Darboux’s Theorem and Quantisation
نویسنده
چکیده
It has been established that endowing classical phase space with a Riemannian metric is sufficient for describing quantum mechanics. In this letter we argue that, while sufficient, the above condition is certainly not necessary in passing from classical to quantum mechanics. Instead, our approach to quantum mechanics is modelled on a statement that closely resembles Darboux’s theorem for symplectic manifolds. Pacs codes: 03.65.-w, 03.65.Ca, 04.60.Ds. 2000 MSC codes: 81S10, 81P05. Preprint no. OUTP-01-59P.
منابع مشابه
Contact Geometry
2 Contact manifolds 4 2.1 Contact manifolds and their submanifolds . . . . . . . . . . . . . . 6 2.2 Gray stability and the Moser trick . . . . . . . . . . . . . . . . . . 13 2.3 Contact Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Darboux’s theorem and neighbourhood theorems . . . . . . . . . . 17 2.4.1 Darboux’s theorem . . . . . . . . . . . . . . . . . . . . . . . 17...
متن کاملDarboux theory of integrability in the sparse case
Darboux’s theorem and Jouanolou’s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context t...
متن کاملThat Strange Procedure Called Quantisation
This is a pedagogical and (almost) self-contained introduction into the Theorem of Groenewold and van Howe, which states that a naive transcription of Dirac’s quantisation rules cannot work. Some related issues in quantisation theory are also discussed. First-class constrained systems a briefly described in a slightly more ‘global’ fashion.1
متن کاملA Generalization of an Integrability Theorem of Darboux
In Chapter I, Livre III of his monograph “Systèmes Orthogonaux” Darboux stated three integrability theorems. They provide local existence and uniqueness of solutions to systems of first order Partial Differential Equations of the type: ∂xiuα(x) = f α i (x, u(x)), i ∈ Iα ⊆ {1, . . . , n}. Here x = (x1, . . . , xn) , u = (u1, . . . , um), and f α i (x, u(x)) are given functions. For each dependen...
متن کاملRelative Formality Theorem and Quantisation of Coisotropic Submanifolds
We prove a relative version of Kontsevich’s formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich’s theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to th...
متن کامل