Maxima Manual: Graßmann-Algebra

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#### 20.2.2.7 Graßmann-Algebra

The `itensor` package can perform operations on totally antisymmetric covariant tensor fields. A totally antisymmetric tensor field of rank (0,L) corresponds with a differential L-form. On these objects, a multiplication operation known as the exterior product, or wedge product, is defined.

Unfortunately, not all authors agree on the definition of the wedge product. Some authors prefer a definition that corresponds with the notion of antisymmetrization: in these works, the wedge product of two vector fields, for instance, would be defined as

                a a  - a a
                 i j    j i
     a  /\ a  = -----------
      i     j        2

More generally, the product of a p-form and a q-form would be defined as

                           1     k1..kp l1..lq
    A       /\ B       = ------ D              A       B
     i1..ip     j1..jq   (p+q)!  i1..ip j1..jq  k1..kp  l1..lq

where `D` stands for the Kronecker-delta.

Other authors, however, prefer a “geometric” definition that corresponds with the notion of the volume element:

    a  /\ a  = a a  - a a
     i     j    i j    j i

and, in the general case

                           1    k1..kp l1..lq
    A       /\ B       = ----- D              A       B
     i1..ip     j1..jq   p! q!  i1..ip j1..jq  k1..kp  l1..lq

Since `itensor` is a tensor algebra package, the first of these two definitions appears to be the more natural one. Many applications, however, utilize the second definition. To resolve this dilemma, a flag has been implemented that controls the behavior of the wedge product: if `igeowedge_flag` is `false` (the default), the first, "tensorial" definition is used, otherwise the second, "geometric" definition will be applied.

Operator: ~

  The wedge product operator is denoted by the tilde `~`. This is a binary operator. Its arguments should be expressions involving scalars, covariant tensors of rank one, or covariant tensors of rank `l` that have been declared antisymmetric in all covariant indices.

  The behavior of the wedge product operator is controlled by the `igeowedge_flag` flag, as in the following example:

      (%i1) load("itensor");
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i])~b([j]))$
                                       a  b  - b  a
                                        i  j    i  j
      (%t2)                            -------------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(a([i,j])~b([k]))$
                                a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j
      (%t4)                     ---------------------------
                                             3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(a([i])~b([j]))$
      (%t6)                            a  b  - b  a
                                        i  j    i  j
      (%i7) ishow(a([i,j])~b([k]))$
      (%t7)                     a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j

Operator: |

  The vertical bar `|` denotes the "contraction with a vector" binary operation. When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction. Thus, it is possible to define the contraction operation in an index-free manner.

  In the `itensor` package, contraction with a vector is always carried out with respect to the first index in the literal sorting order. This ensures better simplification of expressions involving the `|` operator. For instance:

      (%i1) load("itensor");
      (%o1)      /share/tensor/itensor.lisp
      (%i2) decsym(a,2,0,[anti(all)],[]);
      (%o2)                                done
      (%i3) ishow(a([i,j],[])|v)$
                                          %1
      (%t3)                              v   a
                                              %1 j
      (%i4) ishow(a([j,i],[])|v)$
                                           %1
      (%t4)                             - v   a
                                               %1 j

  Note that it is essential that the tensors used with the `|` operator be declared totally antisymmetric in their covariant indices. Otherwise, the results will be incorrect.

Function: extdiff (expr, i)

  Computes the exterior derivative of expr with respect to the index i. The exterior derivative is formally defined as the wedge product of the partial derivative operator and a differential form. As such, this operation is also controlled by the setting of `igeowedge_flag`. For instance:

      (%i1) load("itensor");
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(extdiff(v([i]),j))$
                                        v    - v
                                         j,i    i,j
      (%t2)                             -----------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(extdiff(a([i,j]),k))$
                                 a      - a      + a
                                  j k,i    i k,j    i j,k
      (%t4)                      ------------------------
                                            3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(extdiff(v([i]),j))$
      (%t6)                             v    - v
                                         j,i    i,j
      (%i7) ishow(extdiff(a([i,j]),k))$
      (%t7)                    - (a      - a      + a     )
                                   k j,i    k i,j    j i,k

Function: hodge (expr)

  Compute the Hodge-dual of expr. For instance:

      (%i1) load("itensor");
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric(g);
      (%o2)                            done
      (%i3) idim(4);
      (%o3)                            done
      (%i4) icounter:100;
      (%o4)                             100
      (%i5) decsym(A,3,0,[anti(all)],[])$

      (%i6) ishow(A([i,j,k],[]))$
      (%t6)                           A
                                       i j k
      (%i7) ishow(canform(hodge(%)))$

                                %1 %2 %3 %4
                     levi_civita            g        A
                                             %1 %102  %2 %3 %4
      (%t7)          -----------------------------------------
                                         6

      (%i8) ishow(canform(hodge(%)))$
                       %1 %2 %3 %8            %4 %5 %6 %7
      (%t8) levi_civita            levi_civita            g
                                                           %1 %106
                                   g        g        g      A         /6
                                    %2 %107  %3 %108  %4 %8  %5 %6 %7
      (%i9) lc2kdt(%)$

      (%i10) %,kdelta$

      (%i11) ishow(canform(contract(expand(%))))$
      (%t11)                     - A
                                    %106 %107 %108

Option variable: igeowedge_flag

  Default value: `false`

  Controls the behavior of the wedge product and exterior derivative. When set to `false` (the default), the notion of differential forms will correspond with that of a totally antisymmetric covariant tensor field. When set to `true`, differential forms will agree with the notion of the volume element.

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