Dynamical theory of a bidimensional system with an infinite number of degrees of freedom

The dynamical theory of a bidimensional model of hard squares with elastic collisions is presented. The time evolution is shown to exist on a large class of infinite configurations. Moreover, it is proved that any equilibrium state, that is any solution of the equilibrium equations, is concentrated on this set of allowed initial configurations and is invariant under the time evolution.     

Propagator of a particle characterized by an internal symmetry

We examine the propagators associated with particles characterized by an internal symmetry. Of particular interest is the case of spontaneously broken Kac-Moody algebra.     

Relativistic quantum mechanics with non conserved number of particles

A covariant scheme is given for the second quantization of directly interacting particles.In a model, the space of interacting state vectors is explicitly constructed, at the price of an invariant alteration of the original framework, which is necessary owing to the presence of divergences.     

Symmetry transformations in Batalin-Vilkovisky formalism

Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation e ^s = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying _p (He ^s) = 0) determines a symmetry of the theory with the action functionalS. Research supported in part by NSF grant No. DMS-9201366.     

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model (A-model) in the case when the target space is an (m ₀|m ₁)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m ₀–m ₁)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Research supported in part by NSF grant No. DMS-9201366.     

On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non-autonomous potential.     

Quantization of gauge theories with global anomalies

Global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization.     

Quantization of first-class constraints with structure functions

As part of a program to desuperize BRST symmetries, we show how to translate the BRST construction completely in terms of standard symplectic geometry in the case where the constraints are derived from a foliation on a configuration space. The consequences of this approach on quantization are investigated. As a corollary, we solve (in a restricted setting) the problem of how to deal in quantization with (globally independent) first-class constraints with structure functions rather than structure constants.     

Universal deformations of simple Lie algebras

This Letter examines the question of the structure of the Hopf algebra deformations of the universal enveloping algebras of the simple Lie algebras. Deformations of a complex algebra A are viewed as algebras defined over formal power series rings that specialize to A when the parameters go to 0. Only the case of U(sl(2,C)) is treated but the methods are general. Under the Ansatz that the two Borel subalgebras are deformed as Hopf algebras but possibly differently, we construct a universal two-parameter deformation.     

Frame-like action and unfolded formulation for massive higher-spin fields

Unfolded equations of motion for symmetric massive bosonic fields of any spin in Minkowski and (A)dS$(A)\mathit{dS}$ spaces are presented. Manifestly gauge invariant action for a spin s?2$s?2$ massive field in any dimension is constructed in terms of gauge invariant curvatures.     

Hamiltonian operators and associative algebras with a derivation

We prove that an algebraic structure proposed by Gel'fand and Dorfman in studying Hamiltonian operators is equivalent to an associative algebra with a derivation under a unitary condition.     

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenberg model to achieve topological stability on the lattice.     

Shannon number and degrees of freedom of an image

The problem of the determination of the number of significant degrees of freedom of an image is considered. The equivalence with the Shannon number is shown to hold with a great generality.     

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK ²ⁿ is raised.In memory of Julian Schwinger     

Absence of species doubling in finite-element quantum electrodynamics

In this Letter, it will be demonstrated explicitly that the finite-element formulation of quantum electrodynamics is free from fermion doubling. We do this by (1) examining the lattice fermion propagator and using it to compute the one-loop vacuum polarization on the lattice, and (2) by an explict computation of vector and axial-vector current anomalies for an arbitrary rectangular lattice in the Schwinger model. There it is shown that requiring that the vector current be conserved necessitates the use of a square lattice, in which case the axial-vector current is anomalous.     

An example of an infinite system with zero quantum dynamical entropy

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained new improved tree graph inequalities that have led to the extra 1/N! factors. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory.This work was supported in part by the National Science Foundation under grant no. PHY-87-01329.     

Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.