \(\((\(-1\))\)\^b\) \((1\/\(b\ + \ \ c\))\)\ \ \((b\&\(\(-1\)\/2\))\)\)\)\)\)]], " (Remove term that sums to zero, factor)\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\(-1\)\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \((\(c\ - \ 1\)\&\(\(-1\)\/2\))\) \ \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(1\/\(c\ + \ a\ - \ 1\/2\)\) \(\(\[CapitalGamma](1\/2)\)\ \ \(\[CapitalGamma](c)\)\)\/\(\[CapitalGamma](c\ + \ 1\/2)\)\)\)\)]], " (Identity from Gross and Jewicki, page 22)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\(-1\)\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \(\(\(\((\(-1\))\)\^\(c\ - \ 1\)\) \(\[CapitalGamma]( c\ - \ 1\ + \ 1)\)\)\/\(\(\[CapitalGamma]( c\ - \ 1\/2 + \ 1)\) \(\[CapitalGamma](\(-\(1\/2\)\) + \ 1)\)\)\) \(\((\(-1\))\)\^\(\(a\)\(\ \)\)\) \(1\/\(c\ + \ a\ - \ 1\/2\)\) \(\(\[CapitalGamma](1\/2)\)\ \ \(\[CapitalGamma](c)\)\)\/\(\[CapitalGamma](c\ + \ 1\/2)\)\)\)\)]], " (Identity from Gross and Jewicki, page 22)\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \(\(\[CapitalGamma](c)\)\/\(\[CapitalGamma]( c\ + \ 1\/2)\)\) \(\((\(-1\))\)\^\(\(a\)\(\ \)\(+\)\(\ \)\(c\)\(\ \ \)\)\) \(1\/\(c\ + \ a\ - \ 1\/2\)\) \(\(\ \ \)\(\[CapitalGamma](c)\)\)\/\(\[CapitalGamma](c\ + \ 1\/2)\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \(\(\[CapitalGamma](c\ - \ 1\/2)\)\/\(\[CapitalGamma]( c)\)\) \(\((\(-1\))\)\^\(\(a\)\(\ \)\(+\)\(\ \)\(c\)\(\ \ \)\)\) \(1\/\(c\ + \ a\ - \ 1\/2\)\) \(\(\ \ \)\(\[CapitalGamma](c)\)\)\/\(\[CapitalGamma](c\ + \ 1\/2)\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \(\(\[CapitalGamma](c\ - \ 1\/2)\)\/\(\[CapitalGamma]( c\ + \ 1\/2)\)\) \(\((\(-1\))\)\^\(a\ + \ c\)\) 1\/\(c\ + \ a\ - \ 1\/2\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(1\/\(2\ \[Pi]\)\ \(\@\(2\ a\)\) \ \(\@\(2\ c\ - \ 1\)\) \(1\/\(c\ - \ 1\/2\)\) \(\((\(-1\))\)\^\(\(a\)\(\ \)\(+\)\(\ \)\(c\)\(\ \ \)\)\) 1\/\(c\ + \ a\ - \ 1\/2\)\)\)\)]], "\\\n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\(1\/\(\(\ \)\(\[Pi]\)\)\ \(\@\(2\ \ a\)\/\@\(2\ c\ - \ 1\)\) \(\((\(-1\))\)\^\(\(a\)\(\ \)\(+\)\(\ \)\(c\)\(\ \)\ \)\) 1\/\(a\ + \ \((c\ - \ 1\/2)\)\)\)\(=\)\(\ \)\(M\_\(\((2)\)\ a, \ c\)\)\(\ \ \)\)\)\)]], " (Thus the inverse is verified.)\n\nAnd so \n\n", StyleBox["0", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], TraditionalForm]]], "\n\n\nFor an example, I will now determine the half-string representation \ of the Tachyon.\n\nIn the full-string representation, the Tachyon is defined \ as \n\n[25]\n\n", StyleBox["T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ x\_0\)\), " ", StyleBox["0", FontWeight->"Bold"]}], TraditionalForm]]], " (Full-string)\n\nFirst, I'll simplify ", Cell[BoxData[ \(TraditionalForm\`p\ x\_0\)]], " and get it in terms of the half-string creation and annihilation \ operators, b and ", Cell[BoxData[ \(TraditionalForm\`\(b\^+\)\)]], "\n\n[25a]\n\nFrom [14],\n\n\[ImaginaryI] p", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(x\_0\)\(\ \)\(=\)\)\)\)]], " \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`p\ \(x(\[Pi]\/2)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`\(\(\@2\) \[ImaginaryI]\ p\)\/\[Pi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n\nFrom [18a], [18b], and [23b], \n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\(-\ \[ImaginaryI]\)\/\@2\), SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]]}], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "+ \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\(2\/\(2\ n\ - \ 1\)\) P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n\nand\n\n", Cell[BoxData[ \(TraditionalForm\`\(b\_\(Q\ n\)\^+\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\/\@2\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"(", FormBox[\(\(2 n\ - \ 1\)\/2\), "TraditionalForm"], ")"}]], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "- \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`2\/\(2\ n\ - \ 1\)\)]], Cell[BoxData[ \(TraditionalForm\`P\_\(Q\ 2 n\ - \ 1\)\)]], ")\n\nso\n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(Q\ n\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(Q\ n\)\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ FormBox[\(\(2 n\)\(\ \)\(-\)\(\ \)\), "TraditionalForm"], "1"}]], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(Q\ 2 n\ - \ 1\)\)]], "\n\nand \n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ n\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\((+)\)\ n\)\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ FormBox[\(\(2 n\)\(\ \)\(-\)\(\ \)\), "TraditionalForm"], "1"}]], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\((\[Chi]\_\(L\ 2 n\ - \ 1\)\ + \ \[Chi]\_\(R\ 2 \ n\ - \ 1\))\)\)]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Chi]\_\(L\ 2 n\ - \ 1\)\ + \ \[Chi]\_\(R\ 2 n\ - \ 1\ \)\), " ", "=", " ", RowBox[{ FractionBox[\(-\ \[ImaginaryI]\), SqrtBox[ RowBox[{ FormBox[\(\(2 n\)\(\ \)\(-\)\(\ \)\), "TraditionalForm"], "1"}]]], RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ n\)\), "-", FormBox[\(\(b\_\(\((+)\)\ n\)\^+\)\()\)\), "TraditionalForm"]}]}]}]}], TraditionalForm]]], "\n\nRecombining,\n\n[25b]\n\n\[ImaginaryI] p", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(x\_0\)\(\ \)\(=\)\)\)\)]], "\[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`p\ \(x(\[Pi]\/2)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\(\(\@2\) p\)\/\[Pi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\((2\ m\ - \ 1)\)\^\(3\/2\)\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\), "-", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\()\)\), "TraditionalForm"]}]}]}]}], TraditionalForm]]], "\n\nDefining ", Cell[BoxData[ \(TraditionalForm\`p\&_\_m\ = \(\(\(\(\@2\) p\)\/\[Pi]\) \(\((\(-1\))\)\^m\/\((2\ m\ - \ \ 1)\)\^\(3\/2\)\)\(\ \ \ \)\)\)]], ", \n\n\[ImaginaryI] p", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(x\_0\)\(\ \)\(=\)\)\)\)]], " \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`p\ \(x(\[Pi]\/2)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ \(TraditionalForm\`p\&_\_m\)]], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\), "-", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\()\)\), "TraditionalForm"]}]}], TraditionalForm]]], " \n\nAnd rewriting [25],\n\n", StyleBox["T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\ p\&_\_c\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ c\)\), "-", FormBox[\(\(b\_\(\((+)\)\ c\)\^+\)\()\)\), "TraditionalForm"]}]}]}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], " (summing over repeated indices)\n\nWhich should hopefully be an \ eigenstate of the momentum operator.\n\nNow looking for a suitable momentum \ operator P, \n\nfrom [1b], \n\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " = ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "\[ImaginaryI] (", Cell[BoxData[ \(TraditionalForm\`a\_0\)]], "- ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[\(a\_0\), "TraditionalForm"], "+"], TraditionalForm]]], ")\n", Cell[BoxData[ \(TraditionalForm\`\(x\_0\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\/2\)]], "\[ImaginaryI] (", Cell[BoxData[ \(TraditionalForm\`\(a\_0\^+\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`a\_0\)]], ")\n", Cell[BoxData[ \(TraditionalForm\`p\_0\)]], " = (", Cell[BoxData[ \(TraditionalForm\`a\_\(\(0\)\(\ \)\) + \ \(a\_0\^+\)\)]], ")\n\nA commutator exists\n\n[", Cell[BoxData[ \(TraditionalForm\`x\_0, \ P\)]], "] = \[ImaginaryI]\n\nand\n\nP = ", Cell[BoxData[ \(TraditionalForm\`p\_0\)]], "= - \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_0\)\)]], "= - \[ImaginaryI] ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x(\[Pi]\/2)\)\)]], ", \n\nand since\n", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x(\[Pi]\/2)\)\)\)]], "- ", Cell[BoxData[ \(TraditionalForm\`\@2\/\[Pi]\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\)]], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n\n\nWorking with the commutator, \n\n[", Cell[BoxData[ \(TraditionalForm\`x\_0, \ P\)]], "] = \[ImaginaryI]\n \n Inserting the most general form,\n \n[", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\(x(\[Pi]\/2)\), "-", RowBox[{ FormBox[\(\@2\/\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(\[Chi]\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(\[Chi]\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], "TraditionalForm"]}]}]}], ",", " ", RowBox[{\(c\ P\), " ", "+", " ", RowBox[{"d", " ", FormBox[\(\@2\/\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], FormBox[ RowBox[{" ", RowBox[{\(\((\(-1\))\)\^m\/\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{ FormBox[\(\(P\_\(L\ 2\ m - 1\)\)\(\ \)\), "TraditionalForm"], "+", " ", FormBox[\(P\_\(R\ \ 2 m - 1\)\), "TraditionalForm"]}], ")"}]}]}], "TraditionalForm"], " "}]}]}]}]}], TraditionalForm]]], "] = \[ImaginaryI]\n \nExpanding and removing the cross terms which do not \ contribute (terms whose subscript does not match)\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(c\ \ [x(\[Pi]\/2), \ P\ \ ]\), " ", "+", " ", RowBox[{"d", " ", FormBox[\(2\/\[Pi]\^2\), "TraditionalForm"], RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\(1\/\((2\ m\ - \ 1)\)\^2\), RowBox[{"(", " ", RowBox[{ RowBox[{"[", RowBox[{\(\[Chi]\_\(L\ 2\ m - 1\)\), ",", FormBox[\(P\_\(L\ 2\ m - 1\)\), "TraditionalForm"]}], "]"}], "+", " ", RowBox[{"[", RowBox[{\(\[Chi]\_\(R\ 2\ m - 1\)\), ",", FormBox[\(P\_\(R\ 2\ m - 1\)\), "TraditionalForm"]}], "]"}]}], " ", ")"}]}]}]}]}], " ", "=", " ", "\[ImaginaryI]", " "}], TraditionalForm]]], "\n \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(c\ \[ImaginaryI]\), "+", " ", RowBox[{"d", " ", FormBox[\(\(\(\ \)\(4\)\(\ \)\)\/\[Pi]\^2\), "TraditionalForm"], \(\[Sum]\+\(m = 1\)\%\[Infinity]\( 1\/\((2\ m\ - \ 1)\)\^2\) \[ImaginaryI]\)}]}], " ", "=", " ", "\[ImaginaryI]", " "}], TraditionalForm]]], "\n \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(c\ \[ImaginaryI]\), "+", " ", RowBox[{"d", " ", FormBox[\(\(\(\ \)\(4\)\)\/\[Pi]\^2\), "TraditionalForm"], \(\[Pi]\^2\/8\), "\[ImaginaryI]"}]}], " ", "=", " ", "\[ImaginaryI]", " "}], TraditionalForm]]], "\n \n So, guessing that c = 0 (otherwise I'd get extra terms in the \ Tachyon commutation)\n \n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(d\ 1\/2\)\(\ \)\(=\)\(\ \)\(1\)\(\ \ \)\)\)\)]], "\nd = 2\n \nfor \n\n", Cell[BoxData[ \(TraditionalForm\`p\_0\ = \ P\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(2\ \@2\)\/\[Pi]\ \(\[Sum]\+\(n = \ 1\)\%\[Infinity]\(\(\((\(-1\))\)\^\(n\ - \ 1\)\/\(2\ n\ - \ 1\)\) \((P\_\(\(L\ 2\ n\)\(\ \)\(-\)\(\ \)\(1\)\(\ \)\) + \ P\_\(R\ 2\ n\ - \ 1\))\)\(\ \)\)\)\)]], "\n\n\n[25d]\n\nand acting on the vacuum, this gives \n\nP ", StyleBox["T = ", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(b\_\(\((+)\)\ m\)\^+\), "TraditionalForm"]}], ")"}]}]}]}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_m\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_m\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_m\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}], TraditionalForm]]], " (Half-String Summing over repeated indicies)\n\nNow simplifying P ", StyleBox["T", FontWeight->"Bold"], StyleBox[", with the goal of commuting the b terms through to act on the \ vacuum and leave the number p as the result of the momentum operator.\n\n", FontVariations->{"CompatibilityType"->0}], "P ", StyleBox["T =", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(b\_\(\((+)\)\ m\)\^+\), "TraditionalForm"]}], ")"}]}]}]}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}], TraditionalForm]]], StyleBox["\n", FontVariations->{"CompatibilityType"->0}], "P ", StyleBox["T =", FontWeight->"Bold"], " ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(b\_\(\((+)\)\ m\)\^+\), "TraditionalForm"]}], ")"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) \ n\)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\n\nNow evaluating the commutator\n\n[", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\ n\)\), "+", FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_m\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_m\^+\)\(]\)\), "TraditionalForm"]}], TraditionalForm]]], ",\n\n(", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\ n\)\), "+", FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}], TraditionalForm]]], ")( ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_m\), "-", FormBox[\(\(\(\(b\_\(\((+)\)\(\ \)\)\)\_m\^+\)\()\)\)\(\ \)\(-\)\(\ \ \)\), "TraditionalForm"]}], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\ m\)\), "-", FormBox[\(b\_\(\((+)\)\ m\)\^+\), "TraditionalForm"]}], TraditionalForm]]], ")( ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_n\), "+", FormBox[\(\(\(\(b\_\(\((+)\)\(\ \)\)\)\_n\^+\)\()\)\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], "\n\n - ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\ n\)\) \(\(b\_\((+)\)\)\_m\^+\)\), "+", FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_m\), FormBox[\(\(\ \)\(\(+\)\(\ \)\)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(b\_\(\((+)\)\ m\)\^+\), "TraditionalForm"], \(\(b\_\(\((+)\)\(\ \)\)\)\_n\)}], " ", "-", " ", \(\(b\_\(\((+)\)\ m\)\) \(\(b\_\(\((+)\)\(\ \ \)\)\)\_n\^+\)\)}], TraditionalForm]]], "\n \n since [", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\((+)\) m\)\^+\)\)]], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(m, \ n\)\)]], ", this leaves zero if n \[NotEqual] m\n \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FormBox[ RowBox[{\(\(b\_\(\((+)\)\ n\)\) \(b\_\(\((+)\) n\)\^+\)\), "+", FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}], "TraditionalForm"]}], " ", FormBox[ RowBox[{\(b\_\(\((+)\)\ n\)\), FormBox[\(\(\ \)\(\(+\)\(\ \)\)\), "TraditionalForm"]}], "TraditionalForm"], FormBox[ RowBox[{ RowBox[{ FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"], \(b\_\(\((+)\)\ n\)\)}], " ", "-", " ", \(\(b\_\(\((+)\)\ n\)\) \(b\_\(\((+)\)\ n\)\^+\)\)}], "TraditionalForm"]}], TraditionalForm]]], " \n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FormBox[ RowBox[{\(2 \( b\_\(\((+)\)\ n\)\) \(b\_\(\((+)\) n\)\^+\)\), "+", FormBox[\(2 \( b\_\(\((+)\)\ n\)\^+\)\), "TraditionalForm"]}], "TraditionalForm"]}], " ", FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"]}], TraditionalForm]]], " \n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", FormBox[ RowBox[{\(2 \( b\_\(\((+)\)\ n\)\) \(b\_\(\((+)\) n\)\^+\)\), "+", FormBox[\(2 \( b\_\(\((+)\)\ n\)\^+\)\), "TraditionalForm"]}], "TraditionalForm"]}], " ", FormBox[\(b\_\(\((+)\)\ n\)\), "TraditionalForm"]}], " ", "-", \(2 \[Delta]\_\(m, \ n\)\)}], TraditionalForm]]], "\n \n So [", Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\((+)\)\ n\)\), "+", FormBox[\(b\_\(\((+)\)\ n\)\^+\), "TraditionalForm"]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_m\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_m\^+\)\(]\)\), "TraditionalForm"]}], TraditionalForm]]], " = -2", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Delta]\_\(m, \ n\)\)\)\)]], "\n \n (sum over repeated indices below)\n \nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{\(\[PartialD]\_\[Rho]\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[Rho]", "(", " ", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\()\)\), "TraditionalForm"]}]}]]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) \ n\)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\n \nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{\(\[PartialD]\_\[Rho]\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]]}], SuperscriptBox["\[ExponentialE]", RowBox[{"\[Rho]", "(", " ", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\()\)\), "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\ 2\)\ \[Delta]\_\(m, \ q\)\ \[Rho]\)\)\), StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) \ n\)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\nP ", StyleBox["T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{\(\[PartialD]\_\[Rho]\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\ \), "TraditionalForm"]}]}], "TraditionalForm"], " "}]]}], SuperscriptBox["\[ExponentialE]", RowBox[{"\[Rho]", "(", " ", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\()\)\), "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\ 2\)\ \(\(p\_q\)\&_\) \[Delta]\_\(m, \ q\)\ \[Rho]\)\)\ \), StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) \ n\)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"]}]}]}], StyleBox[" ", FontWeight->"Bold"], \( | \_\(\[Rho]\ = \ 0\)\)}], TraditionalForm]]], "\nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\), " ", RowBox[{ SuperscriptBox[ SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\ \), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " "], "(", RowBox[{\(b\_\(\((+)\)\ m\)\), "+", FormBox[\(\(b\_\(\((+)\)\ m\)\^+\)\ - 2\ \(p\_m\)\&_\), "TraditionalForm"]}], " ", ")"}], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) \ n\)\^+\) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\n\nand using the commutator identity found earlier to commute ", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ m\)\)]], ", \n\nP ", StyleBox["T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(2 \@ 2\)\/\[Pi]\), "TraditionalForm"]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\(\(\((\(-1\))\)\^\(m + 1\)\/\@\(2\ m\ - \ 1\)\)\(\ \ \ \)\)\^\ \), RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ m\)\^+\), FormBox[\(\(-\ \[Phi]\_\(m, \ C\)\)\ \((\(\(b\_\((+)\)\)\_C\ \^+\))\)\ - 2\ \(p\_m\)\&_\), "TraditionalForm"]}], " ", ")"}], " ", StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\n\nNow examining the b terms, they will cancel one another if\n\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\ \((\(-1\))\)\^\(m + \ 1\)\/\@\(2\ m\ - \ 1\)\ \[Phi]\_\(m, \ n\)\ = \ \((\(-1\))\)\^\(n + 1\)\/\@\ \(2\ n\ - \ 1\)\)]], "\n\nI know that\n", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_\(n, \ m\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\@\(2\ n\ - \ 1\)\) \(\@\(2\ m\ - \ 1\)\) \(1\/\(2\ \((n\ + \ m\ - \ 1)\)\)\) \((\(n\ - \ 1\)\&\(\(-1\)\/2\))\) \((\(m\ - \ \ 1\)\&\(\(-1\)\/2\))\)\)]], "\nso\n\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^\(m + 1\)\)\(\ \)\(\@\(2\ n\ - \ 1\)\) \(1\/\(2\ \((n\ + \ m\ - \ 1)\)\)\) \((\(n\ - \ 1\)\&\(\(-1\)\/2\))\) \((\(m\ - \ \ 1\)\&\(\(-1\)\/2\))\)\(\ \)\)\)]], " should equal ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^\(n + 1\)\/\@\(2\ n\ - \ 1\)\)]], "\n\nwhich is verifiable through ", StyleBox["Mathematica", FontSlant->"Italic"], ". Generally, this sum equals ", Cell[BoxData[ \(\(-\(Cos[n\ \[Pi]]\/\@\(\(-1\) + 2\ n\)\)\)\)]], ".