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arXiv:1211.7351 · Modern Physics Letters A

Revisiting canonical quantization

John R. Klauder

Published Jun 30, 2014• 1 citation
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Abstract

Conventional canonical quantization procedures directly link various c-number and q-number quantities. Here, we advocate a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory with ℏ > 0, in conformity with the real world wherein nature has chosen ℏ > 0 rather than ℏ = 0. While keeping the good results of conventional procedures, some examples are presented for which the new procedures offer better results than the conventional ones.

DOI
10.1142/s0217732314300201

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Conventional canonical quantization procedures directly link various c-number and q-number quantities. Here, we advocate a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory with ℏ > 0, in conformity with the real world wherein nature has chosen ℏ > 0 rather than ℏ = 0. While keeping the good results of conventional procedures, some examples are presented for which the new procedures offer better results than the conventional ones.
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.
Abstract While canonical quantization solves many problems there are some problems where it fails. A close examination of the classical/quantum connection leads to a new connection that permits quantum and classical realms to coexist, as is the case in the real world. For those problems for which conventional quantization works well, the new procedures yield identical results. However, we offer an example that fails when quantized conventionally but succeeds when quantized with the new procedures.
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.
Following Dirac, the rules of canonical quantization include classical and quantum contact transformations of classical and quantum phase space variables. While arbitrary classical canonical coordinate transformations exist that is not the case for some analogous quantum canonical coordinate transformations. This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a ccc-number to a qqq-number. A different relationship of ccc-numbers and qqq-numbers in the procedures of Enhanced Quantization shows the compatibility of all quantum operators with all classical canonical coordinate transformations.
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.
Canonical quantization relies on Cartesian, canonical, phase-space coordinates to promote to Hermitian operators, which also become the principal ingredients in the quantum Hamiltonian. While generally appropriate, this procedure can also fail, e.g., for covariant, quartic, scalar fields in five-and-more spacetime dimensions (and possibly four spacetime dimensions as well), which become trivial; such failures are normally blamed on the `problem' rather than on the `quantization procedure'. In Enhanced Quantization the association of ccc-numbers to qqq-numbers is chosen very differently such that: (i) there is no need to seek classical, Cartesian, phase-space coordinates; (ii) {\it every} classical, contact transformation is applicable and {\it no} change of the quantum operators arises; (iii) a new understanding of the importance of `Cartesian coordinates' is established; and (iv) although discussed elsewhere in detail, the procedures of enhanced quantization offer fully acceptable solutions yielding non-trivial results for quartic scalar fields in four-and-more spacetime dimensions. In early sections, this paper offers a wide-audience approach to the basic principles of Enhanced Quantization using simple examples; later, several significant examples are cited for a deeper understanding. An historical note concludes the paper.