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Communications in Theoretical Physics

A No-Go Theorem for Nonlinear Canonical Quantization

Miroslav Engliš

Published Mar 15, 2002• 3 citations
Publisher

Abstract

We want to point out the following strengthening of the classical theorem of Groenewold and van Hove: There exists no mapping Op from polynomial observables on the phase space into linear operators on which would map Poisson brackets into commutators, the position and momentum observables p and q into the usual (Schrödinger) position and momentum operators, and would obey the von Neumann rule for and . The point is that neither linearity, nor continuity etc. of Op are assumed.

DOI
10.1088/0253-6102/37/3/287

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We want to point out the following strengthening of the classical theorem of Groenewold and van Hove: There exists no mapping Op from polynomial observables on the phase space into linear operators on which would map Poisson brackets into commutators, the position and momentum observables p and q into the usual (Schrödinger) position and momentum operators, and would obey the von Neumann rule for and . The point is that neither linearity, nor continuity etc. of Op are assumed.
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.
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.
Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of ``constants of motion" is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.
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.
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S(K)(q,p,t)S^{(K)}(q,p,t)S(K)(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing ∂∂p\frac{\partial }{\partial p}∂p∂​ by 0 and p by ℏı∂∂q\frac{\hbar }{\imath } \frac{\partial }{\partial q}ℏ​∂q∂​ , thus providing an explanation for the common quantization rules.