Infinite volume limits of the canonical free Bose gas states on the Weyl algebra

It was shown by Araki and Woods that the infinite free Bose gas can be described by states on the Weyl algebra; they conjectured a certain family of states parameterized by temperature and density to be the infinite volume limit of the Gibbs canonical states. We show here that this conjecture is correct. We show that the volume dependent canonical states are equicontinuous in the density by a detailed calculation and a combinatorial result that gives cancellations. This allows us to develop a method of Kac that connects the canonical states explicitly with the grand canonical states which are more easily controlled in the infinite volume limit.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

Finite nuclei in the reggeon “toy model”

Hadron–nucleus amplitudes at high energies are studied in the “toy” Regge model in zero transverse dimension for finite nuclei, when the standard series of fan diagrams is converted into a finite sum and loses physical sense at quite low energies. Taking into account all the loop contributions by numerical methods we find a physically meaningful amplitudes at all energies. They practically coincide with the amplitudes for infinite nuclei. A surprising result is that for finite nuclei and small enough triple pomeron coupling the infinite series of fan diagrams describes the amplitude quite well in spite of the fact that in reality the series should be cut and as such deprived of any physical sense at high energies.     

Phase transitions in quantum field theory. I. Phases of the Thirring model

In this series of papers we exhibit and analyse phase transitions in quantum field theory. In this paper we consider the Thirring model. We show that when the interaction becomes sufficiently attractive there is a transition to a vacuum that is ‘dead” in the sense there are no finite energy excitations. Nevertheless the corresponding continuum Green's functions exist. We make this demonstration precise by considering the model on a lattice and constructing the continuum limit explicitly on either side of the critical point. For this we extensively use the connection between the $\text{spin}\text{-}\text{1}\text{2}$x-y-z chain and the lattice model. We also show a new continuum theory with four fermion interactions exists in 1 + 1 dimensions. This theory corresponds to taking the continuum limit of the spin chain in absence of any external magnetic field. Its Hamiltonian differs from that of the Thirring model by addition of fermion number operator with an infinite coefficient and is not renormalizable in the conventional sense. It has more interesting critical properties and a different spectrum.     

On the Stationary BBGKY Hierarchy for Equilibrium States

We consider infinite classical systems of particles interacting via a smooth, stable and regular two-body potential. We establish a new direct integration method to construct the solutions of the stationary BBGKY hierarchy, assuming the usual Gaussian distribution of momenta. We prove equivalence between the corresponding infinite hierarchy and the Kirkwood–Salsburg equations. A problem of existence and uniqueness of the solutions of the hierarchy with appropriate boundary conditions is thus solved for low densities. The result is extended in a milder sense to systems with a hard core interaction.     

Are nonrenormalizable gauge theories renormalizable?

We raise the issue whether gauge theories, that are not renormalizable in the usual powercounting sense, are nevertheless renormalizable in the modern sense that all divergences can be cancelled by renormalization of the infinite number of terms in the bare action. We find that a theory is renormalizable in this sense if the a priori constraints that we impose on the form of the bare action correspond to the cohomology of the BRST-transformations generated by the action. Recent cohomology theorems of Bamich, Brandt, and Henneaux are used to show that conventionally nomenormalizable theories of Yang-Mills fields (such as quantum chromodynamics with heavy quarks integrated out) and/or gravitation are renormalizable in the modern sense.     

Pathologies in the sticky limit of hard sphere Yukawa models for colloidal fluids: a possible correction

A known ‘sticky hard sphere’ model, starting from a hard sphere Yukawa potential and taking the limit of infinite amplitude and vanishing range with their product remaining constant, is shown to be ill-defined. This is because its Hamiltonian (which we call SHS2) leads to an exact second virial coefficient which diverges, unlike that of Baxter's original model (SHS1). This deficiency has never been observed so far, since the linearization implicit in the ‘mean spherical approximation’ (MSA), within which the model is analytically solvable, partly masks such a pathology. To overcome this drawback and retain some useful features of SHS2, we propose both a new model (SHS3) and a new closure (‘modified MSA’), whose combination yields an analytical solution formally identical with the SHS2-MSA solution. This mapping allows the recovery of many results derived from SHS2, after a re-interpretation within a correct framework. Possible developments are indicated.     

Influence of the Pseudopotential Nonlocality on the Calculated Optical Characteristics of Crystals

The standard procedure used to calculate the optical characteristics of crystals is the replacement of the coordinate operator, which is not well defined in the case of infinite periodic systems, by the momentum operator. This is correct only for the local potential, and in general, the replacement leads to the occurrence of the velocity operator. In the present study, the influence of the appropriate corrections, analyzed previously by other researchers only for atoms, is examined for optical spectra of a wide class of crystals.     

