7

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a representation, is it not possible for Grassmann Variables? The reason for a representation is, then probably it will be easier to derive some of the properties.

Qmechanic
  • 220,844
Jaswin
  • 1,855

2 Answers2

6

I think that this Wikipedia article will tells this all.

The only problem is that for $n$ (I mean $\theta_1,\theta_2,...\theta_n$) Grassmann numbers you will need to use $2^n\times 2^n$ matrices.

Kostya
  • 20,288
-2

The following code for Mathematica implements 4-dimensional (with 2 generators) Gressmann numbers:

Clear["Global`*"]
Unprotect[Log]; Log[0] = \[Lambda]; Protect[Log];
Unprotect[Power];
Power[0, 0] = 1;
Protect[Power];
Unprotect[Dot];
Dot[x_?NumberQ, y_] := x y;
Protect[Dot];
Matrix /: Matrix[x_?MatrixQ] := 
  First[First[x]] /; x == First[First[x]] IdentityMatrix[Length[x]];
Matrix /: NonCommutativeMultiply[Matrix[x_?MatrixQ], y_] := 
  Dot[Matrix[x], y];
Matrix /: NonCommutativeMultiply[Matrix[y_, x_?MatrixQ]] := 
  Dot[y, Matrix[x]];
Matrix /: Dot[Matrix[x_], Matrix[y_]] := Matrix[x . y];
Matrix /: Matrix[x_] + Matrix[y_] := Matrix[x + y];
Matrix /: x_?NumericQ + Matrix[y_] := 
  Matrix[x IdentityMatrix[Length[y]] + y];
Matrix /: x_?NumericQ  Matrix[y_] := Matrix[x y];
Matrix /: Matrix[x_]*Matrix[y_] := Matrix[x . y] /; x . y == y . x;
Matrix /: Power[Matrix[x_ ?MatrixQ], y_] := 
  Matrix[MatrixPower[x, y]];
Matrix /: Power[Matrix[x_?MatrixQ], Matrix[y_?MatrixQ]] := 
  Exp[Matrix[y] . Log[Matrix[x]]];
Matrix /: Im[Matrix[x_?MatrixQ]] := Matrix[Im[x]]
Matrix /: Re[Matrix[x_?MatrixQ]] := Matrix[Re[x]]
Matrix /: Arg[Matrix[x_?MatrixQ]] := Matrix[Arg[x]]

$Post2 = 
  FullSimplify[FullSimplify[# /. Subscript[[Theta], 1] -> Matrix[( {
               {0, 0, 0, 0},
               {1, 0, 0, 0},
               {0, 0, 0, 0},
               {0, 0, 1, 0}
              } )] /. Subscript[[Theta], 2] -> Matrix[( {
              {0, 0, 0, 0},
              {0, 0, 0, 0},
              {1, 0, 0, 0},
              {0, -1, 0, 0}
             } )] /. [CurlyEpsilon] -> Matrix[( {
             {0, 0, 0, 0},
             {0, 0, 0, 0},
             {0, 0, 0, 0},
             {1, 0, 0, 0}
            } )] /. 
        f_[args1___?NumericQ, Matrix[mat_], args2___?NumericQ] :> 
         Matrix[MatrixFunction[f[args1, #, args2] &, mat]]] /. 
      Matrix[( {
          {a_, 0, 0, 0},
          {b_, a_, 0, 0},
          {c_, 0, a_, 0},
          {d_, , b, a_}
         } )] :> 
       a + b Subscript[[Theta], 1] + c Subscript[[Theta], 2] + 
        d [CurlyEpsilon]] /. Matrix[( {
        {a_, 0, 0, 0},
        {b_, a_, 0, 0},
        {c_, 0, a_, 0},
        {d_, , b, a_}
       } )] :> 
     a + b Subscript[[Theta], 1] + c Subscript[[Theta], 2] + 
      d [CurlyEpsilon] &;
$Post = Nest[$Post2, #, 3] /. Dot -> NonCommutativeMultiply &;

Test:

In:=Sqrt[Subscript[\[Theta], 1] + Subscript[\[Theta], 
  2] + \[CurlyEpsilon] + 5]

Out:=(10+[CurlyEpsilon]+Subscript[[Theta], 1]+Subscript[[Theta], 2])/(2 Sqrt[5])
Anixx
  • 11,524