Quantal physics is established as a manifestation of symmetry more far-reaching than hitherto appears to have been recognized. In this primary manifestation, the coordinate transformations of spacetime invariance are themselves the elementary variables, which define their own properties without appeal to an assumed quantal formalism. In irreducible representations, the symmetry variables are inherently indeterminate, and the probabilistic laws invoked in the interpretation of traditional quantum physics are found to originate in geometric relations between these variables. Completeness is, therefore, not an issue, and the quantum of action is not part of the theory of symmetry variables. Quantal physics thus emerges as but an implication of relativistic invariance, liberated from a substance to be quantized and a formalism to be interpreted. A symmetry variable appears in a measurement with one of its eigenvalues, but does not have a value (cannot be represented by a number) in an irreducible representation, which combines sets of eigenvalues. It is this generalized significance of a measurement that allows for correlations that cannot arise for classical variables. The observation of symmetry variables is illustrated by an interferometer experiment measuring reflection symmetry and by the equivalent coincidence experiment registering the polarization of two quanta. The measurement process becomes a matter of following the state of affairs of the symmetry variables in their unitary evolution. For the resolution of the dilemmas that quantal phenomena have been felt to pose, it appears crucial to recognize that indeterminacy, as an inherent property of a symmetry variable in a multidimensional representation, is not affected by subsequent observations. A position variable and the canonical commutator with momentum, which are basic elements of nonrelativistic quantum mechanics, emerge from spacetime symmetry, but require the link between space and time of relativistic invariance. The transition to the classical regime is analyzed in terms of a quenching of nonlocality in the state of affairs of the multidimensional symmetry variables. While the elementary variables constitute individual quanta in irreducible representations, product representations of spacetime symmetry describe systems of bosons and fermions, which form local fields with canonical properties. The discussion is focused on spacetime invariance (noninteracting quanta), but gauge invariance is itself a primary manifestation of symmetry and is as such encompassed by the theory of symmetry variables.

In this paper the authors present a general analysis of resonant elastic scattering of x rays. The approach exploits crystal symmetry, scattering geometry, and polarization. Extinction rules, their breaking, and the observation of forbidden reflections are discussed. As an application, a detailed description of the experiment performed by Finkelstein, Shen, and Shastri at the iron $K$ edge in $\alpha$ ${\mathrm{Fe}}_2$${\mathrm{O}}_3$ (hematite) is reported.

The reliable development of highly complex organisms is an intriguing and fascinating problem. The genetic material is, as a rule, the same in each cell of an organism. How then do cells, under the influence of their common genes, produce spatial patterns? Simple models are discussed that describe the generation of patterns out of an initially nearly homogeneous state. They are based on nonlinear interactions of at least two chemicals and on their diffusion. The concepts of local autocatalysis and of long-range inhibition play a fundamental role. Numerical simulations show that the models account for many basic biological observations such as the regeneration of a pattern after excision of tissue or the production of regular (or nearly regular) arrays of organs during (or after) completion of growth. Very complex patterns can be generated in a reproducible way by hierarchical coupling of several such elementary reactions. Applications to animal coats and to the generation of polygonally shaped patterns are provided. It is further shown how to generate a strictly periodic pattern of units that themselves exhibit a complex and polar fine structure. This is illustrated by two examples: the assembly of photoreceptor cells in the eye of Drosophila and the positioning of leaves and axillary buds in a growing shoot. In both cases, the substructures have to achieve an internal polarity under the influence of some primary pattern-forming system existing in the fly's eye or in the plant. The fact that similar models can describe essential steps in organisms as distantly related as animals and plants suggests that they reveal some universal mechanisms.

Momentum-conserving lattice gases are simple, discrete, microscopic models of fluids. This review describes their hydrodynamics, with particular attention given to the derivation of macroscopic constitutive equations from microscopic dynamics. Lattice-gas models of phase separation receive special emphasis. The current understanding of phase transitions in these momentum-conserving models is reviewed; included in this discussion is a summary of the dynamical properties of interfaces. Because the phase-separation models are microscopically time irreversible, interesting questions are raised about their relationship to real fluid mixtures. Simulation of certain complex-fluid problems, such as multiphase flow through porous media and the interaction of phase transitions with hydrodynamics, is illustrated.

