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arXiv:2010.03984 · Journal of High Energy Physics Gravitation and Cosmology

A Unified Combination of Classical and Quantum Systems

John R. Klauder

Published Jan 1, 2021• 4 citations
arXivHTMLPDF

Abstract

Any particular classical system and its quantum version are normally viewed as separate formulations that are strictly distinct. Our goal is to overcome the two separate languages and create a smooth and common procedure that provides a clear and continuous passage between the conventional distinction of either a strictly classical or a strictly quantized state. While path integration, among other procedures, provides an alternative route to connect classical and quantum expressions, it normally involves complicated, model-dependent, integrations. Our alternative procedures involve only model-independent procedures, and use more natural and straightforward integrations that are universal in kind. To introduce the basic procedures our presentation begins with familiar methods that are limited to basic, conventional, canonical quantum mechanical examples. In the final sections we illustrate how alternative quantization procedures, e.g., spin and affine quantizations, can also have smooth paths between classical and quantum stories, and with a few brief remarks, can also lead to similar stories for non-renormalizable covariant scalar fields as well as quantum gravity.

DOI
10.4236/jhepgc.2021.71012

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Any particular classical system and its quantum version are normally viewed as separate formulations that are strictly distinct. Our goal is to overcome the two separate languages and create a smooth and common procedure that provides a clear and continuous passage between the conventional distinction of either a strictly classical or a strictly quantized state. While path integration, among other procedures, provides an alternative route to connect classical and quantum expressions, it normally involves complicated, model-dependent, integrations. Our alternative procedures involve only model-independent procedures, and use more natural and straightforward integrations that are universal in kind. To introduce the basic procedures our presentation begins with familiar methods that are limited to basic, conventional, canonical quantum mechanical examples. In the final sections we illustrate how alternative quantization procedures, e.g., spin and affine quantizations, can also have smooth paths between classical and quantum stories, and with a few brief remarks, can also lead to similar stories for non-renormalizable covariant scalar fields as well as quantum gravity.
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.
A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The new tools are applied to several examples including: (1) A quantum formulation that is invariant under arbitrary classical canonical transformations of coordinates; (2) A toy model that for all positive energy solutions has singularities which are removed at the classical level when the correct quantum corrections are applied; (3) A fairly simple model field theory with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (4) A model of scalar field theories with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (5) A viable formulation of the kinematics of quantum gravity that respects the strict positivity of the spatial metric in both its classical and quantum versions; and (6) A proposal for a non-trivial quantization of [Formula: see text] that is ripe for study by Monte Carlo computational methods. All of these examples use fairly general arguments that can be understood by a broad audience.
Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck’s constant ℏ > 0, meaning that the classical and quantum world views must actually coexist . Traditionally, canonical quantization procedures postulate a direct linking of various c -number and q -number quantities that lie in disjoint realms, along with the quite distinct interpretations given to each realm. In this paper we propose a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory letting them coexist as required. This proposal also shines light on alternative linking assignments of classical and quantum quantities that offer different perspectives on the very meaning of quantization. In this paper we focus on elaborating the general principles, while elsewhere we have published several examples of what this alternative viewpoint can achieve; these examples include removal of singularities in classical solutions to certain models, and an alternative quantization of several field theory models that are trivial when quantized by traditional methods but become well defined and nontrivial when viewed from the new viewpoint.
This text describes novel treatments of quantum problems using enhanced quantization procedures. When treated conventionally, certain systems yield trivial and unacceptable results. This book describes enhanced procedures, generally involving extended correspondence rules for the association of a classical and a quantum theory, which, when applied to such systems, yield nontrivial and acceptable results. The text begins with a review of classical mechanics, Hilbert space, quantum mechanics, and scalar quantum field theory. Next, analytical skills are further developed, a special class of models is studied, and a discussion of continuous and discontinuous perturbations is presented. Later chapters cover two further classes of models both of which entail discontinuous perturbations. The final chapter offers a brief summary, concluding with a conjecture regarding interacting covariant scalar quantum field theories. Symmetry is repeatedly used as a tool to help develop solutions for simple and complex problems alike. Challenging exercises and detailed references are included.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates xnx_nxn​. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.