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arXiv:2111.10035

PT and Anti-PT Symmetry Arising in Time Dependent Weak Value Measurements

Allen Parks, John Edward Gray, Garrett K. Josemans

Published Nov 19, 2021• 0 citations
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Abstract

Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time dependent Hamiltonians that arise from time dependent weak values. This allows one to generalize some work on non-Hermitian variables in quantum mechanics (QM) due to Bender related to weak values and weak measurement. We also speculate how these symmetries might apply to distinguishing between the various two-time interpretations of QM in the Conclusions of this paper.

DOI
10.48550/arxiv.2111.10035

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Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time dependent Hamiltonians that arise from time dependent weak values. This allows one to generalize some work on non-Hermitian variables in quantum mechanics (QM) due to Bender related to weak values and weak measurement. We also speculate how these symmetries might apply to distinguishing between the various two-time interpretations of QM in the Conclusions of this paper.
Quantum mechanical weak values and their measurement have been a focus of theoretical, experimental, and applied research for more than two decades. The concept of PT symmetry was also introduced into quantum mechanics during this time. This paper defines the notion of a weak value measurement pointer Hamiltonian and establishes equivalences between its Dirac symmetries, its PT symmetries, its eigenvalues, and the associated weak value. The affect of these symmetries upon measurement pointer observables is also identified.
In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) I ( t ) for a TD non-Hermitian Hamiltonian H(t) H ( t ) . Assuming H(t) H ( t ) to be simultaneously \rho(t) ρ ( t ) -pseudo-Hermitian and \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 I 2 k = 1 with k k odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , the symmetry-metric relation finally allow us to derive the \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond
A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.
We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under PT\mathcal{P}\mathcal{T}PT, where P\mathcal{P}P is the parity and T\mathcal{T}T denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (T2=1{\mathcal{T}}^{2}=1T2=1). We discover an analog of Kramer's theorem for PT\mathcal{P}\mathcal{T}PT quantum mechanics, present a prototypical example of a PT\mathcal{P}\mathcal{T}PT quantum system with odd time reversal, and discuss potential applications of the formalism.
In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterpart are important for the comprehension of posed problems in quantum optics and quantum chemistry, which consists of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. We obtain the propagator for a general timedependence of the parameters and the wave-functions are obtained explicitly for constant frequency and mass and a linear time-dependence in the potential. The wave-functions are used to obtain the expectation value of the Hamiltonian. Although not PT symmetric the case of study exhibits real values of energy.
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.