\n\nleaving \n\nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], \(\(4 \@ 2\)\/\[Pi]\), RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\(\(\((\(-1\))\)\^m\/\@\(2\ m\ - \ 1\)\)\(\ \ \)\)\^\ \ \), FormBox[\(\(\ \)\(\(p\_m\)\&_\)\), "TraditionalForm"], " ", StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\n\nWhich, evaluating, leaves \n\nP ", StyleBox["T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], \(\(4 \@ 2\)\/\[Pi]\), RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\(\(\((\(-1\))\)\^m\/\@\(2\ m\ - \ 1\)\)\(\ \ \)\)\^\ \ \), FormBox[\(\(\(\(\@2\) 1\ p\)\/\[Pi]\) \((\(-1\))\)\^m\/\((2\ m\ - \ 1)\)\^\ \(3\/2\)\), "TraditionalForm"], " ", StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], \(\(p\ 8\)\/\[Pi]\^2\), RowBox[{ FormBox[\(\[Sum]\+\(m = 1\)\%\[Infinity]\), "TraditionalForm"], RowBox[{\(\((1\/\((2\ m\ - \ 1)\)\^2)\)\^\ \), " ", StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}]}]}], TraditionalForm]]], "\nP ", StyleBox["T ", FontWeight->"Bold"], "= ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{ FormBox[\(\(p\_q\)\&_\), "TraditionalForm"], FormBox[ RowBox[{"(", RowBox[{\(\(b\_\(\((+)\)\(\ \)\)\)\_q\), "-", FormBox[\(\(\(b\_\(\((+)\)\(\ \)\)\)\_q\^+\)\()\)\), "TraditionalForm"]}]}], "TraditionalForm"], " "}]], " ", StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), FontWeight->"Bold"], " ", \(p\^\ \), " ", StyleBox[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"]}], TraditionalForm]]], "\n\n[26]\n\nNow I seek to normalize the vacuum:\n\nDefine ", StyleBox["A", FontWeight->"Bold", FontVariations->{"Underline"->True}], " to be the traditional bra notation for A and ", StyleBox["A ", FontWeight->"Bold"], "to be the traditional ket notation.\n\nGiven that ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontVariations->{"Underline"->True}], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], FontVariations->{"Underline"->True}], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], TraditionalForm]]], "= 1, ", StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" 0", FontWeight->"Bold"], " = 1 and ", StyleBox["0 ", FontWeight->"Bold"], StyleBox[" = \[Eta] ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\(-1\)\/2\) \(b\_\((+)\)\^+\)\ \[Phi]\ \ \(b\_\((+)\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], TraditionalForm]]], ", I seek to solve for \[Eta].\n\nStarting by calculating the expecation \ value \n\n", StyleBox["0 ", FontWeight->"Bold"], StyleBox[" = \[Eta] ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\(-1\)\/2\) \(b\_\((+)\)\^+\)\ \[Phi]\ \ \(b\_\((+)\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], TraditionalForm]]], "\n", StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], " ", StyleBox["0 ", FontWeight->"Bold"], StyleBox[" = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], \(\[ExponentialE]\^\(\(\(-1\)\/2\) b\_\((+)\)\ \[Phi]\ b\_\((+)\)\)\), " ", "\[Eta]", " ", StyleBox["\[Eta]", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\(-1\)\/2\) \(b\_\((+)\)\^+\)\ \ \[Phi]\ \(b\_\((+)\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}], TraditionalForm]]], "\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], \(\[ExponentialE]\^\(\(\(-1\)\/2\) b\_\((+)\)\ \[Phi]\ b\_\((+)\)\)\), StyleBox[ RowBox[{" ", StyleBox[" ", FontVariations->{"CompatibilityType"->0}]}]], FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\(-1\)\/2\) \(b\_\((+)\)\^+\)\ \ \[Phi]\ \(b\_\((+)\)\^+\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}], TraditionalForm]]], "\n\nNow utilizing the gaussian identity ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\(w\^T\)[A]\^\(-1\)\) w\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ A)\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i = 1\)\%N\( \ \[ExponentialE]\^\(\(-\ x\^T\) A\ x\)\) \(\[ExponentialE]\^\(\(w\^T\) x\)\) \[DifferentialD]x\_i\)\)]], "\n\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], RowBox[{ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", "[", \(\(\(\(\[Pi]\^\(N\/2\)\)( det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\ \[Product]\+\(i = 1\)\%N\( \[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\) \ \(\[ExponentialE]\^\(\(b\_\((+)\)\) x\)\) \[DifferentialD]x\_i\)\), "]"}], " ", "[", " ", \(\(\(\(\[Pi]\^\(M\/2\)\)( det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\ \[Product]\+\(i = 1\)\%M\( \[ExponentialE]\^\(\(-\ y\^T\) \[Phi]\^\(-1\)\ y\)\) \ \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) y\)\) \[DifferentialD]y\_i\)\), "]"}], " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}], TraditionalForm]]], "\n\nSince ", Cell[BoxData[ \(TraditionalForm\`b\_\((+)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(b\_\((+)\)\^+\)\)]], " = 1, \n\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", \(\(\(\[Pi]\^\(N\/2\)\)( det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\), \ \(\(\(\[Pi]\^\(M\/2\)\)(det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{\(\[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\), \(\[ExponentialE]\^\ \(\(-\ y\^T\) \[Phi]\^\(-1\)\ y\)\), \(\[ExponentialE]\^\(\(b\_\((+)\)\) x\)\), \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\) y\)\), \(\[DifferentialD]x\_i\), " ", \(\[DifferentialD]y\_i\), " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}]}]}]}]}]}], TraditionalForm]]], "\n\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", \(\(\(\[Pi]\^\(N\/2\)\)( det\ 2 \[Phi]\^\(-1\))\)\^1\), \(\[Pi]\^\(M\/2\)\), RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{\(\[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\), \(\[ExponentialE]\^\ \(\(-\ y\^T\) \[Phi]\^\(-1\)\ y\)\), \(\[ExponentialE]\^\(\(\ \)\(x\ y\)\)\), \ \(\[DifferentialD]x\_i\), " ", \(\[DifferentialD]y\_i\), " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}]}]}]}]}]}], TraditionalForm]]], "(As a result of the commutator, allowing ", Cell[BoxData[ \(TraditionalForm\`b\_\((+)\)\)]], "to annihilate the RHS vacua, and ", Cell[BoxData[ \(TraditionalForm\`\(b\_\((+)\)\^+\)\)]], "to annihilate the LHS vacua.\n\nNow consuming y in the identity, \n\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", \(\(\(\[Pi]\^\(N\/2\)\)(det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{"\[Integral]", RowBox[{\(\[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\), \ \(\[ExponentialE]\^\(\(-\ x\^T\)\ \[Phi]\ x\)\), \(\[DifferentialD]x\_i\), " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}]}]}]}], TraditionalForm]]], "\n\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", \(\(\(\[Pi]\^\(N\/2\)\)(det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{"\[Integral]", RowBox[{\(\[ExponentialE]\^\(\(-\ \(\(x\^T\)(\[Phi]\^\(-1\) - \ \ \[Phi])\)\)\ x\)\), \(\[DifferentialD]x\_i\), " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}]}]}]}], TraditionalForm]]], "\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^2\), " ", SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], StyleBox[" ", FontVariations->{"Underline"->False}], SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "L"], " ", \(\(\(\[Pi]\^\(N\/2\)\)(det\ 2 \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{"\[Integral]", RowBox[{\(\[ExponentialE]\^\(\(-\ \(\(x\^T\)(\[Phi]\^\(-1\) - \ \ \[Phi])\)\)\ x\)\), \(\[ExponentialE]\^\(0\ x\)\), \(\[DifferentialD]x\_i\), " ", FormBox[ RowBox[{ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], StyleBox[" ", FontVariations->{"Underline"->False}]}], "TraditionalForm"]}]}]}]}], TraditionalForm]]], "\n", StyleBox["1 = ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`\(\(\[Eta]\^2\)\(\ \)\(\((det\ 2 \ \[Phi]\^\(-1\))\)\^\(1\/2\)\/\((det\ \((\[Phi]\^\(-1\) - \ \ \[Phi])\))\)\^\(1\/2\)\)\(\ \)\)\)]], "\n", StyleBox[" ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`\(\(\[Eta]\^2\)\(=\)\(\ \)\(\((det\ \ \((\[Phi]\^\(-1\) - \ \[Phi])\))\)\^\(1\/2\)\/\((det\ 2 \ \[Phi]\^\(-1\))\)\^\(1\/2\)\)\(\ \)\)\)]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/2\)], TraditionalForm]]], "\n \[Eta] = ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{\(1\/2\), RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/4\)], TraditionalForm]]], " ", StyleBox["(This differs by a factor of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`1\/2\)], FontWeight->"Bold"], StyleBox[")", FontWeight->"Bold"], "\n \n[27] \n \n Now normalizing the Tachyon with normalization factor \ \[Eta]\n \n", StyleBox["T", FontWeight->"Bold"], " = \[Eta] ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\ p\&_\_c\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ c\)\), "-", FormBox[\(\(b\_\(\((+)\)\ c\)\^+\)\()\)\), "TraditionalForm"]}]}]}]], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], " (From [25])\n\nManipulating the Tachyon expression, I wish to express it \ in terms of ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\((+)\)\(\ \)\)\^+\)\)]], " only.\n\nSince [", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ c\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\((+)\)\ d\)\^+\)\)]], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(c, \ d\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\(\(\[ExponentialE]\^\(\(A\)\(\ \)\(+\)\)\)\(\ \ \)\)\^B\ = \[ExponentialE]\^A\ \[ExponentialE]\^B\ \[ExponentialE]\^\(\(-1\)\ \/2\)[A, \ B]\)]], "\n\n[27a]\n\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{\(-\ p\&_\_c\), " ", RowBox[{"(", RowBox[{\(b\_\(\((+)\)\ c\)\), "-", FormBox[\(\(b\_\(\((+)\)\ c\)\^+\)\()\)\), "TraditionalForm"]}]}]}]], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\(-\ p\&_\_c\)\ b\_\(\((+)\)\ c\)\), "+", " ", RowBox[{\(p\&_\_c\), " ", FormBox[\(b\_\(\((+)\)\ c\)\^+\), "TraditionalForm"]}]}]}]], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{ RowBox[{\(\(\(\ \)\(p\)\)\&_\_c\), " ", FormBox[\(b\_\(\((+)\)\ c\)\^+\), "TraditionalForm"]}], "-", " ", \(p\&_\_c\ b\_\(\((+)\)\ c\)\)}]}]], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(p\&_\_d\), " ", FormBox[\(b\_\(\((+)\)\ d\)\^+\), "TraditionalForm"]}]], \(\[ExponentialE]\^\(\(\ \)\(\(-\ p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{\(\(-1\)\/2\), \((p\&_\_d)\), " ", RowBox[{\((\(-\(\(\ \)\(p\)\)\&_\_\(\(c\)\(\ \)\)\))\), " ", "[", RowBox[{ FormBox[\(b\_\(\((+)\)\ d\)\^+\), "TraditionalForm"], ",", " ", FormBox[\(b\_\(\((+)\)\ c\)\), "TraditionalForm"]}], "]"}], " "}]]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(p\&_\_d\), " ", FormBox[\(b\_\(\((+)\)\ d\)\^+\), "TraditionalForm"]}]], \(\[ExponentialE]\^\(\(\ \)\(\(-\ p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\)\), \(\[ExponentialE]\^\(\(\ \(-1\)\/2\) \((p\&_\_c)\)\(\ \)\((\(\(\ \)\(p\)\)\&_\_\(\(c\)\(\ \)\))\)\(\ \ \ \)\)\)}], TraditionalForm]]], "\n\nSubstituting,\n\n", StyleBox["T", FontWeight->"Bold"], " = \[Eta] ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{\(p\&_\_d\), " ", FormBox[\(b\_\(\((+)\)\ d\)\^+\), "TraditionalForm"]}]], \(\[ExponentialE]\^\(\(\ \)\(\(-\ p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\)\), \(\[ExponentialE]\^\(\(\ \(-1\)\/2\) \((p\&_\_c)\)\(\ \)\((\(\(\ \)\(p\)\)\&_\_\(\(c\)\(\ \)\))\)\(\ \ \ \)\)\), " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \(\ \[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], "\n\n", StyleBox["T", FontWeight->"Bold"], " = \[Eta] ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ \(b\_\(\((+)\)\ c\)\^+\)\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(p\&_\_c\)\(\ \ \)\(p\&_\_c\)\(\ \)\)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\ \)\), " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\) \ \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], " \n\n[27b] \n\nNow commuting so that the remaining ", Cell[BoxData[ \(TraditionalForm\`b\_\((+)\)\)]], " may annihilate the vacuum,\n\n", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\ \ b\_\(\((+)\)\ c\)\)\)\ \[ExponentialE]\^\(\(-\(1\/2\)\) \(b\_\(\((+)\) n\)\^+\ \) \(\[Phi]\_\(n\ m\)\) \(b\_\(\((+)\) \(m\)\(\ \)\)\^+\)\)\)]], "\n\ngiven that ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\(w\^T\)[A]\^\(-1\)\) w\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ A)\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i = 1\)\%N\( \ \[ExponentialE]\^\(\(-\ x\^T\) A\ x\)\) \(\[ExponentialE]\^\(\(w\^T\) x\)\) \[DifferentialD]x\_i\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\ \ b\_\(\((+)\)\ c\)\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i \ = 1\)\%N\( \[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\) \(\[ExponentialE]\^\(\(b\_\ \((+)\)\^+\)\ x\)\) \[DifferentialD]x\_i\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i \ = 1\)\%N\( \[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\) \(\[ExponentialE]\^\(\(\ \ \)\(\(-p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\)\) \ \(\[ExponentialE]\^\(\(b\_\((+)\)\^+\)\ x\)\) \[DifferentialD]x\_i\)\)]], "\n\nSince [", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ c\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\((+)\)\ d\)\^+\)\)]], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(c, \ d\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^A\ \[ExponentialE]\^B\ = \ \[ExponentialE]\^B\ \[ExponentialE]\^A\ \[ExponentialE]\^\([A, \ B]\)\)]], ",\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i \ = 1\)\%N\( \[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\) \(\[ExponentialE]\^\(\(b\_\ \((+)\)\^+\)\ x\)\) \[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\ b\_\(\((+)\)\ \ c\)\)\)\ \ \[ExponentialE]\^\(\(-p\&_\)\ x\)\ \ \[DifferentialD]x\_i\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i \ = 1\)\%N\( \[ExponentialE]\^\(\(-\ x\^T\) \[Phi]\^\(-1\)\ x\)\) \ \(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ - p\&_\ )\) x\)\) \[DifferentialD]x\_i \[ExponentialE]\^\(\(\ \ \)\(\(-p\&_\_c\)\ b\_\(\((+)\)\ c\)\)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ - p\&_\ )\) \(\[Phi](\(b\_\((+)\)\^+\)\ - p\&_\ )\)\(\ \)\)\) \[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\ \ b\_\(\((+)\)\ c\)\)\)\)]], "\n\nSubstituting,", StyleBox["\n\n[27c]\n", FontWeight->"Bold"], "\n", StyleBox["T", FontWeight->"Bold"], " = \[Eta] ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ \(b\_\(\((+)\)\ c\)\^+\)\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(p\&_\_c\)\(\ \ \)\(p\&_\_c\)\(\ \)\)\)\), " ", 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\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ \(-p\&_\)\ )\) \(\[Phi](\ \ \(b\_\((+)\)\^+\)\ )\)\)\), FormBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ \(-p\&_\)\ )\) \(\ \[Phi](\ \(-p\&_\)\ )\)\)\), "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\(-\(1\/2\)\) \ \((\(b\_\((+)\)\^+\)\ )\) \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\) \(\ \[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \(\[Phi](\ p\&_\ )\)\)\) \[ExponentialE]\^\(\(-\(1\/2\)\) \((\ p\&_\ )\) \ \(\[Phi](\ p\&_\ )\)\)\)]], "\n\nFor \n\n", StyleBox["[NOTE! This differs only by the SIGN on the p b terms. This \ originates because I have -p wherever he has P. With this definition, I can \ replace -", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`p\&_\)]], StyleBox[" with ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`p\&_\)]], StyleBox[" EVERYWHERE and redefine ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`p\&_\)]], StyleBox[" where it appears without affecting the results. See 27c]", FontWeight->"Bold"], "\n\n", StyleBox["T", FontWeight->"Bold"], " = \[Eta] ", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ \(b\_\(\((+)\)\ c\)\^+\)\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(p\&_\_c\)\(\ \ \)\(p\&_\_c\)\(\ \)\)\)\), " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\)\ )\) \ \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\), \ \(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \(\[Phi](\ p\&_\ )\)\)\), FormBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ p\&_\ )\) \(\[Phi](\ p\&_\ )\)\)\), "TraditionalForm"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], " \n\nNow normalizing,\n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontWeight->"Bold"], " ", StyleBox[ SubscriptBox[ StyleBox["0", FontVariations->{"Underline"->True}], "L"], FontWeight->"Bold"], StyleBox[" ", FontWeight-> "Bold"], \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ p\&_\ )\) \(\[Phi](\ p\&_\ )\)\)\), \(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \(\ \[Phi](\ p\&_\ )\)\)\), \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((b\_\((+)\)\ )\) \ \(\[Phi](b\_\((+)\)\ )\)\)\), \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \ \(p\&_\_c\)\(\ \)\(p\&_\_c\)\(\ \)\)\)\), \(\[ExponentialE]\^\(\(\ \ \)\(p\&_\_c\ b\_\(\((+)\)\ c\)\)\)\), StyleBox[\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ p\ \ \(x(\[Pi]\/2)\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\[ImaginaryI]\ p\ \(x(\[Pi]\/2)\)\)\), FontColor->RGBColor[1, 0, 0]], " ", \(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ \(b\_\(\((+)\)\ c\)\^+\)\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(p\&_\_c\)\(\ \ \)\(p\&_\_c\)\(\ \)\)\)\), " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\)\ )\) \ \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\), \ \(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \(\[Phi](\ p\&_\ )\)\)\), FormBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ p\&_\ )\) \(\[Phi](\ p\&_\ )\)\)\), "TraditionalForm"], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], "\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontWeight->"Bold"], " ", StyleBox[ SubscriptBox[ StyleBox["0", FontVariations->{"Underline"->True}], "L"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(\[ExponentialE]\^\(\(-\((\ p\&_\ )\)\) \(\[Phi](\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\ \)\(\(-p\&_\_c\)\(\ \)\(p\&_\_c\)\ \(\ \)\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \(\[Phi](\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]], \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((b\_\((+)\)\ )\) \(\ \[Phi](b\_\((+)\)\ )\)\)\), StyleBox[\(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ b\_\(\((+)\)\ \ c\)\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ \(b\_\(\((+)\)\ \ c\)\^+\)\)\)\), FontColor->RGBColor[1, 0, 0]], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\)\ )\) \ \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\), StyleBox[\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \(\[Phi](\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], " (Eliminating ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ p\ \ \(x(\[Pi]\/2)\)\)\)]], " terms)\n\nSimplifying the ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \ \(\(\[Phi]\_\(n\ m\)\)(\ p\&_\ )\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(p\&_\_c\ b\_\(\((+)\)\ \ c\)\)\)\)]], " and conjugate terms,\n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[\(\[ExponentialE]\^\(\(-\((\ p\&_\ )\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontWeight->"Bold"], " ", StyleBox[ SubscriptBox[ StyleBox["0", FontVariations->{"Underline"->True}], "L"], FontWeight->"Bold"], StyleBox[" ", FontWeight-> "Bold"], \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((b\_\((+)\)\ )\) \ \(\[Phi](b\_\((+)\)\ )\)\)\), StyleBox[\(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \((I\ + \ \[Phi])\ \) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\)\ )\) \ \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\), StyleBox[" ", FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], "\n\nwhich is fairly obvious from the gaussian identity.\n\nSince [", Cell[BoxData[ \(TraditionalForm\`b\_\(\((+)\)\ a\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\(b\_\(\((+)\) b\)\)\(\ \)\)\^+\)\)]], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(a, \ b\)\)]], ",\n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[\(\[ExponentialE]\^\(\((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight-> "Bold"], \(\[ExponentialE]\^\(\(-\((\ p\&_\ )\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontWeight->"Bold"], " ", StyleBox[ SubscriptBox[ StyleBox["0", FontVariations->{"Underline"->True}], "L"], FontWeight->"Bold"], StyleBox[" ", FontWeight-> "Bold"], \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((b\_\((+)\)\ )\) \ \(\[Phi](b\_\((+)\)\ )\)\)\), " ", StyleBox[\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \((I\ + \ \[Phi])\ \) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]], " ", \(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\(b\_\((+)\)\^+\)\ )\) \ \(\[Phi](\(b\_\((+)\)\^+\)\ )\)\)\), StyleBox[" ", FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], "\n\nFrom the Gaussian Identity, ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\(w\^T\)[A]\^\(-1\)\) w\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Pi]\^\(N\/2\)\)( det\ A)\)\^\(1\/2\)\) \(\[Integral]\[Product]\+\(i = 1\)\%N\( \ \[ExponentialE]\^\(\(-\ x\^T\) A\ x\)\) \(\[ExponentialE]\^\(\(w\^T\) x\)\) \[DifferentialD]x\_i\)\)]], "\nI am left commuting \n\n", Cell[BoxData[ FormBox[ StyleBox[\(\(\[ExponentialE]\^\(\((\(\[ImaginaryI]\/\@2\) b\_\((+)\)\ )\) x\)\) \[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\)\)\), FontWeight->"Plain"], TraditionalForm]]], "\n\nleaving me with\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \((I\ \ + \ \[Phi])\) \((\ p\&_\ )\)\)\) \(\[ExponentialE]\^\(\((\(\[ImaginaryI]\/\@2\) b\_\((+)\)\ )\) x\)\) \[ExponentialE]\^\(\(\[ImaginaryI]\/\@2\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\) x\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\((\(b\_\((+)\)\^+\)\ )\) \((I\ \ + \ \[Phi])\) \((\ p\&_\ )\)\)\) \[ExponentialE]\^\(\(\[ImaginaryI]\/\@2\) \ \((b\_\((+)\)\ + \ \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) x\)\)]], "\n\nand\n\n", Cell[BoxData[ FormBox[ StyleBox[\(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \((I\ + \ \[Phi])\) \ \((\ p\&_\ )\)\)\ \ \[ExponentialE]\^\(\((\(\[ImaginaryI]\/\@2\) \(b\_\((+)\)\ \^+\)\ )\) x\)\), FontWeight->"Plain"], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\((\(\[ImaginaryI]\/\@2\) \ \(b\_\((+)\)\^+\)\ )\) x\)\), StyleBox[\(\[ExponentialE]\^\(\((b\_\((+)\)\ )\) \((I\ + \ \[Phi])\ \) \((\ p\&_\ )\)\)\), FontWeight->"Plain"], StyleBox[" ", FontWeight-> "Plain"], \(\[ExponentialE]\^\(\(\[ImaginaryI]\/\@2\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\) x\)\), StyleBox[" ", FontWeight->"Plain"]}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\(\[ImaginaryI]\/\@2\) \((\(b\_\ \((+)\)\^+\) + \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) x\)\) \[ExponentialE]\^\(\((b\_\((+)\)\ )\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\)\)\)]], "\n\nfor \n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), \(\[ExponentialE]\^\(\((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\)\)\), StyleBox[" ", FontWeight-> "Bold"], \(\[ExponentialE]\^\(\(-\((\ p\&_\ )\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\)\), StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold", FontVariations->{"Underline"->True}], "R"], FontWeight->"Bold"], " ", StyleBox[ SubscriptBox[ StyleBox["0", FontVariations->{"Underline"->True}], "L"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \((b\_\((+)\)\ + \ \ \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\)\ \[Phi]\ \((b\_\((+)\)\ + \ \ \((I\ \ + \ \[Phi])\) \((\ p\&_\ )\))\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\ \(b\_\((+)\)\^+\) \ + \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\)\ \[Phi]\ \((\ \(b\_\((+)\)\^+\) + \ \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(0\_L\), FontWeight->"Bold"], StyleBox[ SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"], FontWeight->"Bold"]}], TraditionalForm]]], "\n\nafter annihilation. The gaussian identity, then, becomes (pulling the \ ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], " out for later recombination)\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(-x\)\ \[Phi]\^\(-1\)\ \ x\)\)\), StyleBox[\(\[ExponentialE]\^\(\(\[ImaginaryI]( b\_\((+)\)\ + \ \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) x\)\), FontColor->RGBColor[1, 0, 0]], \(\[DifferentialD]x\_i\), \(\(\(\[Pi]\^\(M\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(-y\)\ \(\(\(\[Phi]\ \)\(\ \)\)\^\(-1\)\) y\)\)\), StyleBox[\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI](\ \ \(b\_\((+)\)\^+\) + \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) y\)\)\), FontColor->RGBColor[1, 0, 0]], \(\[DifferentialD]y\_i\)}]}]}]}]}]}]}], TraditionalForm]]], "\n\nCommuting, this leaves me with \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(N\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), \(\(\(\[Pi]\^\(M\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{\(\[ExponentialE]\^\(\(-\ y\)\ \(\(\(\[Phi]\)\(\ \)\)\^\(-1\)\) y\)\), StyleBox[\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI](\ \(b\_\ \((+)\)\^+\) + \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) y\)\)\), FontColor->RGBColor[0, 0, 1]], RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%N\), RowBox[{\(\[ExponentialE]\^\(\(-\ x\)\ \[Phi]\^\(-1\)\ x\)\), StyleBox[\(\[ExponentialE]\^\(\(\[ImaginaryI]( b\_\((+)\)\ + \ \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\))\) x\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\(-\ x\)\ y\)\), FontColor->RGBColor[0, 0, 1]], \(\[DifferentialD]y\_i\), \(\[DifferentialD]x\_i\ \)}]}]}]}]}]}]}], TraditionalForm]]], "\n\nNow I can exploit this opportunity to allow the b terms to annihilate \ the vacuum and incorporate the commutator into one of the gaussians, leaving\n\ \n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\(\(\[Pi]\^\(N\/2\)\)(det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), FontColor->RGBColor[1, 0, 0]], \(\(\(\[Pi]\^\(M\/2\)\)(det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{\(\[ExponentialE]\^\(\(-\ y\)\ \(\(\(\[Phi]\)\(\ \)\)\^\(-1\)\) y\)\), StyleBox[\(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]( I\ + \ \[Phi])\) \((\ p\&_\ )\) y\)\)\), FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["\[Integral]", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(\[Product]\+\(i = 1\)\%N\), FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(\[ExponentialE]\^\(\(-\ x\)\ \[Phi]\^\(-1\)\ x\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\ \ \ \)\(\(\[ImaginaryI](\((I\ + \ \[Phi])\) \((\ p\&_\ )\) + \ \[ImaginaryI]\ y)\) x\)\)\), FontColor->RGBColor[0, 0, 1]], \(\[DifferentialD]y\_i\), StyleBox[\(\[DifferentialD]x\_i\), FontColor->RGBColor[1, 0, 0]]}]}]}]}]}]}]}], TraditionalForm]]], "\n\nConsuming a gaussian,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(M\/2\)\)(det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{\(\[ExponentialE]\^\(\(-\ y\)\ \(\(\(\[Phi]\)\(\ \)\)\^\(-1\)\) y\)\), \(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]( I\ + \ \[Phi])\) \((\ p\&_\ )\) y\)\)\), StyleBox[\(\[ExponentialE]\^\(\(-\(1\/2\)\) \((\((I\ + \ \ \[Phi])\) \((\ p\&_\ )\) + \ \ \[ImaginaryI]\ y)\)\ \[Phi]\ \ \ \((\((I\ + \ \[Phi])\) \((\ p\&_\ )\) + \ \[ImaginaryI]\ y)\)\)\), FontColor->RGBColor[0, 0, 1]], \(\[DifferentialD]y\_i\)}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Pi]\^\(M\/2\)\)(det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{ SuperscriptBox["\[ExponentialE]", StyleBox[\(\(-\ y\)\ \(\(\(\[Phi]\)\(\ \)\)\^\(-1\)\) y\), FontColor->RGBColor[1, 0, 0]]], \(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]( I\ + \ \[Phi])\) \((\ p\&_\ )\) y\)\)\), SuperscriptBox["\[ExponentialE]", StyleBox[\(\(-\ \(1\/2\)\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\ - \ \[ImaginaryI]\ y\ \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\) - \ y\ \[Phi]\ y\), FontColor->RGBColor[0, 0, 1]]], \(\[DifferentialD]y\_i\)}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\(\ \)\)\), \(\(\(\[Pi]\^\(M\/2\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), RowBox[{"\[Integral]", RowBox[{\(\[Product]\+\(i = 1\)\%M\), RowBox[{ StyleBox[\(\[ExponentialE]\^\(\(-\ \(y(\ \(\(\[Phi]\)\(\ \)\)\ \^\(-1\) - \[Phi])\)\) y\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]( I\ + \ \[Phi])\) \((\ p\&_\ )\) y\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\ \)\(\(-\(\[Phi]( I\ + \ \[Phi])\)\) \((\ p\&_\ )\) y\)\)\), FontColor->RGBColor[1, 0, 0]], \(\[DifferentialD]y\_i\)}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\(\ \)\)\), StyleBox[ SuperscriptBox[ RowBox[{ StyleBox[\(\[Pi]\^\(M\/2\)\), FontColor->RGBColor[1, 0, 0]], StyleBox["(", FontColor->GrayLevel[0]], StyleBox[\(det\ \[Phi]\^\(-1\)\), FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\(1\/2\), FontColor->GrayLevel[0]]], FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox["\[Integral]", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(\[Product]\+\(i = 1\)\%M\), FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(\[ExponentialE]\^\(\(-\ \(y(\ \(\(\[Phi]\)\(\ \)\)\ \^\(-1\) - \[Phi])\)\) y\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]( I\ - \ \[Phi])\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\) y\)\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[DifferentialD]y\_i\), FontColor->RGBColor[1, 0, 0]]}]}]}]}], TraditionalForm]]], "\n\nNow consuming the gaussian, \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\(\ \)\)\)( det\ \[Phi]\^\(-1\))\)\^\(1\/2\)\), StyleBox[\(\((det\ \((\(\(\[Phi]\)\(\ \)\)\^\(-1\) - \ \[Phi])\)\^\(-1\))\)\^\(\(-1\)\/2\)\), FontColor->RGBColor[0, 0, 1]], StyleBox[\(\[ExponentialE]\^\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((I\ - \[Phi])\) \(\((\ \ \(\(\[Phi]\)\(\ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((I\ - \ \[Phi])\) \((I\ \ + \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "\n\nPulling the constants into the normalization factor, the entire \ expression is: \n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[\(\[ExponentialE]\^\(\((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(-\((\ p\&_\ )\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\(\ \)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[\(\[ExponentialE]\^\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((I\ - \[Phi])\) \(\((\ \ \(\(\[Phi]\)\(\ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((I\ - \ \[Phi])\) \((I\ \ + \ \[Phi])\) \((\ p\&_\ )\)\)\), FontColor->RGBColor[1, 0, 0]]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{\(1\/2\), RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/2\)], TraditionalForm]]], "\n\n", StyleBox["T", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" T", FontWeight->"Bold"], " =", Cell[BoxData[ FormBox[ RowBox[{\(\[Eta]\^\(\(2\)\(\ \)\)\), StyleBox[\(\[ExponentialE]\^\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((I\ - \[Phi])\) \(\((\ \ \(\(\[Phi]\)\(\ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((I\ - \ \[Phi])\) \ \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\ + \ \((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\) - \ \(1\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\ - \(\((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\(\ \ \ \)\)\)\), FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{\(1\/2\), RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/2\)], TraditionalForm]]], "\n\nExpanding the (I - \[Phi]) terms to be (\[Phi] - 1)(\[Phi] - 1) and \ then re-writing, I get\n\n=", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ StyleBox["\[ExponentialE]", FontColor->GrayLevel[0]], RowBox[{\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((\[Phi])\) \(\((\ \(\(\ \[Phi]\)\(\ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((\[Phi])\) \((I\ + \ \ \[Phi])\) \((\ p\&_\ )\)\), StyleBox[" ", FontColor->GrayLevel[0]], "+", " ", RowBox[{\((\ p\&_\ )\), StyleBox[\((I\ + \ \[Phi])\), FontColor->RGBColor[0, 0, 1]], \((\ \[Phi])\), \(\((\ \(\(\[Phi]\)\(\ \)\)\^\(-1\) \ - \[Phi])\)\^\(-1\)\), StyleBox[\((I\ + \ \[Phi])\), FontColor->RGBColor[0, 0, 1]], \((\ p\&_\ )\)}], " ", "+", " ", RowBox[{\(\(-1\)\/2\), \((\ p\&_\ )\), \((I\ + \ \[Phi])\), \(\((\ \(\(\[Phi]\)\(\ \ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\), RowBox[{ StyleBox["(", FontColor->RGBColor[0, 0, 1]], \(I\ + \ \[Phi]\), ")"}], \((\ p\&_\ )\)}], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(\((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\)\), FontColor->GrayLevel[0]], StyleBox["-", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(\(1\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["-", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->GrayLevel[0]], " "}]]}]}]], FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{\(1\/2\), RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/2\)], TraditionalForm]]], "\n\nNow, since\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox[ RowBox[{"\[Phi]", "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\[Phi]\ - \ \[Phi]\^\(-1\))\), SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)]}], " ", "+", " ", SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " "}]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ RowBox[{\(-1\), " ", "+", " ", SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " "}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ StyleBox["\[ExponentialE]", FontColor->GrayLevel[0]], RowBox[{ RowBox[{ StyleBox[\(\(-1\)\/2\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], RowBox[{ StyleBox["(", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{\(-1\), " ", "+", " ", SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)]}], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\((\[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]]}], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], RowBox[{ StyleBox["(", FontColor->GrayLevel[0]], StyleBox[ RowBox[{\(-1\), " ", "+", " ", SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)]}], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]]}], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\((\ \(\(\[Phi]\)\(\ \ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["+", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\((\ p\&_\ )\) \(\((I\ + \ \[Phi])\)\^2\) \((\ p\&_\ )\)\), FontColor->RGBColor[1, 0, 0]], StyleBox["-", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\(1\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\[Phi]( I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["-", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->GrayLevel[0]], " "}]]}]}]], FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "\n\nNow cancelling these exponents, I get \n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ StyleBox["\[ExponentialE]", FontColor->GrayLevel[0]], RowBox[{ RowBox[{ StyleBox[\(\(-1\)\/2\), FontColor->RGBColor[1, 0, 0]], StyleBox[\((\ p\&_\ )\), FontColor->RGBColor[1, 0, 0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->RGBColor[1, 0, 0]], StyleBox[\((\ p\&_\ )\), FontColor->RGBColor[1, 0, 0]]}], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["+", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\(\(-1\)\/2\) \((\ p\&_\ )\) \((I\ + \ \[Phi])\) \(\((\ \(\(\[Phi]\)\(\ \ \)\)\^\(-1\) - \[Phi])\)\^\(-1\)\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["-", FontColor->GrayLevel[0]], StyleBox[\(\((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->GrayLevel[0]], StyleBox[\(\(\ \)\(\ \)\), FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], RowBox[{ StyleBox["(", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[ SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->GrayLevel[0]], " "}]]}]}]], FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "\n\nAnd now I notice that the first two terms in the exponent sum \n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ StyleBox["\[ExponentialE]", FontColor->GrayLevel[0]], RowBox[{ RowBox[{\(-\((\ p\&_\ )\)\), \((I\ \.01 + \.01\ \[Phi])\), " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ \.01 + \.01\ \[Phi])\), \((\ p\&_\ )\)}], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["-", FontColor->RGBColor[1, 0, 0]], StyleBox[\(\((\ p\&_\ )\) \((I\ + \ \[Phi])\) \((\ p\&_\ )\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->GrayLevel[0]]}]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], RowBox[{ StyleBox["(", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[ SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->GrayLevel[0]], " "}]]}]}]], FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "\n\nAnd again,\n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], StyleBox[ SuperscriptBox[ StyleBox["\[ExponentialE]", FontColor->GrayLevel[0]], RowBox[{ RowBox[{\(-\((\ p\&_\ )\)\), RowBox[{"(", RowBox[{"I", " ", "+", " ", "\[Phi]", " ", "+", " ", RowBox[{\((I\ + \ \[Phi])\), " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ + \ \[Phi])\)}]}], ")"}], \((\ p\&_\ )\)}], StyleBox[\(\(\ \ \)\(\ \)\), FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], RowBox[{ StyleBox["(", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[ SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[")", FontColor->GrayLevel[0]]}], StyleBox[\((I\ + \ \[Phi])\), FontColor->GrayLevel[0]], StyleBox[\((\ p\&_\ )\), FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox[" ", FontColor->GrayLevel[0]], " "}]]}]}]], FontColor->RGBColor[0, 0, 1]]}], TraditionalForm]]], "\n\nFinally, noticing that \n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"I", " ", "+", " ", "\[Phi]", " ", "+", " ", RowBox[{\((I\ + \ \[Phi])\), " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ + \ \[Phi])\)}]}], " ", "=", " "}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", \(\((I\ + \ \[Phi])\)\^\(-1\)\), RowBox[{"(", RowBox[{"I", " ", "+", " ", "\[Phi]", " ", "+", " ", RowBox[{\((I\ + \ \[Phi])\), " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ + \ \[Phi])\)}]}], ")"}]}], " ", "=", " "}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", RowBox[{"I", " ", "+", " ", RowBox[{ SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ + \ \[Phi])\)}]}], ")"}]}], " ", "="}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", RowBox[{"I", " ", "+", " ", RowBox[{ SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], " ", \((I\ + \ \[Phi])\)}]}], ")"}]}], " ", "="}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", RowBox[{\(\((I\ + \ \[Phi])\)\^\(-1\)\), " ", "+", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)]}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "="}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", RowBox[{\(\((I\ + \ \[Phi])\)\^\(-1\)\), " ", "+", " ", SuperscriptBox[ RowBox[{"(", FormBox[\(\((\[Phi]\^\(-1\)\ - \ I)\) \((I\ + \ \[Phi])\)\), "TraditionalForm"], ")"}], \(-1\)]}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "="}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", RowBox[{\(\((I\ + \ \[Phi])\)\^\(-1\)\), " ", "+", " ", RowBox[{ SuperscriptBox[ FormBox[\((I\ + \ \[Phi])\), "TraditionalForm"], \(-1\)], \(\((\[Phi]\^\(-1\)\ - \ \ I)\)\^\(-1\)\)}]}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "=", " "}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", " ", RowBox[{ SuperscriptBox[ FormBox[\((I\ + \ \[Phi])\), "TraditionalForm"], \(-1\)], \((\((\[Phi]\^\(-1\)\ - \ \ I)\)\^\(-1\)\ + \ 1)\)}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "=", " "}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", " ", RowBox[{ SuperscriptBox[ FormBox[\((I\ + \ \[Phi])\), "TraditionalForm"], \(-1\)], \((\((\[Phi]\^\(-1\)\ - \ \ I)\)\^\(-1\)\ + \ \(\((\[Phi]\^\(-1\)\ - \ I)\)\^\(-1\)\) \((\[Phi]\^\(-1\)\ - \ I)\))\)}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "=", " "}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", " ", RowBox[{ SuperscriptBox[ FormBox[\((I\ + \ \[Phi])\), "TraditionalForm"], \(-1\)], \((\(\(\[Phi]\^\(-1\)\)(\ \[Phi]\^\(-1\)\ - \ I)\)\^\(-1\))\)}], ")"}], " ", \((I\ + \ \[Phi])\)}], " ", "=", " "}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\((\ I\ + \ \[Phi])\), " ", RowBox[{"(", StyleBox[ SuperscriptBox[ RowBox[{\(\[Phi]\^\(-1\)\), "(", FormBox[\(\[Phi]\^\(-1\)\ - \ \[Phi]\), "TraditionalForm"], ")"}], \(-1\)], FontColor->GrayLevel[0]], ")"}], " ", \((I\ + \ \[Phi])\), " "}]}], TraditionalForm]]], "\n\nSo the terms in the exponential cancel, leaving \n\n", Cell[BoxData[ FormBox[ StyleBox[\(\[Eta]\^\(\(2\)\(\ \)\)\), FontColor->GrayLevel[0]], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{\(1\/2\), RowBox[{"det", "(", RowBox[{"1", " ", "-", " ", FormBox[\(\[Phi]\^2\), "TraditionalForm"]}], ")"}]}], ")"}], \(1\/2\)], TraditionalForm]]], "\n\nJust like the vacuum.\n\n", StyleBox["Alternative to Linear Algebra", FontSize->18], "\n\n[28]\n\nFrom [1a],\n\nx(\[Sigma]) =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x\_0\)\)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\)]], "cos(n \[Sigma])\n\nand so\n\nx(\[Sigma]) Cos[m \[Sigma]] =", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(x\_0\)\)\)]], "Cos[m \[Sigma]] + ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\)]], "cos(n \[Sigma]) Cos[m \[Sigma]]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Pi]\( x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\), " ", "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Integral]\_0\%\[Pi] x\_0\)\), "TraditionalForm"], \(Cos[ m\ \[Sigma]]\), \(\[DifferentialD]\[Sigma]\)}], " ", "+", RowBox[{ FormBox[\(\[Integral]\_0\%\[Pi]\@ 2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\), "TraditionalForm"], \(cos(n\ \[Sigma])\), " ", \(Cos[ m\ \[Sigma]]\), \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Pi]\( x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\), " ", "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", RowBox[{ FormBox[\(\[Integral]\_0\%\[Pi]\@ 2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity] x\_n\), "TraditionalForm"], \(cos(n\ \[Sigma])\), " ", \(Cos[ m\ \[Sigma]]\), \(\(\[DifferentialD]\[Sigma]\)\(\ \)\)}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\(x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\ + \ \ \[Integral]\_\ \(\[Pi]\/2\)\%\[Pi]\( x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", RowBox[{ FormBox[\(\(\@2\) \(x\_n\) \[Integral]\_0\%\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\), "TraditionalForm"], \(\(cos(n\ \[Sigma])\)\ Cos[ m\ \[Sigma]] \(\(\[DifferentialD]\[Sigma]\)\(\ \ \)\)\)}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\(x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\ + \ \ \[Integral]\_\ \(\[Pi]\/2\)\%0\( x(\[Pi]\ - \[Sigma])\)\ Cos[ m\ \((\[Pi]\ - \ \[Sigma])\)] \[DifferentialD]\[Sigma]\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", " ", RowBox[{ FormBox[\(\(\@2\) \(x\_n\) \[Integral]\_0\%\[Pi]\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\), "TraditionalForm"], \(\(cos(n\ \[Sigma])\)\ Cos[ m\ \[Sigma]] \(\(\[DifferentialD]\[Sigma]\)\(\ \ \)\)\)}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\(x(\[Sigma])\)\ Cos[ m\ \[Sigma]] \[DifferentialD]\[Sigma]\ - \ \ \ \[Integral]\_0\%\(\[Pi]\/2\)\(x(\[Pi]\ - \[Sigma])\)\ Cos[ m\ \((\[Pi]\ - \ \[Sigma])\)] \[DifferentialD]\[Sigma]\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", " ", RowBox[{ FormBox[\(\(\@2\) x\_n\), "TraditionalForm"], RowBox[{ FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\), "TraditionalForm"], \(\(\[Pi]\/2\) \[Delta]\_\(m, \ \ n\)\)}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((x(\[Sigma])\ - \ x(\[Pi]\/2)\ \ + \ x(\[Pi]\/2))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ - \ \ \[Integral]\ \_0\%\(\[Pi]\/2\)\((x(\[Pi]\ - \[Sigma]) - \ x(\[Pi]\/2)\ \ + \ x(\[Pi]\/2)\ )\) Cos[m\ \((\[Pi]\ - \ \[Sigma])\)] \[DifferentialD]\[Sigma]\), "=", RowBox[{ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", \(x\_m\)}], "TraditionalForm"], \(\[Pi]\/\@2\), \((\(-\(\[Delta]( m)\)\)\ \ + \ 1)\)}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((\(\[Chi]\_L\)(\[Sigma]) + \ x(\[Pi]\/2))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ - \ \ \[Integral]\ \_0\%\(\[Pi]\/2\)\((\(\[Chi]\_R\)(\[Sigma])\ \ + \ x(\[Pi]\/2)\ )\) Cos[m\ \((\[Pi]\ - \ \[Sigma])\)] \[DifferentialD]\[Sigma]\), "=", RowBox[{ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", " ", \(x\_m\)}], "TraditionalForm"], \(\[Pi]\/\@2\), \((\(-\(\[Delta]( m)\)\)\ \ + \ 1)\)}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((\(\[Chi]\_L\)(\[Sigma]) + \ x(\[Pi]\/2))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ - \ \ \[Integral]\ \_0\%\(\[Pi]\/2\)\((\(\[Chi]\_R\)(\[Sigma])\ \ + \ x(\[Pi]\/2)\ )\) \(\((\(-1\))\)\^m\) Cos[m\ \ \[Sigma]] \[DifferentialD]\[Sigma]\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", " ", RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(\[Pi]\/\@2\), \((\(-\(\[Delta]( m)\)\)\ \ + \ 1)\)}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((\(\[Chi]\_L\)(\[Sigma]) - \ \ \(\((\(-1\))\)\^m\) \(\(\[Chi]\_R\)(\[Sigma])\))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ + \ \ \[Integral]\ \_0\%\(\[Pi]\/2\)\((x(\[Pi]\/2)\ + \ \(\((\(-1\))\)\^m\) \(x(\[Pi]\/2)\)\ \ )\) Cos[m\ \ \[Sigma]] \[DifferentialD]\[Sigma]\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(\[Pi]\/\@2\), \((\(-\(\[Delta]( m)\)\)\ \ + \ 1)\)}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((\(\[Chi]\_L\)(\[Sigma]) - \ \ \(\((\(-1\))\)\^m\) \(\(\[Chi]\_R\)(\[Sigma])\))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ + \ \ \[Pi]\ \(x(\ \[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ RowBox[{ FormBox[\(\(\ \)\(\[Pi]\ x\_0\)\), "TraditionalForm"], \(\[Delta](m)\)}], " ", "+", RowBox[{ FormBox[\(x\_m\), "TraditionalForm"], \(\[Pi]\/\@2\), \((\(-\(\[Delta]( m)\)\)\ \ + \ 1)\)}]}]}], TraditionalForm]]], "\n\nLetting ", Cell[BoxData[ FormBox[ RowBox[{\(\(x'\)\_m\), "=", FormBox[ FormBox[\(x\_m\), "TraditionalForm"], "TraditionalForm"]}], TraditionalForm]]], " if m = 0 and ", Cell[BoxData[ FormBox[ RowBox[{\(\(x'\)\_m\), "=", FormBox[ FormBox[\(x\_m\/\@2\), "TraditionalForm"], "TraditionalForm"]}], TraditionalForm]]], " otherwise,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\((\(\[Chi]\_L\)(\[Sigma]) - \ \ \(\((\(-1\))\)\^m\) \(\(\[Chi]\_R\)(\[Sigma])\))\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\ + \ \ \[Pi]\ \(x(\ \[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n\n\nI recall that\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma])\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( 1\/2\) \[Chi]\_\(L\ n\)\)]], "(1 - ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^n\)]], ")Cos(n \[Sigma]) \n\n", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_R\)]], "(\[Sigma]) = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( 1\/2\) \[Chi]\_\(R\ n\)\)]], "(1 - ", Cell[BoxData[ \(TraditionalForm\`\((\(-1\))\)\^n\)]], ")Cos(n \[Sigma]) \n\nso\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\), RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/2\), RowBox[{\(\[Chi]\_\(L\ n\)\), "(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], \(Cos( n\ \[Sigma])\)}]}], "-", " ", RowBox[{\(\((\(-1\))\)\^m\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/2\), RowBox[{\(\[Chi]\_\(R\ n\)\), "(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], \(Cos( n\ \[Sigma])\)}]}]}]}], ")"}], \(Cos[ m\ \[Sigma]]\), \(\[DifferentialD]\[Sigma]\)}]}], " ", "+", " ", \(\[Pi]\ \(x(\[Pi]\/2)\)\ \(\[Delta](m)\)\)}], "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ n - 1\) - \ \ \(\((\(-1\))\)\^m\) \[Chi]\_\(R\ 2 n\ - \ 1\))\) \ \(\[Integral]\_0\%\(\[Pi]\/2\)\(Cos(\((2\ n\ - \ 1)\)\ \[Sigma])\) Cos[m\ \[Sigma]] \[DifferentialD]\[Sigma]\)\ + \ \ \[Pi]\ \ \(x(\[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/2\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \ \(\((\(-1\))\)\^m\) \[Chi]\_\(R\ 2 n\ - \ 1\))\)\ \((Sin[1\/2\ \ \((1 + m - 2\ n)\)\ \[Pi]]\/\(1 + m - 2\ n\) + Sin[1\/2\ \((\(-1\) + m + 2\ n)\)\ \[Pi]]\/\(\(-1\) + m \ + 2\ n\))\)\)\ + \ \ \[Pi]\ \(x(\[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/2\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \ \(\((\(-1\))\)\^m\) \[Chi]\_\(R\ 2 n\ - \ 1\))\)\ \((\(\(1\/2\) \ \((1\ + \ \((\(-1\))\)\^m)\) \((\(-1\))\)\^\(\(m\ - \ 2 n\)\/2\)\)\/\(1 + \ m - 2\ n\) + \(\(\(-1\)\/2\) \((1\ + \ \((\(-1\))\)\^m)\) \ \((\(-1\))\)\^\(\(m + \ 2 n\)\/2\)\)\/\(\(-1\) + m + 2\ n\))\)\)\ + \ \ \ \[Pi]\ \(x(\[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/4\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \ \(\((\(-1\))\)\^m\) \[Chi]\_\(R\ 2 n\ - \ 1\))\)\ \((1\ + \ \ \((\(-1\))\)\^m)\) \(\((\(-1\))\)\^\(\(\(m\)\(\ \)\)\/2 - \ n\)\) \((1\/\(1 + m - 2\ n\) - 1\/\(\(-1\) + m + 2\ n\))\)\)\ + \ \ \[Pi]\ \ \(x(\[Pi]\/2)\)\ \(\[Delta](m)\)\), "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(\(x'\)\_m\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\n\nEvidently, ", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], " comes into play when calculating the zero-mode full string coordinate \ from half-string coordinates as:\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/2\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \[Chi]\_\(R\ 2 n\ - \ 1\))\) \(\((\(-1\))\)\^\(-n\)\) \((1\/\(1 - \ 2\ n\) - 1\/\(\(-1\) + 2\ n\))\)\)\ + \ \ \[Pi]\ \(x(\[Pi]\/2)\)\), " ", "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], FormBox[\(x\_0\), "TraditionalForm"]}]}], TraditionalForm]]], "\n\nWhich is just [14]. \n\nfor\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/4\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \[Chi]\_\(R\ 2 n\ - \ 1\))\) \((1\ + \ \((\(-1\))\)\^m)\) \ \(\((\(-1\))\)\^\(\(\(m\)\(\ \)\)\/2 - \ n\)\) \((1\/\(1 + m - 2\ n\) - 1\/\(\(-1\) + m + 2\ n\))\)\)\), " ", "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], RowBox[{"(", FormBox[\(\(\(x\_0 - x(\[Pi]\/2)\)\()\)\)\(\ \)\), "TraditionalForm"]}]}]}], TraditionalForm]]], "when m = 0.