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arXiv:1204.1858

Classical/Quantum=Commutative/Noncommutative?

Vladimir V. Kisil

Published Apr 9, 2012• 2 citations
arXivHTMLPDF

Abstract

In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac's proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial rôle in quantum theory shall be attributed to the usage of complex numbers. The paper provides English and Russian versions.

DOI
10.48550/arxiv.1204.1858

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In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac's proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial rôle in quantum theory shall be attributed to the usage of complex numbers. The paper provides English and Russian versions.
In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Diracs proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit ϵ such that ϵ2 = 0. Thus, the crucial role in quantum theory shall be attributed to the usage of complex numbers.
Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That was Dirac's appreciation of the mathematical properties of the physical quantities as presented in Heisenberg's new quantum theory. After all, the familiar real, or complex, number system only came into existence through the history of mathematics. Values of physical quantities having a commutative product is an assumption that is not compatible with quantum physics. The revolutionary idea of Heisenberg and Dirac was pulled back to a much more conservative setting by the work of Schrödinger, followed by Born and Bohr. What Bohr missed is that the real number values we obtained from our measurements are only a consequence of the design of the kind of experiments and our using real numbers to calibrate the output scales of our apparatus. It is only our modeling of the information obtained about the physical quantities rather than what Nature dictates. We have proposed an explicit notion of definite noncommutative values of observables that gives a picture of quantum mechanics as realistic as the classical theory. In this article, we illustrate how matrices can be taken as noncommutative (q-)numbers serving as the values of physical quantities, each to be seen as a piece of quantum information. Our main task is to clarify the subtle issues involved in setting up a conventional scheme assigning matrices as values to the physical quantities.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue—the square of Dirac delta distributions.
Abstract. Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated with-out reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an un-derlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. PACS number: 03.65.Ta 1 Why Quantum Mechanics without Classical Spacetime? Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a clas-
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.