Quantization on the Virasoro group
type: Item
instance of: scholarly article
title: Quantization on the Virasoro group
zbMATH Open document ID: 0703.22013
DOI: 10.1007/BF02125701
author: Víctor Aldaya
author: José Navarro-Salas
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: The group approach to quantization developed by the authors is applied to quantize the Virasoro group. This approach differs from the standard method of geometric quantization by taking as a starting point the entire group manifold rather than the co-adjoint orbit. First, the formal group law of the centrally extended group \(diff^{\sim} S^ 1\) is found. The next step is to find the quantization form, i.e. the vertical component of the left-invariant canonical 1-form. This is done using duality with respect to the left- invariant vector fields which in turn can be quite simply calculated from the composition law. Finally, the quantization is accomplished by defining the wave functions as the U(1)-equivariant functions on \(diff^{\sim} S^ 1\) satisfying certain polarization conditions and finding the group representations by calculating the action of left translations on polarized functions. Some properties of the representations obtained in such a way are discussed. In particular, the absence of null vectors is established.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 4152738
zbMATH Keywords: Virasoro group
zbMATH Keywords: geometric quantization
zbMATH Keywords: formal group law
zbMATH Keywords: centrally extended group
zbMATH Keywords: left-invariant canonical 1-form
zbMATH Keywords: vector fields
zbMATH Keywords: wave functions
zbMATH Keywords: representations
zbMATH Keywords: polarized functions
reviewed by: Paweł Maślanka
MaRDI profile type: Publication
cites work: Coadjoint orbits of the Virasoro group
cites work: Deformation theory and quantization. I: Deformations of symplectic structures
cites work: Quantization as a consequence of the symmetry group: An approach to geometric quantization
cites work: Group manifold analysis of the structure of relativistic quantum dynamics
cites work: A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group
cites work: Formal group laws for affine Kac-Moody groups and group quantization
cites work: Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra
cites work: Algebraic quantization on a group and nonabelian constraints
cites work: On unitary ray representations of continuous groups
cites work: Q5731354
cites work: A class of non-polarizable symplectic manifolds
description: scientific article; zbMATH DE number 4152738