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arXiv:2303.11112

A Completion of Quantum Mechanics

Jürg Martin Fröhlich, Zhou Gang, Alessandro Pizzo

Published Mar 20, 2023• 1 citation
arXivHTMLPDF

Abstract

A proposal of how to complete non-relativistic quantum mechanics to a physically meaningful, mathematically precise and logically coherent theory is reviewed. Our proposal leads to a general, non-linear stochastic law for the time-evolution of states of individual physical systems. An application of the general formalism to the quantum theory of fluorescence of an atom coupled to the radiation field is sketched. Some remarks on relativistic quantum theory conclude our review.

DOI
10.48550/arxiv.2303.11112

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A proposal of how to complete non-relativistic quantum mechanics to a physically meaningful, mathematically precise and logically coherent theory is reviewed. Our proposal leads to a general, non-linear stochastic law for the time-evolution of states of individual physical systems. An application of the general formalism to the quantum theory of fluorescence of an atom coupled to the radiation field is sketched. Some remarks on relativistic quantum theory conclude our review.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
This note starts with a recapitulation of what people call the ``Measurement Problem'' of Quantum Mechanics (QM). The dissipative nature of the quantum-mechanical time-evolution of averages of states over large ensembles of identical isolated systems consisting of matter interacting with the radiation field is discussed and shown to correspond to a stochastic time-evolution of states of individual systems. The importance of dissipation for the successful completion of measurements is highlighted. To conclude, a solution of the ``Measurement Problem'' is sketched in an idealized model of a double-slit experiment.
To begin with, some of the conundrums concerning Quantum Mechanics and its interpretation(s) are recalled. Subsequently, a sketch of the "ETH-Approach to Quantum Mechanics" is presented. This approach yields a logically coherent quantum theory of "events" featured by physical systems and of direct or projective measurements of physical quantities, without the need to invoke "observers". It enables one to determine the stochastic time evolution of states of physical systems. We also briefly comment on the quantum theory of indirect or weak measurements, which is much easier to understand and more highly developed than the theory of direct (projective) measurements. A relativistic form of the ETH-Approach will be presented in a separate paper.
It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated (open) systems featuring events. A precise general law for the time evolution of states replacing the Schrödinger equation is formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous "measurement problem." Our general results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field in a limit where the speed of light tends to infinity. The discussion of these models is the main subject of our paper.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.