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arXiv:1011.3709

Conceptual Problems Related to Time and Mass in Quantum Theory

Daniel M. Greenberger

Published Nov 16, 2010• 12 citations
arXivHTMLPDF

Abstract

The concept of proper time cannot be just taken over from classical theory and applied to quantum theory. There are a number of serious ambiguities related to it. Similarly, the concept of mass has some inconsistencies attached to it. We argue that both of these concepts only make sense if they are included in the theory as dynamical variables, and subsequently as conjugate operators. We discuss the necessary uncertainty principle between them.

DOI
10.48550/arxiv.1011.3709

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•Jan 1, 2003•1 citation
The concept of proper time cannot be just taken over from classical theory and applied to quantum theory. There are a number of serious ambiguities related to it. Similarly, the concept of mass has some inconsistencies attached to it. We argue that both of these concepts only make sense if they are included in the theory as dynamical variables, and subsequently as conjugate operators. We discuss the necessary uncertainty principle between them.
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.
We review Bohr’s reasoning in the Bohr–Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.
Abstract We present a ‘proof’ that the standard treatment of the mass and proper time as parameters, rather than dynamical variables, is actually inconsistent, and we show how they can be introduced as a generalization of the standard Hamiltonian Dynamics. We discuss various topics such as the equivalence principle and particle decay and the uncertainty principle from this point of view. I am honored to be able to give this talk as part of the 60th birthday festival for professor. Wolfgang Schleich, a long-term colleague, a very good friend, and a beloved and highly respected member of the quantum community, and I hope our collaboration can continue for many years to come. I extend my best wishes to him and his wonderful wife Kathy on this occasion.
Abstract It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the "twin-paradox" effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis-tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics.
Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.
One of the most profound and difficult problems of physics in the last century is the construction of a simple theory which unites the ideas of quantum mechanics and relativity. The concepts which are used in quantum mechanics related to the measurements of positions, momenta, energy and time, are the same concepts in terms of which the special relativistic postulates are formulated. Hence, it is at least possible to formulate the requirement of special relativistic invariance for quantum theories and to ascertain whether these requirements are satisfied. The fact that the answer is nearer to no than to yes, that quantum mechanics has not yet been fully adjusted to the postulates of the special theory, is perhaps irritating. In the general theory of relativity, the situation is different: the space-time coordinates are only auxiliary quantities to which arbitrary values for every event could be given. Hence, the measurement of position, that is, of the space coordin ates, is certainly not a significant measurement if the postulates of the general theory are adopted: any arbitrary value can be given to the coordinates. The same holds for momenta. The purpose of this brief paper is to make in principle some personal and particular remarks on the fundamental concepts of time and mass. Some comments about the consequences on the relativistic dynamics of quantum systems are also formulated.