The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916736868024320 |
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| author | McRae, Craig |
| author_facet | McRae, Craig |
| contents | In the representation theory of Lorentzian orthogonal groups, there are well known arguments as to why the parity inversion operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$, should be realized as linear and anti-linear operators respectively (Wigner 1932). Despite this, standard constructions of double covers of the Lorentzian orthogonal groups naturally build time reversal operators in such a manner that they are linear, and the anti-linearity is put in ad-hoc after the fact. This article introduces a viewpoint naturally incorporating the anti-linearity into the construction of these double covers, through what Wigner called co-representations, a kind of semi-linear representation. It is shown how the standard spinoral double covers of the Lorentz group -- $\operatorname{Pin}(1,3)$ and $\operatorname{Pin}(3,1)$ -- may be naturally centrally extended for this purpose, and the relationship between the $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ operators is discussed. Additionally a mapping is constructed demonstrating an interesting equivalence between Majorana and Weyl spinors. Finally a co-representation is built for the de Sitter group $\operatorname{Pin}(1,4)$, and it is demonstrated how a theory with this symmetry has no truly scalar fermion mass terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups McRae, Craig Mathematical Physics Representation Theory In the representation theory of Lorentzian orthogonal groups, there are well known arguments as to why the parity inversion operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$, should be realized as linear and anti-linear operators respectively (Wigner 1932). Despite this, standard constructions of double covers of the Lorentzian orthogonal groups naturally build time reversal operators in such a manner that they are linear, and the anti-linearity is put in ad-hoc after the fact. This article introduces a viewpoint naturally incorporating the anti-linearity into the construction of these double covers, through what Wigner called co-representations, a kind of semi-linear representation. It is shown how the standard spinoral double covers of the Lorentz group -- $\operatorname{Pin}(1,3)$ and $\operatorname{Pin}(3,1)$ -- may be naturally centrally extended for this purpose, and the relationship between the $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ operators is discussed. Additionally a mapping is constructed demonstrating an interesting equivalence between Majorana and Weyl spinors. Finally a co-representation is built for the de Sitter group $\operatorname{Pin}(1,4)$, and it is demonstrated how a theory with this symmetry has no truly scalar fermion mass terms. |
| title | The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups |
| topic | Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2505.09041 |