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Finite-dimensional PT-symmetric Hamiltonians with an application to neutrino oscillations

Alexander Fagerlund, Chistopher Ekman

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

In this report, we first briefly summarize Hermitian quantum mechanics before moving on to the non-Hermitian case. We then review PT-symmetric quantum mechanics with a focus on finite-dimensional systems, and include a novel generalization of a perturbative calculation of the C-operator. After briefly covering the basics of neutrino oscillations, we perturbatively examine a PT-symmetric addition to the neutrino oscillation Hamiltonian. We examine the effects of the addition with two different definitions of transition probabilities. However, probability is not conserved to first order with either definition. Further, we note that the effect of the chosen perturbation is to shift the transition probabilities by some phase, and to change the amplitudes of the transition probabilities.

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Adam Millican-Slater

PT-symmetric quantum mechanics is an alternative to the usual hermitian quantum mechanics. We will start this thesis by taking an overview of the subject, seeing some of the elementary consequences of this different approach. The main part of the work will be an depth study of a specific Hamiltonian(diagram) This is a generalisation of the well understood harmonic oscillator with angular momentum. By making this generalisation we break the hermiticity of the problem. This leads to some intriguing results. We will be particularly interested in the merging of eigenvalues for M <1.We study the problem using a number of techniques. First the Hamiltonian is studied at the classical level and the behaviour of a particle moving in the corresponding potential is studied. Having seen the consequences at the classical level we return to the quantum case. The Hamiltonian is first solved perturbativly. This method is shown to be valid for PT-symmetric quantum mechanics. It is shown that asymptotic limits of the matrix do not capture the full behaviour of the energy levels. We then move on to study the problem considering techniques arising from the ODE/IM correspondence. Using this approach we are able to give an analytic description of the phenomena and explain the merging of eigenvalues.

•Jan 1, 2004•1 citation
In this report, we first briefly summarize Hermitian quantum mechanics before moving on to the non-Hermitian case. We then review PT-symmetric quantum mechanics with a focus on finite-dimensional systems, and include a novel generalization of a perturbative calculation of the C-operator. After briefly covering the basics of neutrino oscillations, we perturbatively examine a PT-symmetric addition to the neutrino oscillation Hamiltonian. We examine the effects of the addition with two different definitions of transition probabilities. However, probability is not conserved to first order with either definition. Further, we note that the effect of the chosen perturbation is to shift the transition probabilities by some phase, and to change the amplitudes of the transition probabilities.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
We investigate how to define in a consistent way the probabilities of the transitions between the “flavor” states of the two-level quantum system, which is described by a non-Hermitian but parity and time-reversal (PT) symmetric Hamiltonian. Explicit calculations are carried out to demonstrate the conservation of probability if a proper definition of the final state is adopted. Finally, this formalism is applied to two-flavor neutrino oscillations νμ → νμ and νμ → ντ in vacuum, where the exact PT symmetry requires the vacuum mixing angle to be maximal, which is compatible with current neutrino oscillation experiments. A possible generalization to the three-flavor case is briefly discussed.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT\mathcal{PT}PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT\mathcal{PT}PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT\mathcal{PT}PT symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of PT\mathcal{PT}PT symmetry by focusing on elementary one-dimensional PT\mathcal{PT}PT-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of PT\mathcal{PT}PT-symmetric quantum theory.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.
PT-symmetry, invariance under the combined operations of spatial reflection and time reversal, arises naturally in non-Hermitian physics and has been attracting an increasing amount of attention in the literature. Here, we examine probabilistic structures within the unbroken and broken PT-symmetric regimes from a C*-algebraic perspective. We show that PT-unbroken probability is directly compatible with the operator-algebraic formulation of standard QM. However, though PT-broken Hamiltonians may generate well-defined non-unitary dynamics, basic structural properties required for quantum probability typically fail. We analyze this failure via relation to the GNS construction, ultimately demonstrating that an extension to a C*-algebraic probability theory on a Hermitian supersystem is required to define a meaningful probability calculus within the PT-broken system itself. Thereby, consistent probability assignment must be extrinsic to the PT-broken phase, and must be inherited from a dilation to a Hermitian system. We offer a novel interpretation of the broken PT-symmetric regime as a non-ordered, non-probabilistic subalgebra of a larger Hermitian, fully quantum system. Additionally, our work clarifies the structural importance of positivity in standard quantum mechanics.
PT-symmetric quantum mechanics is an alternative to the usual hermitian quantum mechanics. We will start this thesis by taking an overview of the subject, seeing some of the elementary consequences of this different approach. The main part of the work will be an depth study of a specific Hamiltonian(diagram) This is a generalisation of the well understood harmonic oscillator with angular momentum. By making this generalisation we break the hermiticity of the problem. This leads to some intriguing results. We will be particularly interested in the merging of eigenvalues for M <1.We study the problem using a number of techniques. First the Hamiltonian is studied at the classical level and the behaviour of a particle moving in the corresponding potential is studied. Having seen the consequences at the classical level we return to the quantum case. The Hamiltonian is first solved perturbativly. This method is shown to be valid for PT-symmetric quantum mechanics. It is shown that asymptotic limits of the matrix do not capture the full behaviour of the energy levels. We then move on to study the problem considering techniques arising from the ODE/IM correspondence. Using this approach we are able to give an analytic description of the phenomena and explain the merging of eigenvalues.