On the Mickelsson-Faddeev extension and unitary representations (Q1263685)

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.