Volume 39, Issue 2, February 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Diffeomorphism invariant integrable field theories and hypersurface motions in Riemannian manifolds
View Description Hide DescriptionWe discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a secondorder partial differential equation for the corresponding time function at which the hypersurface passes the point Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphisminvariant field theories. At least in some (infinite class of) cases, which could be viewed as a largevolume limit of Euclidean branes moving in an arbitrary dimensional Riemannian manifold, the models are integrable: In the timefunction formulation the equation becomes linear [with a harmonic function on the embedding Riemannian manifold]. We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in by methods used in electrostatics and point out an additional gradient flow structure in In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.

New integrable generalizations of Calogero–Moser quantum problem
View Description Hide DescriptionA oneparameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrödinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value.

Augmented temporal logic formalism for historiesbased generalized quantum mechanics
View Description Hide DescriptionAn axiomatic development of dynamics of systems in the framework of histories is given which contains the history versions of classical and traditional quantum mechanics as special cases. We consider theories which admit a quasitemporal structure (a generalization of the concept of time using partial semigroups) and whose “single time” propositions have the mathematical structure of a logic; isomorphism of logics at different “instants of time” is not assumed. The concept of directed partial semigroup is introduced to incorporate the concept of direction of flow of time in quasitemporal theories. Starting with a few simple axioms, the space of history propositions is explicitly constructed and shown to be an orthoalgebra (as envisaged in the scheme of Isham and Linden [J. Math. Phys. 35, 5452–5476 (1994)]); its subspace consisting of history filters (“homogeneous histories”) is a meet semilattice. The partial semigroups employed allow semiinfinite irreducible decompositions.

Shadow of noncommutativity
View Description Hide DescriptionWe analyze the structure of the limit of a family of algebras describing noncommutative versions of space–time, with κ a parameter of noncommutativity. Assuming the Poincaré covariance of the limit, we show that, besides the algebra of functions on Minkowski space, must contain a nontrivial extra factor which is Lorentz covariant and which does not commute with the functions whenever it is not commutative. We give a general description of the possibilities and analyze some representative examples.

The role of a form of vector potential — normalization of the antisymmetric gauge
View Description Hide DescriptionResults obtained for the antisymmetric gauge by Brown and Zak are compared with those based on pure grouptheoretical considerations and corresponding to the Landau gauge . Imposing the periodic boundary conditions one has to be very careful since the first gauge leads to a factor system which is not normalized. A period introduced in Brown’s and Zak’s papers should be considered as a magnetic one, whereas the crystal period is in fact . The “normalization” procedure proposed here shows the equivalence of Brown’s, Zak’s, and other approaches. It also indicates the importance of the concept of magnetic cells. Moreover, it is shown that factor systems (of projective representations and central extensions) are gaugedependent, whereas a commutator of two magnetic translations is gaugeindependent. This result indicates that a form of the vector potential (a gauge) is also important in physical investigations.

Renormalized contact potential in two dimensions
View Description Hide DescriptionWe obtain for the attractive Dirac δfunction potential in twodimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in degrees of freedom (in the center of momentum frame) when there are particles. The effective Hamiltonian for particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The threebody case is examined in detail, and in this case a variational estimate of the groundstate energy is given.

Splitting in large dimension and infrared estimates. II. Moment inequalities
View Description Hide DescriptionThis is the continuation of notes written for the NATOASI conference in Il Ciocco (September 96) consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the Schrödinger operator and the proof of infrared estimates for quantities attached to Gaussiantype measures. These notes were mainly reporting on the “old” contributions of Dyson, Fröhlich, Glimm, Jaffe, Lieb, Simon, and Spencer (in the 1970s) in connection with more recent contributions of Pastur, Khoruzhenko, Barbulyak, and Kondrat’ev which treat in general more sophisticated models. Here we concentrate on the simplest model related to field theory and extend the results of Barbulyak and Kondrat’ev by mixing ideas coming from Pastur and Khozurenko related to the use of Bogolyubov’s inequality with classical inequalities due to Ginibre, Lebowitz, Sokal, and others, or, in the case when the temperature is zero, by applying rather elementary estimates on Schrödinger operators, in order to find lower bounds for secondorder moments attached to the measure with This question was “left to the reader” in lectures given by J. Fröhlich in 1976 [Acta Phys. Austriaca, Suppl. XV, 133–269 (1976)], but we think that it is worthwhile to do this “homework” carefully.

Unitary implementations of oneparameter squeezing groups
View Description Hide DescriptionFor the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic oneparameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.

Quantum theory of the real and the complexified projective line
View Description Hide DescriptionQuantum deformations of sets of points of the real and the complexified projective line are constructed. These deformations depend on the deformation parameter and certain further parameters The deformations for which the subspace of polynomials of degree three has a basis of ordered monomials are selected. We show that the corresponding algebras of three points have “polynomiality.” Invariant elements which turn out to be cross ratios in the classical limit are defined. For the special case a quantum cross ratio with properties similar to the classical case is presented. As an application a quantum version of the real Euclidean distance is given.

