A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.

A procedure is described whereby the result of the measurement of any of the three Cartesian components of the spin of a single spin-(1/2) particle at a single time can be inferred with certainty from the result of two other measurements, one of which is carried out before, and the other after, the time in question.

It is well known that the resonant tori of an integrable, classical Hamiltonian system break up under a small perturbation into new tori which wind around the remaining nonresonant tori. It turns out that the Maslov indices of one of these satellite tori depend on the Maslov indices of its unperturbed, resonant parent, and on the integers which enter into the resonance condition. They do not, however, depend on the functional form of the Hamiltonian or the perturbation. Our results are valid for any number of degrees of freedom.

The problem of linear mode conversion in nonuniform media is solved for general geometry, in that the dispersion functions ${\mathrm{D}}_{\mathrm{a}}$(k,x),${\mathrm{D}}_{\mathrm{b}}$(k,x) of the two, orthogonally polarized, coupled modes a,b have spatial gradients ∂${\mathrm{D}}_{\mathrm{a}}$/∂${\mathrm{x}}^{\mathrm{\mu }}$,∂${\mathrm{D}}_{\mathrm{b}}$/∂${\mathrm{x}}^{\mathrm{\mu }}$ which need not be parallel. The transmission ratio is found to be exp(-2π‖η${\mathrm{\Vert }}^2$/‖B‖), where η is the coupling coefficient, and B is the Poisson Bracket {${\mathrm{D}}_{\mathrm{a}}$,${\mathrm{D}}_{\mathrm{b}}$}=(∂${\mathrm{D}}_{\mathrm{a}}$/${\mathrm{\partial }}_{\mathrm{X}}^{\mathrm{\mu }}$)(∂${\mathrm{D}}_{\mathrm{b}}$/∂${\mathrm{k}}_{\mathrm{\mu }}$)${\mathit{D}}_{\mathit{b}}$) -(∂${\mathit{D}}_{\mathit{a}}$/∂${\mathit{K}}_{\mathrm{\mu }}$)(∂${\mathit{D}}_{\mathit{b}}$/${\mathrm{\partial }}_{\mathit{X}}^{\mathrm{\mu }}$). /∂${\mathrm{x}}^{\mathrm{\mu }}$). The further generalization to weak dissipation and ray divergence is included.

A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator ${\mathrm{L}}_0$ is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.

The time evolution of an initially localized state of a quantum particle coupled to a dissipative environment with spectral density I(ω)∝${\mathrm{\omega }}^{\mathrm{\alpha }}$ for low frequencies is discussed. At finite temperatures, the width of the state can grow subdiffusively or superdiffusively, depending on α. For α>2 damping becomes ineffective for long times and the state spreads kinematically. At zero temperature the spreading is slower and for α<1 an initially localized state remains localized for all times.

To study properties of self-avoiding surfaces, we consider D-dimensional elastic manifolds embedded in d-dimensional space. In the (d,D) plane there is a line separating ideal (Gaussian), and self-avoiding (interacting) behaviors. We develop the techniques for systematic perturbation expansions, and renormalization-group analysis, about any point on this line. The usual ε expansion for polymers about d=4 is a special case. The exponents ν and γ are calculated to lowest order, providing very good values for polymers (D=1), and reasonable indications for surfaces (D=2).

The Rydberg constant $R_{\infty }$ was determined to 3 parts in ${10}^{10}$ by direct comparison of the four H,D Balmer-β transitions with a National Bureau of Standards standard laser. This is the most precise value: $R_{\infty }$=109 737.315 73(3) ${\mathrm{cm}}^{\mathrm{-}1}$; it approaches the limits of accuracy for optical measurements. The fine-structure splittings and isotope shift are in excellent agreement with theory. The result agrees with less precise experiments by Zhao et al. and Biraben and Julien but disagrees with the result of Hildum et al.

Random superpositions of plane waves, designed to mimic the local behavior of completely random eigenfunctions of classically chaotic Hamiltonian systems, are shown to have surprising properties, including structures which may be precursors of periodic orbit scar localization.

We describe the selection mechanism for novel pattern formation in a driven chain of nonlinear oscillators. Phase organization arises as a consequence of dynamical selection of minimally stable states.

Experimental results are presented which provide the first measurement of the effects of a gravitational field on charged particles, equivalent to the red shift for photons. Two Josephson-effect batteries (V≃300 μV) having a vertical separation of 7.2 cm are connected in opposition by superconducting wires. A voltage difference of 2.35×${10}^{\mathrm{-}21}$ V is maintained between these batteries by means of the gravitational red shift. The emf around this loop is, however, measured to be less than 1×${10}^{\mathrm{-}22}$ V, consistent with the predicted invariance of the gravito-electrochemical potential along the wires.

Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of one-dimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions.

We present a general sum rule in the Born approximation governing neutrino scattering from local potentials in a crystal. This relation facilitates the placing of stringent bounds on the force exerted on a crystal by a flux of neutrinos. The force is found to depend linearly on the number of scattering centers and no exotic coherent effects are predicted.

The total intensity averaged over all angles of short-wavelength radiation either scattered or spontaneously emitted nearly isotropically from each of a system of N radiators or scatterers is shown to vary linearly with N for large N. Although coherent effects can give intensities proportional to ${\mathrm{N}}^2$ in certain directions (e.g., Bragg angles), this enhancement is mainly at the expense of radiation otherwise emitted in other directions and does not appear in the overall angular distribution.

A nonlinear Hamiltonian model is presented for which exact analytic results can be obtained. It is shown that macroscopically distinguishable quantum states can be generated and detected by means of a homodyne detector. The vacuum fluctuations may inhibit their observability; however, this contribution can be subtracted.

The transition from quasiperiodicity to chaos is studied for the situation where the sine circle map becomes supercritical. The numerical results show convincingly that scaling behavior is present in this transition.

According to general relativity, the calculated rate of motion of lunar perigee should include a contribution of 19.2 msec/yr from geodetic precession. We show that existing analyses of lunar–laser-ranging data confirm the general-relativistic rate for geodetic precession with respect to the planetary dynamical frame. In addition, the comparison of Earth-rotation results from lunar laser ranging and from very-long-baseline interferometry (VLBI) shows that the relative drift of the planetary dynamical frame and the extragalactic VLBI reference frame is small. The estimated accuracy is about 10%.

A well-balanced hollow copper sphere freely floating and almost totally submerged in water is observed to move toward the nearby edge of a cliff. This observation is consistent with a substance-dependent, medium-range force postulated in a recent reanalysis of the Eötvös experiment. Further measurements with different elements and geometries will be required to establish definitely the existence, source, and description of such a new force.

We have placed a torsion balance (containing two Be and two Cu test bodies) on a hillside to search for intermediate-range forces that couple to baryon number. Our results constrain (at 1σ) the strength of such an interaction to be ‖α̃‖≤1×${10}^{\mathrm{-}4}$ for ranges 250≤λ≤1400 m and ‖α̃‖≤1×${10}^{\mathrm{-}3}$ for ranges 30<λ<250 m.

The problem of two magnetic moments in a Fermi gas is studied with the numerical renormalization group used by Wilson for the Kondo problem. Even when the interaction energy of the moments is much smaller than the Kondo energy, the asymptotic low-temperature behavior is that of a correlated Kondo effect. An effective Hamiltonian for the low-temperature properties, which are nonuniversal, is deduced.