A new formulation is presented for the thermodynamics of the massive Thirring model in the attractive-coupling regime. A certain controversy between previous theories is resolved and an overall understanding of the thermodynamics is reached. In particular, it is shown that the soliton mass is discontinuous with the coupling constant at finite temperatures.

It is proposed that one should admit in the path integral for the quantized gravitational field only those space-times for which the final three-geometry is located in the future of the initial one. As a consequence, and unlike the situation for the Yang-Mills field, the resulting causal amplitude is not annihilated by all the gauge (surface deformation) generators. In supergravity the causal amplitude turns out not to be annihilated by the local supersymmetry generators either.

A preliminary experimental determination by Hill, Bos, and Goode of the interior rotation of the sun leads to a nonzero value for the quadrupole moment coefficient $J_2$. This produces a deviation of 1.6% from Einstein's prediction of the precession of the perihelion of Mercury. A nonsymmetric gravitational theory can fit the measured precession with this $J_2$ and all other solar-system relativity experiments for one value of a post-Newtonian parameter in the theory. A prediction is made for the perihelion precession of Icarus.

Chaotic oscillations due to delay-induced instabilities are easily seen with a simple and familiar acoustic system. The measured bifurcation diagram to chaos, which is very different from that for a one-dimensional map model, is compared with numerical results. The bifurcation structure is dominated by the ratio of the response time to the delay time of the system.

Expanding the Dirac resolvent around its nonrelativistic limit, the authors propose a new method for calculating relativistic corrections to bound-state energies. The present results are more powerful than the expressions obtained by Foldy and Wouthuysen and remain applicable for Coulomb as well as nonlocal interactions.

Measurements of ion cyclotron-resonance frequencies in a Penning trap, by a laser fluorescence technique, are described. This technique has been applied to indirect measurements of the proton-to-electron mass ratio and the ${}_{}{}^9{}_{}{}^{}\mathrm{Be}_{}^{+}{}_{}{}^{}$ electron $g_J$ factor. It is found that $\frac{m_p}{m_e}=1836.152\mathrm{}38\left(62\right)\mathrm{}$ (0.34 ppm) and $g_J\left({}_{}{}^9{}_{}{}^{}\mathrm{Be}_{}^{+}{}_{}{}^{}\right)=2.002\mathrm{}26206\left(42\right)\mathrm{}$ (0.21 ppm). Ultimately, ion cyclotron-resonance accuracies near 1 part in ${10}^{13}$ should be possible.

The object of this Letter is to show that except in the case of canonically conjugate observables, the generalized Heisenberg inequality does not properly express the quantum uncertainty principle. It is, in general, too weak. An inequality is obtained which does express the principle.

An exact solution of the eigenvalue problem $\{-\frac{d^2}{d{x}^2}+\Delta [1-a_0\rho \mathrm{}\left(x\right)\left]\right\}{\psi }_n\mathrm{}\left(x\right)=E_n{\psi }_n\mathrm{}\left(x\right)\mathrm{}$ with $\rho \mathrm{}\left(x\right)={\Sigma }_{n=1\mathrm{}}^N{\left|{\psi }_n\mathrm{}\right(x\left)\right|}^2$ and with periodic boundary condition is presented. The solution gives rise to a density wave with [const-$\rho \mathrm{}\left(x\right)\mathrm{}$] proportional to $\mathrm{s}{\mathrm{n}}^2\left[\right(\frac{x}{\lambda }\left)\right|m]\mathrm{}$ for suitable values of the parameters $\lambda$ and $m$. The solution rests upon some remarkable properties of the solutions of Lamé's equation.

The change in total energy-momentum of $N$ classically scattering charged point particles is shown to equal a flux integral depending on the asymptotic retarded fields of the particles. This flux integral is the energy-momentum flux of the electromagnetic field if the charges scatter according to the Lorentz-Dirac equation. All integrals are finite.

The authors prove, by considering a simple, exactly solvable model, that quantum systems under external perturbations may display a chaotic motion analogous to the corresponding classical ones.

Quantum statistics can be reconciled with such classical ideas as distinguishable particles. Bose-Einstein and Fermi-Dirac statistics are derived for distinguishable particles by making an assumption regarding probabilities which is different from the traditional assumption but equally reasonable.

It is shown that eigenvalues of two-dimensional angular momentum remain integer valued in the magnetic field of a solenoid, contrary to published assertions that they are modified by the flux. For a vortex, flux does contribute, and the angular momentum can fractionize, as asserted in the literature, provided phases of wave functions are chosen consistently with the solenoid problem. Long-range effects of flux, the distinction between orbital and canonical angular momentum, and interactions with Cooper pairs are essential to this argument.

When a series of aluminum slabs are placed in one leg of a perfect-silicon-crystal neutron interferometer, a continuous and significant loss of contrast is observed. This observation is interpreted as being due to the finite length of the neutron wave packets.

The proposition that the coherence length of de Broglie wave packets remains unchanged even though the length of the packets increases upon propagation is discussed and demonstrated.

A conjecture, based on a comparison of different mean-field-like approximations, is proposed for determining the first- or second-order character of phase transitions. The $q$-state Potts model and several gauge theories [$Z_2$, $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$, $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$, SO(3)] on a lattice are considered. For $Z_2$ gauge theory, in particular, a critical dimensionality $d_c=3.66\mathrm{}$ is predicted, whereas for the Potts model $q_c=2.75\mathrm{}$ in $d=3\mathrm{}$, $q_c=2.4\mathrm{}$ in $d=4\mathrm{}$, and $q_c\propto \exp \left[\frac{1}{\mathrm{}(d-1\mathrm{})\mathrm{}}\right]$ for $d\to 1$ are obtained.

Pattern selection is discussed in regard to a situation where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. The velocity of the propagating front and the wavelength of the pattern formed behind the front are determined by a marginal-stability criterion. The special system studied here has a Lyapunov functional, but the periodic state which propagates is not the one which minimizes the functional.

The conventional thermodynamic fluctuation theory does not include local correlations and, hence, fails at volumes of the order of the correlation volume ${\xi }^d$. In this paper, an attempt is made to include local correlations in a thermodynamic theory by considering fluctuations in a sequence of concentric systems of decreasing volume. Tests are made on fluctuations in the magnetization within the one-dimensional Ising model. The results are a clear improvement over those of the conventional theory.

A statistical-mechanical model is proposed and solved exactly for the pair correlation function of planar uniaxial ferromagnets in the scaling limit, both with and without external magnetic field.

The field equations of $N=1\mathrm{}$ supergravity in $d=11\mathrm{}$ dimensions admit a spontaneous compactification on the seven-sphere to an $N=8\mathrm{}$ theory in $d=4\mathrm{}$ with local SO(8) invariance. A spontaneously broken version of this theory is provided by another solution of the field equations for which the metric on $S^7$ is distorted. The isometry group of this new solution is SO(5) ⊗ SU(2), the relevant holonomy group is $G_2$, and the $N=8\mathrm{}$ supersymmetry is broken down to $N=1\mathrm{}$ at the Planck scale. The implications for grand unified theories are briefly discussed.

A nonlinear differential equation for the asymptotic decay of the pair correlation function of a fluid at its critical point is obtained from the square-gradient theory (and its extension to fourth order), and analyzed when the critical exponent $\eta$ is either zero or nonzero. Its solutions are shown to be consistent with the correct power-law decay if and only if the ordinary scaling relations together with hyperscaling (for $\eta >0\mathrm{}$) are valid.