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MathGroup Archive 2005

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Re: Grassmann Calculus / Indexed Objects / Simplify

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60811] Re: Grassmann Calculus / Indexed Objects / Simplify
  • From: dh <dh at metrohm.ch>
  • Date: Thu, 29 Sep 2005 05:41:31 -0400 (EDT)
  • References: <dhdank$8ae$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hello Robert,
I assume you are using conventions from relativity about repeated indices.
1. Question:
Not that I understand everything in your code, but
as far as I can see you are using unique indices only in CSF and ACSF. 
And there you are using them only to denote contractions. Well, the name 
of the index in a contraction is unimportant and you can as well name 
them always the same, e.g. \alpha

2. Question: e[a,b]theta[b]=theta[a]
this can be easily achieved by a rule, e.g.:

e/: e[a_,b_]theta[b_]:=theta[a]

here e/: is used to assigne the rule to e.

sincerely, Daniel


Robert Schoefbeck wrote:
> Hello all,
> 
> I want to implement superspace calculations in a mathematica package.
> I define (code below) grassmannian theta-variables with upper and lower, 
> dotted and undotted indices. Having declared (anti)- commutativity
> with DeclareOdd i use myTimes to handle multiplications of Superfields.
> 
> Dummy indices are generated by use of Unique[]. All this multiplication 
> stuff works pretty well but looking at the one but last output i have 
> the problem that contractions with different dummies are not recognized 
> as equal.
> 
> My questions are: Is there a way to tell mathematica to keep the 
> information that indices are equal but forgetting about the name of the 
> dummy?
> 
> I'd also like to pull indices with metrics, something like
> 
> e[a,b]theta[b]=theta[a]
> 
> How should i implement the defining relations for index pulling
> (i have vector, and chiral/antichiral spinor indices, color indices and 
> so on) such that 1) Simplify can handle them?
> 
> any help is welcome, does maybe someone have a sample application of how
> to handle indices?
> i'd also appreciate any comment on the code.
> 
> robert schoefbeck
> 
> 
> 
> 
> short description:
> 
> OddPQ tests whether an object has odd parity or not
> 
> mytimes handles ,multiplications
> 
> CSF[phi,lambda,F]  ... chiral superfield defined by component fields 
> (lambda is odd)
> ACSF ... antichiral Superfield
> 
> 
> Code:
> 
> 
> 
> Notebook[{
> Cell[BoxData[
>      \(\(Clear[myTimes];\)\)], "Input"],
> 
> Cell[BoxData[
>      \(\(\(Clear[DeclareOdd, OddPQ, myTimes]\[IndentingNewLine]
>      \(DeclareOdd[
>            var__] := \(\((OddPQ[#] =
>                  True)\) &\) /@ {var};\)\[IndentingNewLine]
>      \(\(DeclareEven[
>            var__] := \(\((OddPQ[#] =
>                  False)\) &\) /@ {var};\)\(\[IndentingNewLine]\)
>      \)\)\( (*OddPQ[myTimes[x__]] :=
>          Mod[Count[OddPQ /@ {x}, True], 2] \[Equal]
>            1*) \)\)\)], "Input"],
> 
> Cell[BoxData[
>      \(\(DeclareOdd[\[Theta][\[Alpha]_], \[Theta]b[\[Alpha]_]];\)\)], \
> "Input"],
> 
> Cell[BoxData[
>      \(\(OddPQ[x_?NumericQ] := False;\)\)], "Input"],
> 
> Cell[BoxData[
>      \(\(DeclareOdd[\[Theta][\[Alpha]_], \[Theta]b[\[Alpha]_]];\)\)], \
> "Input"],
> 
> Cell["rearranging of elementary Odd variables", "Text"],
> 
> Cell[BoxData[{
>      \(myTimes[a___, x_, b___, x_, c___] /; OddPQ[x] :=
>        0\), "\[IndentingNewLine]",
>      \(myTimes[x___] /; \((And @@ \(OddPQ /@ {x}\))\) &&
>            Not[OrderedQ[{x}]] :=
>        Signature[{x}]*myTimes[Sequence @@ Sort[{x}]]\)}], "Input"],
> 
> Cell["linearity", "Text"],
> 
> Cell[BoxData[
>      \(myTimes[x___, y_Plus, z___] := \(myTimes[x, #, z] &\) /@
>          y\)], "Input"],
> 
> Cell[BoxData[
>      \(myTimes[x___, y_Times, z___] :=
>        myTimes[x, Apply[Sequence, y], z]\)], "Input"],
> 
> Cell[BoxData[
>      \(myTimes[x___, y_myTimes, z___] :=
>        myTimes[x, Apply[Sequence, y], z]\)], "Input"],
> 
> Cell["a single expression in mytimes whose head is not myTimes", \
> "Text"],
> 
> Cell[BoxData[
>      \(myTimes[a_] /; Head[a] =!= Times := a\)], "Input"],
> 
> Cell["\<\
> after linearity downvalues: myTimes\[Rule]Times if possible\
> \>", "Text"],
> 
> Cell[BoxData[
>      \(myTimes[a_, b_] /; \(! OddPQ[a]\) || \(! OddPQ[b]\) :=
>        a*b\)], "Input"],
> 
> Cell[BoxData[
>      \(\(myTimes[] = 1;\)\)], "Input"],
> 
> Cell["\<\
> split off avars in myTimes (corrected version: Length[coms>0])\
> \>", "Text"],
