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Proper time as an operator

Shuichi Matsumoto, 修一 松本

Published Dec 30, 1999• 0 citations
OpenAlex

Abstract

When we quantize a system consisting of a single particle, the proper time τ and the rest mass m are usually dealt with as parameters. In the present article, however, we introduce a new quantization rule by which these quantities are regarded as operators in addition to the position and the momentum. Applying this new rule to a scalar particle and to a particle with spin 1/2, we analyze the time evolution of the operator τ. In the former case, the evolution of the proper time perfectly matches several well-established classical formulae. In the case of the particle with spin 1/2, our new rule implies that an oscillation appears in the time evolution of the operator τ. This oscillation is similar to Zitterbewegung which is well-known in the ordinary Dirac theory. We formulate one physical effect of this oscillation by considering the interaction with a gravitational field, and estimate how small it is.

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When we quantize a system consisting of a single particle, the proper time τ and the rest mass m are usually dealt with as parameters. In the present article, however, we introduce a new quantization rule by which these quantities are regarded as operators in addition to the position and the momentum. Applying this new rule to a scalar particle and to a particle with spin 1/2, we analyze the time evolution of the operator τ. In the former case, the evolution of the proper time perfectly matches several well-established classical formulae. In the case of the particle with spin 1/2, our new rule implies that an oscillation appears in the time evolution of the operator τ. This oscillation is similar to Zitterbewegung which is well-known in the ordinary Dirac theory. We formulate one physical effect of this oscillation by considering the interaction with a gravitational field, and estimate how small it is.
When we quantize a system consisting of a single particle, the proper time τττ and the rest mass mmm are usually dealt with as parameters. In the present article, however, we introduce a new quantization rule by which these quantities are regarded as operators in addition to the position and the momentum. Applying this new rule to a scalar particle and to a particle with spin $ 1/2 $, we analyze the time evolution of the operator τττ. In the former case, the evolution of the proper time perfectly matches several well-established classical formulae. In the case of the particle with spin 1/2, our new rule implies that an oscillation appears in the time evolution of the operator τττ. This oscillation is similar to Zitterbewegung which is well-known in the ordinary Dirac theory. We formulate one physical effect of this oscillation by considering the interaction with a gravitational field, and estimate how small it is.
We study the quantum properties of an oscillator in proper time. This proper time oscillator is a particle model with mass that is on shell. Its internal time can be treated as a self-adjoint operator. The displaced time and displaced time rate of the oscillator obey an uncertainty relation resembling the one between position and momentum, which is different from the usual energy-time uncertainty relation. In addition, we demonstrate that a matter field with proper time oscillators satisfies the Klein-Gordon equation. It has properties of a zero-spin quantum field. The formulations adopted permits a more symmetrical treatment between time and space in a matter field.
Abstract The Pryce (e) spin and position operators of the quantum theory of Dirac's free field were re-defined and studied recently with the help of a new spin symmetry and suitable spectral representations [Eur. Phys. J. C 82, 1073 (2022)]. This approach is generalized here, associating a pair of integral operators acting directly on particle and antiparticle wave spinors in momentum representation to any integral operator in configuration representation, acting on mode spinors. This framework allows an effective quantization procedure, giving a large set of one-particle operators with physical meaning as the spin and orbital parts of the isometry generators, the Pauli-Lubanski and position operators, or other spin-type operators proposed to date. Special attention is paid to the operators that mix the particle and antiparticle sectors whose off-diagonal associated operators have oscillating terms producing Zitterbevegung. The principal operators of this type, including the usual coordinate operator, are derived here for the first time. As an application, it is shown that an apparatus measuring these new observables may prepare and detect one-particle wave packets moving uniformly without Zitterbewegung or spin dynamics, spreading in time normally as any other relativistic or even non-relativistic wave packet.
A satisfactory proper time operator is found for Dirac's quantum relativistic electron. A self-consistent scheme is developed in which this operator and Beck's proper time derivative have the properties they should have. As an application of the proper time formalism as a research tool, properties of three position operators (which are generally considered as covariant) are studied in regard to their behaviour under the inhomogeneous Lorentz group and under ordinary and proper time evolution. They are Pryce's class d, Bacry's and R. J. Finkelstein's operators. It results that all three have some unreasonable properties for Dirac's electron. The reason for the difficulty in the case of Finkelstein's operator can be traced back to an incorrect definition of it. By making some changes a satisfactory position operator is obtained but it turns out to be equivalent to Bunge's one (up to an arbitrary and unimportant constant of motion). The results of this paper reinforce those of previous work which show that (for Dirac's electron) Bunge's position operator seems to be in a privileged place as regards the solution of the localization problem. Incidentaly, the interpretation of some relations previously used as auxiliary formulas for computations, allows us to obtain a manifestly covariant expression for Dirac's Hamiltonian and a second manifestly covariant form of Dirac's equation.
Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.
Several quantum proper time derivatives are obtained from the Beck one in the usual framework of relativistic quantum mechanics (spin-1/2 case). The ``scalar Hamiltonians'' of these derivatives should be thought of as the conjugate variables of the proper time. Then, the Hamiltonians would play the role of mass operators, suggesting the formulation of an adequate extended indefinite mass framework. We propose and briefly develop the framework corresponding to the Feynman parametrization of the Dirac equation. In such a case we derive the other parametrizations known in the literature, linking the extension of the different proposals of quantum proper time derivatives again.