SimTeEx
What is it?
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.
This page describes the Mathematica code SimTeEx which can be used to put in canonical
form such polynomial expressions with tensors. It can handle any kind of tensor symmetry. The program contains the following functions:
- CanonicalForm (the main function)
- YoungSymmetrizeTensor
- SnIrrepsInTensor
- SingleProjector
- SameEquationsQ
- SymmetriesOfNumericalTensor
They are explained in the program's paper; additionally the SimTeEx package comes with a .nb file with several examples on how to use these functions.
References
The SimTeEx program is described in the paper "Renato M. Fonseca, Using SimTeEx to simplify polynomial expressions with tensors,
arXiv:2412.14390 [hep-ph]".
Some functions of this package require the GroupMath code, which was published in "Renato M. Fonseca, GroupMath: A Mathematica package for group theory calculations,
arXiv:2011.01764 [hep-th], Computer Physics Communications 267 (2021)"
Installing the code
The main function of SimTeEx (CanonicalForm) does not need any extra software to run. However, this package includes other functions which do require the group theory code of GroupMath, so in order to run them both programs must be downloaded and installed correctly. The GroupMath code can be found here 
while SimTeEx can be obtained from this page:
(SimTeEx 1.0.1)
The GroupMath and SimTeEx folders should be placed in a directory visible to Mathematica. A good folder to put them is in
(Mathematica base directory)/AddOns/Applications
Linux, Mac OS
(Mathematica base directory)\AddOns\Applications
That's it. To load SimTeEx, type in the front end
<<SimTeEx`
For questions, comments or bug reports, please contact me at
renatofonseca@ugr.es
Author
Renato Fonseca
Emails
renatofonseca@gmail.com
or
renatofonseca@ugr.es
Last updated
19 December 2024
