Laurence G. Yaffe

Adiabatic Theorems and Applications to the Quantum Hall Effect

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.

The density of instantons at finite temperature

Abstract The temperature dependence of the complete SU( N ) instanton density is evaluated. Large scale instantons are found to be suppressed at high temperature.

Finite temperature SU(2) lattice gauge theory

We discuss SU(2) lattice gauge theories at non-zero temperature and prove several rigorous results including i) the absence of confinement for sufficiently high temperature in the pure gauge theory, and ii) the absence of spontaneous chiral symmetry breaking for sufficiently high temperature in the theory with massless fundamental representation fermions.

Spherically symmetric classical solutions in SU(2) gauge theory with a Higgs field

Abstract A consistent ansatz for time dependent classical solutions in an SU(2) gauge theory with a doublet Higgs field is presented. The (3+1)-dimensional field equations are reduced to those of an effective (1+1)-dimensional theory. This ansatz describes solutions which travel between topologically distinct classical vacua of the non-abelian gauge theory. The real time version of these solutions describes the creation and decay of the unstable static “sphaleron”, the imaginary time version describes a euclidean instanton.

The coherent state variational algorithm: A numerical method for solving large-N gauge theories

This paper presents a new approach for studying large-N gauge theories which directly exploits the classical nature of the N → ∞ limit. This method supplies a practical algorithm for computing and minimizing the classical hamiltonian (or effective action) which governs N = ∞ dynamics, and allows one to calculate physical quantities such as the mass spectrum or scattering amplitudes of glueballs or mesons. Two different implementations of the basic ideas are discussed; one variant provides an algorithm for constructing N = ∞ master field matrices, while the other works directly with a list of expectation values of physical operators. Algorithms are developed for both the hamiltonian and euclidean formulations of lattice gauge theories. The inclusion of fermions in the hamiltonian version is also described. Detailed tests of the method in the context of the exactly solvable one-plaquette model are presented.

Quantizing gauge theories: Non-classical field configurations, broken symmetries and gauge copies

Abstract The quantization of non-Abelian gauge theories around arbitrary field configurations is considered. We find that the functional integral may be consistently expanded about any field which is sufficiently close to a constrained minimum. A unified treatment of gauge fixing, zero modes and constraints is presented in detail. All problems associated with gauge copies and similar double-counting phenomena are formally removed by further restrictions placed on the space of fluctuations; these restrictions have no effect on the perturbative expansion about a given field.

The coherent state variational algorithm: (II). Implementation and testing☆

The coherent state variational algorithm provides a method for solving the large-N limit of non-abelian gauge theories. An implementation of this algorithm, capable of minimizing the large-N effective action and computing meson and glueball spectra, has recently been completed. Hamiltonian or euclidean formulations of lattice gauge theories, in any dimension, may be studied. Bose or Fermi fundamental representation matter fields may be included. This paper discusses the design and testing of this implementation. The method involves explicit manipulation of expectation values of physical operators and may be applied directly in infinite volume. The error introduced by the truncation of the set of physical observables (necessary to obtain a finite procedure) is studied by applying the algorithm to a variety of exactly soluble model theories. These include φ4 scalar field theories, (ψψ)2 fermion theories, 2-dimensional euclidean pure gauge theory, and 1 + 1 dimensional QCD. Modest size calculations are shown to yield accurate results, even in theories possessing asymptotic freedom, spontaneous symmetry breaking, or large-N phase transitions.

Global symmetry restoration in high temperature Higgs theories

A simple high temperature expansion is developed for lattice gauge theories with scalar matter fields. The expansion is used to prove the absence of global symmetry breaking for sufficiently high temperature.

QCD and Hadronic Structure

Quantum chromodynamics (QCD) is generally believed to be the true theory of strong interactions. It is supposed to predict correctly all observed hadronic behavior. In particular, the inner dynamics of QCD are believed to produce the fundamental properties of confinement and dynamical chiral symmetry breaking. Confinement, or the statement that all observed hadrons are colorless bound states of quarks, is thought to be due to the effective coupling of QCD rising indefinitely as one probes increasing distances. Dynamical chiral symmetry breaking refers to the belief that the true vacuum state of QCD is not chirally invariant (even in the chirally symmetric limit where all light quark masses are neglected.) As a consequence one should find dynamical quark mass generation and formation of composite Goldstone bosons such as the pion. Unfortunately it has proven extremely difficult to derive these properties directly from QCD.