### Citations

1146 |
Teitelboim ”Quantization of Gauge System
- Henneaux, C
- 1992
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Citation Context ...in the (linear) spaces of states and observables. Dirac’s ideas have been reviewed, refined, and discussed many times in the literature; see e.g. the comprehensive textbook by Henneaux and Teitelboim =-=[11]-=-. Here we shall merely give a brief coordinate-free description of how to construct the right classical Poisson algebra of functions (the ‘physical observables’). 3.1 First-class constraints Let (P,ω)... |

409 |
Lectures on quantum mechanics (Belfer graduate School
- Dirac
- 1964
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Citation Context ... very difficult to explicitly do the quotient construction ˆ P → ˆ P/∼ = ¯ P , i.e. to solve the constraints classically. Dirac has outlined a strategy to implement the constraints after quantisation =-=[4]-=-. The basic mathematical reason why this is considered a simplification is seen in the fact that the whole problem is now posed in linear spaces, i.e. the construction of sub- and quotient spaces in t... |

241 |
Local Quantum Physics - Fields, Particles, Algebras
- Haag
- 1996
(Show Context)
Citation Context ... Quantum Field Theory (for more discussion on this point, that also covers interesting technical issues, I recommend [12]). As an exception—to a certain degree—I would list Local Quantum Field Theory =-=[10]-=-, which axiomatically starts with a general kinematical framework for Poincaré invariant quantum field theories without any a priori reference to classical theories. Although this can now be generalis... |

113 |
On the Principles of elementary quantum mechanics
- Groenewold
- 1946
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Citation Context ...ion will facilitate the technical presentation of the following arguments, but we remark that it can be dispensed with [8]. 2.3 The theorem of Groenewold and van Howe In a series of papers Groenewold =-=[9]-=- and van Hove [19, 18] showed that a canonical quantisation satisfying requirements 1.–8. does not exist. The proof is instructive and therefore we shall present it in detail. For logical clarity it i... |

91 |
Topological and global aspects of quantum theory
- Isham
- 1983
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Citation Context ...he way we traditionally understand Quantum Mechanics and also a substantial part of Quantum Field Theory (for more discussion on this point, that also covers interesting technical issues, I recommend =-=[12]-=-). As an exception—to a certain degree—I would list Local Quantum Field Theory [10], which axiomatically starts with a general kinematical framework for Poincaré invariant quantum field theories witho... |

32 |
Foundations of mechanics, Benjamin/Cummings
- Abraham, Marsden
- 1978
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Citation Context ...estrict to them. 7One might wonder what is actually implied by the last condition and whether it is not unnecessarily restrictive. This is clarified by the theorem of Stone and von Neumann (see e.g. =-=[1]-=-), which says that if the 2n operators Q(q i ) and Q(pi) are represented irreducibly up to finite multiplicity (to allow for finitely many internal quantum numbers) and satisfy the required commutatio... |

28 | BRST cohomology and phase space reduction in deformation quantization
- Bordemann, Herbig, et al.
- 2000
(Show Context)
Citation Context ... may become more ‘wild’ in infinitesimal neighbouring level sets, thereby forbidding most of these functions to be extended to some invariant functions in a neighbourhood of ˆ P in P . See Sect. 3 of =-=[2]-=- for an example and more discussion of this point. Appendix 1: Geometry of Hamiltonian Systems A symplectic manifold is a pair (P,ω), where P is a differentiable manifold and ω is a closed (i.e. dω = ... |

19 | Lectures on loop quantum gravity
- Thiemann
(Show Context)
Citation Context ...um Gravity will finally turn out to be are still diverse, though more precise pictures now definitely emerge within the individual approaches, as you will hopefully be convinced in the other lectures =-=[14, 15, 17]-=-, so that reliable statements about similarities and differences on various points can now be made. The present contribution deliberately takes focus on a very particular and seemingly formal point, i... |

17 | 1995] On a full quantization of the torus
- Gotay
(Show Context)
Citation Context ... C ∞ functions is called full quantisation in the literature.) Surprisingly there is a non-trivial counterexample to this conjecture: it has been shown that a full quantisation exists for the 2-torus =-=[6]-=-. 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... |

