Mathematical Physics in the New Millennium

This article is the C18 chapter from the IUPAP Year 2000 book, Physics 2000.

Introduction

Mathematical physics spans every subfield of physics. Its aim is to apply the most powerful mathematical techniques available to the formulation and solution of physical problems. Mathematics is the language of theoretical physics and, like other languages, it provides a means of organizing thought and expressing ideas in a precise consistent manner. Physicists who are articulate in the language of mathematics have made the greatest contributions to the modern formulations of physics. The list is long and includes such names as Newton, Maxwell, Einstein, Schrödinger, Heisenberg, Weyl, Wigner and Dirac.

Mathematical physics is both interdisciplinary and in the mainstream of physics. Whereas experimental physicists make use of engineering and electronics techniques in their investigations, theoretical physicists make extensive use of mathematics. What is special about mathematical physicists is that they communicate and interact with both physicists and mathematicians. Some address mathematical problems that arise in physics. Others are primarily theoretical physicists who invoke mathematical methodology in the interpretation of physical phenomena, e.g., by the development and solution of physical models. What they have in common is an interest in understanding the exciting systems and mathematical challenges that physics uncovers. The results advance physics as a whole and make contributions to mathematics and new technolology. Physics is not an isolated activity. It both feeds on and enriches many related areas.

The value of strong interactions between scientists and mathematicians is underscored by the fact that many scientific revolutions have been linked to corresponding developments in mathematics. In this short essay, it is possible to mention only a few examples selected from the suggestions and paragraphs received from many contributors.

General relativity; the theory of gravity

In his general theory of relativity, Einstein introduced the revolutionary idea that gravity could be understood as a warping of space-time. This science fiction-like idea emerged as a logical consequence of a simple principle: the principle of equivalence.

For centuries, the study of geometry was restricted to flat spaces. Then, as a result of investigating the consequences of discarding Euclid's axiom about parallel lines never meeting, new curved-space geometries were discovered. These provided a natural, and even simple, language for the conceptually difficult task of describing the curvature of four-dimensional space-time.

General relativity was a theory ahead of its time due to a paucity of experimental contacts. Even by mid-century, it remained unclear if the gravitational waves implied by the linear approximation to general relativity were real and the notion of what we now call a black hole was dismissed as an absurdity.

Then, under the stimulus of advances in astronomy and cosmology, general relativity entered a golden age. Penrose and Hawking applied global techniques to the causal structure of space-time to prove very generally that singularities are inevitable at the endpoint of gravitational collapse and the big bang. Within a few years, understanding of gravitational collapse and black holes progressed from its inchoate beginnings to a sophisticated discipline comparable in elegance, rigour and generality to thermodynamics, a subject that it turned out unexpectedly to resemble.

Insight into the origin of this resemblance came in 1974 with Hawking's discovery that, at the quantum level, black holes are thermodynamical black bodies with characteristic temperatures. Many see this result as a clue to some future synthesis of general relativity, quantum theory and thermodynamics. Its exact nature remains, at century's end, a mystery.

A new discipline, numerical relativity, has mushroomed in the last decade thanks to advances in digital technology and refinements in the numerical integration of hyperbolic differential equations. Within the next decade it should become possible to compute the pattern of gravitational waves emitted from a pair of inspiralling black holes (the "binary black hole grand challenge") and to link these predictions with observational results from several gravitational-wave detectors now going on line.

By geometrizing the gravitational field, general relativity introduced a new viewpoint: the aim of describing all the basic forces and elementary particles in terms of some structure more general than Riemannian geometry. Noncommutative geometry, twistors and loop variables all represent gropings in this direction. The most actively pursued program is superstring (or M-) theory. It has been said that string theory is a part of twenty-first century physics that fell by chance into the twentieth century, before the appropriate mathematics had been developed. M-theory calls on the full arsenal of modern mathematics - topology, homotopy, Calabi-Yau spaces, Riemann surfaces, moduli spaces, Kac-Moody algebras, ..., a symbiosis, which can only get closer as the field develops.

