Volume 39, Issue 4, April 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Gauge fixing in the partition function for generalized quantum dynamics
View Description Hide DescriptionWe discuss the problem of gauge fixing for the partition function in generalized quantum (or trace) dynamics, deriving analogs of the De Witt–Faddeev–Popov procedure and of the BRST invariance familiar in the functional integral context.

Geometric approach to inverse scattering for hydrogenlike systems in a homogeneous magnetic field
View Description Hide DescriptionWe consider the Hamiltonian of one quantum particle in a homogeneous magnetic field and a scalar potential in three space dimensions. For a given magnetic field the high velocity limit of the scattering operator uniquely determines the scalar potential if it is of short range. If, in addition, longrange potentials are present, some knowledge of (the far out tail of) the longrange part is needed to define a modified Dollard wave operator and a scattering operator. Again its high velocity limit uniquely determines the total scalar potential for a given magnetic field. We generalize our results to a system of two interacting quantum particles with opposite electric charges.

Phasespace representation of quantum state vectors
View Description Hide DescriptionPhasespace representation of quantum state vectors is obtained within the framework of the relativestate formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relativeposition state and relativemomentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phasespace functions which represent a state vector of the relevant quantum system are obtained in terms of the relativeposition states and the relativemomentum states. The absolutesquare of the phasespace function represents the probability distribution of the phasespace variables. Timeevolution of a quantum system is investigated in terms of the phasespace functions. The relations to the phasespace representations formulated by the other methods are obtained.

Feynman integral in regularized nonrelativistic quantum electrodynamics
View Description Hide DescriptionWe express the unitary time evolution in nonrelativistic regularized quantum electrodynamics at zero and positive temperature by a Feynman integral defined in terms of a complex Brownian motion. An average over the quantum electromagnetic field determines the form of the quantum mechanics in an environment of a quantum black body radiation. In this nonperturbative formulation of quantum electrodynamics we prove the existence of the classical limit We estimate an error to some approximations commonly applied in quantum radiation theory.

Semidirect product of CCR and CAR algebras and asymptotic states in quantum electrodynamics
View Description Hide DescriptionA algebra containing the CCR and CAR algebras as its subalgebras and naturally described as the semidirect product of these algebras is discussed. A particular example of this structure is considered as a model for the algebra of asymptotic fields in quantum electrodynamics in which Gauss’ law is respected. The appearance in this algebra of a phase variable related to electromagnetic potential leads to the universal charge quantization. Translationally covariant representations of this algebra with energymomentum spectrum in the future lightcone are investigated. It is shown that vacuum representations are necessarily nonregular with respect to total electromagnetic field. However, a class of translationally covariant, irreducible representations is constructed explicitly, which remain as close as possible to the vacuum, but are regular at the same time. The spectrum of energymomentum fills the whole future lightcone, but there are no vectors with energymomentum lying on a mass hyperboloid or in the origin.

Continuous time and consistent histories
View Description Hide DescriptionWe discuss the use of histories labelled by a continuous time in the approach to consistenthistories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

Classical limits for quantum maps on the torus
View Description Hide DescriptionWe provide a rigorous canonical quantization for the following toral automorphisms: cat maps, generalized kicked maps, and generalized Harper maps. For each of these systems we construct a unitary propagator and prove the existence of a welldefined classical limit. We also provide an alternative derivation of Hannay and Berry results for the cat map propagator on the plane.

Operator product expansions and stress operators
View Description Hide DescriptionWe explain in detail how to derive the operator product expansions (OPEs) among generating functionals of physical vertex operators (GFPVO) of fermionic particles (i.e., GFPVO in the Ramond sector of superstring) and the stress operators. As for OPEs among GFPVO of bosonic particles (i.e., in the Neveu–Schwarz sector of superstring) and the stress operators, they are those simply obtained by supergeneralizing OPEs among GFPVO (in bosonic string) and the stress operator.

Lieseries approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics
View Description Hide DescriptionA version of a normalform method of Kummer and Gompa, analogous to the Lieseries methods of classical mechanics, is used to rigorously construct uniform longtime approximations to the quantum evolution of a system of resonant or nonresonant anharmonic oscillators. The Hamiltonian governing the system is a selfadjoint operator in the Hilbert space given formally by Here, denotes the Hamiltonian of ν onedimensional simple harmonic oscillators and their anharmonic coupling, being a small parameter and an operator of multiplication by a smooth function of polynomial growth at infinity. We treat the case in which an arbitrary number of the frequencies of these oscillators are rationally independent, imposing a standard diophantine condition on the independent frequencies if Under these assumptions, stated precisely in the paper, an thorder approximant to the exact solution of the Schrödinger equation satisfying the initial condition is constructed recursively, where the initial state belongs to a family of smooth functions dense in H. The main result is that differs in Hnorm from by for and ε in an arbitrary compact interval This result is obtained by an approach simpler and quite different from that of Kummer and Gompa, and extends their work to oscillators with nonpolynomial couplings, under very general resonance conditions.

Generalized Pöschl–Teller, Toda, Morse potentials and group
View Description Hide DescriptionQuantum integrable systems related with group manifold in various coordinate systems defined by the various decompositions: KHK, HAH, HKH, NHN, NHK of the group on one parameter subgroups are considered. The explicit expressions for waves functions, spectra, and matrices are given.

Duality, partial supersymmetry, and arithmetic number theory
View Description Hide DescriptionWe find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry, in which some, but not all, of the operators of a theory have superpartners, and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analog of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function.

