SURVEY

An overview is provided of the singularity theorems in cosmological contexts at a level suitable for advanced graduate students. The necessary background from tensor and causal geometry to understand the theorems is supplied, the mathematical notion of a cosmology is described in some detail and issues related to the range of validity of general relativity are also discussed.

In this article we review the framework for spontaneous replica symmetry breaking. Subsequently that is applied to the example of the statistical mechanical description of the storage properties of a McCulloch¯ Pitts neuron, i.e., simple perceptron. It is shown that in the neuron problem, the general formula that is at the core of all problems admitting Parisi's replica symmetry breaking ansatz with a one-component order parameter appears. The details of Parisi's method are reviewed extensively, with regard to the wide range of systems where the method may be applied. Parisi's partial differential equation and related differential equations are discussed, and the Green function technique is introduced for the calculation of replica averages, the key to determining the averages of physical quantities. The Green function of the Fokker¯Planck equation due to Sompolinsky turns out to play the role of the statistical mechanical Green function in the graph rules for replica correlators. The subsequently obtained graph rules involve only tree graphs, as appropriate for a mean-field-like model. The lowest order Ward¯Takahashi identity is recovered analytically and shown to lead to the Goldstone modes in continuous replica symmetry breaking phases. The need for a replica symmetry breaking theory in the storage problem of the neuron has arisen due to the thermodynamical instability of formerly given solutions. Variational forms for the neuron's free energy are derived in terms of the order parameter function x(q), for different prior distribution of synapses. Analytically in the high temperature limit and numerically in generic cases various phases are identified, among them is one similar to the Parisi phase in long-range interaction spin glasses. Extensive quantities like the error per pattern change slightly with respect to the known unstable solutions, but there is a significant difference in the distribution of non-extensive quantities like the synaptic overlaps and the pattern storage stability parameter. A simulation result is also reviewed and compared with the prediction of the theory.

As a result of advances in experimental and theoretical physics, many interesting problems have arisen in condensed-matter physics, typically as a result of the quantum-mechanical nature of a system. Areas of interest include Anderson localization, universal conductance fluctuations, normal electron persistent currents, and the properties of quasicrystals. Understanding such systems is challenging because of complications arising from the large number of particles involved, intractable symmetries, the presence of time-dependent or nonlinear terms in the Schrödinger equation, etc. Some progress has been made by studying large scale classical analog experiments which may accurately model the salient quantum-mechanical features of a condensed-matter system. This paper describes research with a number of acoustical systems which have addressed contemporary problems in condensed-matter physics.

We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).

One of the major developments of twentieth-century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this article the authors review the early history of gauge theory, from Einstein's theory of gravitation to the appearance of non-Abelian gauge theories in the fifties. The authors also review the early history of dimensional reduction, which played an important role in the development of gauge theory. A description is given of how, in recent times, the ideas of gauge theory and dimensional reduction have emerged naturally in the context of string theory and noncommutative geometry.

The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high-Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.

A relatively new class of materials has been found in which the basic assumption of Landau Fermi-liquid theory?that at low energies the electrons in a metal should behave essentially as a collection of weakly interacting particles?is violated. These non-Fermi-liquid systems exhibit unusual temperature dependences in their low-temperature properties, including several examples in which the specific heat divided by temperature shows a singular log T temperature dependence over more than two orders of magnitude, from the lowest measured temperatures in the milliKelvin regime to temperatures over 10 K. These anomalous properties, with their often pure power-law or logarithmic temperature dependences over broad temperature ranges and inherent low characteristic energies, have attracted active theoretical interest from the first experimental report in 1991. This article first describes the various theoretical approaches to trying to understand the source of strong temperature- and frequency-dependent electron-electron interactions in non-Fermi-liquid systems. It then discusses the current experimental body of knowledge, including a compilation of data on non-Fermi-liquid behavior in over 50 systems. The disparate data reveal some interesting correlations and trends and serve to point up a number of areas where further theoretical and experimental work is needed. ©2001 The American Physical Society

A longstanding goal in chemical physics has been the control of atoms and molecules using coherent light fields. This paper provides a brief overview of the field and discusses experiments that use a programmable pulse shaper to control the quantum state of electronic wavepackets in Rydberg atoms and electronic and nuclear dynamics in molecular liquids. The shape of Rydberg wavepackets was controlled by using tailored ultrafast pulses to excite a beam of caesium atoms. The quantum state of these atoms was measured using holographic techniques borrowed from optics. The experiments with molecular liquids involved the construction of an automated learning machine. A genetic algorithm directed the choice of shaped pulses which interacted with the molecular system inside a learning control loop. Analysis of successful pulse shapes that were found by using the genetic algorithm yield insight into the systems being controlled.