Quantum mechanics

Quantum mechanics has long been a subject of intense investigation in both physics and mathematics. The first steps towards a quantum theory came from the pioneering work of physicists such as Planck, Einstein, and Bohr. The formulation of quantum mechanics, by Heisenberg, Schrödinger, Born, Dirac and others as the fundamental theory of nonrelativistic matter, invoked abstract mathematics: linear algebra, infinite-dimensional complex (Hilbert) spaces, noncommmutative algebra, and group theory. Quantum mechanics became a subfield of mathematics in 1949 with its rigorous formulation by von Neumann.

Object like atoms are characterized in quantum mechanics by sets of wave functions and energies, which satisfy a differential equation, called the Schrödinger equation. However, following initial success in deriving the energy levels of the hydrogen atom, it soon became clear that exact solutions of the Schrödinger equation are possible only for systems with few particles or for models with special symmetries. Approximate solutions are obtained by separating the problem into two parts such that one part has simple properties and its Schrödinger equation is exactly solvable. It is then assumed that the wave functions for the complete problem can be expanded in terms of the known solutions for the simpler problem. Moreover, by ordering the solutions of the simple equation by increasing energy, one can obtain a sequence of approximations to the full problem which, if they converge, give accurate results as found, for example, in numerous applications to atomic physics.

In considering the validity of such an approach, mathematical physicists investigate the properties of differential equations and the completeness of solutions to the Schrödinger equation. A set of wave functions is said to be complete if an arbitrary wave function can be expanded in terms of them. However, completeness is hard to prove for general many-particle systems for which some of the particles may be unbound to the others. It has recently been proved under special conditions.

Fortunately for physics, the physical world has a hierarchical structure which enables condensed matter to be explained in terms of atoms and molecules which, in turn, are explained in terms of nuclei and electrons, etc. This stratification suggests the construction of sequences of models in which the building blocks of one model are the objects of study of a more microscopic model. Furthermore, models of complex systems can be constructed, with restricted degrees of freedom, to explain particular phenomena. Useful models are ones with clear mathematical structures, which enable the properties of a model to be inferred from basic assumptions (axioms). Enormous progress has made during the past twenty years using dynamical symmetry (group theory) for such purposes.

Group theory; dynamical systems

In physics, group theory provides a mathematical formalism to classify and study the symmetry properties of a system. The fundamental books of Weyl and Wigner on the theory of groups and quantum mechanics established group theory as an essential tool of quantum mechanics. Between 1928 and 1938, Wigner (sometimes with students) introduced time reversal invariance and made applications to: atomic and molecular spectra, properties of crystals and their energy bands, nuclear structure and spectroscopy. In 1929, Bethe introduced the use of finite crystal point groups to the analysis of degeneracies of quantum states in crystals; this led to crystal field theory which in turn played a fundamental role in the design of lasers and fibre-optic communication systems. Of particular significance for later developments was Wigner's 1930 paper on vibrational spectra of molecules. This paper introduced to physics the "Frobenius method", later called "the method of induced representations". In 1938, Wigner used this method to construct all irreducible unitary representations of the Poincaré group for positive mass particles in what is recognized to be a landmark of twentieth century physics. In the forties, Racah laid the foundations for calculating the properties of many-particle quantum systems such as occur in atoms, nuclei and subatomic particles.

The method of induced representations is now a widely used tool in both mathematics and physics, with applications, for example, to the theory of crystal space groups and rotations of molecules and nuclei. Fundamental applications to the theory of Fermi surfaces and the related band structures that underlie the theory of conductors (and semiconductors) have been made in recent years. The theory of induced representations has also been applied extensively in nuclear and other areas of mathematical physics within the framework of coherent-state theory.

An example of a subject investigated initially for purely mathematical reasons is that of Penrose tiling. The problem was to tile a plane aperiodically with tiles of at most two shapes. Solutions to this problem were later invoked to explain the so-called "quasi-crystals" observed in 1984. Such crystals are now known to be quasi-periodic and described by functions with a number of periods greater than the dimension of space.

