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arXiv:1107.0501 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

P T symmetry in relativistic quantum mechanics

Carl M. Bender, Philip D. Mannheim

Published Nov 29, 2011• 16 citations
arXivHTMLPDF

Abstract

In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate $t$ is a parameter and thus the time-reversal operator $\mathcal{T}$ does not actually reverse the sign of $t$. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time $t$ is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend $\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as $\mathcal{P}$ effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with $\mathcal{P}\mathcal{T}$ thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but $\mathcal{P}\mathcal{T}$ symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.

DOI
10.1103/physrevd.84.105038

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In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate ttt is a parameter and thus the time-reversal operator T\mathcal{T}T does not actually reverse the sign of ttt. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time ttt is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend T\mathcal{T}T and PT\mathcal{P}\mathcal{T}PT symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects T tT=\ensuremath−t\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}tT tT=\ensuremath−t just as P\mathcal{P}P effects PxP=\ensuremath−x\mathcal{P}x\mathcal{P}=\ensuremath{-}xPxP=\ensuremath−x, with PT\mathcal{P}\mathcal{T}PT thus effecting PT x\ensuremathμPT=\ensuremath−x\ensuremathμ\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}PT x\ensuremathμ​PT=\ensuremath−x\ensuremathμ​. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT\mathcal{P}\mathcal{T}PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p 2 + ix 3 , which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p 2 + ix 3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p 2 − x 4 , which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.
In 1998 it was discovered that the requirement that a Hamiltonian be Dirac Hermitian (H = H†) can be weakened and generalized to the requirement that a Hamiltonian be PT symmetric ([H,PT] = 0); that is, invariant under combined space reflection and time reversal. Weakening the constraint of Hermiticity allows one to consider new kinds of physically acceptable Hamiltonians and, in effect, it amounts to extending quantum mechanics from the real (Hermitian) domain into the complex domain. Much work has been done on the analysis of various PT‐symmetric quantum‐mechanical models. However, only very little analysis has been done on PT‐symmetric quantum‐field‐theoretic models. Here, we describe some of what has been done in the context of PT‐symmetric quantum field theory and describe some possible fundamental applications.
We describe a novel class of quantum mechanical particle oscillations in both relativistic and nonrelativistic systems based on P T symmetry and T 2 = − 1 (relevant for fermions), where P is parity and T is time reversal. The Hamiltonians are chosen at the outset to be self-adjoint with respect to a P T inner product. The quantum mechanical time evolution is based on a modified C P T inner product constructed in terms of a suitable C operator. The resulting quantum mechanical evolution is shown to be unitary and probability is conserved by the oscillations. Published by the American Physical Society 2024
Abstract In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric—the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.
This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.