FCSchoutenBruteForce
FCSchoutenBruteForce[exp, {}, {}]
can be used to show that certain terms are zero by repeatedly applying, Schouten’s identity in a brute force way.
The algorithm tries to find replacements which follow from the Schouten’s identity and make the length of the given expression shorter.
It is not guaranteed to terminate and in general can often get stuck. Still, with some luck it is often possible to show that certain terms vanish by a sequence of transformations that would be otherwise very difficult to find.
See also
Overview, Schouten.
Examples
One may not recognize it easily, but the following expression is zero by Schouten’s identity
FCClearScalarProducts[]
exp = LC[][p1, p2, p3, p4] SP[p5, p6] + LC[][p2, p3, p4, p5] SP[p1, p6] +
LC[][p3, p4, p5, p1] SP[p2, p6] + LC[][p4, p5, p1, p2] SP[p3, p6] -
LC[][p1, p2, p3, p5] SP[p4, p6]
(p5⋅p6)ϵˉp1p2p3p4−(p4⋅p6)ϵˉp1p2p3p5+(p1⋅p6)ϵˉp2p3p4p5+(p2⋅p6)ϵˉp3p4p5p1+(p3⋅p6)ϵˉp4p5p1p2
FCSchoutenBruteForce[exp, {}, {}]
FCSchoutenBruteForce: The following rule was applied: ϵˉp2p3p4p5(p1⋅p6):→ϵˉp1p3p4p5(p2⋅p6)−ϵˉp1p2p4p5(p3⋅p6)+ϵˉp1p2p3p5(p4⋅p6)−ϵˉp1p2p3p4(p5⋅p6)
FCSchoutenBruteForce: The numbers of terms in the expression decreased by: 5
FCSchoutenBruteForce: Current length of the expression: 0
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