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Springer tracts in modern physics

Non-Abelian and Abelian Symmetry Groups Containing Time-Reversal Operators

Ion I. Geru

Published Jan 1, 2018• 0 citations
Publisher

Abstract

On the basis of the group-theoretic approach, the existence of six new time reversal operators is proved, along with the well-known anti-unitary time-reversal operator introduced into quantum mechanics by Wigner. Among the new time-reversal operators, three are anti-unitary and three are unitary. A characteristic feature of the new time-reversal operators is that under their action the signs do not change for all three spin projection operators, but only for two or only for one of them. For this reason, such operators should be called operators of incomplete time reversal, in contrastIncomplete time reversal to the Wigner operator, which in this context is an operator of complete time reversal.

DOI
10.1007/978-3-030-01210-6_8

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On the basis of the group-theoretic approach, the existence of six new time reversal operators is proved, along with the well-known anti-unitary time-reversal operator introduced into quantum mechanics by Wigner. Among the new time-reversal operators, three are anti-unitary and three are unitary. A characteristic feature of the new time-reversal operators is that under their action the signs do not change for all three spin projection operators, but only for two or only for one of them. For this reason, such operators should be called operators of incomplete time reversal, in contrastIncomplete time reversal to the Wigner operator, which in this context is an operator of complete time reversal.
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.
A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner's theorem. Wigner's theorem shows that every symmetry of the quantum state space must be either a unitary transformation, or a antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolution associated to quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. We propose one such modification, wherein the evolutions are restricted to a suitable set of time symmetric operations, including both unitary evolution and the state reductions associated to projective measurements.
A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.
In Newtonian physics, the equation of motion is invariant when the direction of time (t→−t) is flipped. However, in quantum physics, flipping the direction of time changes the sign of the Schrödinger equation. An anti-unitary operator is needed to restore time reversal in quantum physics, but this is at the cost of not having a consistent definition of time reversal applicable to all fundamental theories. On the other hand, a quantum system composed of a pair of entangled particles behaves in such a manner that when the state of one particle is measured, the second particle ‘simultaneously’ acquires a determinate state. A notion of absolute simultaneity seems to be inferred by quantum mechanics, even though it is forbidden by the postulates of relativity. We aim to point out that the above two problems can be overcome if the wavefunction is defined with respect to proper time, which in fact is the real physical time instead of ordinary time.
The discussion on time-reversal in quantum mechanics exists at least since Wigner's ``Uber die Operation der Zeitumkehr in der Quantenmechanik'' paper in 1932. If and how the dynamics of the quantum world is time-reversible has been the subject of many controversies. Some have seen quantum mechanics as fundamentally time-irreversible, see for example von Neumann, and some have seen in that the ultimate cause of time's arrow and second law behavior. In his best-selling book ``The Emperor's New Mind'', Roger Penrose argues similarly and concludes that ``our sought-for quantum gravity must be a time-asymmetric theory'' [p.351]. Not so long ago, we read yet about another project in Physicalia 26: to extend quantum mechanics into new fundamentally irreversible equations, thus proposing a new theory giving ``... une description fondamentale irreversible de tout systeme physique''.