The understanding of chaotic systems can be considerably improved with the knowledge of their periodic-orbit structure. The identification of the low-order unstable periodic orbits embedded in a strange attractor induces a hierarchical organization of the dynamics which is invariant under smooth coordinate changes. The applicability of this technique is by no means limited to analytical or numerical calculations, but has been recently extended to experimental time series. As a specific example, the authors review some of the major results obtained on a nuclear-magnetic-resonance laser which have led to an extension of the conventional (Bloch-Kirchhoff) equations of motion, to the construction of approximate generating partitions, and to an efficient control of the chaotic system around various unstable periodic orbits. The determination of the symbolic dynamics, with the precision achieved by recording all unstable cycles up to order 9, improves the topological and metric characterization of a heteroclinic crisis. The periodic-orbit approach permits detailed study of chaotic motion, thereby leading to an improved classification scheme which subsumes the older ones, based on estimates of scalar quantities such as fractal dimensions and metric entropies.

With the high-temperature superconductors a qualitatively new regime in the phenomenology of type-II superconductivity can be accessed. The key elements governing the statistical mechanics and the dynamics of the vortex system are (dynamic) thermal and quantum fluctuations and (static) quenched disorder. The importance of these three sources of disorder can be quantified by the Ginzburg number $Gi=\frac{{\left(\frac{T_c}{H_c^2}\varepsilon {\xi }^3\right)}^2}{2}$, the quantum resistance $Qu=\left(\frac{e^2}{\hslash }\right)\left(\frac{{\rho }_n}{\varepsilon \xi }\right)$, and the critical current-density ratio $\frac{j_c}{j_o}$, with $j_c$ and $j_o$ denoting the depinning and depairing current densities, respectively (${\rho }_n$ is the normal-state resistivity and ${\varepsilon }^2=\frac{m}{M}<1\mathrm{}$ denotes the anisotropy parameter). The material parameters of the oxides conspire to produce a large Ginzburg number $\mathrm{Gi}\sim {10}^{-2\mathrm{}}$ and a large quantum resistance $\mathrm{Qu}\sim {10}^{-1\mathrm{}}$, values which are by orders of magnitude larger than in conventional superconductors, leading to interesting effects such as the melting of the vortex lattice, the creation of new vortex-liquid phases, and the appearance of macroscopic quantum phenomena. Introducing quenched disorder into the system turns the Abrikosov lattice into a vortex glass, whereas the vortex liquid remains a liquid. The terms "glass" and "liquid" are defined in a dynamic sense, with a sublinear response $\rho ={\frac{\partial E}{\partial j}|}_{j\to 0}$ characterizing the truly superconducting vortex glass and a finite resistivity $\rho (j\to 0)>0$ being the signature of the liquid phase. The smallness of $\frac{j_c}{j_o}$ allows one to discuss the influence of quenched disorder in terms of the weak collective pinning theory. Supplementing the traditional theory of weak collective pinning to take into account thermal and quantum fluctuations, as well as the new scaling concepts for elastic media subject to a random potential, this modern version of the weak collective pinning theory consistently accounts for a large number of novel phenomena, such as the broad resistive transition, thermally assisted flux flow, giant and quantum creep, and the glassiness of the solid state. The strong layering of the oxides introduces additional new features into the thermodynamic phase diagram, such as a layer decoupling transition, and modifies the mechanism of pinning and creep in various ways. The presence of strong (correlated) disorder in the form of twin boundaries or columnar defects not only is technologically relevant but also provides the framework for the physical realization of novel thermodynamic phases such as the Bose glass. On a macroscopic scale the vortex system exhibits self-organized criticality, with both the spatial and the temporal scale accessible to experimental investigations.

Most of our knowledge of the electronic structure of atoms and molecules is derived from excitation energies and transition probabilities. These observable quantities are related to the electronic wave functions by integrals over unmeasured variables. Another observable more directly related to the wave function than energy or transition probability is the single-electron momentum density, the probability that an electron in a well-defined orbital has a given value of momentum. Over the last twenty years a technique has been developed for measuring momentum densities in atoms and molecules. The technique, ($e$,$2e$) spectroscopy, is based on electron-impact ionization with complete determination of the momentum of both incoming and outgoing electrons. The conditions necessary to extract momentum-density information from the ionization experiments are examined and related to general theories of electron scattering. Different experimental arrangements are reviewed and momentum-density results from selected examples are discussed.

