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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.14390 |
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| _version_ | 1866916605520248832 |
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| author | Fonseca, Renato M. |
| author_facet | Fonseca, Renato M. |
| contents | 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 $αT_{ia}U_{aj}+βT_{ib}U_{jb}=\left(α-β\right)T_{ia}U_{aj}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Using SimTeEx to simplify polynomial expressions with tensors Fonseca, Renato M. High Energy Physics - Phenomenology Symbolic Computation General Relativity and Quantum Cosmology High Energy Physics - Theory 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 $αT_{ia}U_{aj}+βT_{ib}U_{jb}=\left(α-β\right)T_{ia}U_{aj}$. 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. |
| title | Using SimTeEx to simplify polynomial expressions with tensors |
| topic | High Energy Physics - Phenomenology Symbolic Computation General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2412.14390 |