A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916683832098816 |
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| author | McRae, Craig |
| author_facet | McRae, Craig |
| contents | A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown how the entire Lorentz group may be understood as a semi-direct product between its identity component and the Klein four group of spacetime reflections: $\operatorname{O}(1,3) = \operatorname{SO}^+(1,3) \rtimes \operatorname{K}_4$. This gives way to a convenient classification of tensors transforming under $\operatorname{O}(1,3)$, namely that there are four representations of $\operatorname{O}(1,3)$ for each representation of $\operatorname{SO}^+(1,3)$, and it is shown how the representation theory of the Klein group $\operatorname{K}_4$ allows for simple book keeping of the spacetime reflection properties of general Lorentzian tensors, and combinations thereof, with several examples given. There is a brief discussion of the time reversal of the electromagnetic field, concluding in agreement with standard texts such as Jackson, and works by Malament. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05400 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$ McRae, Craig Mathematical Physics Representation Theory A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown how the entire Lorentz group may be understood as a semi-direct product between its identity component and the Klein four group of spacetime reflections: $\operatorname{O}(1,3) = \operatorname{SO}^+(1,3) \rtimes \operatorname{K}_4$. This gives way to a convenient classification of tensors transforming under $\operatorname{O}(1,3)$, namely that there are four representations of $\operatorname{O}(1,3)$ for each representation of $\operatorname{SO}^+(1,3)$, and it is shown how the representation theory of the Klein group $\operatorname{K}_4$ allows for simple book keeping of the spacetime reflection properties of general Lorentzian tensors, and combinations thereof, with several examples given. There is a brief discussion of the time reversal of the electromagnetic field, concluding in agreement with standard texts such as Jackson, and works by Malament. |
| title | A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$ |
| topic | Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2501.05400 |