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Malaysian Journal of Fundamental and Applied Sciences

No-Go theorems and quantization

Hishamuddin B Zainuddin, Toh Sing Poh, Nurisya Mohd Shah, Mazlinda Zainy, Zuriati Ahmad Zukarnain, Jumiah Hassan, Zainul Abidin Hassan

Published Jun 16, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.11113/mjfas.v3n1.21

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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.
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.
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 Bell 1964 theorem states that nonlocality is a necessary feature of hidden variable theories that reproduce the statistical predictions of quantum mechanics. In view of the no-go theorems for non-contextual hidden variable theories already existing up to 1964, and due to Gleason and Bell, one is forced to acknowledge the contextual character of the hidden variable theory which the Bell 1964 theorem refers to. Both the mathematical and the physical justifications of this contextualism are reconsidered. Consequently, the role of contextualism in recent no-hidden-variables proofs and the import of these proofs are investigated. With reference to the physical intuition underlying contextualism, the possibility is considered whether a context-dependence of individual measurement results is compatible with context-independence of the statistics of measurement results.
One of the recent no-go theorems on Ψ-epistemic interpretations of quantum proves that there are no 'maximally epistemic' interpretations of quantum theory. The proof utilises similar arrangements to Clifton's quantum contextuality proof and has parallels to Harrigan and Rudolph's quantum deficiency no-go theorem, itself based on the Kochen-Specker quantum contextuality proof. This paper shows how the Kochen-Specker theorem can also be turned into a no 'maximally epistemic' theorem, but of a more limited kind.
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.