Particles and scaling for lattice fields and Ising models

The conjectured inequality ⁽⁶⁾0 leads to the existence of _d ⁴ fields and the scaling (continuum) limit ford-dimensional Ising models. Assuming ⁽⁶⁾0 and Lorentz covariance of this construction, we show that ford6 these _d ⁴ fields are free fields unless the field strength renormalizationZ ^–1 diverges. Let be the bare charge and the lattice spacing. Under the same assumptions (⁽⁶⁾0, Lorentz covariance andd6) we show that if ^4–d is bounded as 0, thenZ ^–1 is bounded and the limit field is free.Supported in part by the National Science Foundation under Grant MPS 74-13252Supported in part by the National Science Foundation under Grant MPS 75-21212     

Non-translation invariant Gibbs states with coexisting phases III: Analyticity properties

We consider the spatially inhomogeneous Gibbs states for the three dimensional Ising and Widom-Rowlinson models. We prove the analyticity inz=exp(–2J) for small |z| of the spin correlation functions of these Gibbs states and of the surface tension.Supported in part by N.S.F. Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Critical exponents and elementary particles

Particles are shown to exist for a.e. value of the mass in single phase ⁴ lattice and continuum field theories and nearest neighbor Ising models. The particles occur in the form of poles at imaginary (Minkowski) momenta of the Fourier transformed two point function. The new inequalitydm ²/dZ, where =m ₀ ² is a bare mass² andZ is the strength of the particle pole, is basic to our method. This inequality implies inequalities for critical exponents.Supported in part by the National Science Foundation under grant PHY 76-17191Supported in part by the National Science Foundation under grant MPS 75-21212     

Soliton mass and surface tension in the (λ|Ø|4)2 quantum field model

The spectrum of the mass operator on the soliton sectors of the anisotropic (|ø|⁴)₂—and the (ø⁴)₂—quantum field models in the two phase region is analyzed. It is proven that, for small enough >0, the mass gapm _s() on the soliton sector is positive, andm _s()=0(^–1). This involves estimatingm _s() from below by a quantity () analogous to the surface tension in the statistical mechanics of two dimensional, classical spin systems and then estimating () by methods of Euclidean field theory. In principle, our methods apply to any two dimensional quantum field model with a spontaneously broken, internal symmetry group.A Sloan Foundation Fellow; Research supported in part by the U.S. National Science Foundation under Grant No. MPS 75-11864.Supported in part by the National Science Foundation under Grant No. PHY 76-17191     

Semiclassical quantum mechanics

We consider the 0 limit of the quantum dynamics generated by the HamiltonianH()=–(²/2m)+V. We prove that the evolution of certain Gaussian states is determined asymptotically as 0 by classical mechanics. For suitable potentialsV inn3 dimensions, our estimates are uniform in time and our results hold for scattering theory.Supported in part by the National Science Foundation under Grant PHY 78-08066     

Instantons in aU(1) lattice gauge theory: A Coulomb dipole gas

We study the decompositionA=A _I +A _SW of aU(1) lattice gauge field into instanton and spin wave parts. The action also decomposes,A=A _I +A _SW +R. HereA _I is a Coulomb dipole gas,A _SW is a zero mass free field, andR is a higher order remainder. We studyA _I in detail, ford4, in the dilute gas case (which corresponds to the low temperature limit of the gauge field theory). We establish the leading behavior of the free energy:f ^–d a. Here is the lattice spacing,a is a geometrical constant and is an activity defined in terms of a small number of instanton configurations. Our methods suggest the absence of screening in the dilute dipole gas,d4, in contrast to Debye screening for the dilute monopole gas.Supported in part by the National Science Foundation under Grant PHY 76-17191Supported in part by the National Science Foundation under Grant PHY 75-21212     

Theory of hyperfine anomalies in muonic atoms

Negative muon spin precession experiments by Yamazaki et al. have found giant hyperfine anomalies in muonic atoms ranging from a few percent up to 36%. In order to understand their results, we present Breit interaction calculations based on atomic self-consistent unrestricted Dirac-Fock solutions which explicitly include all electrons and the negative muon. The Breit interaction results (including the relativistic correction for the bound muong-factor), vary from near zero for ^–O/N to –5% for ^–Pd/Rh; this latter is much larger than the calculated muonic or nuclear Bohr-Weisskopf anomalies and much smaller than the 36% measured value. For ^}Ni/Co we find a calculated range of results (depending on assumed electronic configurations) of –2.3 to –2.7% in excellent agreement with recent measurements of the Yamazaki group. This excellent agreement in ^}Ni/Co provides strong support for the earlier suggestions that the discrepancy in the case of ^–Pd/Rh is due to experimental factors.This work was supported by the U.S. National Science Foundation (DMR Grant No. 82-16543) and the U.S. Department of Energy.     