\n\nNow letting ", Cell[BoxData[ \(TraditionalForm\`\(x''\)\_m\)]], "= ", Cell[BoxData[ \(TraditionalForm\`x\_0 - x(\[Pi]\/2)\)]], " when m = 0 and ", Cell[BoxData[ \(TraditionalForm\`\(x''\)\_m\)]], "= ", Cell[BoxData[ \(TraditionalForm\`x\_m\/\@2\)]], "otherwise,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(1\/4\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\((\[Chi]\_\(L\ 2\ \ n - 1\) - \[Chi]\_\(R\ 2 n\ - \ 1\))\) \((1\ + \ \((\(-1\))\)\^m)\) \ \(\((\(-1\))\)\^\(\(\(m\)\(\ \)\)\/2 - \ n\)\) \((1\/\(1 + m - 2\ n\) - 1\/\(\(-1\) + m + 2\ n\))\)\)\), " ", "=", RowBox[{ FormBox[\(\(\ \)\(\[Pi]\)\(\ \)\), "TraditionalForm"], SubscriptBox[ RowBox[{"(", FormBox[\(x''\), "TraditionalForm"], ")"}], "m"]}]}], TraditionalForm]]], "\n\nNote that [1a] can easily be written in terms of these coordinates and \ x(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ")\n\nFrom [8] and [10], we have seen that the \[Chi] modes can be \ determined without knowing ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], " or ", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], "\n\nThis is, of course, obvious. CM coordinates inherently are \ independent of the \"world-space\" position of the body, so I have a set of \ coordinates that give me the CM system in terms of the full-string \ coordinates ", Cell[BoxData[ \(TraditionalForm\`x(\[Pi]\/2)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`x\_m\)]], " where m\[Element]Z\[Epsilon][0, \[Infinity]), and I can reconstruct these \ full-string coordinates mostly from the half-string coordinates and the \ world-space position x(", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], "). There is no ambiguity - all CM coordinates \"forget\" this.\n\n[29]\n\n\ ", StyleBox["3-String Annihilation", "Subsection"], "\n\n", Cell[OLEData["\<\ d@1403D0@Di900000780 K`1C06T0K@0P05@0I@1h07@0801506@0J@1d06l0LP0P04@0K`1S07D0K@1U06h0 M00907/0@@0`03<0>00g0300>00`02d0@`0`0440<`0]0340<@140300;@0i04@0 <00e02d0<00`0300<0130300@`150300@`1504D0O@00000000000000000820P8 20P820P820P820P820P820P820P000000000000000000000000000000000P020 0800P0200820P820P820P1@03`0:0040J@0?00<000000000000h0010lOl203P0 300604h0K`1b06d0H@1/00000P0001P0@dXH05m80@AQBQP0KDP917=82@AdB0T4 000000000000000000000000?011@?;oX@0l00`05P1406D0IP1Q07D0K01d0200 D01Q0780H@1W0780H@1`06P0801606l0KP1d00000000000000000000000?0000 10003000003ooooo0@0000@Pool1041F``000000000000l00000000000000000 4@0009X00000<000000000200000P004000?10000`000004000?100010000004 000?10001@0000l00?0h00000006l1P0000A10000P000100000100000@000140 001001ka40000?oo00000?l0P8200?L001003`02l7@7000@00S`200000l0000@ 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oPg0Ud3X`9M0kL2 @@?;0R@3^@8:0jL2l0:F000090=904D33`10bd3P`P>30d<3O`=<0gP3 E0=`0eX3I0=M0eH3GP=>0eh3B0=M0cd3F`"], "Graphics", ImageSize->{216, 144}], "\n\nThe image above diagrams the interaction of 3 strings, ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\_1, \[Chi]\_2\ and\ \(\(\[Chi]\_3\)\(\ \ \)\(.\)\)\)]], " In half-string coordinates, I assert that\n\n[29a]\n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(V\_3\), FontWeight->"Bold"], " ", "=", " ", \(\(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(i = 1\)\%3\(\[Sum]\ \+\(j = 1\)\%3\(\[Sum]\+\(s = {L, \ R}\)\(\[Sum]\+\(t = {L, \ \ R}\)\(\[Sum]\+\(m = 1\)\%\[Infinity]\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\)\)\)\ \)\)\)\) 0\_\(1 L\)\ 0\_\(1\ R\)\ 0\_\(2\ L\)\ 0\_\(2\ R\)\ 0\_\(3\ L\)\ 0\_\ \(3\ R\)\)}], TraditionalForm]]], "\n\nis the most general form for the 3-string vacuum\n\nWhere k and l \ select elements within a matrix, S and T select Left and Right, and A and B \ select strings 1 through 3.\n\n[29b]\n \nNow, allowing my strings to \ annihilate,\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\[Chi]\_\(1\ L\)\), "TraditionalForm"], "(", "\[Sigma]", ")"}], "-", \(\(\[Chi]\_\(3\ R\)\)(\[Sigma])\)}], ")"}], " ", \(V\_3\)}], " ", "=", " ", "0"}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\[Chi]\_\(2\ L\)\), "TraditionalForm"], "(", "\[Sigma]", ")"}], "-", \(\(\[Chi]\_\(1\ R\)\)(\[Sigma])\)}], ")"}], " ", \(V\_3\)}], " ", "=", " ", "0"}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\[Chi]\_\(3\ L\)\), "TraditionalForm"], "(", "\[Sigma]", ")"}], "-", \(\(\[Chi]\_\(2\ R\)\)(\[Sigma])\)}], ")"}], " ", \(V\_3\)}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\nWhich can also be expressed as the constraint\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ FormBox[\(\[Chi]\_\(i\ L\)\), "TraditionalForm"], "(", "\[Sigma]", ")"}], "-", \(\(\[Chi]\_\(2 \((\((i\ + \ 1)\)\ Mod\ 3)\)\ + \ 1\ R\)\)(\[Sigma])\)}], ")"}], " ", \(V\_3\)}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\nRecalling from [5] that\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Chi]\_L\)(\[Sigma])\), "=", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/2\), RowBox[{\(\[Chi]\_\(L\ n\)\), "(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], \(Cos(n\ \[Sigma])\)}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Chi]\_R\)(\[Sigma])\), "=", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(1\/2\), RowBox[{\(\[Chi]\_\(R\ n\)\), "(", RowBox[{"1", "-", FormBox[\(\((\(-1\))\)\^n\), "TraditionalForm"]}], ")"}], \(Cos(n\ \[Sigma])\)}]}]}], TraditionalForm]]], "\n\nso\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_L\)(\[Sigma]) = \[Sum]\+\(n = 1\)\%\ \[Infinity]\( \[Chi]\_\(L\ 2 n - 1\)\) \(Cos(n\ \[Sigma])\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\_R\)(\[Sigma]) = \[Sum]\+\(n = 1\)\%\ \[Infinity]\( \[Chi]\_\(R\ 2 n - 1\)\) \(Cos(n\ \[Sigma])\)\)]], "\n\nand from [25a]\n\n", Cell[BoxData[ \(TraditionalForm\`b\_\(\(\ \)\(i\ s\ n\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\(\ \)\(i\ s\ n\)\)\^+\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ImaginaryI]\)\)\)]], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ FormBox[\(\(2 n\)\(\ \)\(-\)\(\ \)\), "TraditionalForm"], "1"}]], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Chi]\_\(\(\ \)\(i\ s\ 2 n\ - \ 1\)\)\)]], "\n\nso\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Chi]\_\(\(i\)\(\ \)\(s\)\(\ \)\)\)(\[Sigma])\), "=", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n\ - \ 1\)\), RowBox[{"(", RowBox[{\(b\_\(i\ s\ n\)\), "-", FormBox[\(b\_\(i\ s\ n\)\^+\), "TraditionalForm"]}], ")"}], \(Cos(\((2 n\ - \ 1)\)\ \[Sigma])\)}]}]}], TraditionalForm]]], "\n\nSubstituting,\n\n[29c]\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n\ - \ 1\)\), RowBox[{"(", RowBox[{\(b\_\(i\ s\ n\)\), "-", FormBox[\(b\_\(i\ s\ n\)\^+\), "TraditionalForm"], "-", RowBox[{"(", RowBox[{\(b\_\(j\ t\ n\)\), "-", FormBox[\(b\_\(j\ t\ n\)\^+\), "TraditionalForm"]}], ")"}]}], ")"}], \(Cos(\((2\ n\ - \ 1)\)\ \[Sigma])\), " ", \(V\_3\)}]}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\n(henceforth sum over repeated indices)\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n' - \ 1\)\), RowBox[{"(", RowBox[{\(b\_\(\(\ \)\(i'\ s'\ n'\)\)\), "-", FormBox[\(b\_\(\(\ \)\(i'\ s'\ n'\)\)\^+\), "TraditionalForm"], "-", RowBox[{"(", RowBox[{\(b\_\(\(\ \)\(j'\ t'\ n'\)\)\), "-", FormBox[\(b\_\(j'\ t'\ n'\)\^+\), "TraditionalForm"]}], ")"}]}], ")"}], \(Cos(\((2 n\ - \ 1)\)\ \[Sigma])\), " ", \(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ \ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\), \(0\_\(1 L\)\), " ", \(0\_\(1\ R\)\), " ", \(0\_\(2\ L\)\), " ", \(0\_\(2\ R\)\), " ", \(0\_\(3\ L\)\), " ", \(0\_\(3\ R\)\)}]}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\nChoosing \[Sigma] = 0,\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n'\ - \ 1\)\), RowBox[{"(", RowBox[{\(b\_\(\(\ \)\(i'\ s'\ n'\)\)\), "-", FormBox[\(b\_\(\(\ \)\(i'\ s'\ n'\)\)\^+\), "TraditionalForm"], "-", RowBox[{"(", RowBox[{\(b\_\(\(\ \)\(j'\ t'\ n'\)\)\), "-", FormBox[\(b\_\(j'\ t'\ n'\)\^+\), "TraditionalForm"]}], ")"}]}], ")"}], \(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ \ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\), \(0\_\(1 L\)\), " ", \(0\_\(1\ R\)\), " ", \(0\_\(2\ L\)\), " ", \(0\_\(2\ R\)\), " ", \(0\_\(3\ L\)\), " ", \(0\_\(3\ R\)\)}]}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\[Divide]\nCommuting with the formula from [24f], ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FormBox[ RowBox[{\(b\_\(\(\ \)\(n\)\(\ \)\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(-\(1\/2\)\) \(b\^+\) \ \[Phi]\ \(b\^+\)\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}], "TraditionalForm"], " ", "=", RowBox[{\(-\ \[Phi]\_\(n\ m\)\), " ", \((\(b\_m\^+\))\), \(\[ExponentialE]\^\(\(-\(1\/2\)\) \ \((\(b\^+\))\)\ \(\[Phi](\(b\^+\))\)\)\), SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "L"], SubscriptBox[ StyleBox["0", FontWeight->"Bold"], "R"]}]}]}], TraditionalForm]]], " (summing over repeated indices)\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n'\ - \ 1\)\), RowBox[{\(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ \ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\), "(", RowBox[{\(-\(\[Sum]\+\(m' = 1\)\%\[Infinity]\ \((H\_\(\(\ \ \)\(\(n'\)\(\ \)\(m'\)\(\ \)\(i'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\(s'\)\(\ \ \)\)\)\ \((\(b\_\(\(m'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\)\^+\))\))\)\)\), "-", FormBox[\(b\_\(\(\ \)\(\(n'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\ \)\)\^+\), "TraditionalForm"], "-", RowBox[{"(", RowBox[{\(-\ \(\[Sum]\+\(m' = 1\)\%\[Infinity]\((H\_\(\(\ \ \)\(\(n'\)\(\ \)\(m'\)\(\ \)\(j'\)\(\ \)\(j'\)\(\ \)\(t'\)\(\ \)\(t'\)\(\ \ \)\)\)\ \((\(b\_\(\(m'\)\(\ \)\(j'\)\(\ \)\(t'\)\(\ \)\)\^+\))\))\)\)\), "-", FormBox[\(b\_\(\(\ \)\(n'\ j'\ t'\)\)\^+\), "TraditionalForm"]}], ")"}]}], ")"}], " ", \(0\_\(1 L\)\), " ", \(0\_\(1\ R\)\), " ", \(0\_\(2\ L\)\), " ", \(0\_\(2\ R\)\), " ", \(0\_\(3\ L\)\), " ", \(0\_\(3\ R\)\)}]}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\ndigressing for the moment,\n[29d]\n\nNoting that for all terms where \ the i and s does not match the j and t in the exponent:\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(b\_\(\(\ \)\(\(n'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \ \)\)\)\) \[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ m\ i\ j\ s\ \ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\)\(=\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Delta]\_\(\[Rho]\_n\)\ \(\[ExponentialE]\^\(\ \[Rho]\ b\_\(\(\ \)\(\(i'\)\(\ \)\(s'\)\(\ \)\)\)\)\) \[ExponentialE]\^\(1\/2\ \ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\)\( \ | \_\(\[Rho]\ = \ 0\)\)\)\(=\)\)\)]], "(missing indices becoming vectors)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Delta]\_\(\[Rho]\_n\)\ \ \(\[ExponentialE]\^\(1\/2\ \[Rho]\ H\_\(n\ m\ i'\ j\ s'\ t\)\ \(b\_\(m\ j\ \ t\)\^+\)\)\) \(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ m\ i\ \ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\) \[ExponentialE]\^\(\[Rho]\ b\_\(\(\ \)\(\ \(i'\)\(\ \)\(s'\)\(\ \)\)\)\)\)\( | \_\(\[Rho]\ = \ 0\)\)\)\(=\)\)\)]], "\n\n(For all non-matching j and t)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(1\/2\) \(\((H\_\(\(\ \)\(i'\ j\ s'\ t\)\)\ \(b\ \_\(\(\ \)\(j\ t\)\)\^+\))\)\_n\) \(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\ \^+\)\ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\) b\_\(\(\ \)\(\(n'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\)\)\)\(=\)\)\)]], "\n\n(allowing the right term to annihilate the vacuum)\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(1\/2\) \(\((H\_\(\(\ \)\(i'\ j\ s'\ t\)\)\ \(b\ \_\(\(\ \)\(j\ t\)\)\^+\))\)\_n\) \[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ \ s\)\^+\)\ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\)\(=\)\)\)]], "\n\n\nWhich leaves the possibility of a contribution from these terms. \ However, the string that this is paired with in the vacuum will produce \ similar negative terms, cancelling two of them. This means that \n\n", Cell[BoxData[ \(TraditionalForm\`H\_\(n\ m\ i\ j\ s\ t\)\)]], "= 0 where i \[NotEqual] j and s \[NotEqual] t except possibly for the i s \ j t pairs {1L, 3R}, {2L, 1R} and {3L, 2R}\n\nReturning to [29c]\n\n[29e]\n\n\ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\(-\ \[ImaginaryI]\)\/\@\(2\ n'\ - \ 1\)\), RowBox[{\(\[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ \ H\_\(n\ m\ i\ j\ s\ t\)\ \(b\_\(m\ j\ t\)\^+\)\)\), "(", RowBox[{\(\((H\_\(\(\ \)\(i'\ j'\ s'\ t'\)\)\ \(b\_\(\(\ \ \)\(j'\ t'\)\)\^+\))\)\_\(n'\)\), "-", \(\[Sum]\+\(m' = 1\)\%\[Infinity]\ \((H\_\(\(\ \ \)\(\(n'\)\(\ \)\(m'\)\(\ \)\(i'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\(s'\)\(\ \ \)\)\)\ \((\(b\_\(\(m'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\)\^+\))\))\)\), "-", FormBox[\(b\_\(\(\ \)\(\(n'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \)\ \)\)\^+\), "TraditionalForm"], " ", "-", \(\((H\_\(\(\ \)\(j'\ i'\ t'\ s'\)\)\ \(b\_\(\(\ \ \)\(i'\ s'\)\)\^+\))\)\_\(n'\)\), " ", "+", " ", \(\[Sum]\+\(m' = 1\)\%\[Infinity]\((H\_\(\(\ \ \)\(\(n'\)\(\ \)\(m'\)\(\ \)\(j'\)\(\ \)\(j'\)\(\ \)\(t'\)\(\ \)\(t'\)\(\ \ \)\)\)\ \((\(b\_\(\(m'\)\(\ \)\(j'\)\(\ \)\(t'\)\(\ \)\)\^+\))\))\)\), "+", FormBox[\(b\_\(\(\ \)\(n'\ j'\ t'\)\)\^+\), "TraditionalForm"]}], ")"}], " ", \(0\_\(1 L\)\), " ", \(0\_\(1\ R\)\), " ", \(0\_\(2\ L\)\), " ", \(0\_\(2\ R\)\), " ", \(0\_\(3\ L\)\), " ", \(0\_\(3\ R\)\)}]}], " ", "=", " ", "0"}], TraditionalForm]]], "\n\nSince the + terms do not annihilate the vacuum, and ", Cell[BoxData[ \(TraditionalForm\`\(\(-\ \[ImaginaryI]\)\/\@\(2\ n\ - \ 1\)\) \ \[ExponentialE]\^\(1\/2\ \(b\_\(n\ i\ s\)\^+\)\ H\_\(n\ m\ i\ j\ s\ t\)\ \ \(b\_\(m\ j\ t\)\^+\)\)\)]], "is never zero, to make the above expression zero, the terms arising from \ the gaussian must drop out, so\n\n\n", Cell[BoxData[ \(TraditionalForm\`H\_\(\(\ \)\(\(n'\)\(\ \)\(m'\)\(\ \)\(j'\)\(\ \ \)\(j'\)\(\ \)\(t'\)\(\ \)\(t'\)\(\ \)\)\)\)]], "= 0 for all n' and m'\n\nand the terms that arose from the \ non-cancellation of mismatched i s j t pairs must cancel the remaining b \ creation operators, so \n\n", Cell[BoxData[ \(TraditionalForm\`\((H\_\(\(\ \)\(i'\ j'\ s'\ t'\)\)\ \(b\_\(\(\ \ \)\(j'\ t'\)\)\^+\))\)\_\(n'\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\(\ \)\(j'\ t'\)\)\^+\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((H\_\(\(\ \)\(j'\ i'\ t'\ s'\)\)\ \(b\_\(\(\ \ \)\(i'\ s'\)\)\^+\))\)\_\(n'\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(b\_\(\(\ \)\(\(n'\)\(\ \)\(i'\)\(\ \)\(s'\)\(\ \ \)\)\)\^+\)\)]], "\n\nmeaning that ", Cell[BoxData[ \(TraditionalForm\`H\_\(\(\ \)\(n'\ m'\ i'\ j'\ s'\ t'\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(n'\ m'\)\)]], "for the i s j t pairs {1L, 3R}, {2L, 1R} and {3L, 2R}\n\nSo collapsing H \ into one infinite matrix instead of six,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(H\_\(\(\ \)\(n\ m\ i\ j\ s\ t\)\)\), "=", FormBox[ RowBox[{"-", " ", RowBox[{ FormBox[\(\[Delta]\_\(m\ n\)\), "TraditionalForm"], "(", RowBox[{ RowBox[{ FormBox[\(\[Delta]\_\(j\ \ \((\((i\ + \ 1)\)\ Mod\ 3)\)\ + \ 1\)\), "TraditionalForm"], " ", \(\[Delta]\_\(s\ L\)\), \(\[Delta]\_\(t\ R\)\)}], "+", " ", RowBox[{ FormBox[\(\[Delta]\_\(i\ \ \((\((j + \ 1)\)\ Mod\ 3)\)\ + \ 1\)\), "TraditionalForm"], " ", \(\[Delta]\_\(s\ R\)\), \(\[Delta]\_\(t\ L\)\)}]}], ")"}]}], "TraditionalForm"]}], TraditionalForm]]], "\n\nfor \n\n(Letting L = R + 1 and R = L + 1)\n\n", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(V\_3\), FontWeight->"Bold"], " ", "=", " ", \(\(\[ExponentialE]\^\(\(-1\)\/2\ \(b\_\(\(\ \)\(i\ s\)\)\^+\)\ \ \ \(b\_\(\((\((i\ + \ 1)\)\ Mod\ 3)\)\ + \ 1, \ s\ + \ 1\)\^+\)\)\) 0\_\(1 L\)\ 0\_\(1\ R\)\ 0\_\(2\ L\)\ 0\_\(2\ R\)\ 0\_\(3\ L\)\ \ 0\_\(3\ R\)\)}], TraditionalForm]]], " (b is a vector)\n\n", StyleBox["Ward Identities\n\n", FontSize->16, FontWeight->"Bold"], StyleBox["For now I return to the full-string language to go over the Ward \ identities, which act on my vacuum to provide a family of constraints. \n\n\ They are given by\n\n[30a]\n\n", FontSize->16, FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`L\_\(m\ x\ r\)\ = \ \[Sum]\+\(n = \ 1\)\%\[Infinity]\( \[Alpha]\_\(r\ n\)\^+\)\ \[Alpha]\_\(r\ n + m\)\ + \ \ \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\ + \ p\_\(r\ 0\)\ \[Alpha]\_\(r\ m\)\)]], " (m > 0)\n", Cell[BoxData[ \(TraditionalForm\`L\_\(0\ x\ r\)\ = \ \(1\/2\) p\_\(r\ 0\)\^2\ + \ \[Sum]\+\(n = 1\)\%\[Infinity]\(\( \ \[Alpha]\_\(r\ n\)\^+\)\(\ \)\(\[Alpha]\_\(r\ n\)\)\(\ \)\)\)]], " (m = 0)\n\nWhere r selects the string being acted on and x designates \ that these identities are meant for the matter sector, which is invisible to \ the ghost sector.