> 
> Cell[BoxData[
>      \(myTimes[vars__] /; Length[{vars}] > 2 :=
>        Module[{vl = {vars}, antis, coms, exponents},
>          antis = Select[vl, OddPQ]; \[IndentingNewLine]coms =
>            Complement[vl,
>              antis]; \[IndentingNewLine]exponents = \(Count[
>                  vl, #] &\) /@
>              coms; \[IndentingNewLine]Inner[Power, coms, exponents,
>                Times]*Apply[myTimes, antis] /;
>            Length[coms] > 0]\)], "Input"],
> 
> Cell["4-dim chiral thetas", "Text"],
> 
> Cell[BoxData[
>      \(myTimes[
>            x___] /; \((Length[{x}] > 2 &&
>              Max\ [\(Count[{x}, #[_]] &\) /@
>                    Union[Select[{x}, OddPQ] /.
>                        f_[_] \[RuleDelayed] f]] > 2)\) :=
>        0\)], "Input"],
> 
> Cell["END", "Text"],
> 
> Cell[BoxData[
>      \(\(DeclareOdd[\[Theta][_], \[Theta]b[_]];\)\)], "Input"],
> 
> Cell[BoxData[{
>      \(\(Format[\[Theta]b[
>              Index[\[Alpha]_,
>                down]]] := \[Theta]\&_\_\[Alpha]\& . ;\)\), "\
> \[IndentingNewLine]",
>      \(\(Format[\[Theta]b[
>              Index[\[Alpha]_,
>                up]]] := \[Theta]\&_\^\[Alpha]\& . ;\)\)}], "Input"],
> 
> Cell[BoxData[{
>      \(\(Format[\[Theta][
>              Index[\[Alpha]_,
>                down]]] := \[Theta]\_\[Alpha];\)\), \
> "\[IndentingNewLine]",
>      \(\(Format[\[Theta][
>              Index[\[Alpha]_,
>                up]]] := \[Theta]\^\[Alpha];\)\)}], "Input"],
> 
> Cell[BoxData[
>      \(Format[myConjugate[x_]] := x\&_\)], "Input"],
> 
> Cell[BoxData[{
>      \(\(Clear[CSF];\)\), "\[IndentingNewLine]",
>      \(\(\(CSF[0, 0, F_Symbol] :=
>        Module[{ad1, au1}, indid1 = Unique[\[Alpha]];
>          ad1 = Index[indid1, down];
>          au1 = Index[indid1,
>              up]; \[IndentingNewLine]F\ myTimes[\[Theta][
>                ad1], \[Theta][au1]]]\)\(\[IndentingNewLine]\)
>      \)\)}], "Input"],
> 
> Cell[BoxData[
>      \(\(CSF[\[Phi]_Symbol, \[Lambda]_Symbol, F_Symbol] :=
>          Module[{ad1, ad2, au1, au2, indid1, indid2},
>            indid1 = Unique[\[Alpha]]; indid2 = Unique[\[Alpha]];
>            ad1 = Index[indid1, down]; au1 = Index[indid1, up];
>            ad2 = Index[indid2, down];
>            au2 = Index[indid2,
>                up]; \[IndentingNewLine]DeclareOdd[\[Lambda][_]]; \
> \[IndentingNewLine]DeclareEven[\[Phi]]; \[IndentingNewLine]Format[\
> \[Lambda][Index[\[Alpha]_,
>                    down]]] := \[Lambda]\_\[Alpha]; \
> \[IndentingNewLine]Format[\[Lambda][
>                  Index[\[Alpha]_,
>                    up]]] := \[Lambda]\^\[Alpha]; \[IndentingNewLine]\
> \[Phi] + Sqrt[2] myTimes[\[Theta][au1], \[Lambda][ad1]] +
>              F\ myTimes[\[Theta][au2], \[Theta][
>                    ad2]]];\)\)], "Input"],
> 
> Cell[BoxData[
>      \(\(ACSF[\[Phi]_Symbol, \[Lambda]_Symbol, F_Symbol] :=
>          Module[{ad1, ad2, au1, au2, indid1, indid2},
>            indid1 = Unique[\[Alpha]]; indid2 = Unique[\[Alpha]];
>            au1 = Index[indid1, up]; ad1 = Index[indid1, down];
>            au2 = Index[indid2, up];
>            ad2 = Index[indid2,
>                down]; \[IndentingNewLine]DeclareOdd[\[Lambda][_]]; \
> \[IndentingNewLine]DeclareEven[\[Phi]]; \
> \[IndentingNewLine]Format[\(myConjugate[\[Lambda]]\)[
>                  Index[\[Alpha]_,
>                    down]]] := \[Lambda]\&_\_\[Alpha]\& . ; \
> \[IndentingNewLine]Format[\(myConjugate[\[Lambda]]\)[
>                  Index[\[Alpha]_,
>                    up]]] := \[Lambda]\&_\^\[Alpha]\& . ; \
> \[IndentingNewLine]myConjugate[\[Phi]] +
>              Sqrt[2]
>                myTimes[\[Theta]b[ad1], \(myConjugate[\[Lambda]]\)[
>                    au1]] +
>              myConjugate[
>                  F]\ myTimes[\[Theta]b[ad2], \[Theta]b[
>                    au2]]];\)\)], "Input"],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
>      \(CSF[\[Phi], \[Lambda], F]\)], "Input"],
> 
> Cell[BoxData[
>      FormBox[
>        RowBox[{"\[Phi]", "+",
>          RowBox[{\(\@2\), " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$792\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$792\),
>                  "TraditionalForm"]}], ")"}]}], "-",
>          RowBox[{"F", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$793\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$793\),
>                  "TraditionalForm"]}], ")"}]}]}],
>        TraditionalForm]], "Output"]
> }, Open  ]],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
>      \(myTimes[CSF[\[Phi], \[Lambda], F],
