Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).

It is shown that in a collision between any two aggregates of particles the points of stationary phase for the scattering amplitude lie on the hypersurface determined by the condition that the potential energy of the interaction betwen the two aggregates vanishes.

We show the near-universal existence of a second invariant that is polynomial in the momenta for integrable Hamiltonian systems in two dimensions. Specifically, Hietarinta's three "counterexamples" are converted to polynomial form.

We report a new type of anomalous diffusion in chaotic systems with periodic symmetry. It is characterized by mean square displacements diverging faster than linearly in time. By a comparison of power spectra we conclude that the phenomenon has occurred in a recent observation of $\frac{1}{f}$ noise in Josephson junctions.

The critical dimension of string theories in which the background metric is a product of Minkowski space and an $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ or $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ group manifold is derived. A consistent string theory can be constructed only in the presence of a Wess-Zumino term associated with the compactified dimension. This implies that the compactified radius is quantized in units of the string tension. A generalization to the supersymmetric case is discussed.

It is shown that a linearized time-reversal-invariant and manifestly nondissipative equation of the Schrödinger form is directly derivable from the Kolmogorov-Fokker-Planck stochastic diffusion equation in a single space dimension under certain conditions. The conditions under which this correspondence is achieved suggest the possibility of fluctuations of the space-time metric itself at short (quantum mechanical) distances as the cause of the quantum Brownian motion of microscopic particles.

A method is given for finding the shifts in position of the solitons for the case of nonzero reflection coefficient. Expressions for boost generators in terms of scattering data play a prominent role in the analysis. Phase-shift formulas which show the effect of the radiation component on the soliton motion are deduced for the nonlinear Schrödinger equation, the Korteweg-de Vries equation, and the sine-Gordon equation.

In the context of stochastic quantization of field theories we propose a method to make analytic computations of critical exponents and we evaluate them for ${\left({\phi }^2\right)}_3^2$. It consists of regularization of the theory by use of a convenient non-Markovian process, where nonlocality in time is measured by the regularizing parameter $\sigma$. For a fixed dimensionality there is a value of $\sigma$ where the theory is renormalizable and asymptotically free in the infrared, allowing a perturbative expansion around it.

Noting that quantum measurements are in general incomplete, we develop, starting from a recent entropic formulation of uncertainty, a maximum uncertainty principle to define the statistical mechanics of microscopic systems. The resulting ensemble entropy coincides with the expression of von Neumann, thus providing a unified, quantum basis for statistical physics of all systems. Examples involving momentum and position measurements are discussed.

The relativistic Doppler shift is measured by the counting of the frequency difference between two cw dye lasers. One laser is locked to a two-photon transition in a fast beam of neon, and the other is locked to the same two-photon transition in thermal neon. The experimental result is compared to the prediction of special relativity. The result is in excellent agreement with this theory. An accuracy of 4×${10}^{-5\mathrm{}}$ is obtained, which provides the most accurate direct verification of time dilatation to date.

We describe a precise mathematical theory of the Laughlin argument for the quantization of the Hall conductance for general multiparticle Schrödinger operators with general background potentials. The quantization is a consequence of the geometric content of the conductance, namely, that it can be identified with an integral over the first Chern class. This generalizes ideas of Thouless et al., for noninteracting Bloch Hamiltonians to general (interacting and nonperiodic) ones.

The phase diagram of a Hamiltonian which describes a particle moving in a periodic potential and coupled to an external heat bath has been analyzed in detail. By renormalization-group methods, it is proved that this system exhibits an unusual transition between diffusive behavior and a self-trapped phase. The dynamics along the transition line is shown to be integrable and the order parameter, the mobility, is computed exactly.

We introduce a new random walk with the property that it is strictly self-avoiding and grows forever. It belongs to a different universality class from the usual self-avoiding walk. By definition the critical exponent $\gamma$ is equal to 1. To calculate the exponent $\nu$ of the mean square end-to-end distance we have performed exact enumerations on the square lattice up to 22 steps. This gives the value $\nu =0.57\mathrm{}\pm 0.01$.

It is shown that the three-cocycle arises when a representation of a transformation group is nonassociative, so that the Jacobi identity fails. A physical setting is given: When the translation group in the presence of a magnetic monopole is represented by gauge-invariant operators, a (trivial) three-cocycle occurs. Insisting that finite translations be associative leads to Dirac's monopole quantization condition. Attention is called to the possible relevance of three-cocycles in the quark model's U(6) ⊗ U(6) algebra.

We formulate the Ising model with competing interactions on a Cayley tree, in the infinite-coordination limit, as a two-dimensional nonlinear mapping. The phase diagram displays a Lifshitz point and many modulated phases. We perform calculations to show the existence of a complete devil's staircase at low temperatures. Also, we give strong numerical evidence for the existence of chaotic phases associated with strange attractors.

The fast-atom-laser collinear beams technique has been used to measure the relativistic Doppler effect to an accuracy comparable to the best previous experiments (5 parts in ${10}^3$). Improved calibration data and a similar technique, but employing two lasers, is proposed, which should improve the test of time dilatation by at least another factor of 10.

A differential equation is proposed to describe the evolution of a screw dislocation in a continuous medium under an externally applied displacement field. The derivation is based on the conservation of energy and momentum and the neglect of radiation reaction and renormalizability: The equation of motion must be such that it allows for a renormalization of the infinite self-energy and momentum of the infinitely thin dislocation. The extent to which the result might hold for general dislocation loops is discussed.

Whether quantum mechanical systems, within the interval between two measurements, can be assigned definite, simultaneous values of noncommuting observables has been the subject of a very old dispute. A certain widely held assumption about such systems has been crucial to that dispute. That assumption turns out to be wrong: A previously unknown prediction of quantum mechanics, which fails to satisfy that assumption, is described here, and the consequences of that failure are considered.

We introduce an analog, in classical phase space, of the fundamental building block of time dependence in quantum mechanics, i.e., the superposition state. Applications to two-level systems and quantum beats illustrate this conceptually pleasing phase-space picture of the phase interference between energy eigenstates which comprises pure-state quantum dynamics.

This work is concerned with the Rayleigh-Schrödinger perturbation expansions associated with Hamiltonians of the form $\hat{H}\left(\lambda \right)=\frac{1}{2}{\hat{p}}^2-\frac{Z}{r}+\lambda r^m$, $m=1,2,3,\dots$. These Hamiltonians are important in atomic and molecular physics as well as in confinement studies of quantum chromodynamics. A method of calculating the Rayleigh-Schrödinger coefficients $E^{\mathrm{}\left(n\right),m}$, based on an so(4,2) Lie algebraic reformulation of the Schrödinger equation, is outlined. Nonrigorous Bender-Wu WKB methods are employed to determine the high-order behavior of the $E^{\mathrm{}\left(n\right),m}$.