The last twenty-five years has seen an explosion of uses of dynamical groups and spectrum generating algebras for the construction and solution of models. Advances have followed the recognition that solvable models can usually be expressed in algebraic terms so that the powerful methods of group representation theory can be exploited in their solution. The idea goes back (at least) to a famous 1946 paper of Tomonaga. Modern developments followed the SU(3) model of nuclear rotations and the quark model of baryons and mesons.

That the theory of Lie groups should apply to quantum mechanical problems is natural, considering the fact that Sophus Lie founded his theory for the purpose of solving differential equations like the Schrödinger equation. However, in contrast to the Schrödinger equation, most of the other basic equations of physics are nonlinear (e.g., Einstein equations, Yang-Mills equations, Navier-Stokes equations, ...). This has inspired many developments during the last 30 years on the solution of nonlinear differential equations. Technical tools have been provided by advances in computer science, i.e., algebraic computing.

A crucial development was the discovery of "solitons", i.e., waves that are stable with respect to mutual collisions. Solitons were found in numerical simulations and, subsequently, for a large class of nonlinear equations, now called nonlinear integrable partial differential equations. They are widespread in nature. In oceans they are destructive and can interfere with oil drilling. In optical fibers they are used to carry undistorted information.

Another important discovery in nonlinear mathematical physics is that of "deterministic chaos". It was shown that systems governed by simple nonlinear differential equations, that should be entirely deterministic, actually behave in an unpredictable way over large time periods. The reason for this "nonintegrable" behavior is a very sensitive (exponential in time) dependence of the solutions on the initial data. An example of such a chaotic system is given by the Lorenz equations from meteorology. Deterministic chaos is observed in virtually all complex nonlinear systems. Integrable (soliton) systems typically have infinitely many symmetries and a very simple singularity structure. Chaotic ones have little symmetry and exceedingly complicated singularity structures.

Many-body theory; statistical physics

Many-body theory and statistical physics are attempts to describe the properties of extended matter, which might comprise as many as 1024 atoms per cubic centimetre and for which a detailed microscopic description is inconceivable. A central idea of many-body theory is to make expansions which become exact as the inverse of the particle number approaches zero. In statistical physics the behavior of particles is described by probability theory. In equilibrium statistical mechanics the assumption is that all states of the particles in some fixed volume with a fixed number of particles and fixed energy are equally likely.

A fascinating challenge is to understand and predict properties of phase transitions, such as melting and boiling. A classic discovery of Onsager was that it is possible to find an exact solution for a two-dimensional (Ising) model of a phase transition. Recent discoveries have placed this discovery in a much larger context, relating it to conformal symmetry of two-dimensional systems. Another development is the renormalization group, whose applications to statistical mechanics were promoted by such pioneers as Ken Wilson. This is an attempt to explain certain universal properties of phase transitions. In particular, it is helping explain the extraordinary similarities (critical exponents) of phase transitions in very different systems.

Many-body theory and statistical mechanics take on new features when combined with quantum mechanics. In particular, the notion of equally-likely state has to be revised for systems of identical particles. There are two kinds of elementary particle: bosons and fermions. Quantum mechanical wave functions are symmetric under exchange of coordinates of a pair of bosons and antisymmetric under exchange of fermions. A consequence of this is that two fermions are precluded from occupying the same state; this is the exclusion principle.

One of the early predictions for bosons was Bose-Einstein condensation (BEC), a phenomenon that has recently been observed. BEC raises many interesting questions; e.g., does it depend on the symmetry under exchange of bosons or on an absence of the exclusion principle? The observation of BEC for a system of hydrogen atoms would appear to favour the former. This is because a hydrogen atom is not quite a boson; it is a pair of fermions (a proton and an electron). Thus, while hydrogen atoms are symmetric under exchange, they also obey an exclusion principle. The evidence is that correlated pairs of fermions behave in suitable limits as bosons. A dramatic example of this is superconductivity in which electrons combine to form Cooper pairs. Thus, whereas systems of bosons form superfluids at low temperatures, electrons in metals form superconductors. These phenomena are among the few manifestations of quantum mechanics at a macroscopic level. They have many fascinating properties and commercial applications; they are also of intense mathematical interest. For example, much is learned from them about the nature of phase transitions. Some insights into the nature of superconductivity and the approach to a phase transition in a finite many-body system comes from nuclear physics. The suppression of superconductivity in rotating nuclei with increasing angular momentum is particularly relevant because of the mathematical similarity between Coriolis forces and magnetic fields. The problem of explaining superconductivity at a high critical temperature is a current challenge.