13 |
certaines représentations unitaires d’un groupe infini de transformations. Memoires de la Académie Royale Belgique, Classe de Sciences
- Sur
- 1951
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Citation Context ...te the technical presentation of the following arguments, but we remark that it can be dispensed with [8]. 2.3 The theorem of Groenewold and van Howe In a series of papers Groenewold [9] and van Hove =-=[19, 18]-=- showed that a canonical quantisation satisfying requirements 1.–8. does not exist. The proof is instructive and therefore we shall present it in detail. For logical clarity it is advantageous to divi... |

12 | A discrete history of the Lorentzian path integral
- Loll
(Show Context)
Citation Context ...um Gravity will finally turn out to be are still diverse, though more precise pictures now definitely emerge within the individual approaches, as you will hopefully be convinced in the other lectures =-=[14, 15, 17]-=-, so that reliable statements about similarities and differences on various points can now be made. The present contribution deliberately takes focus on a very particular and seemingly formal point, i... |

10 |
Functorial geometric quantization and van hove’s theorem
- Gotay
- 1980
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Citation Context ...tructions are in some sense generic, though, to my knowledge, there is no generally valid formulation and corresponding theorem to that effect. (For an interesting early attempt in this direction see =-=[5]-=-.) Hence we face the problem to determine F irr and F quant within F ∞. There is no general theory how to do this. If P is homogeneous, i.e. if there is a finite dimensional Lie group G (called the ‘c... |

10 |
Functional Analysis I. Methods of modern mathematical physics
- Reed, Simon
- 1972
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Citation Context ...state that classical observables should be mapped to self adjoint operators. Instead I only required the operators to be symmetric, which is a weaker requirement. This important distinction (see e.g. =-=[16]-=-) is made for the following reason (see e.g. sect. VIII in [16] for the mathematical distinction): let ˆ f be the operator corresponding to the phase-space function f. If ˆ f were self adjoint, then t... |

6 |
The principles of quantum mechanics, International series of monographs on physics
- Dirac
- 1987
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Citation Context ...focus on the traditional ‘canonical’ approaches to quantisation. 2 Canonical Quantisation Historically the rules for ‘canonical quantisation’ where first spelled out by Dirac in his famous book on QM =-=[3]-=-. His followers sometimes bluntly restated these rules by the symbolic line, {·, ·} ↦→ −i � [·,·], (1) which is to be read as follows: map each classical observable (function on phase space) f to an o... |

6 | Introduction to string theory
- Mohaupt
- 207
(Show Context)
Citation Context ...um Gravity will finally turn out to be are still diverse, though more precise pictures now definitely emerge within the individual approaches, as you will hopefully be convinced in the other lectures =-=[14, 15, 17]-=-, so that reliable statements about similarities and differences on various points can now be made. The present contribution deliberately takes focus on a very particular and seemingly formal point, i... |

5 |
le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique. Académie royale de Belgique, bulletin de la classe des sciences
- Sur
- 1951
(Show Context)
Citation Context ...te the technical presentation of the following arguments, but we remark that it can be dispensed with [8]. 2.3 The theorem of Groenewold and van Howe In a series of papers Groenewold [9] and van Hove =-=[19, 18]-=- showed that a canonical quantisation satisfying requirements 1.–8. does not exist. The proof is instructive and therefore we shall present it in detail. For logical clarity it is advantageous to divi... |

1 |
A Groenewold - van Howe theorem for S 2
- Gotay
- 1996
(Show Context)
Citation Context ... for the 2-torus [6]. 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. The general expectation is indeed that GvH-like obstructions are in some sense generic, though, to my knowledge, there is no gener... |

1 |
On the Groenewold - van Howe problem for R 2n
- Gotay
- 1999
(Show Context)
Citation Context ...rong sense, i.e. their families of spectral projectors commute. This extra condition will facilitate the technical presentation of the following arguments, but we remark that it can be dispensed with =-=[8]-=-. 2.3 The theorem of Groenewold and van Howe In a series of papers Groenewold [9] and van Hove [19, 18] showed that a canonical quantisation satisfying requirements 1.–8. does not exist. The proof is ... |