Abstract
The Wigner time-reversal operator \(\mathbf T \) is represented as the product of two or three so-called operators of incomplete time-reversal, under the action of which not all the angular momentum projection operators change sign. It is shown that when the symmetry of time reversal is violated (reduced) in systems with Kramers degeneracy of energy levels, a violation of the Kramers theorem occurs, with the exception of one case when such reducing is insufficient to remove the Kramers degeneracy. The commutation and anticommutation relations between operators of incomplete time reversal, as well as between these operators and the operator \(\mathbf T \), are found. It is shown that these relations are different for Kramers and non-Kramers systems.
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- 1.
In many books and articles the term “metamaterials” is widely used [438–453], but it has no relation to the concept of meta-matter introduced by Wigner in [437]. Unlike this (as well as unlike the definition of meta-matter introduced in [457], we will use the Wigner conception of meta-particles).
- 2.
Since the matrices \(\Sigma _y\) and \(\Sigma _z\) coincide with \(S_y\) and \(S_z\), the sign of the matrix \(\Sigma _x\) differs from that of the matrix \(S_x\), and the commutation relations for the matrices \(\Sigma _\chi \) and, accordingly, \(S_\chi (\chi =x,\ y,\ z)\) are different, then the own angular momentum of the meta-particle could be not called a spin, but otherwise (for example, meta-spin).
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Geru, I.I. (2018). Factorization of Wigner Time-Reversal Operator and Reduction of Time-Reversal Symmetry. In: Time-Reversal Symmetry. Springer Tracts in Modern Physics, vol 281. Springer, Cham. https://doi.org/10.1007/978-3-030-01210-6_9
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DOI: https://doi.org/10.1007/978-3-030-01210-6_9
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Publisher Name: Springer, Cham
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