Often matter is not in equilibrium. There is a flow of particles or of energy, and often the flow itself is changing in time. This requires a nonequilibrium statistical mechanics. Currently nonequilibrium statistical mechanics is beginning to provide an understanding of the limits of cruder theories of nature, such as macroscopic theories of fluids or plasmas.

Symplectic geometry; symplectic groups

Loosely speaking, symplectic geometry is the mathematics of phase spaces. These are the basic spaces of classical mechanics in terms of the positions and momenta of particles. They also provide the appropriate framework for the quantization of classical systems. Because of its practical applications, symplectic geometry is an area where the interaction between mathematics and physics is especially close.

An example is the study of the stability of the solar system. This is a system for which all the relevant laws of physics are known to great precision. However, working out the consequences of those laws is nontrivial. One can integrate the equations of motion step by step to obtain accurate predictions of planetary orbits over time periods of billions of years. Yet it remains an open question as to whether the solar system is bound or not; i.e., if a planet might not some day in the future acquire enough energy to leave the system. Recent developments have shown how the motion of an ideal fluid can be described. This has led to important applications in geophysical fluid dynamics as well as in fluid mechanics.

A fundamental concept of symplectic geometry is the so-called "symplectic form". This is a concept that expresses the pairing of position and momenta as "canonical variables". The fundamental relationship between pairs of canonical variables gives rise to the famous conservation law, known as Liouville's theorem, for the volume of phase space occupied by a system of particles. The content of the theorem is illustrated by the observation that, without removing energy from a system of particles, it is impossible to squeeze them into in a small space by application of a force field without giving them large momenta. A related implication for quantum mechanics is the uncertainty principle; one cannot simultaneously measure the values of a pair of canonical variables, like position and momentum, to better than prescribed limits of accuracy.

Closely linked to symplectic geometry and equally important is the group theory of linear canonical (i.e., symplectic) transformations. These groups have been widely used, for example, for the description of electromagnetic and particle beam optics. An optical lens or a magnetic focusing device corresponds to an element of a group of symplectic transformations. Thus, the effect of a sequence of lenses or focusing devices can be inferred by the standard mathematical rules for composition of group elements. By such means, the aberration effects of optical and particle-beam transport systems can be computed and corrections made to a high level of accuracy. The development of symplectic techniques has revolutionized the design of such systems.

During the past thirty years, the links between classical and quantum mechanics and the routes from one to the other have been explored in great depth using the methods of "geometric quantization". Geometrical quantization is a series of methods which associate unitary group representations ("quantum mechanics") to certain Hamiltonian group actions ("classical mechanics"). The approach takes advantage of insights gained in physics to advance the mathematical study of group representations. It has also been used effectively to study the quantization of collective models, notably in nuclear physics.

Coherent states; optics

Coherent states are used in many areas of physics and mathematics. They were first defined by Schrödinger in 1926 as minimal uncertainty wave packets and used to exhibit the classical behavior of a harmonic oscillator within the framework of quantum mechanics. Because of this special "classical behavior" of quantum harmonic oscillators, coherent states have been used widely, following an influential 1963 paper of Glauber, for describing the coherence properties of electromagnetic radiation; e.g., the light emitted by a laser.

Analysis of optical systems in terms of coherent states is appropriate because, while much of optics can be described by classical theory, some problems require a quantum treatment. Coherent state theory lets one have it both ways. This is particularly important for the theory of quantum interferometers. An interferometer superposes optical beams and the resultant interference effects can be used to infer information about the beams, such as their wavelengths, or about the interferometer, such as the relative phase shift between paths of the interferometer. The minimal wave-packet character of coherent states makes them useful for considering the quantum limits to phase measurements in interferometry. Quantum effects become important for light beams of finite photon number. Interferometers can also be constructed for matter waves, including neutrons and neutral atoms.

