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Journal of Mathematical Physics

The universal propagator for affine [or SU(1,1)] coherent states

John R. Klauder, Wolfgang A. Tomé

Published Nov 1, 1992• 2 citations
Publisher

Abstract

It is shown that it is possible to introduce a universal propagator for a general Hamiltonian appropriate to a single affine degree of freedom. This universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates the coherent state Hilbert space representatives correctly. Furthermore, the universal propagator for several examples is explicitly constructed.

DOI
10.1063/1.529865

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It is shown that it is possible to introduce a universal propagator for a general Hamiltonian appropriate to a single affine degree of freedom. This universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates the coherent state Hilbert space representatives correctly. Furthermore, the universal propagator for several examples is explicitly constructed.
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 601h birthdayIn earlier work universal propagators were introduced for the Heisenberg -Weyl group, the affine group, and the rotation group.By generalizing these constructions we show here that it is possible to introduce a universal propagator for a rather general unitary Lie group.In the context of coherent-state representations, the universal propagator is a single function independent of any particular choice of fiducial vector, which nonetheless, propagates all coherent state Hilbert space representatives correctly.
Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present work studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead, a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.
The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G 0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.