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arXiv:hep-th/9201002 · Physical Review Letters

Inhomogeneous quantum groups as symmetries of phonons

Francesco Bonechi, E. Celeghini, R. Giachetti, E. Sorace, M. Tarlini

Published Jun 22, 1992• 65 citations
arXivHTMLPDF

Abstract

The quantum-deformed (1+1) Poincar\'e algebra is shown to be the kinematical symmetry of the harmonic chain, whose spacing is given by the deformation parameter. Phonons with their symmetries as well as multiphonon processes are derived from the quantum group structure. Inhomogeneous quantum groups are thus proposed as kinematical invariances of discrete systems.

DOI
10.1103/physrevlett.68.3718

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The quantum-deformed (1+1) Poincar'e algebra is shown to be the kinematical symmetry of the harmonic chain, whose spacing is given by the deformation parameter. Phonons with their symmetries as well as multiphonon processes are derived from the quantum group structure. Inhomogeneous quantum groups are thus proposed as kinematical invariances of discrete systems.
Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems. The method is illustrated on the one-dimensional Heisenberg ferromagnet whose symmetry is investigated by means of the quantum Galilei group ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{q}}$(1) here introduced. Both the single magnon and the s=1/2 bound states of n magnons are completely described by the algebra. Therefore, some of the results provided by the Bethe-ansatz method emerge as a natural consequence of the quantum symmetry of the discrete chain.
The formalism of non-Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space-time realization of the group, casts the Schrödinger operator into the familiar time-dependent space-time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time-dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space-time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.
A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.
It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.
This paper rigorously investigates the profound interplay between quantum group symmetries and the deformed spectral transform, particularly in the context of soliton theory. We delve into how the introduction of a deformation parameter, characteristic of quantum groups, modifies the fundamental algebraic structures underlying integrable systems and, consequently, the spectral properties of their soliton solutions. By employing the framework of the quantum inverse scattering method (QISM), we construct deformed Lax pairs and analyze their impact on the scattering data. Our analysis reveals that quantum group symmetries induce a non-trivial deformation of the scattering matrix and its associated Jost functions, leading to quantized spectral parameters and modified conservation laws. We demonstrate how this deformation manifests in the properties of solitons, suggesting a departure from classical integrability and opening avenues for understanding quantum integrable models. The implications extend to a deeper understanding of quantum field theories and statistical mechanics models where these symmetries are paramount.