Compact quantum systems
Robert Fawcett · Bulletin of the Australian Mathematical Society · 1993
This thesis is primarily concerned with compact quantum systems, their relationship to non-compact quantum systems and the classical limit of these systems.A compact quantum system is a quantum mechanical system whose underlying kinematical Lie algebra is a compact algebra instead of the usual non-compact Weyl-Heisenberg algebra w n (of the canonical hermitian position and momentum operators) which can be recovered by a contraction of the compact algebra.It is shown that compact quantum systems can be described in terms of the special orthogonal, unitary and compact sympletic algebras and their representations.Boson realisations of the compact Lie algebras ao{n + 2), u(n + 1) and sp(2n + 2) are given, modified slightly from realisations given by Exner, Havlicek and Lassner (Czechoslovak J. Phys.B 26,1213,1976) in terms of canonical coordinate and momentun operators.These boson realisations provide a natural generalisation of the Dyson boson realisation of JO(3).They also have a recurrent structure which in the cases of so(n + 2) and u(n + 1) allows the explicit construction of any desired irreducible representation entirely in terms of bosons.Using these boson realisations, a general formulation for compact quantum systems with n degrees of freedom based on so{n + 2) and u(n + 1) is given in terms of n compact coordinate and momentum operators (2n + 1 degrees of freedom and 2n + 1 coordinate and position operators in the case of sp(2n + 2)).The dynamics of such a system is assumed to be determined in the usual way by a Hamiltonian operator, taken here to be a polynomial in the coordinate and momentum operators.Such compact quantum systems can be contracted to corresponding non-compact quantum systems by means of a contraction of the underlying compact kinematical Lie algebra.A new definition of the contraction of a Lie algebra and its representations by the method of sequences of representations is given, suitable to the purposes of this application.The consequences of the definition are explored at length, in particular, with regard to the reducibility and decomposability of the resultant representations of the contracted