Computations with tensors are ubiquitous in fundamental physics, and so is the usage of Einstein's dummy index convention for the contraction of indices. For instance, $T_{ ia}U_{ aj}$ is readily recognized as the same as $T_{ ib}U_{ bj}$, but a computer does not know that T[i,a]U[a,j] is equal to T[i,b]U[b,j]. Furthermore, tensors may have symmetries which can be used to simply expressions: if $U_{ ij}$ is antisymmetric, then $\alpha T_{ ia}U_{ aj}+\beta T_{ ib}U_{ jb}=\left(\alpha-\beta\right)T_{ ia}U_{ aj}$. The fact that tensors can have elaborate symmetries, together with the problem of dummy indices, makes it complicated to simplify polynomial expressions with tensors. In this work I will present an algorithm for doing so, which was implemented in the Mathematica package SimTeEx (Simplify Tensor Expressions). It can handle any kind of tensor symmetry.
Categories: hep-ph cs.SC gr-qc hep-th