We derive for the first time second-order gradient corrections to the Thomas-Fermi kinetic-energy and entropy density functionals for a fermion system at finite temperature. Density variational calculations for a heated nucleus with a Skyrme interaction lead to excellent agreement with results of Hartree-Fock calculations.

Self-avoiding random surfaces are analyzed by renormalization-group methods. The Hausdorff dimension $\frac{1}{\nu }$ and the critical plaquette fugacity are computed for different dimensionalities $d$; in particular, $\nu =\frac{1}{2}-\frac{\epsilon }{4}+O\left({\epsilon }^2\right)$ for $d=2+\epsilon$. The model describes "sheet polymers" in a good solvent: A Flory type of argument yields $\nu =\frac{3}{\mathrm{}(4+d)\mathrm{}}$, in good agreement with the renormalization results, and a critical dimensionality $d_c=8\mathrm{}$, with $\nu =\frac{1}{4}$.

The Ablowitz-Ramani-Segur conjecture relates the Painlevé property, i.e., polelike local singularity structure, to integrability. As such this conjecture cannot be applied, as integrability detector, to nonlocal integrodifferential equations. We show, however, that, for the physically interesting cases of the Benjamin-Ono and intermediate long-wave equations, the Painlevé property can be recovered despite the apparent nonlocality.

We derive an equation of motion for a fully observed classical system which is coupled to a quantum companion whose degrees of freedom are only partially observed. Our purpose is to discuss the nature of the quantum noise by using a generalization of the Langevin equation.

A purely algebraic procedure is presented for calculating recursion relations for $S$ matrices belonging to problems associated with the group SU(1,1). The procedure involves supplementing a recently introduced group-theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior. Our formulation makes use of the Euclidean group, which contains the symmetry transformations for a free particle. The Coulomb and Pöschl-Teller potentials are discussed to illustrate our methods.

An $\mathrm{XY}$ model of classical spins, on a two-dimensional square lattice, with nearest-neighbor interaction $V\left(\theta \right)=2J[1-{\left(\frac{{cos}^2\theta }{2}\right)}^{p^2}]$ is studied by a Monte Carlo simulation that also monitors the vortex excitations. Both continuous and first-order transitions are found. In the latter the density of vortices sharply increases at the transition point. The transition turns first order, because the shape of $V$ is such that vortex-antivortex pair formation is suppressed at low temperatures for $p\gtrsim 3$.

A novel technique, based on Grassmann path integrals, is shown to provide a powerful new tool in the theory of phase transitions in two-dimensional Ising-spin models. Accurate phase boundaries as well as exact critical singularities are produced for two representative models of disorder: the dilute ferromagnet and the $\pm J$ spin-glass.

It is shown that in order to determine all reaction amplitudes for a reaction containing particles with arbitrary spins without even a discrete ambiguity (except for an overall phase factor), one must measure at least one observable with spin polarization along each of three noncoplanar directions. The result has an impact on experimental programs measuring polarization quantities in elementary particle, nuclear, atomic, and molecular physics.

The problem of the early-time decay rate of a model system is addressed by examining the evolution of the probability current at a detector located a macroscopic distance from the source. If one interprets this current as the flux of decay products, early-time departures from Fermi's "golden rule" are not, in principle, observable.

The WKB cross section for backward glory scattering of a massless polarized wave propagating in curved space-time is given explicitly.

The Kohn-Sham equations are derived from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy, $E_{\mathrm{xc}}$, without explicit recourse to a minimum or extremal principle, thus providing new support for this implementation of density-functional theory in a most common situation. Furthermore, it is shown that for a large class of approximate expressions for $E_{\mathrm{xc}}$, there exist approximate Kohn-Sham equations which give the same density and energy as other means of calculation within the same diagrammatic approximation.

A general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the $d=3\mathrm{}$ Ising model.

A quantum system which can tunnel out of a metastable state, and which interacts with an environment at temperature $T$, is considered. It is found that heat enhances the tunneling probability at $T=0\mathrm{}$ by a factor $\exp \left[A\mathrm{}\right(T\left)M{\omega }_0\frac{q_0^2}{\hslash }\right]$, where $M$ is the mass of the system, ${\omega }_0$ is the frequency of small oscillations about the metastable state, and $q_0$ is the tunneling distance. For an undamped system $A\mathrm{}\left(T\right)\mathrm{}$ is exponentially small, $A\mathrm{}\left(T\right)\mathrm{}\propto \exp \left(\frac{-\hslash {\omega }_0}{k_{B}T}\right)$, whereas for a dissipative system $A\mathrm{}\left(T\right)\mathrm{}$ grows algebraically with temperature.

Invasion-percolation observables which should scale with exponents interrelated through scaling and hyperscaling hypotheses are investigated. Evidence is presented that the hyperscaling relation between $\tau$, the spatial dimension, and the fractal dimension breaks down at the defender threshold. Thus an independent exponent must be introduced to describe the scaling of the finite-defender-cluster distribution.

Because of complicated topology of the configuration space for indistinguishable particles in two dimensions, Feynman's path-integral formulation allows exotic statistics. All possible quantum statistics in two-space are characterized by an angle parameter $\theta$ which interpolates between bosons and fermions. The current formalisms in terms of topological action of multivalued wave functions can be derived in a model-independent way.

A notion of length for quantum mechanical trajectories is introduced within the realm of stochastic mechanics. Using a stochastic calculus of variation, one shows that the geodesic dynamics is not the free one, but the quantum evolution in the time-dependent quadratic potential associated with the Wiener process in stochastic mechanics. The length for the free evolution is also examined.

Generalizing a construction of Berry and Simon, we show that non-Abelian gauge fields arise in the adiabatic development of simple quantum mechanical systems. Characteristics of the gauge fields are related to energy splittings, which may be observable in real systems. Similar phenomena are found for suitable classical systems.

A covariant, symplectic, classical dynamical system is presented whose quantization, by replacement of the Poisson brackets with commutators, gives precisely the Dirac electron theory. For the classical system the velocity and momentum are independent dynamical variables, as in Dirac electron; it undergoes a real Zitterbewegung so that the spin is the orbital angular momentum of the Zitterbewegung. Thus the longstanding problem of representing the configuration space of quantum spin by a classical model is solved. Three different forms of the dynamical equations are given.

The most precise determination of the magnetic moments of the electron and positron come from a University of Washington $g-2\mathrm{}$ experiment. The accuracy of this experiment has now been improved to the stage where a detailed theory of the line shape is required in order to determine the moments with comparable precision. We present here the results of this theory.

The approach to equilibrium in a bistable system is governed by a small eigenvalue which appears to be difficult to compute. We show that supersymmetry makes possible an easy evaluation of this eigenvalue.