Parallel transport in the determinant line bundle: The zero index case

For a product family of invertible Weyl operators on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator onR×X.Supported in party by NSF Grants PHY8605978 and PHY-82-15249 and the Robert A. Welch FoundationSupposed in part by NSF Grant PHY-82-15249     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

A general relation between kink-exchange and kink-rotation

The normal correlation between spin and statistics is shown to be valid for arbitrary kinks, among them theSU(n) Skyrmions forn3. It is assumed in the proof that no gauge-ambiguity attaches to the values of the underlying scalar field, and that conversely each configuration of this field is represented quantum mechanically by a Hilbert subspace of dimension precisely one.Supported in part by NSF Grant No. PHY 83-18350     

A general Lee-Yang theorem for one-component and multicomponent ferromagnets

We show that any measure on ⁿ possessing the Lee-Yang property retains that property when multiplied by a ferromagnetic pair interaction. Newman's Lee-Yang theorem for one-component ferromagnets with general single-spin measure is an immediate consequence. We also prove an analogous result for two-component ferromagnets. ForN-component ferromagnets (N 3), we prove a Lee-Yang theorem when the interaction is sufficiently anisotropic.Research supported in part by NSF grant PHY 78-25390 A01Research supported in part by NSF grant PHY 78-23952     

Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

We study the bifurcation of radially symmetric solutions of +f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.Research supported in part by the NSF under Grant No. MCS-800 2337     

The short distance behavior of (φ 4)3

We consider the ₃ ⁴ quantum field theory on a torus and study the short distance behavior. We reproduce the standard result that the singularities can be removed by a simple mass renormalization. For the resulting model we give anL _p bound on the short distance regularity of the correlation functions. To obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields.Research supported by NSF Grant DMS 9102564Research supported by NSF Grant PHY9200278.Research supported by the Natural Sciences and Engineering Research Council of Canada.     

On the decay of correlations inSO(n)-symmetric ferromagnets

We prove that for low temperaturesT the spin-spin correlation function of the two-dimensional classicalSO(n)-symmetric Ising ferromagnet decays faster than |x|^–constT providedn2. We also discuss a nearest neighbor continuous spin model, with spins restricted to a finite interval, where we show that the spin-spin correlation function decays exponentially in any number of dimensions.Work supported in part by NSF, Grant PHY76-17191A Sloan Fellow     

Modes of internal motion in three-body systems

We propose an approach to investigate the possible internal motions of three-body quantum-mechanical systems held together by pairwise potentials. The method utilizes the shape-density functions which we define and then extract from the eigenwave functions of the system. Through these densities we determine the dominant geometric configurations and modes of internal motion of the 0⁺, 2⁺ and 4⁺ states of a model three-boson system with and without a repulsive core in the interaction.Research supported in part by NSF Grant No. PHY83-06584 and by The Science Fund of the Chinese Academy of Sciences     

Infrared bounds and the peierls argument in two dimensions

We propose a definition of contours for spin systems which leads to improved estimates on the region of parameters where several phases coexist. We discuss as examples anisotropic rotators and a ⁴ lattice field theory. Our contours are estimated using infrared bounds and they are related to those of Euclidean Field Theory.Part of this work was done while Jean Bricmont was at the Mathematics Department of Princeton University and Jean-Raymond Fontaine was at the Mathematics Department of Rutgers University. This work was supported by NSF Grant No. MCS 78-01885 at Princeton University and by NSF Grant No. PHY 78-15920 at Rutgers University     

The non-relativistic limit ofP(ϕ)2 quantum field theories: Two-particle phenomena

It is proved that for two-particle phenomena theP()₂ quantum field theories with speed of lightc converge to non-relativistic quantum mechanics with a function potential in the limitc.Supported by NSF Grant No. PHY 7506746     

Rigorous proof of the high-temperature Josephson inequality for critical exponents

I give a rigorous proof of the high-temperature Josephson inequalitydv 2–, following the original ideas of Josephson. The proof is applicable to a class of models including the ferromagnetic Ising model and the ⁴ lattice field theory.Research supported in part by NSF Grant PHY 78-23952.     

Quantum sets and clifford algebras

The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton. This material is based upon work supported in part by NSF Grant No. PHY8007921.     

The isoperimetric problem for pinwheel tilings

In aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)² arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; ifP andQ are points separated by large Euclidean distanceR, then the shortest path along tile edges fromP toQ has lengthR+o(R).Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113.Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship and Texas ARP Grant 003658-037.     

Analyticity of a hard-core multicomponent lattice gas

A multicomponent anti-Widom-Rowlinson lattice gas is introduced. An arbitrary numberM of particle types is permitted, all having the same activity. The only interactions are nearest-neighbor exclusions oflike particles (analogous to map-coloring problems). For any lattice it is shown that there is a finite numberM ₀ (depending only on the coordination number of the lattice) such that for allMM ₀ the infinite volume correlation functions exist and are analytic functions of the activity, for all positive values of the common activity.Research supported in part by NSF Grant No. GP-33535X, AFOSR Grant No. 73-2430B, and NSF Grant No. MPS75-20638.     

Lattice systems with a continuous symmetry

We prove that the expansion in powers of the temperatureT of the correlation functions and the free energy of the plane rotator model on ad-dimensional lattice is asymptotic to all orders inT. The leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series. We also prove the inverse power decay of the pair correlation at low temperatures ford=3.Supported by NSF Grant No. MCS 78-01885Supported by NSF Grant No. PHY 78-15920Supported by NSF Grant No. DMR 73-04355Supported by NSF Grant No. PHY-7825390 A01On leave from: Institut de Physique Théorique, Université de Louvain, BelgiumAlso: Department of Physics     

1/n expansion for a Quantum Field Model

A nonperturbative study of the 1/n expansion in Euclidean Quantum Field Theory is started. The expansion is shown to be asymptotic to the vacuum energy of the (²) ₂ ² model, for arbitrary coupling constant.Supported in part by the National Science Foundation under Grant No. PHY 79-16812     

Free fermions and the Alexander-Conway polynomial

We show how the Conway Alexander polynomial arises from theq deformation of (Z ₂ graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU _q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU _q> sl(1, 1) algebra describes free fermions propagating on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.Work supported in part by NSF grant no. DMS-8822602Work supported in part by the NSF: grant nos. PYI PHY 86-57788 and PHY 90-00386 and by CNRS, France     

Localization for random Schrödinger operators with correlated potentials

We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|^–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627     

Non-translation invariant Gibbs states with coexisting phases

We investigate the spatially inhomogeneous states of two component,A - B, Widom-Rowlinson type lattice systems. When the fugacity of the two components are equal and large, these systems can exist in two different homogeneous (translation invariant) pure phases oneA-rich and oneB-rich. We consider now the system in a box with boundaries favoring the segregation of these two phases into top and bottom parts of the box. Utilizing methods due to Dobrushin we prove the existence, in three or more dimensions, of a sharp interface for the system which persists in the limit of the size of the box going to infinity. We also give some background on rigorous results for the interface problem in Ising spin systems.Supported in part by NSF Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Singular continuous spectrum for palindromic Schrödinger operators

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x _n =1 _J (₀+n), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational and every half-open intervalJ.Work partially supported by NSERC.This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material.