An overview is presented of methods for time-dependent treatments of molecules as systems of electrons and nuclei. The theoretical details of these methods are reviewed and contrasted in the light of a recently developed time-dependent method called electron-nuclear dynamics. Electron-nuclear dynamics (END) is a formulation of the complete dynamics of electrons and nuclei of a molecular system that eliminates the necessity of constructing potential-energy surfaces. Because of its general formulation, it encompasses many aspects found in other formulations and can serve as a didactic device for clarifying many of the principles and approximations relevant in time-dependent treatments of molecular systems. The END equations are derived from the time-dependent variational principle applied to a chosen family of efficiently parametrized approximate state vectors. A detailed analysis of the END equations is given for the case of a single-determinantal state for the electrons and a classical treatment of the nuclei. The approach leads to a simple formulation of the fully nonlinear time-dependent Hartree-Fock theory including nuclear dynamics. The nonlinear END equations with the ab initio Coulomb Hamiltonian have been implemented at this level of theory in a computer program, ENDyne, and have been shown feasible for the study of small molecular systems. Implementation of the Austin Model 1 semiempirical Hamiltonian is discussed as a route to large molecular systems. The linearized END equations at this level of theory are shown to lead to the random-phase approximation for the coupled system of electrons and nuclei. The qualitative features of the general nonlinear solution are analyzed using the results of the linearized equations as a first approximation. Some specific applications of END are presented, and the comparison with experiment and other theoretical approaches is discussed.

The macroscopic electric polarization of a crystal is often defined as the dipole of a unit cell. In fact, such a dipole moment is ill defined, and the above definition is incorrect. Looking more closely, the quantity generally measured is differential polarization, defined with respect to a "reference state" of the same material. Such differential polarizations include either derivatives of the polarization (dielectric permittivity, Born effective charges, piezoelectricity, pyroelectricity) or finite differences (ferroelectricity). On the theoretical side, the differential concept is basic as well. Owing to continuity, a polarization difference is equivalent to a macroscopic current, which is directly accessible to the theory as a bulk property. Polarization is a quantum phenomenon and cannot be treated with a classical model, particularly whenever delocalized valence electrons are present in the dielectric. In a quantum picture, the current is basically a property of the phase of the wave functions, as opposed to the charge, which is a property of their modulus. An elegant and complete theory has recently been developed by King-Smith and Vanderbilt, in which the polarization difference between any two crystal states—in a null electric field—takes the form of a geometric quantum phase. The author gives a comprehensive account of this theory, which is relevant for dealing with transverse-optic phonons, piezoelectricity, and ferroelectricity. Its relation to the established concepts of linear-response theory is also discussed. Within the geometric phase approach, the relevant polarization difference occurs as the circuit integral of a Berry connection (or "vector potential"), while the corresponding curvature (or "magnetic field") provides the macroscopic linear response.

Various experimental methods based on positron annihilation have evolved into important tools for researching the structure and properties of condensed matter. In particular, positron techniques are useful for the investigation of defects in solids and for the investigation of solid surfaces. Experimental methods need a comprehensive theory for a deep, quantitative understanding of the results. In the case of positron annihilation, the relevant theory includes models needed to describe the positron states as well as the different interaction processes in matter. In this review the present status of the theory of positrons in solids and on solid surfaces is given. The review consists of three main parts describing (a) the interaction processes, (b) the theory and methods for calculating positron states, and (c) selected recent results of positron studies of condensed matter.