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

New super-selection sectors (“soliton-states”) in two dimensional Bose quantum field models

A rigorous construction of new super-selection selectors — so-called soliton-sectors — for the quantum sine-Gordon equation and the (·)²-quantum field models with explicitly broken isospin symmetry in two space-time dimensions is presented. These sectors are eigenspaces of the chargeQdx(grad )(x) with non-zero eigenvalue. The scattering theory for quantum solitons is briefly discussed and shown to have consequences for the physics in the vacuum sector. A general theory is developed which explains why soliton-sectors may exist for theories in two but not in four space-time dimensions except possibly for non-abelian Yang-Mills theories.Supported in part by the National Science Foundation under Grant NSF-GP-39048 and by the ETH, Zürich.     

Existence of the critical point in φ4 field theory

We consider the ⁴ quantum field theory in two and three spacetime dimensions. In the single phase region the physical mass (inverse correlation length)m() decreases continuously to zero as the bare mass parameter approaches a critical value _c from above. In three dimensions the critical point _c is in the single phase region and the physical mass vanishes there,m( _c )=0.A consequence of our results is that the critical exponentv governing the approach to infinite correlations is bounded below (rigorously) by its classical value, 1/2.Supported in part by the National Science Foundation under Grant MPS74-13252     

The statistics of Curie-Weiss models

LetS _n denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofS _n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S _n-nm)/n^1–1/2k has a limiting distribution with density proportional to exp[-_s ^2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.Research supported in part by National Science Foundation Grants MPS 76-06644 (to RSE) and MPS 74-04870 A01 (to CMN).     

A possible constructive approach to ϕ 4 4 III

We continue our investigation into a constructive approach for _d ⁴ and prove under the hypotheses of our previous work that on a lattice and in the single phase region the theory is uniquely determined by the intermediate renormalization conditions. Then the Gaussian theory and the Ising model are the two extremal cases for all _d ⁴ theories.On leave of absence from the Freie Universität BerlinSupported in part by the Deutsche Forschungsgemeinschaft and by the National Science Foundation under Grant MPS 75-21212     

Atomic volume dependence of the153Eu isomer shift in ScxY1−x

Mössbauer studies of dilute¹⁵³Eu in Sm_0.05Sc _x Y_0.95–x (0x0.95) reveal large changes in the isomer shift as a function ofx. The strong dependence of the isomer shift on alloy composition, is interpreted in terms of changes in the charge density at the Eu nucleus ((0)), which result from changes in atomic volume. The isomer shift was found to depend linearly on V/V (V/V changes between 0 and –0.3). A value of (–17.5±2.0)a ₀ ^–3 was obtained for d(0)/d lnV.This research was supported by a grant from the United States—Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

The Schrödinger equation for quantum fields with nonlinear nonlocal scattering

This paper considers perturbationsH=H ₀+V of the Hamiltonian operatorH ₀ of a free scalar Boson field.V is a polynomial in the annihilation creation operators. Terms of any order are allowed inV, but point interactions, such as :0(x)⁴(x)⁴:dx, are not considered. Unnormalized solutions for the Schrödinger equation are found. For 0, these solutions have a partial asymptotic expansion in powers of . The set of all possible pertubation termsV forms a Lie algebra. General properties of this Lie algebra are investigated.This work was supported in part by the National Science Foundation, NSF GP-4364.     

Geometric analysis of φ4 fields and Ising models. Parts I and II

We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) ⁴ Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the ⁴ field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the universality of hyperscaling.A. P. Sloan Foundation Research Fellow. Supported in part by the National Science Foundation Grant PHY-78-25390-A02     

TheN 7/5 law for charged bosons

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfiesE ₀–AN ^7/5. We prove that 7/5 is the correct power by proving thatE ₀–BN ^7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed thatE ₀–CN^1/4. This 1/4 law is also validated here by showing thatE ₀–DN^1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.Work partially supported by U.S. National Science Foundation grant DMS 8600748Work partially supported by U.S. National Science Foundation grant PHY85-15288-A01Work supported by Alfred Sloan Foundation dissertation fellowship     

Search for emission of nucleons and nucleon pairs following muon capture in selected light nuclei

Preliminary results of measurements of proton energy spectra and search for coincidentp, n pair emission following negative muon capture in¹²C,¹⁶O,²⁷Al are given. A very clear signal for proton emission is found, which can be exploited in future detailed studies.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.This work was supported in part by the U.S. National Science Foundation, grant 83-10044-A 2, the German Bundesministerium für Forschung und Technologie, the International Buero Julich, and the Swiss National Science Foundation.     

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     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

Decay of correlations in surface models

A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e ^– In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.Supported in part by the National Science Foundation under grant No. PHY 79-16812.