\n\nSimilarly, but with an extra term, the ghost-sector \ identity is \n\n[30b]\n\n", Cell[BoxData[ \(TraditionalForm\`L\_\(m\ \[Phi]\ r\)\ = \ \[Sum]\+\(n = 1\)\%\ \[Infinity]\( \[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\ \[Alpha]\_\(\[Phi]\ r\ n + m\)\ \ + \ \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\ + \ \((p\_\(\[Phi]\ r\ 0\)\ - \ \ \(3\/2\) n)\) \[Alpha]\_\(\[Phi]\ r\ m\)\)]], " (m > 0)\n", Cell[BoxData[ \(TraditionalForm\`L\_\(0\ \[Phi]\ r\)\ = \ \(1\/2\) p\_\(\[Phi]\ r\ 0\)\^2\ - \ 1\/8 + \ \[Sum]\+\(n = 1\)\%\[Infinity]\(\( \[Alpha]\_\(\[Phi]\ r\ \ n\)\^+\)\(\ \)\(\[Alpha]\_\(\[Phi]\ r\ n\)\)\(\ \)\)\)]], " (m = 0)\n\nAlso, the 3-string matter vertex is expressed by\n\n[30c]\n\n\ ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(V\^x\), FontWeight->"Bold"], StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], RowBox[{ StyleBox[\(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(\[Mu] = 0\)\%5\ \(\[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, n = 0\)\%\[Infinity]\( \ \[Alpha]\_\(r\ \[Mu]\ m\)\^+\) N\_\(\(r\)\(\ \)\(s\)\(\ \)\(m\)\(\ \)\(n\)\(\ \)\ \)\ \(\[Alpha]\_\(s\ \[Mu]\ n\)\^+\)\)\)\)\)\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\nAnd the bose-style representation of the ghost vacuum is expressed by \ \n\n[30d]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(V\^\[Phi]\), StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], RowBox[{\(\[ExponentialE]\^\(\(1\/2\) \[ImaginaryI]\ \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\(\[Phi]\_r\)(\[Pi]\/2)\)\(\ \)\)\)\), \(\ \[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, \ n = 0\ \)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \(N\_\(r\ s\ m\ n\)\) \(\ \[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\), StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\nSo, to get useful constraints from the Ward Identities, I need to \ evaluate\n\n", Cell[BoxData[ FormBox[ RowBox[{\((L\_\(\(x\)\(\ \)\) + \ L\_\[Phi])\), StyleBox[\(V\^\[Phi]\), FontWeight->"Bold"], StyleBox[\(V\^x\), FontWeight->"Bold"]}], TraditionalForm]]], "\n\nDigressing for a moment to rewrite the ghost vacuum,\n\n[30.1a]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(V\^\[Phi]\), StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(1\/2\) \[ImaginaryI]\ \(\(\[Sum]\+\(r \ = 1\)\%3\)\(\(\[Phi]\_r\)(\[Pi]\/2)\)\(\ \)\)\)\), \ \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, \ n = \ 0\)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \(N\_\(r\ s\ m\ n\)\) \(\ \[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\), StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], " =\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(1\/2\), "\[ImaginaryI]", " ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), RowBox[{"(", RowBox[{ FormBox[\(\[Phi]\_\(r\ 0\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2 n'\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], "TraditionalForm"]}]}], ")"}], " "}]}]], \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = \ 1\)\%3\(\[Sum]\+\(m, \ n = 0\)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \ \(N\_\(r\ s\ m\ n\)\) \(\[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\)}], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(\(0\_1\) \(0\_2\) 0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "=\n\nWhich I'd like to write in terms of creation operators only.\n\n\ Noting that \n\n[30.1b]\n\n", Cell[BoxData[ \(TraditionalForm\`\([a\_n, \ a\_m]\)\ = \ \[Delta]\_\(m, \ n\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_n\ = \(\@n\) a\_n\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\([\[Alpha]\_n, \ \[Alpha]\_m]\)\ = \ n\ \[Delta]\_\(m, \ n\)\)]], "\n\nAnd recalling that \n\n[1a]\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Phi]\_r\)(\[Sigma])\), "=", RowBox[{ FormBox[\(\[Phi]\_\(r\ 0\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity] \[Phi]\_\(r\ n\)\), "TraditionalForm"], \(cos(n\ \[Sigma])\)}]}]}], TraditionalForm]]], "\n[1b]\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Phi]\_\(r\ 0\)\), "=", RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], "\[ImaginaryI]", " ", RowBox[{"(", RowBox[{ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " \n", Cell[BoxData[ FormBox[ RowBox[{\(p\_0\), "=", RowBox[{"(", FormBox[\(a\_\(\(0\)\(\ \)\) + \ \(a\_0\^+\)\), "TraditionalForm"], ")"}]}], TraditionalForm]]], " \n[1c]\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Phi]\_\(\(\ \)\(r\ n\)\)\), "=", RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(\[ImaginaryI]\ \@\(2\/n\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(\[Phi]\ r\ n\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ n\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n\nso\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Phi]\_r\)(\[Pi]\/2)\), "=", RowBox[{ FormBox[\(\[Phi]\_\(r\ 0\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\(\((\(-1\))\)\^n\) \ \[Phi]\_\(r\ \ 2 n\)\), "TraditionalForm"]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[ImaginaryI]\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), " ", RowBox[{"(", RowBox[{\(\[Phi]\_\(r\ 0\)\), " ", "+", FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], "TraditionalForm"]}], ")"}], " "}]}]}]], \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ \ s = 1\)\%3\(\[Sum]\+\(m, \ n = 0\)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ \ m\)\^+\) \(N\_\(r\ s\ m\ n\)\) \(\[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\)}], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(\(0\_1\) \(0\_2\) 0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], "=\n\nWhich using the identity\n\n", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(A\ + \ B\)\ = \ \(\ \[ExponentialE]\^\(B\ + \ A\)\ = \ \(\[ExponentialE]\^B\) \(\[ExponentialE]\ \^A\) \[ExponentialE]\^\(\(-1\)\/2\)[B, \ A]\)\)]], "\n\nbecomes\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), RowBox[{"{", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[ImaginaryI]\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], "TraditionalForm"]}]}]}]], "}"}], \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = \ 1\)\%3\(\[Sum]\+\(m, \ n = 0\)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \ \(N\_\(r\ s\ m\ n\)\) \(\[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\), StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "=\n\nNow evaluating the part in braces,\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\(-\[ImaginaryI]\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]}]], SuperscriptBox["\[ExponentialE]", RowBox[{\(\[ImaginaryI]\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]], SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\(-1\)\/2\), "[", RowBox[{ RowBox[{\(\(-\[ImaginaryI]\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}], ",", " ", RowBox[{\(\[ImaginaryI]\/2\), RowBox[{\(\[Sum]\+\(r' = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n'' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n''\), FormBox[\(\[ImaginaryI]\/\(2\ n''\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r'\ 2 n''\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]}], "]"}]}]]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\(-\[ImaginaryI]\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]}]], StyleBox[ SuperscriptBox["\[ExponentialE]", RowBox[{\(\(-\[ImaginaryI]\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(\[ImaginaryI]\/\(2\ n'\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]], FontColor->GrayLevel[ 0]], \(\[ExponentialE]\^\(\(\ \)\(\(1\/8\) \(\[Sum]\+\(r = 1\)\%3\ \(\[Sum]\+\(n' = 1\)\%\[Infinity]\((\(-1\))\)\^n'\/\(2\ n'\)\)\)\)\)\)}], TraditionalForm]]], "\n\nThe commutator being just a number, it can be pulled to the front\n\n\ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(1\/8\) \(\[Sum]\+\(r = \ 1\)\%3\(\[Sum]\+\(n' = 1\)\%\[Infinity]\((\(-1\))\)\^n'\/\(2\ n'\)\)\)\)\)\), \ \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]}]], RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(\(-1\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]], \ \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, \ n = \ 0\)\%\[Infinity]\( \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \(N\_\(r\ s\ m\ n\)\) \(\ \[Alpha]\_\(s\ \[Phi]\ n\)\^+\)\)\)\)\)}], "}"}], StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox["=", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "\n\nNow I wish to commute the bracketed portion of the above.\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(\(-1\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]], \(\[ExponentialE]\^\(\(1\/2\) \(\ \[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, \ n = 0\)\%\[Infinity]\( \ \[Alpha]\_\(r\ \[Phi]\ m\)\^+\) \(N\_\(r\ s\ m\ n\)\) \(\[Alpha]\_\(s\ \[Phi]\ \ n\)\^+\)\)\)\)\)}], TraditionalForm]]], "\n\nNow employing the gaussian identity ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\(w\^T\)[A]\^\(-1\)\) w\) = \ \[Product]\+\(i = 1\)\%N\(\(\( \[Pi]\^\(N\/2\)\)( det\ A)\)\^\(1\/2\)\) \(\[Integral]\(\[ExponentialE]\^\(\(-\ \ x\^T\) A\ x\)\) \(\[ExponentialE]\^\(\(w\^T\) x\)\) \[DifferentialD]x\_i\)\)]], " and breaking up my gaussian,\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(\(-1\)\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", FormBox[\(\[Product]\+\(i = 1\)\%N\(\(\( \[Pi]\^\(N\/2\)\)( det\ N)\)\^\(1\/2\)\) \ \(\[Integral]\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \(x\^T\) N\ x\)\) \(\[ExponentialE]\^\(\(\[Alpha]\^+\)\ \ x\)\) \[DifferentialD]x\_\(s'\ i\)\)\), "TraditionalForm"]}], TraditionalForm]]], "\n\nNow defining ", Cell[BoxData[ FormBox[ RowBox[{\(A\_\(r\ n\)\), " ", "=", RowBox[{\(\(-1\)\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), FormBox[\(1\/\(\(\ \)\(n\)\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], "TraditionalForm"], " "}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(A\_\(r\ n\)\ \[Alpha]\_\(r\ n\)\)\), " ", FormBox[\(\[Product]\+\(i = 1\)\%N\(\(\( \[Pi]\^\(N\/2\)\)( det\ N)\)\^\(1\/2\)\) \ \(\[Integral]\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \(x\^T\) N\ x\)\) \(\[ExponentialE]\^\(\(\[Alpha]\^+\)\ \ x\)\) \[DifferentialD]x\_\(s'\ i\)\)\), "TraditionalForm"]}], TraditionalForm]]], "\n\nI need to commute \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(A\_\(r\ n\)\ \[Alpha]\_\(r\ n\)\)\), " ", FormBox[\(\[ExponentialE]\^\(\(\[Alpha]\_\(r\ t\)\^+\)\ x\)\), "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^A\) \[ExponentialE]\^B\ = \ \(\ \[ExponentialE]\^B\) \(\[ExponentialE]\^A\) \[ExponentialE]\^\([A, \ \ B]\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\(\[Alpha]\_\(r\ t\)\^+\)\ \ x\)\) \[ExponentialE]\^\(A\_\(r\ n\)\ \[Alpha]\_\(r\ n\)\)\ \[ExponentialE]\^\ \([A\_\(r\ n\)\ \[Alpha]\_\(r\ n\), \(\[Alpha]\_\(r\ t\)\^+\)\ x]\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\(\[Alpha]\_\(r\ t\)\^+\)\ \ x\)\) \[ExponentialE]\^\(A\_\(r\ n\)\ \[Alpha]\_\(r\ n\)\)\ \[ExponentialE]\^\ \(\(n\)\(\ \)\(A\_\(r\ n\)\) \(x\)\(\ \)\)\)]], "\n\nAnd letting the annihilator term act on the vacuum,\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Product]\+\(i = 1\)\%N\(\(\( \[Pi]\^\(N\/2\)\)( det\ N)\)\^\(1\/2\)\) \ \(\[Integral]\(\[ExponentialE]\^\(\(-\ \(1\/2\)\) \(x\^T\) N\ x\)\) \(\[ExponentialE]\^\(\((\(\[Alpha]\_\(r\ t\ \)\^+\)\ + \ n\ A\_\(r\ n\))\)\ x\)\) \[DifferentialD]x\_\(s'\ i\)\)\), "TraditionalForm"]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = 1\)\ \%3\(\[Sum]\+\(m, \ n = 0\)\%\[Infinity]\((\(\[Alpha]\_\(r\ t\)\^+\)\ + \ m\ \ \(A\_\(r\ m\)\^+\))\) \(\(N\_\(r\ s\ m\ n\)\)(\(\[Alpha]\_\(s\ n\)\^+\)\ + \ \ n\ A\_\(s\ n\))\)\)\)\)\)]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(1\/8\) \(\[Sum]\+\(r = \ 1\)\%3\(\[Sum]\+\(n' = 1\)\%\[Infinity]\((\(-1\))\)\^n'\/\(2\ n'\)\)\)\)\)\), \ \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Sum]\+\(r, \ s = 1\)\%3\(\ \[Sum]\+\(m, \ n = 0\)\%\[Infinity]\((\(\[Alpha]\_\(r\ m\)\^+\)\ + \ m\ \(A\_\(r\ m\)\^+\))\) \(\(N\_\(r\ s\ m\ n\)\)(\ \(\[Alpha]\_\(s\ n\)\^+\)\ + \ n\ A\_\(s\ n\))\)\)\)\)\), " ", StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox["=", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "\n\nNow expanding my exponential and exploiting the symmetry of N, and \ accumulating real constants into C,\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ "C", " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\), " ", \(\[ExponentialE]\^\(\[Sum]\+\(r, \ s = 1\)\%3\(\[Sum]\+\(m, \ \ n = 0\)\%\[Infinity]\((\(\[Alpha]\_\(r\ m\)\^+\))\) \(\(N\_\(r\ s\ m\ \ n\)\)(\ n\ A\_\(s\ n\))\)\)\)\), " ", StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox["=", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "\n\n\nRecalling my definition of A, ", Cell[BoxData[ FormBox[ RowBox[{\(A\_\(r\ n\)\), " ", "=", RowBox[{\(\(-1\)\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), FormBox[\(1\/\(\(\ \)\(n\)\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], "TraditionalForm"], " "}]}], TraditionalForm]]], ",\n\nIt would be highly convenient if I had a convenient identity for \n\n\ ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(r = 1\)\%3\(\[Sum]\+\(s = 1\)\%3\(\[Sum]\+\ \(m = 1\)\%\[Infinity]\(\[Sum]\+\(n = 1\)\%\[Infinity]\( N\_\(r\ s\ m\ n\)\)(\ n\ A\_\(s\ n\))\)\)\)\)]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(r = 1\)\%3\), RowBox[{\(\[Sum]\+\(s = 1\)\%3\), RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), RowBox[{\(N\_\(r\ s\ m\ n\)\), "(", " ", RowBox[{"n", " ", \(\(-1\)\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), FormBox[\(1\/\(\(\ \)\(n\)\)\), "TraditionalForm"], FormBox["", "TraditionalForm"]}], "TraditionalForm"]}], ")"}]}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(s = 1\)\%3\), RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), " ", RowBox[{\(\(-1\)\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), \(1\/\(\(\ \)\(n\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], \(\[Sum]\+\(r = 1\)\%3\( N\_\(r\ s\ m\ n\)\) n\)}]}]}]}], TraditionalForm]]], "\n\nI do have an identity (Hloused and Jevicki, p. 149) that ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(r = 1\)\%3 m\ N\_\(r\ s\ m\ n\)\ = \ \((\(-1\))\)\^\(m\ + \ 1\)\ \ \[Delta]\_\(m, \ n\)\)]], ", so\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(s = 1\)\%3\), RowBox[{\(\[Sum]\+\(m = 1\)\%\[Infinity]\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), " ", RowBox[{\(\(-1\)\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), \(1\/\(\(\ \)\(n\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], " ", \(\((\(-1\))\)\^\(m\ + \ 1\)\), " ", \(\[Delta]\_\(m, \ n\)\)}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(s = 1\)\%3\), RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), " ", RowBox[{\(1\/2\), FormBox[ RowBox[{\(Cos(\(n\ \[Pi]\)\/2)\), \(1\/\(\(\ \)\(n\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], " ", \(\((\(-1\))\)\^n\), " "}]}]}], TraditionalForm]]], "\n\nLeaving me with\n\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), " ", RowBox[{\(\[Alpha]\_\(r\ m\)\^+\), \(1\/2\), FormBox[ RowBox[{\(Cos(\(m\ \[Pi]\)\/2)\), \(1\/\(\(\ \)\(m\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], " ", \(\((\(-1\))\)\^m\), " "}]}]], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), " ", RowBox[{\(\[Alpha]\_\(r\ 2\ m\)\^+\), FormBox[ RowBox[{\(\((\(-1\))\)\^m\), \(1\/\(\(\ \)\(2\ m\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], " ", \(\((\(-1\))\)\^\(2 m\)\), " "}]}]}]], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(r = 1\)\%3\), " ", RowBox[{\(\[Alpha]\_\(r\ 2\ m\)\^+\), FormBox[ RowBox[{\(\((\(-1\))\)\^m\), \(1\/\(\(\ \)\(2\ m\)\)\), FormBox["", "TraditionalForm"]}], "TraditionalForm"], " "}]}]}]], TraditionalForm]]], "\n\nRecombining with the original, this leaves me with \n\n", Cell[BoxData[ FormBox[ RowBox[{ "C", " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\), " ", StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], TraditionalForm]]], "\n\nNow returning to the Ward identites,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(V\^\[Phi]\), StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], RowBox[{ "C", " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\), " ", StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\nTo find the identity exactly, I start by evaluating the commutation.