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arXiv:1604.07754 · Journal of Physics A Mathematical and Theoretical

Bypassing the Groenewold–van Hove obstruction on ${{\mathbb{R}}}^{2n}$: a new argument in favor of Born–Jordan quantization

Maurice A de Gosson

Published Sep 2, 2016• 2 citations
arXivHTMLPDF

Abstract

There are known obstructions to a full quantization of in the spirit of Dirac's approach, the most known being the Groenewold and van Hove no-go result. We show, following a suggestion of S K Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type . This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.

DOI
10.1088/1751-8113/49/39/39lt01

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Bypassing the Groenewold–van Hove obstruction on R2n{{\mathbb{R}}}^{2n}R2n: a new argument in favor of Born–Jordan quantization
There are known obstructions to a full quantization of in the spirit of Dirac's approach, the most known being the Groenewold and van Hove no-go result. We show, following a suggestion of S K Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type . This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born-Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born--Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.
We want to point out thc following strcngthcning of the classical theorem of Grocnewold and van Hove:There exists no ,napping Op from polynomial observables f (p, q) on the phase space R2n into linear operators on L2(Rn)which would map Poisson brackets into commutators, the position and momentum obscrvablcs p and q into the usual(Schrodinger) position and momentum operators, and would obey the yon Neumann rifle 0p(cfk) = c 0p(f)k for k = 1,2, 3and c R. The point is that neither linearity, nor continuity etc. of Op are assumed.
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.
Motivated by models of mixed quantum-classical hybrid theories, we consider the representation of classical observables as Hilbert-space operators based on van Hove's unitary representation of contact transformations. This leads to a reformulation of classical mechanics where the phase space density satisfies the Liouville equation and, more importantly, classical observables are represented by van Hove operators that satisfy a commutation algebra isomorphic to the classical Poisson algebra of functions in phase space. Thus, this Hilbert space approach captures all the essential features of the phase space formulation of classical mechanics in that it reproduces its algebraic structure -- something that is not achieved by other approaches such as the Koopman-von Neumann theory. As an important feature, the definition of appropriate physical states in this formalism imposes certain natural requirements on the phase of the classical wave function, which ensure correct expectation values for the observables. Furthermore, we discuss how to extend the approach to the hybrid van Hove theory of mixed classical-quantum systems and illustrate its application with the example of interacting classical and quantum oscillators. Finally, we focus on the question of how hybrid van Hove theory evades no-go theorems on mixed classical-quantum systems and point to the various assumptions of the no-go theorems that do not apply to our approach. This demonstrates that these no-go theorems are not universally applicable, contrary to their original claims.
In this review, we would like to highlight the three known no-go theorems in quantum physics in relation to the processof quantization that maps classical observables to quantum ones. The quantization approach considered is a mixture ofIsham’s group-theoretic quantization and geometric quantization with special emphasis on underlying compact phasespace geometry of spheres. The first is Groenewold-van Hove theorem that states the obstruction of quantizing the fullalgebra of observables and in the sphere case, only limited to the spin observables plus the constant functions. Theother two are theorems of Bell and Kochen-Specker stating that the only hidden variable theories allowed by quantumphysics are nonlocal and contextual ones. We give simple examples of these no-go theorems and indicate someinteresting problems arising from them for the field of quantization.