Particles in singular magnetic field
View Description Hide DescriptionAn algebraic formalism for description of quantum states of charged particle with spin moving in twodimensional space under the influence of a singular magnetic field is developed in terms of graded algebras. The fundamental assumption is that the particle is transformed into a composite system which consists of quasiparticles, quasiholes, and magnetic fluxes. Such a system is endowed with generalized statistics determined by a grading group and a commutation factor on it. Composite systems corresponding to the quantum Hall effect and the electronic magnetotransport anomaly are described. The Fock space representation is also given.

Vacuum polarization and the geometric phase: Gauge invariance
View Description Hide DescriptionA nonperturbative approach to the vacuum polarization for quantized fermions in external vector potentials is discussed. It is shown that by a suitable choice of counterterms the vacuum polarization phase is both gauge and renormalization independent, within a large class of nonperturbative renormalizations.

Ladder operators for isospectral oscillators
View Description Hide DescriptionWe present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed “distorted” Heisenberg algebra (including the generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the nonlocal operators concerned are given in terms of pseudodifferential operators. Using the new annihilation operators, new classes of coherent states are constructed for isospectral oscillator Hamiltonians. The corresponding Fock–Bargmann representations are also considered, with specific reference to the order of the entire function family in each case.

Algebraic characterization of vector supersymmetry in topological field theories
View Description Hide DescriptionA cohomological characterization of a class of linearly broken Ward identities is provided. The examples of the topological vector supersymmetry and of the Landau ghost equation are discussed. The existence of these linearly broken Ward identities lies in the BRST exactness of suitable antifield dependent cocycles with negative ghost number. This algebraic set up turns out to be related to the cohomological reformulation of the Noether theorem given by G. Barnich, F. Brandt, and M. Henneaux, Commun. Math. Phys. 174, 57, 93 (1995); M. Henneaux, J. Pure Appl. Algebra100, 3 (1995).

Quantization of a particle in a background Yang–Mills field
View Description Hide DescriptionTwo classes of observables defined on the phase space of a particle are quantized, and the effects of the Yang–Mills field are discussed in the context of geometric quantization.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Hamiltonian systems on cosymplectic manifolds
View Description Hide DescriptionThe Hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories. To do this, the notion of cosymplectic manifold is introduced, and a suitable Hamiltonian formalism is developed so that the field equations for scalar and vector Hamiltonian functions are derived.

On the derivation of boundary integral equations for scattering by an infinite twodimensional rough surface
View Description Hide DescriptionA plane acoustic wave is incident upon an infinite, rough, impenetrable surface. The aim is to find the scattered field by deriving a boundary integral equation over , using Green’s theorem and the freespace Green’s function. This requires careful consideration of certain integrals over a large hemisphere of radius ; it is known that these integrals vanish as if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. It is shown that the wellknown Helmholtz integral equation is not valid in all circumstances. For example, it is not valid when the scattered field includes plane waves propagating away from along the axis of the hemisphere. In particular, it is not valid for the simplest possible problem of a plane wave at normal incidence to an infinite flat plane. Some suggestions for modified integral equations are discussed.

Integrable structure of the new Calogero models
View Description Hide DescriptionWe show that the integrability of the dynamical system recently proposed by Calogero and characterized by the Hamiltonian is due to a simple algebraic structure. It is shown that this system is canonically equivalent to the classical Heisenberg magnet with longrange interactions. Our method shows clearly how these types of systems can be generalized and provides the mechanism of integrability of a large class of similar systems.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Quantum critical fluctuations in a ferroelectric model: Quasiaverage approach
View Description Hide DescriptionWe show that critical fluctuations in a quantum ferroelectric model depend not only on the range of harmonic interactions but also on infinitesimal perturbations breaking symmetry of the model. This means that algebra of fluctuation operators is degenerated for (unique) critical equilibrium state.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Colored noise and a characteristic level crossing problem
View Description Hide DescriptionIn this paper we discuss the activation of a dynamical system of one spatial variable forced by an external Gaussian stationary colored noise. The model is an analog of the Smoluchowski equation when external fluctuations are present. We assume that the underlying deterministic dynamics are derived from a potential forming a well. We enlarge the state space by including the noise variable, and then use halfrange expansion techniques, singular perturbations, and matched asymptotics to solve the boundary value problem for the steadystate probability density function, in the asymptotic limit of small correlation time and over the entire regime of noise strength. We find a uniformly valid expression for the activation rate to the top of the potential barrier (a characteristic point for the unforced dynamical system). We show the important effects of the boundary behavior of the process on its activation rate. If the intensity of the noise is small compared to the autocorrelation of the noise, then the effective activation energy is modified by the noise autocorrelation. We consider an example of the activation process in a double well potential, and compare our uniformly valid results with results reported in the literature.

Variational principles for nonlinear dynamical systems
View Description Hide DescriptionA variational method for Hamiltonian systems is analyzed. Two different variational characterization for the frequency of nonlinear oscillations is also supplied for nonHamiltonian systems.
