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Proper Time Oscillation and a Particle's Arrival Time

Hou Y. Yau

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.13140/rg.2.2.23390.00324

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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.
We demonstrate that a real scalar quantum field has properties that can be interpreted as a field with vibrations of matters in time and space. Based on this interpretation, the formulations have a more symmetrical treatment of time and space in a matter field. A self-adjoint internal time operator can be defined without contradiction with Pauli's theorem. The particles observed are oscillators in proper time. In motion, the proper time oscillation translates to the oscillations of a particle in both time and space. A particle is oscillating back and forth along its trajectory. Therefore, two particles with the same initial average velocity can reach a target at different times depending on the phases of their oscillations. This leads to an uncertainty in the arrival time of a particle. In particular, we study the effects of these oscillations on the neutrinos' arrival time. The arrival time uncertainty measured can also be used to determine the mass of a neutrino.
We investigate surprising consequences of the hypothesis that gravity and proper time obey the quantum superposition principle. We show that, for a massive body prepared in a superposition of two positions and post-selected in a suitable final state, the elapsed proper time in another region of space can assume an anomalous effective value lying outside the range allowed by classical general relativity. Remarkably, the effective elapsed proper time can even become negative, corresponding to an effective evolution backward in time. We also propose an experiment to observe an anomalous effective proper time associated with the internal degrees of freedom of a quantum particle in an interferometer under Earth's gravitational field. The proposed experiment is, in principle, feasible with current technology, and its realization could provide the first experimental evidence of an interplay between quantum mechanics and general relativity beyond the effects of the Newtonian potential in flat spacetime.
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.
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.
Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.