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Reviews in Mathematical Physics

CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS

U. Cattaneo, Walter Felipe Wreszinski

Published Nov 1, 1999• 10 citations
Publisher

Abstract

A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.

DOI
10.1142/s0129055x99000374

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A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.
The classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non-compact kinematical algebra w 1 (the Weyl-Heisenberg algebra) to the three-dimensional Abelian algebra t 3 . An appropriate definition of the contraction of a Lie algebra and of a sequence of its representations is given. For some quantum systems with simple explicitly integrable dynamics, it is shown how the classical Poisson bracket and classical trajectories are obtained in the limit. Each trajectory is associated with a one-dimensional representation of t 3 within a direct integral of such representations in a Hilbert space. Each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics, and a superselection rule arises naturally between any two such subspaces.
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
Compact quantum dynamics, based on the Lie algebra so(n), is known to be relevant to the description of the internal motion of the electron and neutrino. It is shown here that a compact dynamics can also be based on u(n). In both cases there exists a contraction to a noncompact algebra containing a Weyl–Heisenberg subalgebra and consequently, there also exists an approximation scheme defined entirely by the contraction process, in which a sequence of compact systems described in terms of finite square matrices of increasing size, approaches a noncompact system of the usual type, described in terms of operators on a Hilbert space. Particular boson realizations of so(n) and u(n) are shown to be especially convenient for the discussion of the contractions. Some simple illustrative examples of compact systems are treated algebraically and numerically in order to show how their properties approach those of the corresponding noncompact systems.
We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space.
Quantum theory can be formulated as a theory of operations, more specific, of complex represented operations from real Lie groups. Hilbert space eigenvectors of acting Lie operations are used as states or particles. The simplest simple Lie groups have three dimensions. These groups together with their contractions and subgroups contain - in the simplest form - all physically important operations which come as translations for causal time, for space and for spacetime, as rotations, Lorentz transformations and as Euclidean and Poincare transformations with scattering and particle states and also - via the Heisenberg groups - as the operational structure of nonrelativistic quantum mechanics. The classification of all those groups and their contractions is given together with their Hilbert spaces, constituted by energy-momentum functions. The groups representation matrix elements can be written as residues of energy-momentum poles - simple poles for abelian translations, e.g. in Feynman propagators, and dipoles for simple Lie group operations, e.g. in the Schroedinger wave functions for the nonrelativistic hydrogen atom.