Volume 41, Issue 6, June 2000
 SPECIAL ISSUE: MATHEMATICAL PHYSICS—PAST AND FUTURE

 CLASSICAL MECHANICS

Symplectic geometry and topology
View Description Hide DescriptionThis is a review of the ideas underlying the application of symplectic geometry to Hamiltonian systems. The paper begins with symplectic manifolds and their Lagrangian submanifolds, covers contact manifolds and their Legendrian submanifolds, and indicates the first steps of symplectic and contact topology.

The theory of caustics and wave front singularities with physical applications
View Description Hide DescriptionThis is intended as an introduction to and review of the theory of Lagrangian and Legendrian submanifolds and their associated maps developed by Arnold and his collaborators. The theory is illustrated by applications to Hamilton–Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wave fronts and their associated caustics and singularities.

Reduction theory and the Lagrange–Routh equations
View Description Hide DescriptionReduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré, and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics,plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles, along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, to stability theory,integrable systems, as well as geometric phases. This paper surveys progress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on nonAbelian Routh reduction. We develop the geometry of the associated Lagrange–Routh equations in some detail. The paper puts the new results in the general context of reduction theory and discusses some future directions.

Some aspects of the classical threebody problem that are close or foreign to physical intuition
View Description Hide DescriptionWe propose a simple oscillator model for the reduced threebody problem to understand the stability of orbits with small eccentricity of a light planet. It models the main shorttime features for small mass ratios of the other bodies. These results are confronted with the exact mathematical analysis for stability for all times, and with computer simulation results for bigger mass ratios, where chaotic features emerge.
 QUANTUM MECHANICS

The quantum body problem
View Description Hide DescriptionThis selective review is written as an introduction to the mathematical theory of the Schrödinger equation for particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theory of body Hamiltonians and the space–time and phasespace analysis of bound states and scattering states.

Ionization of a model atom by perturbations of the potential
View Description Hide DescriptionWe study the time evolution of the wave function of a particle bound by an attractive δfunction potential when it is subjected to timedependent variations of the binding strength (parametric excitation). The simplicity of this model permits certain nonperturbative calculations to be carried out analytically both in one and three dimensions. Thus the survival probability of bound state following a pulse of strength r and duration t, behaves as with both θ(∞) and α depending on r. On the other hand, a sequence of short pulses produces an exponential decay over an intermediate time scale.

Schrödinger operators in the twentieth century
View Description Hide DescriptionThis paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics.

Classical and quantum probability
View Description Hide DescriptionWe follow the development of probability theory from the beginning of the last century, emphasizing that quantum theory is really a generalization of this theory. The great achievements of probability theory, such as the theory of processes, generalized random fields, estimation theory, and information geometry, are reviewed. Their quantum versions are then described.
 QUANTUM FIELD THEORY

On revolutionizing quantum field theory with Tomita’s modular theory
View Description Hide DescriptionIn the book of Haag [Local Quantum Physics (Springer Verlag, Berlin, 1992)] about local quantum field theory the main results are obtained by the older methods of  and algebra theory. A great advance, especially in the theory of algebras, is due to Tomita’s discovery of the theory of modular Hilbert algebras [Quasistandard von Neumann algebras, Preprint (1967)]. Because of the abstract nature of the underlying concepts, this theory became (except for some sporadic results) a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the halfsided translations and the halfsided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Sec. IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Sec. VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.

The quest for understanding in relativistic quantum physics
View Description Hide DescriptionWe discuss the status and some perspectives of relativistic quantum physics.

Twist fields and broken supersymmetry
View Description Hide DescriptionA twist field on a cylindrical space–time has the defining property that translation about a spatial circle results in multiplying the field by a phase. In this paper we investigate how such multivalued twist fields fit into the framework of constructive quantum field theory. Twisted theories have an interest in their own right; the twists also serve as infrared regulators that partially preserve the underlying symmetries of the Hamiltonian. The main focus of this paper is to investigate the extent that boson–fermion twistfield systems are compatible with the Lie symmetry and with the supersymmetry that one expects in the same examples without twists. We consider free systems and nonlinear boson–fermion interactions that arise from a holomorphic, quasihomogeneous, polynomial superpotential. We choose the twisting angles to lie on a chosen line in twist parameter space (leaving one free twist parameter). Doing this, we can obtain Lie symmetry and half the number of supersymmetry generators that one expects in our examples without the twists. We also show that the Hamiltonians for scalar twist fields yield twisted, positivetemperature expectations with the “twistpositivity” property. This is important because it justifies the existence of a functional integral representation for twisted, positivetemperature trace functionals. We regularize these systems in a way that preserves symmetry to the maximal extent. We pursue elsewhere other aspects and applications of this method, including bounding the extent of supersymmetry breaking.