\n\ \n[30e]\n\n", Cell[BoxData[ FormBox[ RowBox[{\(L\_\(\[Phi]\ r\ m\)\), StyleBox[\(V\^\[Phi]\), FontWeight->"Bold"]}], TraditionalForm]]], "\n\nFor \n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\((\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\[Phi]\ r\ \ n\)\^+\)\ \[Alpha]\_\(\[Phi]\ r\ n + m\)\ + \ \(1\/2\) \(\[Sum]\+\(n = 1\)\%\ \(m - 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\ + \ \((p\_\(\ \[Phi]\ r\ 0\)\ - \ \(3\/2\) n)\) \[Alpha]\_\(\[Phi]\ r\ m\))\), StyleBox[" ", FontWeight->"Plain", FontVariations->{ "CompatibilityType"-> 0}], \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), " ", \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), \(\[ExponentialE]\^\(\(\(-1\)\/2\) \((\ \(\[Alpha]\_\(\(\ \)\(r'\)\)\^+\))\) \(N(\(\[Alpha]\_\(\(\ \)\(s'\)\(\ \ \)\)\^+\))\)\)\), StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\nNoting that \n\n[30f]\n\n\nTHESE ARE WRONG \n\n", Cell[BoxData[ \(TraditionalForm\`\([a\_n, \ a\_m]\)\ = \ \[Delta]\_\(m, \ n\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_n\ = \(\@n\) a\_n\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\([\[Alpha]\_n, \ \[Alpha]\_m]\)\ = \ n\ \[Delta]\_\(m, \ n\)\)]], "\n\nAnd recalling that \n\n[1a]\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Phi]\_r\)(\[Sigma])\), "=", RowBox[{ FormBox[\(\[Phi]\_\(r\ 0\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity] \[Phi]\_\(r\ n\)\), "TraditionalForm"], \(cos(n\ \[Sigma])\)}]}]}], TraditionalForm]]], "\n[1b]\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Phi]\_\(r\ 0\)\), "=", RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], "\[ImaginaryI]", " ", RowBox[{"(", RowBox[{ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " \n", Cell[BoxData[ FormBox[ RowBox[{\(p\_0\), "=", RowBox[{"(", FormBox[\(a\_\(\(0\)\(\ \)\) + \ \(a\_0\^+\)\), "TraditionalForm"], ")"}]}], TraditionalForm]]], " \n[1c]\n", Cell[BoxData[ FormBox[ RowBox[{\(\[Phi]\_\(\(\ \)\(r\ n\)\)\), "=", RowBox[{ FormBox[\(1\/2\), "TraditionalForm"], FormBox[\(\[ImaginaryI]\ \@\(2\/n\)\), "TraditionalForm"], RowBox[{"(", RowBox[{ FormBox[\(a\_\(\[Phi]\ r\ n\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ n\)\), "TraditionalForm"], "+"], "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], "\n\nso\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Phi]\_r\)(\[Pi]\/2)\), "=", RowBox[{ FormBox[\(\[Phi]\_\(r\ 0\)\), "TraditionalForm"], "+", RowBox[{ FormBox[\(\@2\), "TraditionalForm"], FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\(\((\(-1\))\)\^n\) \ \[Phi]\_\(r\ \ 2 n\)\), "TraditionalForm"]}]}]}], TraditionalForm]]], "\n\nSo I have three terms to commute\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\((\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\[Phi]\ r\ \ n\)\^+\)\ \[Alpha]\_\(\[Phi]\ r\ n + m\)\ + \ \(1\/2\) \(\[Sum]\+\(n = 1\)\%\ \(m - 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\ + \ \((p\_\(\ \[Phi]\ r\ 0\)\ - \ \(3\/2\) m)\) \[Alpha]\_\(\[Phi]\ r\ m\))\), \(\[ExponentialE]\ \^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), " ", \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), \(\[ExponentialE]\^\(\(\(-1\)\/2\) \((\ \(\[Alpha]\_\(\(\ \)\(r'\)\)\^+\))\) \(N(\(\[Alpha]\_\(\(\ \)\(s'\)\(\ \ \)\)\^+\))\)\)\), StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\n\nFor the sake of convenience, I'd like to come up with a few general \ commutator formulae to use overall.\n\n[31a]\n[", Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\ \[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\) \ \[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r' = 1\)\%3 \ \[Phi]\_\(r'\ 0\)\)\)\)\ - \ \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\ \) \(\[Sum]\+\(r = 1\)\%3 \[Phi]\_\(r\ 0\)\)\)\)\) \(\[Alpha]\_\(\[Phi]\ r\ n\ \)\^+\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(\[Rho]\_n\)\([\ \(\[ExponentialE]\^\(\ \(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\)\^+\) \[Rho]\)\) \[ExponentialE]\^\ \(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ \ 0\)\)\)\)\ ]\) - \ \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\ \+\(r = 1\)\%3 \[Phi]\_\(r\ 0\)\)\)\)\) \(\[Alpha]\_\(\[Phi]\ r\ \ n\)\^+\)\)]], "\n\nSince ", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^A\) \[ExponentialE]\^B\ = \ \(\ \[ExponentialE]\^B\) \(\[ExponentialE]\^A\) \[ExponentialE]\^\([A, \ \ B]\)\)]], ",\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_n\)\), RowBox[{"[", " ", RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\ \[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\), " ", \(\[ExponentialE]\^\(\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\ \(\ \)\)\^+\) \[Rho]\)\), " ", SuperscriptBox["\[ExponentialE]", RowBox[{"[", RowBox[{\(\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \ \)\(n\)\(\ \)\)\^+\) \[Rho]\_n\), ",", " ", RowBox[{ RowBox[{\(\(-1\)\/4\), RowBox[{\(\[Sum]\+\(r' = 1\)\%3\), " ", FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"]}]}], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ ' 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "]"}]]}], "]"}]}], "-", " ", \(\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\ \(r = 1\)\%3 \[Phi]\_\(r\ 0\)\)\)\)\) \(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\)}], TraditionalForm]]], "\n\nevaluating \n", Cell[BoxData[ FormBox[ RowBox[{"[", RowBox[{\(\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\(n\)\(\ \)\)\^+\ \) \[Rho]\_n\), ",", " ", RowBox[{ RowBox[{\(\(-1\)\/4\), RowBox[{\(\[Sum]\+\(r' = 1\)\%3\), " ", FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"]}]}], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "]"}], TraditionalForm]]], ",\n", Cell[BoxData[ FormBox[ RowBox[{\(\(-1\)\/4\), "[", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\[Phi]\)\(\ \ \)\(r\)\(\ \)\(n\)\(\ \)\)\^+\) \[Rho]\_n\), ",", " ", RowBox[{ RowBox[{\(\[Sum]\+\(r' = 1\)\%3\), " ", FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"]}], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "]"}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\(-1\)\/4\), \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\)\), RowBox[{\(\[Delta]\_\(r, \ r'\)\), " ", "[", RowBox[{\(\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\(0\)\(\ \ \)\)\^+\) \[Rho]\_0\), ",", " ", RowBox[{ RowBox[{\(\[Sum]\+\(r' = 1\)\%3\), " ", FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"]}], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r'\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "]"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\(-1\)\/4\), \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\)\), " ", RowBox[{\(\[Rho]\_0\), " ", "[", RowBox[{\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\(0\)\(\ \)\)\^+\ \), ",", " ", RowBox[{ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "-", FormBox[ SuperscriptBox[ FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "]"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{\(\(-1\)\/4\), \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\)\), " ", \(\[Rho]\_0\), " ", RowBox[{"(", RowBox[{"[", RowBox[{\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\(0\)\(\ \ \)\)\^+\), ",", " ", FormBox[\(a\_\(\[Phi]\ r\ 0\)\), "TraditionalForm"]}], "]"}], " ", ")"}]}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(1\/4\) \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\ \)\)\(\ \)\(\[Rho]\_0\)\(\ \)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(\[Rho]\_n\)\([\ \[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\ \ \[ExponentialE]\^\(\(\[Alpha]\_\(\(\[Phi]\)\(\ \)\(r\)\(\ \)\)\^+\) \[Rho]\)\ \ \[ExponentialE]\^\(\(1\/4\) \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\)\)\(\ \ \)\(\[Rho]\_0\)\(\ \)\)]\) - \ \(\[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r = 1\)\%3 \[Phi]\_\(r\ 0\)\)\)\)\) \(\ \[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\ \[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\ \((\(\[Alpha]\_\(\(\[Phi]\)\(\ \ \)\(r\)\(\ \)\(n\)\(\ \)\)\^+\)\ + \ \(1\/4\) \[Delta]\_\(\(n\)\(,\)\(\ \ \)\(0\)\(\ \)\))\) - \ \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\ \[Sum]\+\(r = 1\)\%3 \[Phi]\_\(r\ 0\)\)\)\)\) \(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\ \)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(1\/4\) \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\ \)\) \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r' = \ 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\)\(\ \)\)\)]], "\n\n[31b]\n[", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_\(\[Phi]\ r\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)]], "]\n\nSimilarly to 31a,\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(1\/4\) \(\[Delta]\_\(\(n\)\(,\)\(\ \)\(0\)\(\ \)\ \)\) \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\[Sum]\+\(r' = \ 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\)\(\ \)\)\)]], "\n\n[31c]\n[", Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\)\)\)]], "] = 0\n\n[31d]\n[", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_\(\[Phi]\ r\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\)\)\)]], "]\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\_\(\[Phi]\ r\ n\)\) \[ExponentialE]\^\(\ \[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \ \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\ - \ \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_n\)\), RowBox[{"[", RowBox[{ FormBox[\(\[ExponentialE]\^\(\(\[Alpha]\_\(\[Phi]\ r\ n\)\) \ \[Rho]\_n\)\), "TraditionalForm"], \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\ \%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)}], " ", "]"}]}], "-", " ", \(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_n\)\), RowBox[{"[", RowBox[{ FormBox[\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\ \) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[ExponentialE]\^\(\(\[Alpha]\_\(\ \[Phi]\ r\ n\)\) \[Rho]\_n\)\), "TraditionalForm"], " ", \(\[ExponentialE]\^\([\(\(\[Alpha]\_\(\[Phi]\ r\ n\)\) \ \[Rho]\_n\)\(, \^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)\ ]\)\)}], "]"}]}], "-", " ", \(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)}], TraditionalForm]]], "\n\nevaluating \n", Cell[BoxData[ \(TraditionalForm\`\([\(\(\[Alpha]\_\(\[Phi]\ r\ n\)\) \[Rho]\_n\)\(, \ \^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n' = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n'\) \(1\/\(2 n'\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \( n'\)\ \(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)\ ]\)\)]], ",\n", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_n\ [\(\[Alpha]\_\(\[Phi]\ r\ n\)\)\(, \^\(\ \[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n' = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n'\) \(1\/\(2 n'\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \( n'\)\ \(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)\ ]\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_n\ [\[Alpha]\_\(\[Phi]\ r\ n\), \[Sum]\+\(n' \ = 1\)\%\[Infinity]\ \(\((\(-1\))\)\^n'\) \(1\/\(2 n'\)\) \((\(\[Alpha]\_\(\(\ \)\(\(r\)\(\ \)\(2\) \( n'\)\(\ \ \)\)\)\^+\))\)\ \ ]\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_n\ \(\(\(\ \)\(\((\(-1\))\)\^n - 1\)\)\/\(-2\)\)[\[Alpha]\_\(\[Phi]\ r\ n\), \[Sum]\+\(n' = \ 1\)\%\[Infinity]\ \(\((\(-1\))\)\^n'\) \(1\/\(2 n'\)\) \((\(\[Alpha]\_\(\(\ \)\(\(r\)\(\ \)\(2\) \( n'\)\ \(\ \)\)\)\^+\))\)\ \ ]\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_n\ \(\(\(\ \)\(\((\(-1\))\)\^n - 1\)\)\/\(-2\)\) \(\[Sum]\+\(n' = 1\)\%\[Infinity]\(\((\(-1\))\)\ \^n'\) \(1\/\(2 n'\)\)[\[Alpha]\_\(\[Phi]\ r\ n\), \ \ \((\(\[Alpha]\_\(\(\ \)\(\(r\)\(\ \)\(2\) \( n'\)\(\ \)\)\)\^+\))\)\ \ \ ]\)\)]], "\nNow I know that n is even, so a match will be found. \n\n", Cell[BoxData[ \(TraditionalForm\`\[Rho]\_n\ \(\(\(\ \)\(\((\(-1\))\)\^n - 1\)\)\/\(-2\)\) \((\(-1\))\)\^n\)]], "\n\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[PartialD]\_\(\[Rho]\_n\)\), RowBox[{"[", RowBox[{ FormBox[\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\ \) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[ExponentialE]\^\(\(\[Alpha]\_\(\ \[Phi]\ r\ n\)\) \[Rho]\_n\)\), "TraditionalForm"], " ", \(\[ExponentialE]\^\(\[Rho]\_n\ \(\(\(\ \)\(\((\(-1\))\)\ \^n - 1\)\)\/\(-2\)\) \((\(-1\))\)\^n\)\)}], "]"}]}], "-", " ", \(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)(\[Alpha]\_\(\[Phi]\ r\ n\)\ + \ \(\(\(\ \ \)\(\((\(-1\))\)\^n - 1\)\)\/\(-2\)\) \((\(-1\))\)\^n)\), "TraditionalForm"], " ", "-", " ", \(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`\(Cos(\(n\ \[Pi]\)\/2)\) \[ExponentialE]\^\(\[Sum]\+\ \(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 \ n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\ \)\)\)]], "\n\n[31e]\n[", Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ExponentialE]\^\(\(1\/2\) \ \((\(\[Alpha]\_\(\(\ \)\(r'\)\)\^+\))\) \(N(\(\[Alpha]\_\(\(\ \)\(s'\)\(\ \ \)\)\^+\))\)\)\)\)\)]], "] = 0\n\n[31f]\n[", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_\(\[Phi]\ r\ n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[ExponentialE]\^\(\(1\/2\) \ \((\(\[Alpha]\_\(\(\ \)\(r'\)\)\^+\))\) \(N(\(\[Alpha]\_\(\(\ \)\(s'\)\(\ \ \)\)\^+\))\)\)\)\)\)]], "]\n\nExamining [24f] and [30.1d], a similar commutation, this should be\n\n\ ", Cell[BoxData[ \(TraditionalForm\`n\ \(\(N\)\(\ \)\)\_\(r, \ s, \ n, \ m\)\ \((\(\(\(\ \[Alpha]\)\(\ \)\)\_\(s\ m\)\^+\))\) \[ExponentialE]\^\(\(1\/2\) \((\(\(\ \[Alpha]\_\ \)\^+\))\) \(N(\(\(\[Alpha]\_\ \ \)\^+\))\)\)\)]], " (summing over repeated indices)\n\nassuming N is symmetric\n\n[32] \n\n\ Now, with these convenient commutators and a creation-only definition of the \ vertex, I'm ready to commute.