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.
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.
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.
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.
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.
W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥ μ (with Ĥ 0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂ μ ψ = Ĥ μ ψ and Heisenberg equation d /d x μ = ∂ μ + i[Ĥ μ , ] are obtained, from which we have: (1) t is to Ĥ 0 as x j is to Ĥ j ( j = 1, 2, 3); likewise, t is to i∂ 0 as x j is to i∂ j ; (2) the proposed time operator is canonically conjugate to i∂ 0 rather than to Ĥ 0 , therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δ x μ Δ p μ ≥ 1/2 and the Mandelstam–Tamm treatment Δ x μ Δ H μ ≥ 1/2, have been formulated.
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.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.