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arXiv:quant-ph/9901031

Uncertainty principle for proper time and mass II

Shoju Kudaka, Shuichi Matsumoto

Published Jan 14, 1999• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.quant-ph/9901031

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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 1/21/21/2, 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.
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.
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.
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.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
A quantum harmonic oscillator oscillates in space. As we have learned from general relativity, spacetime is weaved as unity. If space and time are supposed to be treated on an equal footing, can a quantum harmonic oscillator oscillate in proper time? In this paper, we examine this possibility by assuming matter can oscillate in proper time. We demonstrate that a field with oscillations of matter in proper time can mimic the properties of a zero-spin bosonic field. A particle observed in this field is a proper time oscillator. The assumption also gives rise to properties that can reduce differences between quantum theory and general relativity, e.g., self-adjoint internal time operator and proper time uncertainty relation. Neglecting all quantum effects, a proper time oscillator can mimic a point mass at rest in general relativity. The spacetime outside a 'stationary' proper time oscillator is a Schwarzschild field. To test the idea, we propose to study a particle's arrival time. In motion, the proper time oscillation translates to oscillations in both time and space. These oscillations lead to uncertainty in the arrival time of a particle. Specifically, we study the effects on a neutrino's arrival time. The possible detection of a neutrino's arrival time uncertainty can test the theory proposed in this paper.