Constructive field theory and applications: Perspectives and open problems
View Description Hide DescriptionIn this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with constructive methods well into the 21st century.

The century of the incomplete revolution: Searching for general relativistic quantum field theory
View Description Hide DescriptionIn fundamental physics, this has been the century of quantum mechanics and general relativity. It has also been the century of the long search for a conceptual framework capable of embracing the astonishing features of the world that have been revealed by these two “first pieces of a conceptual revolution.” The general requirements on the mathematics and some specific developments toward the construction of such a framework are discussed. Examples of covariant constructions of (simple) generally relativistic quantum field theories have been obtained as topological quantum field theories, in nonperturbative zerodimensional string theory and its higherdimensional generalizations, and as spin foammodels. A canonical construction of a general relativistic quantum field theory is provided by loop quantum gravity. Remarkably, all these diverse approaches have turned out to be related, suggesting an intriguing general picture of general relativistic quantum physics.

Particle physics and quantum field theory at the turn of the century: Old principles with new concepts
View Description Hide DescriptionThe present state of quantum field theory(QFT) is analyzed from a new viewpoint whose mathematical basis is the modular theory of von Neumann algebras. Its physical consequences suggest new ways of dealing with interactions, symmetries, Hawking–Unruh thermal properties and possibly also extensions of the scheme of renormalized perturbation theory. Interactions are incorporated by using the fact that the matrix is a relative modular invariant of the interacting—relative to the incoming—net of wedge algebras. This new point of view allows many interesting comparisions with the standard quantization approach to QFT and is shown to be firmly rooted in the history of QFT. Its radical “change of paradigm” aspect becomes particularly visible in the quantum measurement problem.
 NONCOMMUTATIVE GEOMETRY

A short survey of noncommutative geometry
View Description Hide DescriptionWe give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann–Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomialequation for geometries on the foursphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It is of the form and determines both the sphere and all its metrics with fixed volume form. The expectation 〈x〉 is the projection on the commutant of the algebra of 4 by 4 matrices. We also show, using the noncommutative analog of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude with some questions related to string theory.

Noncommutative geometry and fundamental physical interactions: The Lagrangian level—Historical sketch and description of the present situation
View Description Hide DescriptionThese notes comprise (i) a descriptive account of the history of the subject showing how physics and mathematics interwove to develop a mathematical concept of quantum manifold relevant to elementary particletheory; (ii) a detailed technical description, from scratch, of the spectral action formalism and computation.

Quantum groups and noncommutative geometry
View Description Hide DescriptionQuantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planckscale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.
 GENERAL RELATIVITY

Some recent progress in classical general relativity
View Description Hide DescriptionIn this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantummechanical particles (Part II).

Discrete structures in gravity
View Description Hide DescriptionDiscrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewiselinear spaces. Models of threedimensional quantum gravity involving symbols are then described, and progress in generalizing these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalizations are explored for the original formulation of discrete gravity using edgelength variables.
 STATISTICAL PHYSICS

Return to equilibrium
View Description Hide DescriptionWe study an atom with finitely many energy levels in contact with a heat bath consisting of photons (blackbody radiation) at a temperature The dynamics of this system is described by a Liouville operator, or thermal Hamiltonian, which is the sum of an atomic Liouville operator, of a Liouville operator describing the dynamics of a free, massless Bose field, and a local operator describing the interactions between the atom and the heat bath. We show that an arbitrary initial state that is normal with respect to the equilibrium state of the uncoupled system at temperature converges to an equilibrium state of the coupled system at the same temperature, as time tends to +∞ (return to equilibrium).