Volume 37, Issue 1, January 1996
Index of content:

Quantum chains with GL_{ q }(2) symmetry
View Description Hide DescriptionUsually quantum chains with quantum group symmetry are associated with representations of quantized universal algebrasU _{ q }(g). Here we propose a method for constructing quantum chains with C _{ q }(G) global symmetry, where C _{ q }(G) is the algebra of functions on the quantum group. In particular we will construct a quantum chain with GL_{ q }(2) symmetry which interpolates between two classical Ising chains. It is shown that the Hamiltonian of this chain satisfies the generalized braid group algebra.

Quasi‐exactly solvable systems and orthogonal polynomials
View Description Hide DescriptionThis paper shows that there is a correspondence between quasi‐exactly solvable models in quantum mechanics and sets of orthogonal polynomials {P _{ n }}. The quantum‐mechanical wave function is the generating function for the P _{ n }(E), which are polynomials in the energy E. The condition of quasi‐exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value J. The zeros of the critical polynomialP _{ J }(E) are the quasi‐exact energy eigenvalues of the system.

A rigorous treatment of conformal blocks in a model of bosonic conformal field theory
View Description Hide DescriptionA free massless field coupled to a background chargeQ is considered. The conformal blocks of the theory are adequately described by the N‐point vertex V _{ N;0} which is, up to a multiplicative constant, completely determined by the Knizhnik–Zamolodchikov equations and a set of Ward identities on the sphere. The N‐point g‐loop vertex V _{ N;g } for compact Riemann surfaces Σ_{ g } of genus g≠0 is constructed by a sewing procedure. The main new result is a rigorous proof of the fact that V _{ N;0} is trace class for nonoverlapping disks. This allows one to show that the set of all N‐point vertices (for all N) is a modular functor.

Integrable systems of group SO(1,2) and Green’s functions
View Description Hide DescriptionIntegrable quantum systems related with two‐dimensional two‐ and one‐sheeted hyperboloid are considered. The explicit expressions of the Green’s functions of the free particles on the those homogenous spaces are given.

Two‐dimensional supersymmetric harmonic oscillator carrying a representation of the GL(21) graded Lie algebra
View Description Hide DescriptionWe study a supersymmetric two‐dimensional harmonic oscillator which carries a representation of the general graded Lie algebra GL(21), formulate it on the superspace, and discuss its physical spectrum.

Vertex normal ordering as a consequence of nonsymmetric bilinear forms in Clifford algebras
View Description Hide DescriptionWe consider Clifford algebras with nonsymmetric bilinear forms. They parametrize the chosen ideal in the isomorphism class of the standard symmetric ones. Since the content of physical theories depends on the injection ⊕^{ n }Λ^{ n }V→CL(V,Q), one has to transform to the standard construction. The injection is described by the antisymmetric part of the bilinear form. This process results in the appropriate vertex normal ordering terms, which are now obtained from the theory itself and not added ad hoc via a regularization argument.

Fermionic matter coupled to higher derivative Chern–Simons theories. II
View Description Hide DescriptionThe diagrammatic and the Feynman rules for the higher derivative Chern–Simons theories in (2+1) dimensions coupled to fermionic matter are constructed. This is done by starting from the path‐integral quantization. Once the diagrammatic and the Feynman rules are given, the regularization and renormalization problem of this higher derivative model is analysed in the framework of the perturbation theory. The unitarity problem related with the possible appearance of ghost states with negative norm is also discussed. Finally, the BRST formalism for the model is constructed and some interesting differences with respect to the formalism applied to usual Chern–Simons models are presented.

Asymptotic algebra for charged particles and radiation
View Description Hide DescriptionA C*‐algebra of asymptotic fields which properly describes the infrared structure in quantum electrodynamics is proposed. The algebra is generated by the null asymptotic of electromagnetic field and the time asymptotic of charged matter fields which incorporate the corresponding Coulomb fields. As a consequence Gauss’ law is satisfied in the algebraic setting. Within this algebra the observables can be identified by the principle of gauge invariance. A class of representations of the asymptotic algebra is constructed which resembles the Kulish–Faddeev treatment of electrically charged asymptotic fields.

General properties between the canonical correlation and the independent‐oscillator model on a partial *‐algebra
View Description Hide DescriptionWe consider a quantum particle in thermal equilibrium with any quantum system in a finite volume under some conditions. For the Heisenberg operator of the momentum operator of the quantum particle, we show that, on a partial *‐algebra, the Heisenberg operator satisfies a quantum Langevin equation, which is similar to the work of Ford et al. [ G. W. Ford, J. T. Lewis, and R. F. O’Connell, Phys. Rev. A 37, 4419 (1988)]. Through the Langevin equation, we show general and mathematical properties between the canonical correlation and the independent‐oscillator model.

Solution to three‐magnon problem for S=1/2 periodic quantum spin chains with elliptic exchange
View Description Hide DescriptionThe method of solving the three‐particle quantum elliptic Calogero–Moser problem is applied to the description of three‐magnon wave functions for the S=1/2 quantum Heisenberg chains with the exchange interaction given by the Weierstrass d function. The Bethe‐like algebraic equations for the three‐magnon case are presented in the explicit form.

