Using SimTeEx to simplify polynomial expressions with tensors

Abstract

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.