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Symmetry

A Quantum Time Coordinate

Armando Martínez-Pérez, Gabino Torres-Vega

Published Feb 11, 2021• 2 citations
Publisher

Abstract

We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.

DOI
10.3390/sym13020306

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We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.
We use a discrete derivative to introduce a time operator for non-relativistic quantum systems with point spectrum. The symmetry requirement on the time operator leads to well-defined time values related to the dynamics of discrete quantum systems. As an illustration, we find travel times between hits with the walls for the quantum particle in a box model. These times suggest a classical analog of time eigenstates. We also briefly consider the Woods–Saxon potential and propose classical analogs for it.
Abstract It is first shown that the Dirac’s equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4 × 4 relativistic time operator for spin-1/2 particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but non-zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli’s objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found .
It is first shown that the Dirac's equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4×44\times 44×4 relativistic time operator for spin-12\frac{1}{2}21​ particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but non-zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli's objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found.
In this paper we wish to explore the possibility of the existence of a quantity that can be regarded as the complement of the Hamiltonian for a quantum system with discrete energy levels, recognizing that if this quantity is represented by an operator the appropriate commutator must be a generalized form of Eq. 2. Although this quantity will have dimensions of time, it will not be appropriate to refer to such an operator as a time operator. As an operator it would represent an observable of the quantum system and not time in the abstract or coordinate sense or as a reading on an external clock. Its eigenstates would represent a state of the quantum system and some measurement on the system, at least in principle, should tell us about the quantity involved. Although this quantity is not time, we would hope that for a particular ideal system its expectation value may change linearly with time, for example, so that a measurement of the quantity would also give a measure of time. In this case, we would be using the system as a clock. Before seeking an operator for the complement of the Hamiltonian we first represent this quantity by a nonorthogonal probability-operator measure, which is a more general concept 6.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.
We explore the properties of quantum states and operators that are conjugate to the Hamiltonian eigenstates and operator when the Hamiltonian spectrum is continuous, i.e., we find time-like operators T^ such that [T^,H^]=iℏ. This is a property expected for a time operator. We explicitly unfold the momentum sign degeneracy of energy states. We consider the free-particle case, and we find, among other things, that the time states are also the solution of the quantized version of the classical motion of the particle.