Holomorphic quantization of linear field theory in the general boundary formulation
type: Item
instance of: scholarly article
title: Holomorphic quantization of linear field theory in the general boundary formulation
zbMATH Open document ID: 1285.81042
DOI: 10.3842/SIGMA.2012.050
author: Robert Oeckl
published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications
publication date: 2013-07-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.5615
review text: A General Boundary Formulation of quantum field theory is a generalization of the, mathematically better known, notion of a topological field theory. It associates a Hilbert space of states to a space manifold, and a linear map to a space-time cobordism between two space like manifolds. It may require extra structure, and the results may depend on more than the topology, but it does not require a background metric. The paper begins by formulating a list of axioms for such a formulation. It then describes how a classical action functional on fields gives rise to a \(1\)-form on the space of germs of classical solutions restricted to space-like slices. From here R. Oeckl, describes the natural symplectic form on the space of these germs, and explains why the restriction of the space of solutions to an initial of final surface is a Lagrangian subspace. He then describes holomorphic quantization in the context of geometric quantization. Basically, the local solutions inherits a sesquilinear form constructed out of the natural symplectic form and a compatible almost complex structure. Completing this space gives a first (pre-quantum) Hilbert space. The state space is the Hilbert space of holomorphic functions on the first Hilbert sapce. The next section of the paper describes a theory of Gaussian integration on the projective limit of subspaces of the pre-quantum Hilbert space endowed with compatible measures. He continues with the definition of holomorphic functions on the projective limit leading to the construction of the hilbert space of states. The next part of the paper describes how to pass from the data of a classical system to a quantum theory satisfying the GBF axioms. It is interesting to note that his formula for the amplitude is a Gaussian integral (with exponential of a negative definite form -- not an imaginary form) over the space of classical solutions. This is similar to, but distinct from the Feynman path integral that one might expect. There are certa…
reviewed by: David Auckly
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 81R30
zbMATH DE Number: 6184277
zbMATH Keywords: geometric quantization
zbMATH Keywords: topological quantum field theory
zbMATH Keywords: coherent states
zbMATH Keywords: foundations of quantum theory
zbMATH Keywords: quantum field theory
Wikidata QID: Q62100594
MaRDI profile type: Publication
arXiv ID: 1009.5615
description: scientific article; zbMATH DE number 6184277