The understanding of time in physics has undergone profound changes from classical mechanics to relativity and quantum mechanics. In Newtonian mechanics, time is regarded as an absolute and uniform passage, independent of external phenomena, and is a fundamental parameter of the equations of motion. Relativity emphasises the relativity of time and its close connection with the motion of matter, proposes an operational definition of the simultaneity of distant events, and develops Minkowski’s view of spacetime and Riemannian curved spacetime. In quantum mechanics, time is often regarded as a parameter external to the microscopic system rather than an observable quantity. Although some scholars have attempted to introduce time operators, they have not yet gained widespread recognition. Overall, the understanding of time in physics continues to deepen with the development of theory. Still, the unity of classical and quantum mechanics in time remains an unsolved problem.
The understanding of time in physics has undergone profound changes from classical mechanics to relativity and quantum mechanics. In Newtonian mechanics, time is regarded as an absolute and uniform passage, independent of external phenomena, and is a fundamental parameter of the equations of motion. Relativity emphasises the relativity of time and its close connection with the motion of matter, proposes an operational definition of the simultaneity of distant events, and develops Minkowski’s view of spacetime and Riemannian curved spacetime. In quantum mechanics, time is often regarded as a parameter external to the microscopic system rather than an observable quantity. Although some scholars have attempted to introduce time operators, they have not yet gained widespread recognition. Overall, the understanding of time in physics continues to deepen with the development of theory. Still, the unity of classical and quantum mechanics in time remains an unsolved problem.
In classical mechanics and standard quantum theory, time is treated as an absolute and external parameter, a universal background against which physical events unfold. The foundational theories of modern physics challenge this view. In special relativity, the passage of time depends on an observer's velocity; in general relativity, it depends on the local gravitational potential. This dynamic, observer-dependent nature of time raises fundamental questions about its status and necessitates a unified framework for performing operational calculations that are consistent across these domains.
In the early to mid-twentieth century, physicists found that they could not describe the observed behavior of the electrons orbiting the nuclei of atoms by use of Newtonian physics. A new mathematical description of that behavior was developed. Called quantum mechanics, it has been tremendously successful in accounting for widely diverse phenomena at both the qualitative and quantitative level and is universally regarded as generally applicable to all matter. However quantum mechanics has some peculiar counter intuitive features and some aspects of its interpretation remain in dispute. This chapter focuses on how quantum mechanics changes the Newtonian picture of the past, present and future and of time itself. The mathematical apparatus governing the time dependence of the wave functions which describe states in quantum mechanics is completely time reversible, in a way closely analogous to the time reversibility of Newtonian mechanics, though the state descriptions are different.
In 1905, Einstein established relativity under the condition of denying the absolute space-time valid in Newtonian mechanics. In those days, the quantum theory was also established by accepting De Broglie’s hypothesis of matter wave, regardless of the causality of Newtonian mechanics. It has thus caused some fundamental problems unsolved in the quantum theory, as the causality of quantum mechanics for Newtonian mechanics is unknown. In that situation, it was recently revealed that the quantum theory is reasonably established in accordance with the causality for Newtonian mechanics, and also that the hypothesis of De Broglie is theoretically valid. The fundamental problems unsolved for a long time were also reasonably elucidated then. Further, we then found that a micro particle has an imaginary mass in a local space, or its behavior depends on an imaginary time. In the present work, we thus review fundamental problems in quantum mechanics from a viewpoint of the theoretical structure of physics in accordance with the causality for Newtonian mechanics.
In Newtonian physics, the equation of motion is invariant when the direction of time (t→−t) is flipped. However, in quantum physics, flipping the direction of time changes the sign of the Schrödinger equation. An anti-unitary operator is needed to restore time reversal in quantum physics, but this is at the cost of not having a consistent definition of time reversal applicable to all fundamental theories. On the other hand, a quantum system composed of a pair of entangled particles behaves in such a manner that when the state of one particle is measured, the second particle ‘simultaneously’ acquires a determinate state. A notion of absolute simultaneity seems to be inferred by quantum mechanics, even though it is forbidden by the postulates of relativity. We aim to point out that the above two problems can be overcome if the wavefunction is defined with respect to proper time, which in fact is the real physical time instead of ordinary time.
Although time is one of our most intuitive physical concepts, its understanding at the fundamental level is still an open question in physics. For instance, time in quantum mechanics and general relativity are two distinct and incompatible entities. While relativity deals with events (points in spacetime), with time being observer dependent and dynamical, quantum mechanics describes physical systems by treating time as an independent parameter. To resolve this conflict, in this work, we extend the classical concept of an event to the quantum domain by defining an event as a transfer of information between physical systems. Then, by describing the universe from the perspective of a certain observer, we introduce quantum states of events with space-time-symmetric wave functions that predict the joint probability distribution of a measurement (observation) at $(t,\stackrel{P\vec}{x})$. Under these circumstances, without assuming collapse, we propose that a well-defined instant of time, like any other observable, arises from a single event, thus being an observer-dependent property. In this manner, we obtain a stationary quantum state written as a sum of a sequence of normalized states of events with increasing memories. As a consequence, a counterfactual asymmetry along this sequence of events gives rise to the flow of time as being successive ``snapshots'' from the observer's perspective. In this proposal, which contrasts strikingly to the view in which time is an illusion, it is the many distinguishable states in which the observer stores information that makes the existence of time possible.
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.