Theoretical ideas and experimental results concerning high-temperature superconductors are reviewed. Special emphasis is given to calculations performed with the help of computers applied to models of strongly correlated electrons proposed to describe the two-dimensional Cu${\mathrm{O}}_2$ planes. The review also includes results using several analytical techniques. The one- and three-band Hubbard models and the $t-J$ model are discussed, and their behavior compared against experiments when available. The author found, among the conclusions of the review, that some experimentally observed unusual properties of the cuprates have a natural explanation through Hubbard-like models. In particular, abnormal features like the mid-infrared band of the optical conductivity $\sigma \left(\omega \right)$, the new states observed in the gap in photoemission experiments, the behavior of the spin correlations with doping, and the presence of phase separation in the copper oxide superconductors may be explained, at least in part, by these models. Finally, the existence of superconductivity in Hubbard-like models is analyzed. Some aspects of the recently proposed ideas to describe the cuprates as having a $d_{x^2-y^2}$ superconducting condensate at low temperatures are discussed. Numerical results favor this scenario over others. It is concluded that computational techniques provide a useful, unbiased tool for studying the difficult regime where electrons are strongly interacting, and that considerable progress can be achieved by comparing numerical results against analytical predictions for the properties of these models. Future directions of the active field of computational studies of correlated electrons are briefly discussed.

Many of the properties of orientationally disordered crystals are profoundly affected by the coupling (known as translation-rotation coupling) between translation displacements and molecular orientation. The consequences of translation-rotation coupling depend on molecular and crystal symmetry, and vary throughout the Brillouin zone. One result is an indirect coupling between the orientations of different molecules, which plays an important role in the order/disorder phase transition, especially in ionic orientationally disordered crystals. Translation-rotation coupling also leads to softening of elastic constants and affects phonon spectra. This article describes the theory of the coupling from the point of view of the microscopic Hamiltonian and the resulting Landau free energy. Considerable emphasis is placed on the restrictions due to symmetry as these are universal and can be used to help one's qualitative understanding of experimental observations. The application of the theory to phase transitions is described. The softening of elastic constants is discussed and shown to be universal. However, anomalies associated with the order/disorder phase transition are shown to be restricted to cases in which the symmetry of the order parameter satisfies certain conditions. Dynamic effects on phonon spectra are described and finally the recently observed dielectric behavior of ammonium compounds is discussed. Throughout the article examples from published experiments are used to illustrate the application of the theory including well known examples such as the alkali metal cyanides and more recently discovered orientationally disordered crystals such as the fullerite, ${\mathrm{C}}_{60}$.

Halo states extend over an unusually large space where many of their properties hinge on the tail of the wave functions. Their levels lie mainly just below the thresholds for neutron-emission channels. This short review covers the main distinguishing features of halos and how they are revealed in experiments. Special emphasis is placed on the large variety of experiments apt to study halo structures. Some aspects of current research are treated in greater detail.

During the past two decades there have been substantial advances in magnetic fusion research. On the experimental front, progress has been led by the mainline tokamaks, which have achieved reactor-level values of temperature and plasma pressure. Comparable progress, when allowance is made for their smaller programs, has been made in complementary configurations such as the stellarator, reversed-field pinch and field-reversed configuration. In this paper, the status of understanding of the physics of toroidal plasmas is reviewed. It is shown how the physics performance, constrained by technological and economic realities, determines the form of reference toroidal reactors. A comparative study of example reactors is not made, because the level of confidence in projections of their performance varies widely, reflecting the vastly different levels of support which each has received. Success with the tokamak has led to the initiation of the International Thermonuclear Experimental Reactor project. It is designed to produce 1500 MW of fusion power from a deuterium-tritium plasma for pulses of 1000 s or longer and to demonstrate the integration of the plasma and nuclear technologies needed for a demonstration reactor.

During the past five years, major progress has been made in the experimental study of solid hydrogen at ultrahigh pressures as a result of developments in diamond-cell technology. Pressures at which metallization has been predicted to occur have been reached (250-300 Gigapascals). Detailed studies of the dynamic, structural, and electronic properties of dense hydrogen reveal a system unexpectedly rich in physical phenomena, exhibiting a variety of transitions at ultrahigh pressures. This colloquium explores the study of dense hydrogen as an archetypal problem in condensed-matter physics.

The authors present an overview of ongoing studies of the rich dynamical behavior of the uniform, deterministic Burridge-Knopoff model of an earthquake fault, discussing the model's behavior in the context of current seismology. The topics considered include: (1) basic properties of the model, such as the distinction between small and large events and the magnitude vs frequency distribution; (2) dynamics of individual events, including dynamical selection of rupture propagation speeds; (3) generalizations of the model to more realistic, higher-dimensional models; and (4) studies of predictability, in which artificial catalogs generated by the model are used to test and determine the limitations of pattern recognition algorithms used in seismology.