On the Mickelsson-Faddeev extension and unitary representations
type: Item
instance of: scholarly article
title: On the Mickelsson-Faddeev extension and unitary representations
zbMATH Open document ID: 0688.22003
DOI: 10.1007/BF01218587
author: Doug Pickrell
published in: Communications in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: The Mickelsson-Faddeev extension \(\hat M\) is a certain distinguished noncentral abelian extension of the Hamiltonian gauge group \(C^{\infty}(X,G):\) \(0\to F\to \hat M\to C^{\infty}(X,G)\to 0\), where the kernel F of this extension consists of some class of functions of a gauge potential. It originates from the regularization process of gauge operators. It is known that if X is one-dimensional, \(\hat M\) is the Kac-Moody extension. The author demonstrates that the Mickelsson-Faddeev extension is a three-dimensional analogue of Kac-Moody group whose central charge is replaced by the space of functions of the gauge potential. Also the author investigates a ``universal model'' of the Mickelsson-Faddeev extension proposed by \textit{J. Mickelsson} and \textit{S. G. Rajeev} [ibid. 116, No.3, 365-400 (1988; Zbl 0648.22013)] who constructed an extension of the restricted group \(U_{(4)}\) which pulls back to the Mickelsson- Faddeev extension if \(C^{\infty}(X,G)\) is embedded into \(U_{(4)}\) and has no interesting unitary representation, however, the Mickelsson-Rajeev extension of \(U_{(4)}\) does it. But the crucial problem arised then, whether the latter one can be unitarized. It is shown in the present paper that any separable unitary representation has to vanish on the topologically nontrivial part of the kernel, and hence, the extension cannot be faithfully represented by unitary operators on a separable Hilbert space.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81T60
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 81T05
zbMATH DE Number: 4127526
zbMATH Keywords: Mickelsson-Faddeev extension
zbMATH Keywords: noncentral abelian extension
zbMATH Keywords: Hamiltonian gauge group
zbMATH Keywords: Kac-Moody extension
zbMATH Keywords: Kac-Moody group
zbMATH Keywords: central charge
zbMATH Keywords: gauge potential
zbMATH Keywords: unitary representation
reviewed by: Eugene Kryachko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4127526