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The quantum measurement of time

Scott Roger Shepard

Published May 1, 1994• 0 citations
OpenAlex

Abstract

Traditionally, in non-relativistic Quantum Mechanics, time is considered to be a parameter, rather than an observable quantity like space. In relativistic Quantum Field Theory, space and time are treated equally by reducing space to also be a parameter. Herein, after a brief review of other measurements, we describe a third possibility, which is to treat time as a directly observable quantity.

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Traditionally, in non-relativistic Quantum Mechanics, time is considered to be a parameter, rather than an observable quantity like space. In relativistic Quantum Field Theory, space and time are treated equally by reducing space to also be a parameter. Herein, after a brief review of other measurements, we describe a third possibility, which is to treat time as a directly observable quantity.
Time – what it is and its nature – has from the beginning held mysteries for layman, philosopher, and physicist alike. Periodic phenomena from motions of heavenly bodies and pendulums to quantal atomic clocks provide the means to measure time intervals, and Newton’s absolute time is a background flow against which classical and non-relativistic quantum physics is played out. While internally consistent, these two fields of physics are incompatible with the requirement of treating space and time on an equal footing. Quantum field theories provide the answer, as both treated as parameters, physics residing instead in the states and operators. But even in classical physics and quantum mechanics, time-independent descriptions provide an alternative to an explicitly time-dependent one. These, including a general method of approach called invariant imbedding, are discussed as alternative representations of an underlying reality that is the concern of physics.
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.
Quantum measurement is a fundamental concept in the field of quantum mechanics. The action of quantum measurement, leading the superposition state of the measured quantum system into a definite output state, not only reconciles contradictions between quantum and classical mechanics but also facilitates quantum state manipulations, including reading and resetting. Despite its significance, four fundamental issues -- randomness, instantaneousness, irreversibility, and preferred-basis -- continue to pose significant challenges to the broader application of quantum measurement and our overall understanding of quantum mechanics. In this work, we employ a new space-time approach to relativistic quantum mechanics to address these issues systematically. Our approach provides a comprehensive elucidation of the intricate connections between quantum measurement and quantum unitary evolution, as well as an in-depth analysis for the interdependence of non-local correlations and relativistic theories. We thereby reveal a more fundamental dynamical theory, beyond the traditional time-evolution equation in quantum mechanics, where the axioms of quantum measurement naturally emerge as a corollary. These findings contribute to the advancement of related fields, and our work holds potential implications for future research and applications in the realm of quantum mechanics.
In quantum mechanics time usually appears as classical parameter which means that it is treated as being essentially different from spatial coordinates that are represented by operators. On the other hand, relativity theory demands to treat space and time on an equal footing and there are many approaches to the problem how to introduce time operator. In order to support these approaches we propose the experiment that will clearly expose the equivalence of space and time regarding the relation towards quantum interference. The proposed experiment can show that it is possible to have superposition of being created in two different moments in time, and by measuring standard interference pattern, it will be shown that there is interference in time. It is similar to the case of verification of the possibility of single quantum system to be in the state that is superposition of two different positions in space. If so, this would mean that there should be exactly the same formal approach to the space coordinates and time in quantum mechanics since they are all quantum in their nature.
Following homogeneous canonical formalism where the time is treated on an equal footing with the space coordinates, we propose an alternative formulation of quantum mechanics in which the time is considered as a dynamical variable. The basic equation corresponding to and being equivalent to the ordinary Schrödinger equation is obtained when we consider the space coordinates x as a parameter corresponding to the time t in the ordinary formulation. The formal solution of this equation is given and examined. By making use of this solution the scattering theory is formulated in our formalism. It is shown that observables corresponding to the time and space coordinates can be defined. A short comment on the energy-time uncertainty relation is also given.
The kinematic time operator can be naturally defined in relativistic and nonrelativistic quantum mechanics (QM) by treating time on an equal footing with space. The space–time position operator acts in the Hilbert space of functions of space and time. Dynamics, however, makes eigenstates of the time operator unphysical. This poses a problem for the standard interpretation of QM and reinforces the role of alternative interpretations such as the Bohmian one. The Bohmian interpretation, despite of being nonlocal in accordance with the Bell theorem, is shown to be relativistic covariant.