Obstructions to Quantization

Abstract

In this paper we continue our study of Groenewold-Van Hove obstructions to quantization. We show that there exists such an obstruction to quantizing the cylinder T * S 1 . More precisely, we prove that there is no quantization of the Poisson algebra of T * S 1 which is irreducible on a naturally defined e(2) × R subalgebra. Furthermore, we determine the maximal "polynomial" subalgebras that can be consistently quantized, and completely characterize the quantizations thereof. This example provides support for one of the conjectures in [GGT], but disproves part of another. Passing to coverings, we also derive a no-go result for R 2 which is comparatively stronger than those originally found by Groenewold [Gr] and Van Hove [vH].

On the Groenewold–Van Hove problem for R2n

“…Already some results have been established along these lines, to the effect that under certain circumstances there are obstructions to quantizing both compact and noncompact phase spaces [GGG,GGra,GG2,GM]. We refer the reader to [Go3] for an up-to-date summary.…”

“…Throughout, the Heisenberg algebra h(2n) is regarded as a "basic algebra of observables," cf. [Go3]. We briefly comment on these conditions; a full exposition along with detailed motivation is given in [Go3].…”

“…[Go3]. We briefly comment on these conditions; a full exposition along with detailed motivation is given in [Go3]. Condition (Q1) is Dirac's famous "Poisson bracket → commutator" rule; here is Planck's reduced constant.…”

“…1], where the bar denotes closure. It now follows from (1), the invariance of the domain D, and Sobolev's lemma that U −1 D ⊆ S(R n ), so that U −1 QU is the restriction of dΠ to U −1 D. Finally, in [Go3] there is a sixth criterion that a quantization must satisfy in general, viz. that Q be faithful when restricted to the given basic algebra of observables.…”

Untitled

“…It is even known to be impossible, in the sense of Dirac canonical quantization, as it follows from the Groenewold-van Hove "nogo" theorem [58,59]. In fact, quantization of a small Heisenberg subalgebra of the canonical brackets often suffices.…”