Using SimTeEx to simplify polynomial expressions with tensors

TL;DR

This paper introduces SimTeEx, a Mathematica package that simplifies tensor polynomial expressions by recognizing dummy indices and tensor symmetries, addressing a complex problem in tensor computations.

Contribution

The paper presents a novel algorithm implemented in SimTeEx that can handle any tensor symmetry to simplify tensor polynomial expressions.

Findings

Successfully simplifies tensor expressions with dummy indices

Handles arbitrary tensor symmetries

Improves computational efficiency in tensor algebra

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.