Path integrals on group manifolds : the representation independent propagator for general Lie groups

228 Pages·20.481 MB·English

by  Wolfgang Tomé.

Path Integrals
on
Group Manifolds
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For Marie-Jacqueline and Anne-Sophie
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"The highest reward for a man's toil is not
what he gets for it, but what he becomes by it."
-John Ruskin
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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