Off-campus UMass Amherst users: To download dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.
Non-UMass Amherst users, please click the view more button below to purchase a copy of this dissertation from Proquest.
(Some titles may also be available free of charge in our Open Access Dissertation Collection, so please check there first.)
Unitary representations of gauge groups
I generalize to the case of gauge groups over non-trivial principal bundles representations that I. M. Gelfand, M. I. Graev and A. M. Versik constructed for current groups.^ The gauge group of the principal G-bundle P over M, (G a Lie group with an euclidean structure, M a compact, connected and oriented manifold), as the smooth sections of the associated group bundle is presented and studied in chapter I. Chapter II describes the symmetric algebra associated to a Hilbert space, its Hilbert structure, a convenient exponential and a total set that later play a key role in the construction of the representation.^ Chapter III is concerned with the calculus needed to make the space of Lie algebra valued 1-forms a Gaussian $L\sp2$-space. This is accomplished by studying general projective systems of finitely measurable spaces and the corresponding systems of $\sigma$-additive measures, all of these leading to the description of a promeasure, a concept modeled after Bourbaki and classical measure theory.^ In the case of a locally convex vector space E, the corresponding Fourier transform, family of characters and the existence of a promeasure for every quadratic form on $E\sp\prime$ are established, so the Gaussian $L\sp2$-space associated to a real Hilbert space is constructed.^ Chapter III finishes by exhibiting the explicit Hilbert space isomorphism between the Gaussian $L\sp2$-space associated to a real Hilbert space and the complexification of its symmetric algebra.^ In chapter IV taking as a Hilbert space H the $L\sp2$-space of the Lie algebra valued 1-forms on P, the gauge group acts on the motion group of H defining in an straight forward fashion the representation desired. ^
Huerfano, Ruth Stella, "Unitary representations of gauge groups" (1996). Doctoral Dissertations Available from Proquest. AAI9619396.