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.
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
A generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Qi and corresponding momentum operators Pi, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.
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.
Abstract. Not unitary transformation based on algebraic contraction is proposed to realize mappings between models Hamiltonians. Such a transformation contracts the algebra of the degrees of freedom underlying the Hamiltonian. The rigorous mapping between the anisotropic XXZ Heisenberg model, the Quantum Phase Model, and the Bose Hubbard Model is identified as the group contractions of the algebra u(2) underlying the dynamics of the XXZ Heisenberg model. PACS numbers: 05.70.Fh, 68.35.Rh, 02.20.-a, 74.50.+r The problem of mapping between not equivalent algebras was solved, in mathematical physics, years ago by Inönü and Wigner [1] and subsequently generalized by Saletan [2] when they founded the concept of contraction of a Lie algebra A [3]. Algebraic contraction is a transformation which may be singular on A’s basis (namely, the kernel of the transformation is non trivial), while it is regular on its commutation brackets [4]. Further application of algebraic contractions in physics traces back to studies of Umezawa and coworkers [5]. They shown, under quite general hypothesis,
The Weyl-Wigner description of quantum mechanical operators and states in classical phase-space language is well known for Cartesian systems. We describe a new approach based on ideas of Dirac which leads to the same results but with interesting additional insights. A way to set up Wigner distributions in an interesting non-Cartesian case, when the configuration space is a compact connected Lie group, is outlined. Both these methods are adapted to quantum systems with finite-dimensional Hilbert spaces, and the results are contrasted.
Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature
(N+,N−), the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group
Sp(2N+,2N−), which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature
(1,4), clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length
ℓ and a maximum length
L, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra
so(1,4) and, in the flat-curvature limit, to the Poincaré algebra
iso(1,3) of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.