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arXiv:hep-ph/9606330

Time and the Evolution of States in Relativistic Classical and Quantum Mechanics

Lawrence Paul Horwitz

Published Jun 14, 1996• 2 citations
arXivHTMLPDF

Abstract

A consistent classical and quantum relativistic mechanics can be constructed if Einstein's covariant time is considered as a dynamical variable. The evolution of a system is then parametrized by a universal invariant identified with Newton's time. This theory, originating in the work of Stueckelberg in 1941, contains many questions of interpretation, reaching deeply into the notions of time, localizability, and causality. Some of the basic ideas are discussed here, and as an example, the solution of the two body problem with invariant action-at-a-distance potential is given. A proper generalization of the Maxwell theory of electromagnetic interaction, implied by the Stueckelberg-Schrödinger dynamical evolution equation and its physical implications are also discussed. It is also shown that a similar construction occurs, applying similar ideas, in the theory of gravitation.

DOI
10.48550/arxiv.hep-ph/9606330

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A consistent classical and quantum relativistic mechanics can be constructed if Einstein's covariant time is considered as a dynamical variable. The evolution of a system is then parametrized by a universal invariant identified with Newton's time. This theory, originating in the work of Stueckelberg in 1941, contains many questions of interpretation, reaching deeply into the notions of time, localizability, and causality. Some of the basic ideas are discussed here, and as an example, the solution of the two body problem with invariant action-at-a-distance potential is given. A proper generalization of the Maxwell theory of electromagnetic interaction, implied by the Stueckelberg-Schrödinger dynamical evolution equation and its physical implications are also discussed. It is also shown that a similar construction occurs, applying similar ideas, in the theory of gravitation.
In our investigation of the time evolution of energy states Einstein’s interpretation from more than a century ago is found to have greater internal consistency than that of non-relativistic quantum mechanics. We choose to align more closely with his methods by abandoning the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a relativistically correct Lagrangian method. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, matrix mechanics, the function of the simplest quantum mechanical system an electron cyclotron, chaos theory, and evolutionary biology.
Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schrödinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given.
In this article we study the nature of time in Mechanics. The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation , implies the existence of an absolute time-duration in the sense of Newton. There is a second notion of time for conservative systems which makes the Hamiltonian action evolves at a constant rate. In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory. Then, we propose two different ways to reach the time dependent Schrödinger equation. One way consists of considering a “time constraint” on a free system. The other way is based on the point of view of Hertz, by considering the system as a projection of a free system. In the later manner, the “time” appearing in the Schrödinger equation is a linear combination of the time-duration with the “time” quotient of the action by the energy on each solution of the Hamilton-Jacobi equation. Both of them are based on a rule of quantization that we explain in Section 4.
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 a physical theory we typically describe a system in terms of its state . We specify the relevant physical quantities or variables and then use dynamical laws to find the time evolution of these variables to predict their values in the future. For example, in classical particle mechanics, the state of a system is specified by the positions and velocities (or, equivalently for us, the momenta) of all of the particles in the system (that is, r ( t ) and v ( t ) for each particle). Newton's second law, F = ma , then determines the time evolution of the variables, or of the state, of the system. For electrodynamics, the state variables are the electric and magnetic fields, E and B , and Maxwell's equations govern the time evolution of these. In classical physics (including relativity here), the state variables that are the central entities of the dynamical equations (typically the positions and momenta of the particles) are also the directly observable physical quantities. That is, the state itself is specified in terms of the observables of the theory. The essential features of the classical world view (as modified by relativity) are that the present state of a system in principle determines its future state (as in the flight of a baseball under the influence of gravity) and that the agency responsible for this event-by-event causal structure must propagate from the cause to the effect at some rate no greater than the speed of light.
This work discloses some inconsistencies of quantum mechanics (QM) in two different contexts. On the one hand the equivalence principle in his two formulations is the starting point of all gravitational theories; yet there is no way to make it consistent with the non-relativistic formulation of QM. Moreover, the evaluation of the Schrödinger equation for a particle in an accelerated reference frame sheds light on a trajectory-dependent phase term subject to proper time. On the other hand, the Bargmann theorem unmistakably proves that the Schrödinger equation does not admit any superposition of different masses if the symmetry chosen for the system is the Galilean one. However, this is incompatible with relativity, since mass and energy are equivalent and one can certainly superimpose different energies. Both inconsistencies support the hypothesis that mass and proper time could be treated as variables. In the third chapter we will describe a theory in which the Galilean symmetry is extended by assuming particle mass and proper time as canonically conjugated classical variables, or conjugated observables in the quantum theory. The first three chapters discuss subjects illustrated in several articles of the professor Daniel M. Greenberger, which are listed in the bibliography. Whereas in the last chapter the results of a recent theory on QW, say a discrete model for quantum particle evolution, will be examined. In particular, the striking result is that, in the most natural physical interpretation of QW dynamics, it is not possible to have a constant and invariant mass in any manner. The long term aim of this work is indeed to study the reasons and consequences of a theory of particles with variable mass in the context of wider studies still in progress on QW.