Unitary dilations of commutation relations associated to alternating bilinear forms
type: Item
instance of: scholarly article
title: Unitary dilations of commutation relations associated to alternating bilinear forms
zbMATH Open document ID: 0813.46068
DOI: 10.1007/BF03025729
author: Palle Jorgensen
published in: Mathematische Zeitschrift
publication date: 1995-05-28T00:00:00Z
full work available at URL: https://eudml.org/doc/174537
review text: We study second quantization of symplectic spaces \(V\) with given alternating bilinear form. An algebraic framework is developed using the associated central extension Lie algebras. We then impose the further condition that \(V\) carries a reflection operator satisfying an Osterwalder-Schrader type positivity. Our construction of representations is based on induction from representations of associated (non-degenerate) Heisenberg algebras obtained as unitary dilations of given \(V\). When Osterwalder-Schrader positivity is assumed, we show that \(V\) has a Heisenberg extension inducing representations of \(V\) if and only if \(V\) decomposes in a specific way such that the given symplectic form satisfies a certain a priori estimate relative to the subspaces of the decomposition. Our induction provides in particular representations of the free chiral quantum field on a higher genus Riemann surface.
reviewed by: Palle Jorgensen
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 17B15
zbMATH DE Number: 561298
zbMATH Keywords: second quantization of symplectic spaces
zbMATH Keywords: central extension Lie algebras
zbMATH Keywords: reflection operator
zbMATH Keywords: representations
zbMATH Keywords: Heisenberg algebras
zbMATH Keywords: unitary dilations
zbMATH Keywords: Osterwalder-Schrader positivity
zbMATH Keywords: Heisenberg extension
zbMATH Keywords: chiral quantum field
zbMATH Keywords: higher genus Riemann surface
MaRDI profile type: Publication
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OpenAlex ID: W2009391773
description: scientific article; zbMATH DE number 561298