In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$ . In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$ , unitarily equivalent to ${\cal L}$ . The situation encountered, typically, in ${\cal PT}-$ symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\cal L}$. The situation encountered, typically, in ${\cal PT}-$symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.

Aug 8, 20190 citations

ON THE CONSTRUCTION OF RELATIVISTIC QUANTUM THEORY: A PROGRESS REPORT

H. Pierre Noyes

Feb 25, 20220 citations

On Relativistic Quantum Theory

Lawrence Paul Horwitz

Springer eBooksJan 1, 1983111 citations

Toward relativistic generalization of collapse models

Anirudh Gundhi, Lajos Diósi, Matteo Carlesso

Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem. While experiments are providing increasingly stronger bounds on their parameters, a full-fledged relativistic extension is still missing. Previous attempts have encountered different obstacles, such as violation of microcausality, infinite energy rate, and particle production from vacuum. Here, we propose a generalization of the collapse master equation that is characterized by a local field collapse operator and a non-Markovian noise with a Lorentz invariant correlation. Our construction is able to overcome previously encountered problems and has the desirable properties in the nonrelativistic limit. A specific choice of the noise correlation function is also introduced and discussed.Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem. While experiments are providing increasingly stronger bounds on their parameters, a full-fledged relativistic extension is still missing. Previous attempts have encountered different obstacles, such as violation of microcausality, infinite energy rate, and particle production from vacuum. Here, we propose a generalization of the collapse master equation that is characterized by a local field collapse operator and a non-Markovian noise with a Lorentz invariant correlation. Our construction is able to overcome previously encountered problems and has the desirable properties in the nonrelativistic limit. A specific choice of the noise correlation function is also introduced and discussed.

arXiv:2507.06954Physical Review AJun 22, 20261 citation

ON THE CONDITIONS OF THE RELATIVISTIC INVARIANCE OF THE QUANTUM THEORY

Uy.M. Shirikov

Dec 21, 19560 citations

The quantized discrete time oscillator

George Jaroszkiewicz

Introduction The quantized harmonic oscillator has proven to be perhaps the most important system in continuous time (CT) quantum mechanics (QM). It was the atomic oscillators assumed to line the containing walls of a black-body radiation field that Planck referred to when he postulated the quantization of energy in 1900 (Planck, 1900, 1901), thereby triggering the development of QM. Subsequently, the quantized harmonic oscillator resurfaced in many contexts, most notably in quantum optics and relativistic quantum field theory, including string theory. There are two reasons for the importance of the quantized oscillator: (i) it can be solved completely; and (ii) it has ladder operators that generate Planck's quantized energy levels in particle QM and particle-like states in relativistic quantum field theory. Temporal discretization does not destroy the properties of the oscillator. Rather, it enhances them. We shall show that the parabolic barrier encountered in the classical theory of the discrete time (DT) oscillator, which was discussed in the previous chapter, provides a natural upper bound for important physical quantities such as energy and momentum. In other words, the quantized DT oscillator has a natural cutoff. This may turn out to be of fundamental importance in the regularization (renormalization) of relativistic quantum field theories, where there exist immense problems related to the fact that the CT oscillator has no upper bound to its energy spectrum.Introduction The quantized harmonic oscillator has proven to be perhaps the most important system in continuous time (CT) quantum mechanics (QM). It was the atomic oscillators assumed to line the containing walls of a black-body radiation field that Planck referred to when he postulated the quantization of energy in 1900 (Planck, 1900, 1901), thereby triggering the development of QM. Subsequently, the quantized harmonic oscillator resurfaced in many contexts, most notably in quantum optics and relativistic quantum field theory, including string theory. There are two reasons for the importance of the quantized oscillator: (i) it can be solved completely; and (ii) it has ladder operators that generate Planck's quantized energy levels in particle QM and particle-like states in relativistic quantum field theory. Temporal discretization does not destroy the properties of the oscillator. Rather, it enhances them. We shall show that the parabolic barrier encountered in the classical theory of the discrete time (DT) oscillator, which was discussed in the previous chapter, provides a natural upper bound for important physical quantities such as energy and momentum. In other words, the quantized DT oscillator has a natural cutoff. This may turn out to be of fundamental importance in the regularization (renormalization) of relativistic quantum field theories, where there exist immense problems related to the fact that the CT oscillator has no upper bound to its energy spectrum.

Cambridge University Press eBooksApr 17, 20140 citations

\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger

induces the change

{\cal L} \to {\cal K}

of the Hilbert space of states, reflected by the loss of the Hermiticity of

H\neq H^\dagger

. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts

{\cal K}

into the third, correct physical Hilbert space

{\cal H}

, unitarily equivalent to

{\cal L}

. The situation encountered, typically, in

{\cal PT}-

symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.