Symmetry and symmetry breaking in quantumchromo (flavor)dynamics
View Description Hide DescriptionSymmetries of flavored quantumchromo(flavor)dynamics (QCD) and their breaking are reinvestigated within the Lagrange–Hamilton frame as well as on the basis of axiomatic gauge quantum field theory aiming at the resolution of some shortcomings by which QCD is still burdened despite all its undisputable successes: The Noether energy momentum tensors arising from those general representations (substitution rules) of the translation group that leave the standard SU(3) local color gauge and global flavor invariant chiral QCDLagrangian (and hence the Euler–Lagrange field equations) form invariant are shown to contain, besides the standard color and flavor symmetric canonical part, additional terms breaking the internal SU(3) symmetries of the Lagrangian down to chiralflavor symmetry, and in case of color confinement also the color group down to color symmetry only. With these general representations (covariance conditions) of the translation group as an input, a strictly Poincaré covariant gaugequantumfield theory with color confinement is formulated and flavor Goldstone states are shown to occur with finite masses due to relative Einstein causality, the spectrum condition in color neutral sectors, translational covariance, and current conservation. Identifying the finite mass Goldstone states with pseudoscalar mesons, the axial vector part of the chiral SU(3) flavor symmetry is spontaneously broken, leaving only the chiral subgroup as a strict flavor symmetry of QCD.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Wave scattering in one dimension with absorption
View Description Hide DescriptionWave scattering is analyzed in a onedimensional nonconservative medium governed by the generalized Schrödinger equation where and are real, integrable potentials with finite first moments. Various properties of the scattering solutions are obtained. The corresponding scattering matrix is analyzed, and its small and large asymptotics are established. The bound states, which correspond to the poles of the transmission coefficient in the upperhalf complex plane, are studied in detail. When the medium is not purely absorptive, i.e., unless it is shown that there may be bound states at complex energies, degenerate bound states, and singularities of the transmission coefficient imbedded in the continuous spectrum. Some explicit examples are provided illustrating the theory.

Geodesic flow on an hyperboloid model and the twobody motion under repulsive Coulomb forces
View Description Hide DescriptionWe show that (in dimension) the flow defined by the Hamiltonian system for two charged particles of the same sign, is mapped into the geodesic flow over the nonEuclidean space: say the hyperboloid with the axis along which besides having its “boundaries at infinite” identified, it is also punctured at the point (corresponding to the collision states) and whose metric is of negative curvature.

Diagrams in classical and semiclassical perturbation theory
View Description Hide DescriptionWe describe a diagrammatic method for the Poincare–Birkhoff normal forms algorithm of classical mechanics, and indicate the use of the diagrams with an example from hydrodynamics. We also present a generalization of the diagrammatic method to quantum mechanics. The quantum diagrams can be used in the semiclassical version of normal forms developed by Graffi, Paul, and others.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Exact solutions for nonconservative twovelocity discrete Boltzmann models
View Description Hide DescriptionExtended discrete kinetic theory (which we call nonconservative), including sources, sinks, creation and annihilation of test particles, inelastic scattering,… added to the elastic collisions, was recently introduced. For instance, the mass conservation law, by adding polynomials of the mass, becomes nonconservative. To the classical twovelocity elastic collisionmodels, we associate two nonconservative models with polynomials of the first and second degree. We obtain exact stationary, similarity, dimensional and periodic solutions. For their stability we study the corresponding nodes and saddles.

Infinite moments of the partition function for random walks in a random potential
View Description Hide DescriptionWe consider a system of random walks (directed polymers) in a potential which is random in space and time. Investigations into untypical behavior and the inverse of the partition function leads to the result that, regardless of the strength of the potential and in all dimensions, all higher moments of the partition function are unbounded as time goes to infinity. One can interpret this result in a manybody quantum mechanical setting as saying that no matter how small the strength of an attractive potential, in any dimension one can find (the number of particles) large enough so that there is a bound state.

Algebraic approach to scattering theory of the Fokker–Planck equation
View Description Hide DescriptionThe onevariable Fokker–Planck equation is studied in a scattering formalism. Scattering processes are represented by paths in onedimensional space, and the paths are treated as algebraic objects that constitute infinitedimensional representations of SL(2, C). Various expressions for the scattering coefficients are derived in a systematic way by means of algebraic methods with considerations on symmetries.

A reformulation of simple liquids theory—Renormalization by one, two, and threeparticle densities
View Description Hide DescriptionWe reformulate the theory of simple liquids in a field theoretical way by taking into account the triplet potential in addition to the external potential and the pair potential The innovation here is the inversion method and the onshell expansion which are the building blocks of a novel use of Legendre transformation developed in field theory. By the inversion method, we renormalize the theory in terms of one, two, and threeparticle densities, and present a diagrammatic representation for a thermodynamical functional, which is the entropy except for a trivial constant, in terms of renormalized variables. In other words, we present an expression for the entropy in terms of only one, two, and threeparticle densities: the particle density where does not appear in the expression. The onshell condition, which is a starting point of the onshell expansion, of the thermodynamical functional thus obtained (the entropy) leads to a set of three selfconsistent equations for one, two, and threeparticle densities. Through one of the selfconsistent equations, we can systematically improve the Kirkwood’s superposition approximation for the threeparticle density. The onshell conditions for other thermodynamical functionals, also obtained in this article, are found to be extentions of various wellknown equations in the theory of simple liquids. The formulation presented here is complementary to the conventional resummation techniques for renormalization of diagrams. In the present formulation, we do not have to care about the topological structure of diagrams, often characterized by the irreducibility of diagrams. Instead, by a perturbative calculation, we can automatically single out the diagrams with the topological structure predicted by the resummation techniques.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Integrable coupled KdV systems
View Description Hide DescriptionWe give the conditions for a system of coupled Korteweg de Vries (KdV) type of equations to be integrable. We find the recursion operators of each subclass and give all examples for