Coherent states were generalized and given group theoretical definitions around 1972. Since that time, the mathematical analysis, generalization, and ever widening set of applications for coherent states has become a field of study in its own right. For example, it has become possible to apply the powerful methods of group representation theory to the analysis of multi-path interferometers. Coherent states associated with a vast number of transformation groups have been constructed; they have been generalized further to quantum groups and even to no groups at all.

Coherent state methods have proved invaluable for contructing representations of Lie groups in the forms needed for applications in many-body quantum mechanics. This use has developed substantially during the past 25 years, including generalizations to vector-valued coherent states.

Spaces of coherent states have interesting geometries and feature in the theory of geometric quantization. They have been used in path integral constructions, inequalities in statistical physics, descriptions of classically chaotic systems; they have been applied to the quantum Hall effect, atomic and molecular physics, nuclear physics, particle physics, etc. Future applications may include quantum communications, quantum cryptography, and quantum computation.

A practical development to emerge from coherent state theory is the use of "wavelets" as alternatives to traditional Fourier analysis. The idea was introduced by Morlet in 1983 for analyzing seismic waves. It has since been set on a rigorous mathematical footing and has become a powerful tool in electrical engineering for signal processing, and in computer science for data compression.

Quantum field theory

Field theory evolved from quantum electrodynamics (QED). It became a theory of particles and fields when it was understood that a relativistic quantum theory of a fixed number of particles is an impossibility.

A problem in QED before mid-century was its prediction of infinite values for a number of physical quantities. It turned out that the infinities where artifacts of expressing the theory in terms of parameters, like the electron mass and charge, which, if computed, would themselves diverge. With the right parameters and techniques for avoiding summations of divergent series, QED achieved extraordinary precision, e.g., in predicting Lamb shifts and the anomalous magnetic moment of the electron. Many names were associated with these adjustments of QED, known as "renormalization"; the list includes French, Weisskopf, Schwinger, Wheeler, Feynman, Dyson, and Tomonaga. QED now ranks as a major achievement of twentieth century physics. It has had many practical applications in areas as diverse as superconductivity, lasers, transistors and microchips.

The latter half of the century has seen many attempts to generalize QED to other force fields of nature. The most successful is the 1954 non-Abelian gauge theory of Yang and Mills. This theory, which uses sophisticated concepts of differential geometry (e.g., connections on fibre bundles), has had a profound impact on mathematical physics. It interfaces with such diverse topics as spontaneous symmetry breaking, nonintegrable phases, integrable systems, solitons, knots, and Kac-Moody algebras. The quantized version of YM theory has been spectacularly successful. It underlies the 1967-68 unified theory of electromagnetic and weak interactions of Glashow, Weinberg, and Salam. Likewise it provides the framework for quantum chromodynamics, a self-consistent theory of the strong interactions that evolved from the 1961-65 quark models of Gell-Man and Ne'eman. These developments show that all the fundamental interactions, including gravity, share a common geometric structure.

A separate remarkable development has been supersymmetric field theory, a theory that unites bosonic and fermionic fields within a common algebraic structure. It is hoped such a field theory will hold the key to the major outstanding problem of the theory of fundamental interactions which is to quantize gravity and provide the sought-after unification of all the fundamental interactions. String theory, which characterizes particles as strings rather than points, presents a promising approach to this difficult problem. But it will be some time in the next century before its success or failure is known.

Acknowledgements

The author is pleased to acknowledge many suggestions and the provision of numerous paragraphs, most of which have been incorporated into the article in one way or another, from the following contributors: Stephen Cook, Rashmi Desai, William Faris, Gordon Drake, Werner Israel, John Klauder, Michael Luke, Louis Michel, Lochlainn O'Raifertaigh, George Rosensteel, Mary Beth Ruskai, Barry Sanders, Ted Shepherd, Shlomo Sternberg, Scott Tremaine, Pavel Winternitz, Djoko Wirosoetisno, and Brian Wybourne.

David Rowe
Department of Physics
University of Toronto
Toronto, ON M5S 1A7
Canada