>        CSF[\[Phi], \[Lambda], F]]\)], "Input"],
> 
> Cell[BoxData[
>      FormBox[
>        RowBox[{\(\[Phi]\^2\), "+",
>          RowBox[{\(\@2\), " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$795\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$795\),
>                  "TraditionalForm"]}], ")"}], " ", "\[Phi]"}], "-",
>          RowBox[{"F", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$796\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$796\),
>                  "TraditionalForm"]}], ")"}], " ", "\[Phi]"}], "+",
>          RowBox[{\(\@2\), " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$798\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$798\),
>                  "TraditionalForm"]}], ")"}], " ", "\[Phi]"}], "-",
>          RowBox[{"F", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$799\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$799\),
>                  "TraditionalForm"]}], ")"}], " ", "\[Phi]"}], "-",
>          RowBox[{"2", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$795\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$798\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$795\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$798\),
>                  "TraditionalForm"]}], ")"}]}]}],
>        TraditionalForm]], "Output"]
> }, Open  ]],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
>      \(myTimes[ACSF[\[Phi], \[Lambda], F],
>        CSF[\[Phi], \[Lambda], F]]\)], "Input"],
> 
> Cell[BoxData[
>      FormBox[
>        RowBox[{
>          RowBox[{"\[Phi]", " ",
>            FormBox[\(\[Phi]\&_\),
>              "TraditionalForm"]}], "+",
>          RowBox[{\(\@2\), " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$816\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$816\),
>                  "TraditionalForm"]}], ")"}], " ",
>            FormBox[\(\[Phi]\&_\),
>              "TraditionalForm"]}], "-",
>          RowBox[{"F", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$817\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$817\),
>                  "TraditionalForm"]}], ")"}], " ",
>            FormBox[\(\[Phi]\&_\),
>              "TraditionalForm"]}], "+",
>          RowBox[{"\[Phi]", " ",
>            FormBox[\(F\&_\),
>              "TraditionalForm"], " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\&_\_\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\^\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"]}], ")"}]}], "+",
>          RowBox[{\(\@2\), " ",
>            FormBox[\(F\&_\),
>              "TraditionalForm"], " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$816\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\_\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\^\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$816\),
>                  "TraditionalForm"]}], ")"}]}], "-",
>          RowBox[{"F", " ",
>            FormBox[\(F\&_\),
>              "TraditionalForm"], " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$817\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$817\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\_\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\^\(\[Alpha]$814\& . \)\),
>                  "TraditionalForm"]}], ")"}]}], "-",
>          RowBox[{"2", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\^\[Alpha]$816\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\_\(\[Alpha]$813\& . \)\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Lambda]\_\[Alpha]$816\),
>                  "TraditionalForm"]}], ")"}], " ",
>            FormBox[\(\[Lambda]\&_\^\(\[Alpha]$813\& . \)\),
>              "TraditionalForm"]}], "-",
>          RowBox[{\(\@2\), " ", "F", " ",
>            RowBox[{"myTimes", "(",
>              RowBox[{
>                FormBox[\(\[Theta]\_\[Alpha]$817\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\^\[Alpha]$817\),
>                  "TraditionalForm"], ",",
>                FormBox[\(\[Theta]\&_\_\(\[Alpha]$813\& . \)\),
>                  "TraditionalForm"]}], ")"}], " ",
>            FormBox[\(\[Lambda]\&_\^\(\[Alpha]$813\& . \)\),
>              "TraditionalForm"]}], "+",
>          RowBox[{\(\@2\), " ", "\[Phi]", " ",
>            FormBox[\(\[Theta]\&_\_\(\[Alpha]$813\& . \)\),
>              "TraditionalForm"], " ",
>            FormBox[\(\[Lambda]\&_\^\(\[Alpha]$813\& . \)\),
>              "TraditionalForm"]}]}], TraditionalForm]], "Output"]
> }, Open  ]]
> }
> ]
> 
> 


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