Energy representations of gauge groups associated with Riemannian flags
type: Item
instance of: scholarly article
title: Energy representations of gauge groups associated with Riemannian flags
zbMATH Open document ID: 0652.53033
DOI: 10.1016/0022-1236(88)90054-7
author: J. Marion
author: Daniel Testard
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let (X,g) be a smooth Riemannian n-manifold and \(Y_ i\) be k submanifolds with C 1 Riemannian structures \(g_ i\), and let \(Y_ i\subset Y_{i-1}\) and dim \(Y_ i<\dim Y_{i-1}\) \((Y_ 0=X)\). This defines a Riemannian flag. If dim \(Y_ i=n-i\), then the Riemannian flag is regular. Let G be a semisimple Lie group and \({\mathfrak g}\) its Lie algebra with the Euclidean structure given by the negative of the Killing form. The Riemannian structures g and \(g_ i\) enable one to define the energy function on the space \(\Omega\) (X,g) of all smooth \({\mathfrak g}\)- valued 1-forms on X. The completion of complexified \(\Omega\) (X,g) defines the Hilbert space \(H_ Y\) identified with the state space of a 1-particle system and its symmetric tensor algebra leads to the Fock space. The so-called energy representation \(U_ Y\) of the gauge group G X of all smooth compactly supported G-valued maps on X is then defined. Two main theorems are proved: I. If \(n\geq 3\) and the Riemannian flag Y is regular, then \(U_ Y\) is irreducible; if \(n=2\), Y is regular and the length of each root of \({\mathfrak g}\) is bounded from below by a positive constant \(K_ Y\), then \(U_ Y\) is irreducible. II. Let Y and Y' be two distinct regular Riemannian flags and let \(n\geq 3\), or \(n=2\) and the length of each root of \({\mathfrak g}\) be greater than or equal to \(Max(K_ Y,K_{Y'})\), where the positive constants \(K_ Y\) and \(K_{Y'}\) are as in I. Then \(U_ Y\) and \(U_{Y'}\) are not unitarily equivalent. The paper is completely self-contained in all its mathematical aspects. The energy representations discussed are on the Fock space and hence there must be some important physical implications (which are not discussed here) for the quantization of gauge fields.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C80
zbMATH DE Number: 4063981
zbMATH Keywords: Gaussian measures
zbMATH Keywords: Riemannian flag
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: energy function
zbMATH Keywords: Fock space
zbMATH Keywords: energy representation
zbMATH Keywords: gauge group
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90054-7
OpenAlex ID: W2043176716
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description: scientific article; zbMATH DE number 4063981