On the quantum deformations of Hamiltonian systems
View Description Hide DescriptionIn this paper we analyze the relationship between operatorial quantization and deformation quantization for Hamiltonian systems on R ^{2n }. We define heuristically generalized symbols for the operators, which make this connection. We construct explicitly deformations which are not equivalent to the Moyal one and show that an infinitesimal, classical canonical transformation does not change the equivalence class of the deformation. The results are applied to the quantum integrability of some two dimensional Hamiltonian systems.

Left–right asymmetry and minimal coupling
View Description Hide DescriptionIn this paper we deal with an alternative approach to the description of massless particles of arbitrary spin. Within this scheme chiral components of a spinor field are regarded as fundamental quantities and treated as independent field variables. The free field Lagrangian is built up from the requirement of chiral invariance. This formulation is parallel to the neutrino theory and allows for a formulation that generalizes, to particles of arbitrary spin, the two‐component neutrino theory. We achieve a spinor formulation of electrodynamics. In the case of the photon, the nonzero helicity components satisfy Weyl’s equations and are associated to observables (electromagnetic fields) whereas the zero helicity components are related to nonobservables (electromagnetic potentials). Within the spinor formulation of electrodynamics the minimal coupling substitution follows as a consequence of the linearity of the interaction and the preference of nature for chiral components, that is, of the left–right asymmetry of nature.

Involutions on the algebra of physical observables from reality conditions
View Description Hide DescriptionSome aspects of the algebraic quantization program proposed by Ashtekar are revisited in this article. It is proven that, for systems with first‐class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, i.e., of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables.

Group quantization on configuration space
View Description Hide DescriptionNew features of a previously introduced group approach to quantization are presented. We show that the construction of the symmetry group associated with the system to be quantized (the ‘‘quantizing group’’) does not require, in general, the explicit construction of the phase space of the system, i.e., does not require the actual knowledge of the general solution of the classical equations of motion; in many relevant cases an implicit construction of the group can be given, directly, on configuration space. To show an application, we construct the symmetry group for the conformally invariant massless scalar and electromagnetic fields and the scalar and Dirac fields evolving in a symmetric curved space‐time or interacting with symmetric classical electromagnetic fields. Further generalizations of the present procedure are also discussed and in particular the conditions under which non‐Abelian groups (mainly affine groups and more general gauge groups) can be included.

Fermion pair production from an electric field varying in two dimensions
View Description Hide DescriptionThe Hamiltonian describing fermion pair production from an arbitrarily time‐varying electric field in two dimensions is studied using a group‐theoretic approach. We show that this Hamiltonian can be encompassed by two, commuting SU(2) algebras, and that the two‐dimensional problem can therefore be reduced to two one‐dimensional problems. We compare the group structure for the two‐dimensional problem with that previously derived for the one‐dimensional problem, and verify that the Schwinger result is obtained under the appropriate conditions.

Hidden local gauge invariance in the one‐dimensional Heisenberg XXZ model with the general boundary terms
View Description Hide DescriptionHidden local gauge invariance in the one‐dimensional Heisenberg XXZmodel with the general boundary terms is studied in the framework of the quantum inverse scattering method. The Bethe ansatz equations are established for the model with the special boundary terms. Our results show that the Hamiltonian and its eigenvectors are explicitly gauge dependent whereas the energy eigenvalues and the Bethe ansatz equations are gauge invariant.

New solution of the wave equation with a uniformly moving point source
View Description Hide DescriptionA new solution of the inhomogeneous d’Alembert equation with the point uniformly moving charge is found. The comparison of the new solution with the Kirchhoff formula solution is performed.

Transitive symplectic manifolds in 1+2 dimensions
View Description Hide DescriptionA complete list of all transitive symplectic manifolds of the Poincaré and Galilei group in 1+2 dimensions is given.

Dielectric screening in a plasma: Some rigorous results
View Description Hide DescriptionIn this article, we give some rigorous results about the properties of the solution of Poisson’s equation describing dielectric screening by electrons trapped and/or untrapped in the Coulomb field of the test charge.

Global cocycle dynamics for infinite mean field quantum systems interacting with the boson gas
View Description Hide DescriptionIn the framework of operator algebraic quantum statistical mechanics, the global nonequilibrium dynamics for a general class of interactions of infinite mean field quantum lattice systems with the boson field is investigated. The associated interaction operators consist of arbitrary powers of the collective density operators of the mean fieldsystem and are linear with respect to the bosonic field operators. Instead of the usual perturbation expansions, here the interactingdynamics are studied by means of cocycle techniques. The cocycle methods are more appropriate to the considered class of interactions, and lead to explicit closed expressions for the dynamical automorphism groups. The cocycle equations connect the classical, collective dynamical behavior of the mean fieldsystem with the one‐boson dynamics. In physical applications such systems are due to collectively ordered finite‐level atoms or the Josephson junction in the thermodynamic limit weakly interacting with the electromagnetic field.