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Contractions of algebras underlying models for superfluidity

Luigi Amico A

Published Feb 25, 2000• 0 citations
OpenAlex

Abstract

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,

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•Jan 1, 1999•8 citations
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,
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
An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ℤ2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
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.
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.