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PIPE

Operator

|

Description

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.

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