A generalized $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. It is shown that even for a finite number of spins the model has a phase transition in the limit $n\to \infty$. The transition features a specific-heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also discussed.

We show that the Hamiltonian of the 1D Hubbard model commutes with a one-parameter family of transfer matrices of a new 2D classical model corresponding to two coupled six-vertex models. Central to this result is a new local algebraic relation, a generalization of the (infinitesimal) startriangle relation.

A spinorial time-space ${\mathfrak{G}}_N$ that supports a Kaluza-Klein theory of gauge potentials can be made from $N$-component spinors of $\mathrm{SL}\mathrm{}(N,C)\mathrm{}$ in the way that the Minkowskian manifold (an ${\mathfrak{G}}_2$) is made from two-component spinors of $\mathrm{SL}\mathrm{}(2,C)\mathrm{}$. ${\mathfrak{G}}_N$ has dimension $n=N^2$ and fundamental form det $\left(k_{\Sigma \dot{\Sigma }}\right)=0\mathrm{}$ of degree $N$, not 2. ${\mathfrak{G}}_N$ unifies the photon and graviton with gauge quanta of all integer spins $s$ with $2<s<N$.

It is shown that a bunch of charged particles following a curved path in a magnetic field is subject to a force due to its own electromagnetic field. One aspect of this is a "centrifugal" force acting on individual particles in the bunch. A resonance mechanism, capable of disrupting the beam at modest currents, is given as an example of the importance of this force. The theory is tested with observations from the Cornell Electron Storage Ring. This force will cause important modifications to existing theories of accelerator stability.

The Floquet solution for the evolution operator of a periodically time-dependent two-level Hamiltonian provides an effective Hamiltonian that can be used to examine the equilibrium properties of the system. In one example, calculations with a model Hamiltonian show that identical equilibrium properties are predicted in the laboratory frame as in a time-dependent interaction frame. A second example considers the changes that arise when the model Hamiltonian contains an explicit periodic time dependence.

Two aspects of the irreversible measurement transformation are explored. The first is the range of allowed values for the joint uncertainty of two operators having no common eigenvector. The second is the constraint on states and operations following from the requirement that the uncertainty resolved by completing a measurement shall be always finite.

The graviton propagator in a de Sitter background is found to be divergent. We show that as a consequence of this divergence, de Sitter space is not a solution of the equations of motion of the complete theory. If we start from de Sitter space as a classical ground state, quantum corrections change it into flat Minkowski space.

The experiment on chaos in surface waves reported by Ciliberto and Gollub is used as a case model for an understanding of the appearance of few-dimensional chaos in systems with an infinite number of degrees of freedom. Center-manifold and normal-form theories are utilized to derive the pertinent dynamical system and to provide an approximate solution of the partial differential equations. Comparisons of theory and experiment are discussed.

We treat the perihelion shift of the planetary motion and the Lamb shift in hydrogen in an arbitrary number of space dimensions. Comparison with experimental data shows that the deviation from dimensionality four of space-time is less than ${10}^{-9\mathrm{}}$ and 3.6 × ${10}^{-11\mathrm{}}$, respectively, on the length scales associated with these phenomena.

A dynamical meaning is given for the phase $\Gamma$ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that $\Gamma$ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase $\Gamma$.

The absorption of energy by highly excited electrically polarized hydrogen atoms in intense microwave fields is studied experimentally, and theoretically in terms of a one-dimensional quantum model. The quantum calculations reveal a sudden onset of basis-set instability that correlates well with the threshold for classical chaos.

A basin boundary can undergo sudden changes in its character as a system parameter passes through certain critical values. In particular, basin boundaries can suddenly jump in position and can change from being smooth to being fractal. We describe these changes ("metamorphoses") and find that they involve certain special unstable orbits on the basin boundary which are accessible from inside one of the basins. The forced damped pendulum (Josephson junction) is used to illustrate these phenomena.

Measurements on the Belousov-Zhabotinskii reaction in a stirred flow reactor reveal, beyond a period-doubling cascade, a sequence of periodic states that is different from the "U (universal) sequence" found in recent experiments on diverse systems. Particular periodic states occur in three different parameter ranges rather than one range as in the U sequence. However, the order of periodic states found here is in accord with a one-dimensional single-parameter map constructed from the laboratory data.

The stability at finite wave numbers of a hard-sphere fluid subjected to a large shear is examined by use of hard-sphere kinetic theory. For a fixed density, for molecular-scale wave numbers, and for a critical shear rate the fluid is found to become unstable with respect to density waves in the direction of the velocity gradient. The critical shear rates obtained are close to those found by Erpenbeck in his nonequilibrium molecular-dynamics simulations of shear-induced ordering.

A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.

It is found that the effective Hamiltonian for nuclear rotation in a diatom is equivalent to that of a charged particle in a background magnetic-monopole field. In certain cases, half-integer orbital angular momentum or non-Abelian fields occur. Furthermore, the effects of magnetic-monopole-like gauge fields can be experimentally observed in spin-resonance experiments with variable magnetic fields.

The usual configuration-space semiclassical approximation to wave equations (e.g., Schrödinger's equation) needs refinement at caustics. When expressed in a phase-space coherent-state representation, such equations admit a semiclassical solution that properly captures the effects of caustics. Transformation of this expression to the configuration-space representation yields a global, uniform, semiclassical approximation.

The low-energy threshold behavior and Levinson's theorem are derived for general, not necessarily rotationally symmetric, two-dimensional scattering systems, with zero-energy resonances and zero-energy bound states explicity taken into account. Surprisingly, $s$-wave-type zero-energy resonances do not contribute at all to Levinson's theorem, while $p$-wave-type zero-energy resonances each contribute a term - $\pi$, exactly like (zero-energy) bound states. Some consequences of this for, e.g., the Witten index in certain supersymmetric quantum-mechanical models are briefly discussed.

We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature $t$ and the soliton density $e^{\frac{-1\mathrm{}}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.

For a general quantum mechanical system undergoing a process, it is shown that one can tell from measurements on this system whether or not the system is characterized by quantum numbers the existence of which is unknown to the observer, even though the detection equipment used by the observer is unable to distinguish among the various possible values of the "secret" quantum number and hence always averages over them.