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Acta Physica Sinica

Time operator in quantum mechanics

Zhiyong Wang, Xiong Cai-Dong, 电子科技大学物理电子学院，成都 610054

Published Jan 1, 2007• 1 citation
Publisher

Abstract

There have been great theoretical and practical interests in investigating time problem in quantum mechanics. Because time enters quantum mechanics as a parameter rather than a dynamical operator, people have to consider how to construct a time operator and calculate average time whenever they are faced with a time problem related to a physical process. In this paper we present a general investigation on time operator and average time.

DOI
10.7498/aps.56.3070

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There have been great theoretical and practical interests in investigating time problem in quantum mechanics. Because time enters quantum mechanics as a parameter rather than a dynamical operator, people have to consider how to construct a time operator and calculate average time whenever they are faced with a time problem related to a physical process. In this paper we present a general investigation on time operator and average time.
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.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
The problem of time in the quantization of gravity arises from the fact that time in Schrödinger’s equation is a parameter. This sets time apart from the spatial coordinates, represented by operators in quantum mechanics (QM). Thus “time” in QM and “time” in general relativity (GR) are seen as mutually incompatible notions. The introduction of a dynamical time operator in relativistic quantum mechanics (RQM), that follows from the canonical quantization of special relativity and that in the Heisenberg picture is also a function of the parameter [Formula: see text] (identified as the laboratory time), prompts to examine whether it can help to solve the disfunction referred to above. In particular, its application to the conditional interpretation of time in the canonical quantization approach to quantum gravity is developed.
Time is routinely treated as a universal parameter applicable across all physical descriptions. This assumption is neither required by existing theory nor supported by physical practice, and it underlies the canonical problem of time in quantum gravity. As a follow-up to Event-Restricted Temporality and the Problem of Time, Event-Restricted Temporality (ERT) is developed into an event-licensed physics: temporal parametrization is a licensed operation whose domain is restricted to descriptions containing irreversible event actualization. Eight independent domain gates are introduced—quantitative, operational, control-theoretic, local, resource-theoretic, informational, thermodynamic, and phase-structural—that jointly constrain the physically admissible ways in which time may enter description. These domain gates convert temporal applicability from a tacit background assumption into a sharply posed physical question. The resulting framework preserves standard dynamics where events occur, explains the absence of time in purely structural and retentive regimes, and dissolves a broad class of temporal paradoxes without modifying quantum mechanics or general relativity.