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corrections to: Re: calculus with Grassmann variables

• To: mathgroup at smc.vnet.net
• Subject: [mg21034] corrections to: Re: [mg20970] calculus with Grassmann variables
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sun, 12 Dec 1999 23:51:04 -0500 (EST)
• Organization: Wolfram Research, Inc.
• References: <199912010650.BAA07735@smc.vnet.net> <8264m8$da7@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau wrote:
> 
> Diego Bellisai wrote:
> >
> > Hi everybody,
> > I am looking for a Mathematica package or notebook which allows me
> > to deal with Grassmann (anticommuting) variables and to perform
> > calculations with both (standard) commuting and anticommuting variables.
> > In particular, I am interested in performing matrix calculus (products,
> > determinants and so on) with "anticommuting entries" matrices and with
> > commuting entries matrices.
> > ...
> 
> Say you encapsulate all anticommuting variables with the head avar[].
> Then you can define product rules to handle the commutation relations
> and furthermore distribute over Plus, pull out "scalars" (everything not
> with head of avar) using ordinary Times, and so on...
> 
> The rules below illustrate this method, though perhaps they could be
> simplified somewhat. They pass some basic sanity tests but I'd not
> guarantee that they are bug-free.
> 
> Clear[myTimes];
> myTimes[a___,avar[x_],b___,avar[x_],c___] := 0
> myTimes[avar[x_],avar[y_],avar[z_]...] /; Not[OrderedQ[{x,y,z}]] :=
>         (Signature[{x,y,z}]*
>         myTimes[Sequence @@ Map[avar,Sort[{x,y,z}]]])
> myTimes[a_] /; Head[a]=!=Times := a
> myTimes[x___,y_Plus,z___] :=
>         myTimes[x,#,z]& /@ y
> myTimes[x___,y_Times,z___] :=
>         myTimes[x,Apply[Sequence,y],z]
> myTimes[x___,y_myTimes,z___] :=
>         myTimes[x,Apply[Sequence,y],z]
> myTimes[] = 1;
> myTimes[vars__] /; Length[{vars}]>2 := Module[{vl={vars},antis,coms},
>         antis = Cases[vl,avar[_]];
>         coms = Complement[vl, antis];
>         Apply[Times,coms]*Apply[myTimes,antis]
>         ]
> myTimes[a_,b_] /; Head[a]=!=avar && Head[b]=!=avar := a*b
> Here is an example. We take a 3x3 matrix with avar[] entries and find
> its inner product with itself using these rules.
> 
> m1 = {{avar[1],x,avar[2]},
>         {x2,avar[1]+2*avar[2],3*avar[1]-avar[2]},
>         {avar[1]-avar[2],x3,-3*avar[1]+avar[2]}};
> 
> In[324]:= InputForm[Inner[myTimes, m1, m1]]
> Out[324]//InputForm=
> {{x*x2 - myTimes[avar[1], avar[2]], 2*x*avar[2] + myTimes[x, avar[1]] +
>    myTimes[avar[1], x] + myTimes[avar[2], x3], 3*x*avar[1] - x*avar[2] +
>    4*myTimes[avar[1], avar[2]]}, {2*x2*avar[2] + myTimes[x2, avar[1]] +
>    myTimes[avar[1], x2] - 2*myTimes[avar[1], avar[2]],
>   x*x2 + 3*x3*avar[1] - x3*avar[2], myTimes[x2, avar[2]] -
>    7*myTimes[avar[1], avar[2]]}, {x2*x3 + 3*myTimes[avar[1], avar[2]],
>   -3*x3*avar[1] - x*avar[2] + 2*x3*avar[2] + myTimes[x3, avar[1]] +
>    myTimes[avar[1], x] + myTimes[avar[2], x3], 3*x3*avar[1] - x3*avar[2]
> +
>    myTimes[avar[1], avar[2]]}}

The example above shows a small flaw wherein we do not correctly change
products involving a single anticommuting variable. In particular the
last rule should have an Or rather than an And.

myTimes[a_,b_] /; Head[a]=!=avar || Head[b]=!=avar := a*b

Moreover the next-to-last rule allowed for an infinite recursion problem
(try doing Inner[myTimes, Inner[myTimes,m1,m1], m1] to see it). The fix
is

myTimes[vars__] /; Length[{vars}]>2 := Module[{vl={vars},antis,coms},
	antis = Cases[vl,avar[_]];
	coms = Complement[vl, antis];
	Apply[Times,coms]*Apply[myTimes,antis] /; Length[coms]>0
	]


> Now for the determinant computation.
> 
> Clear[ncDet]
> ncDet[{{a_}}] := a
> ncDet[mat_?MatrixQ] /; Length[mat]===Length[mat[[1]]] :=
>         With[{len=Length[mat], mat1=Drop[mat,{1}]},
>         Sum[myTimes[mat[[1,j]],ncDet[Drop[mat1,None,{j}]]], {j,len}]]

This needs a correction and an improvement. The correction is to
alternate signs (otherwise we have the permanent rather than the
determinant). The improvement, for efficiency, is to memo-ize results so
that we do not recompute minor expansions but instead look them up after
they are once computed. We get:

ncDet[mat_?MatrixQ] /; Length[mat]===Length[mat[[1]]] :=
  ncDet[mat] = 
    With[{len=Length[mat], mat1=Drop[mat,{1}]},
    Sum[(-1)^(j+1)*myTimes[mat[[1,j]],ncDet[Drop[mat1,None,{j}]]],
{j,len}]]

Daniel Lichtblau
Wolfram Research

• References:
  • calculus with Grassmann variables
    • From: Diego Bellisai <diego.bellisai@pd.infn.it>