Abstract
In this chapter we discuss the properties of the time-reversal operator (introduced into quantum mechanics by Wigner in 1932) for particles without spin, as well as taking into account the spin. There are given Wigner criteria (a), (b), and (c) concerning the absence or presence of an additional degeneracy of energy levels due to the symmetry with respect to time reversal, both in the case when the spin of particles that form the quantum system is taken into account and in the case it is not. Wigner corepresentations of symmetry groups associated with the presence of time-reversal symmetry are also discussed.
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Notes
- 1.
The statistical operator should not be confused with the probability density, denoted earlier also by \(\rho \).
- 2.
Further, if it does not involve misunderstandings, we shall write \(\mathcal {L}\) instead of L (respectively, A instead of \(\mathbf {A}\)) omitting the symbol of operator.
- 3.
Formula (2.119) differs from that given in [222], since our spinor basis of the matrix \(\mathbf {U}\) differs by a permutation of the basis spinors used in [222].
- 4.
In the absence of a magnetic field.
- 5.
The indicated classification of cases (a), (b) and (c) is rather different from that introduced in [78], for which (a) it is related to the case when functions \(\left| \psi \right\rangle \) and \(\mathbf {T}\) \(\left| \psi \right\rangle \) are linearly dependent, (b) it is referred the case when functions \(\left| \psi \right\rangle \) and \(\mathbf {T}\) \(\left| \psi \right\rangle \) are linearly independent and are transformed by nonequivalent representations, and (c) is related to the case when \(\left| \psi \right\rangle \) and \(\mathbf {T}\) \(\left| \psi \right\rangle \) are linearly independent and are transformed by equivalent representations. For ordinary representations both classifications coincide, but for spinor representations the cases (a) and (c) change places.
- 6.
Relation \(\mathbf {SS}^{*}\mathbf {=1}\) corresponds to case (a), when representation D is equivalent to the real representation \(D_{r}\) \((D\sim D^{*}\sim D_{\mathbf {r}})\), while the relation \(\mathbf {SS}^{*}\mathbf {=-1}\) corresponds to the case (c), when a real representation equivalent to D does not exist [76].
- 7.
In the case of Klein–Gordon–Fock equation for a spinless particle \(\mathbf {T}^{2} = +\mathbf {1}.\)
- 8.
Although in 1848 Paster called attention to the fact that some organic compounds are met in biological structures only in a left rotation form [104].
- 9.
Experimental researches, made often the suggestion of theory of two-component neutrino, showed that really the rest mass of neutrino is not equal to zero.
- 10.
Moreover, as noted in [107], in some processes with the participation of neutrinos even the CPT theorem is violated.
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Geru, I.I. (2018). Time Reversal in Quantum Mechanics and Quantized Field Theory. In: Time-Reversal Symmetry. Springer Tracts in Modern Physics, vol 281. Springer, Cham. https://doi.org/10.1007/978-3-030-01210-6_2
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DOI: https://doi.org/10.1007/978-3-030-01210-6_2
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