On the Groenewold–Van Hove problem for R2n

Gotay, Mark J.

doi:10.1063/1.532854

Cited by 34 publications

(44 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One might first guess that this is somehow due to the compactness of the phase space. But this is not true, as a GvH obstruction to full quantisation does exist for the 2-sphere [7]. But the case of the 2-torus seems exceptional, even mathematically.…”

Section: Rulementioning

confidence: 93%

“…Hence (62) gives a quantisation of F pol(∞,1) . It can be shown ( [8], Thm. 8) that F pol(2) and F pol(∞,1) are the only maximal Lie subalgebras of F pol which contain the Heisenberg algebra F pol (1) .…”

Section: Rulementioning

confidence: 99%

“…Then commutativity on D does not imply that Q(f ) and Q(g) commute in the usual (strong) sense of commutativity of self-adjoint operators, namely that all their spectral projectors mutually commute (compare [16], p. 271). This we pose as an extra condition: This extra condition will facilitate the technical presentation of the following arguments, but we remark that it can be dispensed with [8].…”

Section: Defining 'Canonical Quantisation'mentioning

confidence: 99%

“…Following [8], we first observe that the statements (28-30) can actually be generalised: Let P be any real polynomial, then…”

Section: Proof Of Partmentioning

confidence: 99%

“…A Poisson algebra is a vector space V with two maps V × V → V , denoted by '{, }' and '·', which turn V into a Lie algebra (defined by (3)(4)(5)) and a commutative and associative algebra (defined by (8)(9)(10)) respectively, such that (11) holds.…”

Section: The Classical Stagementioning

confidence: 99%

See 4 more Smart Citations

That Strange Procedure Called Quantisation

Giulini

2003

Quantum Gravity

View full text Add to dashboard Cite

show abstract

Section: Rulementioning

confidence: 93%

Section: Rulementioning

confidence: 99%

Section: Defining 'Canonical Quantisation'mentioning

confidence: 99%

“…Following [8], we first observe that the statements (28-30) can actually be generalised: Let P be any real polynomial, then…”

Section: Proof Of Partmentioning

confidence: 99%

Section: The Classical Stagementioning

confidence: 99%

See 3 more Smart Citations

That Strange Procedure Called Quantisation

Giulini

2003

Quantum Gravity

View full text Add to dashboard Cite

show abstract

Problem of time in quantum gravity

2012

View full text Add to dashboard Cite

The Problem of Time occurs because the 'time' of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi-faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as "Dirac-quantize". C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question-types considered in physics. D) The Halliwell-and Gambini-Porto-Pullin-type combined Strategies in the context of semiclassical quantum cosmology.

show abstract

Obstructions to Quantization

Gotay

2000

Mechanics: From Theory to Computation

View full text Add to dashboard Cite

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].

show abstract

On the Groenewold–Van Hove problem for R2n

Cited by 34 publications

References 17 publications

That Strange Procedure Called Quantisation

That Strange Procedure Called Quantisation

Problem of time in quantum gravity

Obstructions to Quantization

Contact Info

Product

Resources

About