\n\n", Cell[BoxData[ FormBox[ RowBox[{\(V\^\[Phi]\), StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\), " ", StyleBox[\(0\_1\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_2\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], StyleBox[\(0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]}], TraditionalForm]]], "\n\nC", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"(", RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\), " ", "+", " ", \(\(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\(\[Alpha]\_\(r\ \ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\), " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", FormBox[\(a\_\(\(r\)\(\ \)\(0\)\(\ \)\) + \ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], ")"}], " ", "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(r\ m\)\)}]}], ")"}], \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\)}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\ \)]], "\n\nFor simplicity's sake, I'll separate this commutation into two parts: \ The quadratic part and the linear part. This gives me\n\nC", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"(", " ", RowBox[{\((\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\ + \ \(1\/2\) \(\[Sum]\+\ \(n = 1\)\%\(m - 1\)\(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\ )\), "+", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", FormBox[\(a\_\(\(r\)\(\ \)\(0\)\(\ \)\) + \ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], ")"}], " ", "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(r\ m\)\)}]}], ")"}], \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], " ", \(\[ExponentialE]\^\(\(1\/2\) \(\[Alpha]\_\(r\ t\)\^+\)\(\ \ \)\(N\_\(r\ s\ m\ n\)\)\(\ \)\(\[Alpha]\_\(s\ n\)\^+\)\(\ \)\)\)}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\ \)]], "\n\nQuadratic Part: ", Cell[BoxData[ \(TraditionalForm\`\((\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\ + \ \(1\/2\) \(\[Sum]\+\ \(n = 1\)\%\(m - 1\)\(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\ )\)\)]], "\nLinear Part:", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", FormBox[\(a\_\(\(r\)\(\ \)\(0\)\(\ \)\) + \ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], ")"}], " ", "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(r\ m\)\)}]}], TraditionalForm]]], "\n\n[32a]\n\nFirst, I'll commute the linear part from [32] through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\(-\[ImaginaryI]\)\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)]], ".\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", FormBox[\(a\_\(\(r\)\(\ \)\(0\)\(\ \)\) + \ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], ")"}], " ", "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(r\ m\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(\(-\[ImaginaryI]\)\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], \(\[Alpha]\_\(r\ m\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(\(-\[ImaginaryI]\)\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}], "+", " ", RowBox[{ FormBox[\(\(\ \)\(a\_\(r\ 0\)\^+\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ m\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\ \+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}], " ", "-", " ", \(\(3\/2\) m\ \[Alpha]\_\(r\ m\)\ \[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\ \/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}], TraditionalForm]]], "\n\nThe third term clearly commutes, so I'll bring it back at the end of \ the calcuation.\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], \(\[Alpha]\_\(r\ m\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\ \+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}], "+", " ", RowBox[{ FormBox[\(\(\ \)\(a\_\(r\ 0\)\^+\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ m\)\), " ", \(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\ \+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)}]}], TraditionalForm]]], "\n\nSince [AB, C] = A[B, C] + [A, C] B,\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), "(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], \(\[Alpha]\_\(r\ m\)\)}], "+", " ", RowBox[{ FormBox[\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ m\)\)}], " ", "+", RowBox[{ RowBox[{"(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], "[", " ", \(\[Alpha]\_\(r\ m\), \ \[ExponentialE]\^\(\(\ \)\(\(\ \[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \), "]"}], " ", "+", " ", \(\([\[Alpha]\_\(r\ 0\), \ \[ExponentialE]\^\(\(\ \)\(\(\ \[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \ \)\)\)\)]\) \[Alpha]\_\(r\ m\)\), " ", "+", RowBox[{ RowBox[{"(", FormBox[\(a\_\(r\ 0\)\^+\), "TraditionalForm"], ")"}], "[", \(\[Alpha]\_\(r\ m\), \ \[ExponentialE]\^\(\(\ \)\(\(\ \[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \), "]"}], " ", "+", " ", \(\([\ \(\[Alpha]\_\(r\ 0\)\^+\), \ \[ExponentialE]\^\(\(\ \ \)\(\(\[ImaginaryI]\/2\) \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\ \)\)\)]\) \[Alpha]\_\(r\ m\)\)}], "\n"}]}], TraditionalForm]]], "\nAnd using my identities [31a] and [31b] along with the fact that [A, B] \ = -[B, A], and that I chose m \[NotEqual] 0\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), "(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], \(\[Alpha]\_\(r\ m\)\)}], "+", " ", RowBox[{ FormBox[\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \(a\_\(r\ \ 0\)\^+\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ m\)\)}], " ", "+", \(\(1\/4\) \[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\ \[Alpha]\_\(r\ m\)\), " ", "+", \(\(1\/4\) \[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\ \[Alpha]\_\(r\ m\)\)}], "\n"}]}], TraditionalForm]]], "\nAnd simplifying leaves me with\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\), RowBox[{"(", RowBox[{ FormBox[ RowBox[{ RowBox[{"(", FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], ")"}], \(\[Alpha]\_\(r\ m\)\)}], "TraditionalForm"], "+", \(\(a\_\(r\ 0\)\^+\) \[Alpha]\_\(r\ m\)\), " ", "+", \(\(1\/2\) \[Alpha]\_\(r\ m\)\)}], " ", ")"}]}], TraditionalForm]]], "\n\nAnd remembering the ", Cell[BoxData[ \(TraditionalForm\`\(-\ \(3\/2\)\) m\)]], StyleBox[" ", FontWeight->"Bold"], "term that commuted earlier,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\[Sum]\+\(r' = 1\)\%3 \[Phi]\_\(r'\ 0\)\)\)\)\), RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), " ", "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(r\ m\)\)}], TraditionalForm]]], "\n\n[32b]\n\nNow I'd like to commute ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\)}], TraditionalForm]]], "through ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\), SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]], "(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\)}], " ", "+", " ", RowBox[{"[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\)}], ",", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], "]"}]}], TraditionalForm]]], "\n\n[32b.1]\n\nevaluating \n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\)}], ",", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], "]"}]}], TraditionalForm]]], "\n\nSince [AB, C] = A[B, C] + [A, C] B,\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], " ", "[", RowBox[{\(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\), ",", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], "]"}], " ", "+", " ", RowBox[{ RowBox[{"[", RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], ",", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], "]"}], \(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\), " "}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], " ", "[", RowBox[{\(\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\)\), ",", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{\(\[Sum]\+\(r = 1\)\%3\), FormBox[ RowBox[{\(\[Sum]\+\(n' = 1\)\%\[Infinity]\), RowBox[{\(\((\(-1\))\)\^n'\), FormBox[\(1\/\(2\ n'\)\), "TraditionalForm"], FormBox[ SuperscriptBox[ FormBox[\(\[Alpha]\_\(\[Phi]\ r\ 2 n'\)\), "TraditionalForm"], "+"], "TraditionalForm"]}]}], "TraditionalForm"]}]}]]}], "]"}], TraditionalForm]]], " \nFrom 31d, \n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \(Cos(\(m\ \[Pi]\)\/2)\)}], TraditionalForm]]], "\n\nAnd combining this result with [32b], I have\n\n[32b.2]\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\), " ", "+", \(1\/2\), "-", " ", \(\(3\/2\) m\)}], ")"}], \((\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\) + \ Cos(\(m\ \[Pi]\)\/2))\)}], TraditionalForm]]], "\n\nas my linear part after commutation through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \ \ \[ExponentialE]\^\ \(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)]], "\n\nNow I'll begin commuting the quadratic parts.\n\n[33]\n\nSince ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Sum]\+\(n = 1\)\%\[Infinity]\( \ \[Alpha]\_\(\[Phi]\ r\ n\)\^+\)\ \[Alpha]\_\(\[Phi]\ r\ n + m\)\)\)\)]], " commutes with ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)]], ", I'll commute ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\[Phi]\ \ r\ n\)\^+\)\ \[Alpha]\_\(\[Phi]\ r\ n + m\)\)]], " through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \ \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \ \)\)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\[Phi]\ \ r\ n\)\^+\)\ \(\[Alpha]\_\(\[Phi]\ r\ n + m\)\) \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\ \+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\)]], " + [", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n' = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \)\ \(r\ n'\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n' + m\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \ \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \ \)\)\)\)\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\ \+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n' = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \)\ \(r\ n'\)\)\^+\)\)]], "[", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Alpha]\_\(\(\ \)\(r\ n' + m\)\)\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \ \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \ \)\)\)\)\)]], "]\n", Cell[BoxData[ \(TraditionalForm\`\ \)]], Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\ \(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\( \ \[Alpha]\_\(\(\ \)\(r\ n\)\)\^+\)\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\)\)\ + \ \ \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(\[Sum]\+\(n' = 1\)\%\[Infinity]\( \ \[Alpha]\_\(\(\ \)\(r\ n'\)\)\^+\) \(Cos(\(\((n' + m)\)\ \[Pi]\)\/2)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\ \(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(\[Sum]\+\(n = 1\)\%\[Infinity]\( \ \[Alpha]\_\(\(\ \)\(r\ n\)\)\^+\)(\ \[Alpha]\_\(\(\ \)\(r\ n + m\)\) + Cos(\(\((n + m)\)\ \[Pi]\)\/2))\)\)]], "\n\n[34]\n\nNow I'd like to commute ", Cell[BoxData[ \(TraditionalForm\`\(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - \ 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\)]], " through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\ \ \ \[ExponentialE]\^\ \(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)]], "\n\nSince ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\(\ \[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\)\)\)]], " commutes with ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \(\(\ \[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\)]], ", I'll commute ", Cell[BoxData[ \(TraditionalForm\`\(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - \ 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \[Alpha]\_\(\[Phi]\ r\ n\)\)\)]], " through ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \ \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \ \)\)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - \ 1\)\(\[Alpha]\_\(\[Phi]\ r\ m\ - \ n\)\) \(\[Alpha]\_\(\[Phi]\ r\ n\)\) \[ExponentialE]\^\(\ \[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \ \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \ \)\)\)\^+\))\)\(\ \)\)\)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\ \(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\ \(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\) + \(1\/2\) \(\(\[Sum]\+\(n = 1\)\%\(m - 1\)\)\(\ \)\([\(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\), \ \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\ ]\)\(\ \)\)\)\)\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\ \(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\ \(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\) + \(1\/2\) \(\(\[Sum]\+\(n = 1\)\%\(m - 1\)\)\(\ \)\((\[Alpha]\_\(r\ m\ - \ n\)[\ \[Alpha]\_\(r\ n\ \), \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\ ]\ \ + \ \([\[Alpha]\_\(r\ m\ - \ n\), \ \ \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)]\) \[Alpha]\_\(r\ n\))\)\(\ \)\)\)\)\)]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = \ 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\ \(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\), "+", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\(m - 1\)\), RowBox[{"(", RowBox[{ RowBox[{\(\[Alpha]\_\(r\ m\ - \ n\)\), " ", \(Cos(\(\((n)\)\ \[Pi]\)\/2)\), FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), "TraditionalForm"]}], " ", "+", " ", RowBox[{\(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\), FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ n\)\)}]}], " ", ")"}]}]}]}]}], TraditionalForm]]], "\n\nWhich begs one more commutation:\n\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\(1\/2\) \(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(\[Sum]\+\(n = 1\)\%\(m - \ 1\)\(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\), "+", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\(m - 1\)\), RowBox[{" ", RowBox[{"(", RowBox[{ RowBox[{\(Cos(\(\((n)\)\ \[Pi]\)\/2)\), FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\ \[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\ \ \[Alpha]\_\(r\ m\ - \ n\)\), "TraditionalForm"]}], " ", "+", " ", RowBox[{\(Cos(\(\((n)\)\ \[Pi]\)\/2)\), FormBox[\(\(\ \)\([\[Alpha]\_\(r\ m\ - \ n\), \ \ \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)]\)\), "TraditionalForm"]}], " ", "+", " ", RowBox[{\(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\), " ", FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ n\)\)}]}], ")"}], " "}]}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\ \)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\) \(1\/2\) \(\[Sum]\+\(n = 1\)\%\(m - 1\)\ \(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\)\), "+", RowBox[{\(1\/2\), RowBox[{\(\[Sum]\+\(n = 1\)\%\(m - 1\)\), RowBox[{"(", " ", RowBox[{ RowBox[{\(Cos(\(\((n)\)\ \[Pi]\)\/2)\), FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \ \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\ \ \[Alpha]\_\(r\ m\ - \ n\)\), "TraditionalForm"]}], " ", "+", " ", FormBox[ RowBox[{" ", RowBox[{\(Cos(\(\((n)\)\ \[Pi]\)\/2)\), " ", \(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\), " ", FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\ \[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), "TraditionalForm"]}]}], "TraditionalForm"], "+", RowBox[{\(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\), " ", FormBox[\(\[ExponentialE]\^\(\[Sum]\+\(s' = \ 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \ \(2\) \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), "TraditionalForm"], \(\[Alpha]\_\(r\ n\)\)}]}], ")"}]}]}]}], TraditionalForm]]], "\n\nLeaving me with, since it doesn't matter whether I add from n = 1 to \ m-1 or the reverse \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\ \%\[Infinity]\ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\), RowBox[{\(\[Sum]\+\(n = 1\)\%\(m - 1\)\), RowBox[{"(", RowBox[{\(\(1\/2\) \(\[Alpha]\_\(r\ m\ - \ n\)\) \[Alpha]\_\(r\ n\)\), " ", "+", " ", \(\(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\)\ \[Alpha]\_\(r\ \ n\)\), " ", "+", FormBox[\(\(1\/2\)\(\ \)\(Cos(\(\((n)\)\ \ \[Pi]\)\/2)\)\(\ \ \)\(Cos(\(\((m\ - \ n)\)\ \[Pi]\)\/2)\)\(\ \ \)\), "TraditionalForm"]}], ")"}]}]}], TraditionalForm]]], " ", StyleBox["\n\n", FontWeight->"Bold"], "So overall, I have\n\n[34a]\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[ExponentialE]\^\(\(\ \)\(\(\[ImaginaryI]\/2\) \ \(\(\[Sum]\+\(r = 1\)\%3\)\(\[Phi]\_\(r\ 0\)\)\(\ \)\)\)\)\), \(\(\ \[ExponentialE]\^\(\[Sum]\+\(s' = 1\)\%3\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \ \(\(\((\(-1\))\)\^n\) \(1\/\(2 n\)\) \((\(\[Alpha]\_\(\(\ \)\(\(s'\) \(2\) \ \(n\)\(\ \)\)\)\^+\))\)\(\ \)\)\)\)\)[ L1\_\(r, \ m\)\ \ + \ \ L2\_\(r, \ m\)\ \ + K\_m\ \ ]\), " ", \(\[ExponentialE]\^\(\(1\/2\) \((\(\[Alpha]\_\(\(\ \ \)\(r'\)\)\^+\))\) \(N(\(\[Alpha]\_\(\(\ \)\(s'\)\(\ \)\)\^+\))\)\)\), FormBox[ StyleBox[\(\(0\_1\) \(0\_2\) 0\_3\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "TraditionalForm"]}], "=", "0"}], TraditionalForm]]], "\n\nWhere, when m > 0,\n\n", Cell[BoxData[ FormBox[ RowBox[{\(L1\_\(r, \ m\)\), " ", "=", RowBox[{\(\((1\/2 - \ \(3\/2\) m)\) \((\[Alpha]\_\(\(r\)\(\ \)\(m\)\(\ \)\))\)\), "+", RowBox[{ RowBox[{"(", " ", RowBox[{ FormBox[ FormBox[\(a\_\(r\ 0\)\), "TraditionalForm"], "TraditionalForm"], "+", \(a\_\(r\ 0\)\^+\)}], ")"}], \(Cos(\(m\ \[Pi]\)\/2)\)}], "+", " ", \(\[Sum]\+\(n = 1\)\%\[Infinity]\( \[Alpha]\_\(\(\ \)\(r\ \ n\)\)\^