Path integrals on group manifolds : the representation independent propagator for general Lie groups PDF
Preview Path integrals on group manifolds : the representation independent propagator for general Lie groups
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IISSBBNN 998811--0022--33335555--88 TThhiiss bbooookk iiss pprriinntteedd oonn aacciidd--ffrreeee ppaappeerr PPrriinntteedd iinn SSiinnggaappoorree bbyy UUttoo-PPiiiinntt For Marie-Jacqueline and Anne-Sophie This page is intentionally left blank "The highest reward for a man's toil is not what he gets for it, but what he becomes by it." -John Ruskin This page is intentionally left blank PREFACE The quantization of physical systems moving on group and symmetric spaces has been an area of active and on-going research over the past three decades. It is shown in this work that it is possible to introduce a repre sentation independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square integrable representations. For a given set of kinematical variables this propagator is a single generalized func tion independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless, correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables. We begin our discussion with a short chapter on some of the standard results from the fields of Algebra, Functional Analysis, and Representation Theory. The results discussed are used freely in the remainder of the book. The reader unfamiliar with these results should refer to chapter 1. In chapter 2 we discuss the relevance and physical interpretation of rep resentation independent propagators. We have chosen the simplest case of a single canonical degree of freedom as a vehicle to introduce the reader to the concept of representation independent propagators. IX X In chapter 3 we discuss the construction of path integrals on group and symmetric spaces. In § 3.1 we review the Feynman path integral on flat, group, and symmetric spaces. § 3.2 is devoted to the study of group coherent states associated with a compact group and the construction of coherent state path integrals based on group coherent states associated with a compact group. In chapter 4 we introduce the notations and basic definitions used through out the volume. The main result of this chapter is Theorem 4.2.1, in which we derive an operator version of the generalized Maurer-Cartan form. Chapter 5 contains the construction of the representation independent prop agator for a real, separable, locally compact, connected and simply connected Lie group with irreducible, square integrable representations. We refer here after to a real, separable, locally compact, connected and simply connected Lie group with irreducible square integrable representations as a general Lie group. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irre ducible representation of the general Lie group generated by these kinematical variables. In § 5.1 we define coherent states for a general Lie group and prove Lemma 5.1.4 and the Corollary 5.1.5 which we apply in the construction of the representation independent propagator and the construction of regularized lattice phase-space path integral representations of the representation indepen dent propagator. Prior to constructing the representation independent propagator for a gen eral Lie group, we construct in § 5.2 the representation independent propagator for any real compact Lie group. It is shown in Theorem 5.2.2 that the repre sentation independent propagator for any compact group correctly propagates
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