具有电子旋转方向的旋转间断的稳定性

用一维混合粒子编码研究了具有电子旋转方向的对称和非对称旋转间断的稳定性。发现在相对窄的过渡层厚度下,具有电子旋转方向的旋转间断是不稳定的,它趋向于发展成离子旋转方向的旋转间断和一些MHD波。这个发展过程部分地受到电子温度的限制。当过渡层厚度增加到足够宽时,电子旋转方向的旋转间断变成稳定的。本文简单地讨论了导致这种旋转间断不稳定的可能原因。     

On the construction of quantized gauge fields

In this paper the construction of the two-dimensional abelian Higgs model begun in two earlier articles is completed. First we show how to remove the remaining ultraviolet cutoff on the gauge field, then we construct the infinite volume limit and verify the axioms of Osterwalder and Schrader for the expectation values of gauge invariant local fields. Finally it is shown that an auxiliary gauge field mass that was introduced to avoid infrared problems can be safely removed.Research supported in part N.S.F. Grant MCS 79-02490Research partly supported by Centre National de la Recherche Scientifique     

BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals

It is common knowledge that in an infinite periodic medium, for instance, an infinite photonic crystal, the direction of propagation of a monochromatic wave packet is given by the normal to the isofrequency diagram. We show that this is no longer true in a finite size medium, due to the existence of evanescent waves near the interfaces of the photonic crystal. We derive a renormalized isofrequency diagram giving the correct direction. We give a physical interpretation, showing that this phenomenon can be considered as a generalized Goos-H?nchen effect.     

Decay of correlations in classical lattice models at high temperature

In classical statistical mechanical lattice models with many body potentials of finite or infinite range and arbitrary spin it is shown that the truncated pair correlation function decays in the same weighted summability sense as the potential, at high temperature.Research partially supported by the National Science Foundation under Grant MCS 78-00680     

The relativistic quantum fields of the D = 2 Ising model

We identify the correct field operators which correspond to the order and disorder variables of the D = 2 Ising model. After infinite resummations we obtain equality between our correlation functions and the correlation functions of McCoy, Tracy and Wu [4].     

A comment on the surface dielectric sum rule

We show that our ?;-sum rule for the electron response function at a surface is not satisfied in the ?-approximation (semi-classical infinite barrier model), and the recent objection to our procedure by Griffin and Harris is shown to be invalid. Although this approximation is useful for many purposes it does not give a correct result for the plasmon contribution to the surface energy, a fact which is directly connected with its failure to satisfy the ?;-sum rule.     

On the classical lump of Yang-Mills fields

It is shown that for pure Yang-Mills fields there is no lump phenomena if the total energy is infinite and diverges mildly in a certain sense. This improves the well-known classical result on the absence of a lump for the finite total energy case. Some exact lump solutions are obtained showing that the energy condition cannot be removed. The results are valid for more general fields.This work was supported by the Chinese Natural Science Foundation, the Chinese Fund of Doctor Programs, and the University of Paris VI, and University of Bourgogne.     

Ladder operator for the one-dimensional Hubbard model

The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.     

On the possibility of an incorrect choice of a quantization volume

The specific features of the choice of an incorrect quantization volume, based on the example of the preparation of a multimode quantum state in parametric scattering, are considered. A small definitely incorrect volume that substantially simplifies the solution of the problem is chosen; it is shown to yield correct results except for an insignificant phase factor, which is proved by comparison with a more accurate correct (in the sense of the quantization volume) solution. This provides hope that the solution of quantum problems can be considerably simplified, at least to a first approximation.     

Infinite symmetry algebras of extended supergravity theories

When extended supergravity theories with noncompact symmetry groups are written in a physical gauge, the noncompact symmetries join with the supersymmetries to generate an infinite-dimensional algebra. The details are worked out explicitly for a two-dimensional theory with an SU(1, 1) internal symmetry. Our analysis confirms the observation of Ellis et al. that the infinite rigid superalgebra should be obtained from the finite-dimensional local superalgebra by replacing scalar fields with their asymptotic values at infinity. The infinite algebra is described by extending the super-Poincaré generators to functions on the coset space defined by the scalar fields at infinity. While mathematically nontrivial, this result is, in a certain sense, trivial from a physical point of view.     

The validity of perturbation theory for the O(N) non-linear sigma models

Recently it has been claimed that ordinary perturbation theory (OPT) gives incorrect weak coupling expansions for lattice O(N) non-linear sigma models in the infinite volume limit, and in particular that the two-dimensional non-abelian models are not asymptotically free, contrary to previous findings. Here it is argued that the problem occurs only for one-dimensional infinite lattices, and that in general, OPT gives correct expansions if physical quantities are first computed on a finite lattice, and the infinite volume limit is taken at the end. In one dimension the expansion is sensitive to boundary conditions because of the severe infrared behavior, but this is not expected to happen in higher dimensions. It is concluded that spin configurations which are far from the perturbative vacuum have too small a measure in the path integral to invalidate OPT, even though they are energetically allowed for non-zero values of the coupling.     

Triviality of a model of particles with point interactions in the thermodynamic limit

We consider a model of fermions interacting via point interactions, defined via a certain weighted Dirichlet form. While for two particles the interaction corresponds to infinite scattering length, the presence of further particles effectively decreases the interaction strength. We show that the model becomes trivial in the thermodynamic limit, in the sense that the free energy density at any given particle density and temperature agrees with the corresponding expression for non-interacting particles.