We derive the geometric quantization program of symplectic manifolds, in the sense of both Kostant-Souriau and Weinstein, from Feynman's path integral formulation on phase space. The state space we use contains states with negative norm and polarized sections determine a Hilbert space. We discuss ambiguities in the definition of path integrals arising from the distinct Riemann sum prescriptions and its consequence on the quantization of symplectomorphisms.

arXiv:2405.17273May 27, 20240 citations

Holomorphic Path Integrals in Tangent Space for Flat Manifolds

Guillermo Capobianco, Walter Reartes

arXiv:1703.10311Mar 30, 20170 citations

Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

Feynman path integrals on manifolds and geometric methods

H. P. Berg

Published Dec 1, 1981• 3 citations

Publisher

DOI: 10.1007/bf02730365

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In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of $S^1$ , the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of $S^1$, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.

arXiv:1703.10311Mar 30, 20170 citations

S^1

, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.