A new gauge-invariant effective action is proposed for quantum gravity, based on older results that go beyond finite-order perturbation theory. Expressed in coordinate space rather than momentum space it should find important applications in theory of the early universe.

A modified droplet model is proposed that incorporates both compact fluctuations that give rise to weak singularities at the first-order transition, and ramified fluctuations that give rise to spinodals in high dimensions. Renormalization-group considerations and studies of the percolation problem allow the use of ramified fluctuations to calculate spinodal exponents for $d>\mathrm{}8\mathrm{}$. The problem in lower dimensions is also discussed together with possible interpretation of these exponents in light of the indeterminacy in locating spinodal points.

The generalized $H$ theorem, ${(-\frac{d}{\mathrm{dt}})}^{n}H>~0$, is verified for $n<~30$ for the Bobylev-Krook-Wu solution of the Boltzmann equation, and for $d$-dimensional generalizations of that solution, $1<~d<~6$.

A self-consistent effective-medium approximation is presented for the problem of diffusion and ac conductivity on a lattice characterized by random values of transfer rates between pairs of nearest-neighbor sites. The approximation is applied to a percolation model in which only a fraction of the bonds are assigned a finite transfer rate. The results reflect the existence of a percolation threshold in the system, and are consistent with the properties of clusters of bonds in the critical region.

An exact algorithm is formulated to calculate the expected walk length $〈n〉$ for a walker (atom, molecule) undergoing random displacements on a finite or infinite (periodic) $d$-dimensional lattice with traps (reactive sites). The method is illustrated for the case of a single deep trap surrounded by shallow traps and the calculated value of $〈n〉$ agrees to within 0.3% of the Monte Carlo result for all lattices considered. The theory introduced is capable of generalization to many new classes of problems in lattice statistics.

The statistics of pure quantum states of the harmonic oscillator is separated into a quantum mechanical and a classical part by associating a classical analog with each quantum state. Several types of states are considered and it is shown that all pure states of the same average energy are equally close to their classical analogs.

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

The Burgers equation is linearized on the Schwartz space $S$. It follows that the Burgers equation can be defined by a Hamiltonian formalism and that it has an infinity of time-independent constants of motion in involution.

A driven anharmonic oscillator is described which exhibits period doubling and chaotic behavior. The measured behavior of the oscillator under successive period doublings is in quantitative agreement with a recent theory which describes the behavior of nonlinear systems. Both the scaling and the convergence rate predicted by the theory are verified by the experiment. The oscillator also exhibits period tripling and quintupling.

For a given statistical state at time $t_0$, the expectation value of some observable at a later time $t_1$ is expressed in a variational form. Different trial choices for the quantities to be varied (state and observable) generate different approximations, in which the evolution of the state is optimally fitted to the measured quantity. Examples are given in the context of mean-field theories.

The zero-temperature limit of quantum statistical mechanics is analyzed from the point of view of Nelson's stochastic quantization. An application is given to the quantum effects in critical phenomena by introducing a description of the static properties based on the dynamics of the equilibrium fluctuations.

In a crossed laser-atomic beam experiment the wavelengths of the $2s-3p$ transitions are measured in H and D to a precision of one part in ${10}^9$. Our value for the Rydberg constant is $R_{\infty }=109\mathrm{}737.31521\left(11\right)\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$. The fine-structure splittings of the $3p$ states in H and D are 3249.8(8) and 3251.7(7) MHz, respectively; the isotope shifts for the $2s-3p_{\frac{1}{2}}$ and $2s-3p_{\frac{3}{2}}$ transitions are 124 260.7(7) and 124 262.6(7) MHz, respectively. Our results largely agree with previous, less precise experiments and with theory.

It is shown that the percolation problem with blocked, one-way, or two-way bonds is self-dual on a square lattice. The usual directed percolation (blocked or one-way bonds) is dual to a simpler process with one- or two-way bonds. The results of a Monte Carlo simulation of the latter are reported and an improved bound on the critical probability is derived. The corresponding resistance problem in which circuit elements have different forward and backward resistances is also shown to be self-dual.

Starting with some observations due to A. B. Zamolodchikov, a short cut to the determination of the Baxter (eight-vertex) -model free energy is developed. The method involves factorizable $S$ matrices in 1 + 1 dimensions, which are briefly reviewed. If the method generalizes, we may associate with each known $S$ matrix a Baxter-like model with a known free energy.

A new variational principle is derived through the construction of the action in terms of a new set of canonical coordinates. The main advantage of this set of coordinates is that they contain exclusively information concerning the Eulerian description of the flow.

A system is proposed for detecting the effect of a weak classical force on a macroscopic mechanical oscillator. The novel feature is a type of ac bridge coupled to a SQUID amplifier. A detailed sensitivity calculation for one type of bridge excitation is reported which shows what is required to reach the standard quantum "limit." Another type of excitation should make it a "back-action evasion" device capable of exceeding the standard quantum "limit".

A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $-\frac{A}{m^y}$ with the distance $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g\left(\mathcal{r}\right)$ is studied for critical bulk coupling. For $y=1\mathrm{}$ the behavior is $g\left(\mathcal{r}\right)\sim \frac{1}{{\mathcal{r}}^{\eta }}$ as $r\to \infty$, with a nonuniversal exponent $\eta$ depending on $A$. For $0<y<\mathrm{}1\mathrm{}$ anomalous decay $g\left(\mathcal{r}\right)\sim \exp [-{\left(\frac{r}{\xi }\right)}^{1-y}]$ is found. Expressions for $\eta$ and $\xi$ in terms of $A$ and $y$ are given.

If the long-distance properties of pure gauge theories are described by thin strings, then quantum fluctuations induce a long-range spin-spin potential between widely separated heavy quarks. This potential falls off as the fifth power of the distance between the quarks and may be important in heavy-quark phenomenology.

Quantum Hamiltonian systems of lattice type are considered which are associated with classical finite-dimensional Lie algebras [like MH(2), SO(3), etc.], and possess complete integrability properties. The canonical Heisenberg $p_n$, $q_n$ variables are found for these Hamiltonian systems and it is shown how to construct their $S$ matrices. These new Hamiltonians are generalizations of the quantum Toda lattice Hamiltonian as well as $\mathrm{XXX}$ models of statistical mechanics.

A new integral equation which linearizes the Korteweg-de Vries and Painlevé II equations, and is related to the potentials of the Schrödinger eigenvalue problem, is presented. This equation allows one to capture a far larger class of solutions than the Gel'fand-Levitan equation, which may be recovered as a special case. As an application this equation, with the aid of the classical theory of singular integral equations, yields a three-parameter family of solutions to the self-similar reduction of Korteweg -de Vries which is related to Painlevé II.