Mathematical physics

Mathematical physics is the scientific discipline concerned with the interface of mathematics and physics. There is no real consensus about what does or does not constitute mathematical physics. A very typical definition is the one given by the Journal of Mathematical Physics: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories."

This definition does, however, not cover the situation where results from physics are used to help prove facts in abstract mathematics which themselves have nothing particular to do with physics. This phenomenon has become increasingly important, with developments from string theory research breaking new ground in mathematics. Eric Zaslow coined the phrase physmatics to describe these developments, although other people would consider them as part of mathematical physics proper.

Important fields of research in mathematical physics include: functional analysis/quantum physics, geometry/general relativity and combinatorics/probability theory/statistical physics. More recently, string theory has managed to make contact with many major branches of mathematics including algebraic geometry, topology, and complex geometry.

Scope of the subject
There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equations (and the related areas of variational calculus, Fourier analysis, potential theory, and vector analysis) are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, and more broadly, functional analysis. These constitute the mathematical basis of another branch of mathematical physics.

The special and general theories of relativity require a rather different type of mathematics. This was group theory: and it played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology in the mathematical description of cosmological as well as quantum field theory phenomena.

Statistical mechanics forms a separate field, which is closely related with the more mathematical ergodic theory and some parts of probability theory.

The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.

Prominent mathematical physicists
The great seventeenth century English physicist and mathematician Isaac Newton [1642-1727] developed a wealth of new mathematics (for example, calculus and several numerical methods (most notably Newton's method)) to solve problems in physics. Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens [1629-1695] (famous for suggesting the wave theory of light), and the German Johannes Kepler [1571-1630] (Tycho Brahe's assistant, and discoverer of the equations for planetary motion/orbit).

In the eighteenth century, two of the great innovators of mathematical physics were Swiss: Daniel Bernoulli [1700-1782] (for contributions to fluid dynamics, and vibrating strings), and, more especially, Leonhard Euler [1707-1783], (for his work in variational calculus, dynamics, fluid dynamics, and many other things). Another notable contributor was the Italian-born Frenchman, Joseph-Louis Lagrange [1736-1813] (for his work in mechanics and variational methods).

In the late eighteenth and early nineteenth centuries, important French figures were Pierre-Simon Laplace [1749-1827] (in mathematical astronomy, potential theory, and mechanics) and Siméon Denis Poisson [1781-1840] (who also worked in mechanics and potential theory). In Germany, both Carl Friedrich Gauss [1777-1855] (in magnetism) and Carl Gustav Jacobi [1804-1851] (in the areas of dynamics and canonical transformations) made key contributions to the theoretical foundations of  electricity, magnetism, mechanics, and fluid dynamics.

Gauss (along with Euler) is considered by many to be one of the three greatest mathematicians of all time. His contributions to non-Euclidean geometry laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann [1826-1866]. As we shall see later, this work is at the heart of general relativity.

The nineteenth century also saw the Scot, James Clerk Maxwell [1831-1879], win renown for his four equations of electromagnetism, and his countryman, Lord Kelvin [1824-1907] make substantial discoveries in thermodynamics. Among the English physics community, Lord Rayleigh [1842-1919] worked on sound; and George Gabriel Stokes [1819-1903] was a leader in optics and fluid dynamics; while the Irishman William Rowan Hamilton [1805-1865] was noted for his work in dynamics. The German Hermann von Helmholtz [1821-1894] is best remembered for his work in the areas of electromagnetism, waves, fluids, and sound. In the U.S.A., the pioneering work of Josiah Willard Gibbs [1839-1903] became the basis for statistical mechanics. Together, these men laid the foundations of electromagnetic theory, fluid dynamics and statistical mechanics.

The late nineteenth and the early twentieth centuries saw the birth of special relativity. This had been anticipated in the works of the Dutchman, Hendrik Lorentz [1853-1928], with important insights from Jules-Henri Poincaré [1854-1912], but which were brought to full clarity by Albert Einstein [1879-1955]. Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity. This was based on the non-Euclidean geometry created by Gauss and Riemann in the previous century.

Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations in four dimensional Minkowski space-time. His general theory of relativity replaced the flat Euclidean geometry with that of a Riemannian manifold, whose curvature is determined by the distribution of gravitational matter. This replaced Newton's scalar gravitational force by the Riemann curvature tensor.

The other great revolutionary development of the twentieth century has been quantum theory, which emerged from the seminal contributions of Max Planck [1856-1947] (on black body radiation) and Einstein's work on the photoelectric effect. This was, at first, followed by a heuristic framework devised by Arnold Sommerfeld [1868-1951] and Niels Bohr [1885-1962], but this was soon replaced by the quantum mechanics developed by Max Born [1882-1970], Werner Heisenberg [1901-1976], Paul Dirac [1902-1984], Erwin Schrodinger [1887-1961], and  Wolfgang Pauli [1900-1958]. This revolutionary theoretical framework is based on a probabilistic interpretation of states,  and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space (Hilbert space, introduced by David Hilbert [1862-1943]). Paul Dirac, for example, used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron.

Later important contributors to twentieth century mathematical physics include Satyendra Nath Bose [1894-1974],  Julian Schwinger [1918-1994], Sin-Itiro Tomonaga [1906-1979], Richard Feynman [1918-1988],  Freeman Dyson [1923- ],   Hideki Yukawa [1907-1981], Roger Penrose [1931- ],  Stephen Hawking [1942- ], and Edward Witten [1951- ].

Mathematically rigorous physics
The term  'mathematical'  physics is also sometimes used in a special sense, to distinguish research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics.

Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.

The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.

The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory. The above are just a few examples. An examination of the current research literature would undoubtedly give other such instances.

The Classics

 * E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge University Press, 1927.
 * E. C. Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, 1939 (reprinted 1985).
 * John von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955.
 * Richard Courant and David Hilbert, Methods of Mathematical Physics. Vols. I and II. John Wiley & Sons, 1989.
 * Hermann Weyl, The Theory of Groups and Quantum Mechanics. 1931.
 * Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics. Parts I and II. McGraw Hill, 1953.
 * Tosio Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1995.
 * Barry Simon and Michael Reed, Methods of Modern Mathematical Physics. Vol. I: Functional Analysis, Academic Press, 1972; Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975; Vol. III: Scattering Theory, Academic Press, 1978; Vol. IV: Analysis of Operators, Academic Press, 1977.
 * Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, 1996.
 * James Glimm and Arthur Jaffe, Quantum Physics: A Functional Integral Point of View. Springer-Verlag, 1987.
 * Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, 1975.
 * Vladimir I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, 1997.
 * Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qualitative Theory of Dynamical Systems. Addison Wesley, 1994.
 * Walter Thirring, A Course in Mathematical Physics I-IV. Springer-Verlag, 1998.
 * Henry Margenau and George M. Murphy, The Mathematics of Physics and Chemistry. Van Nostrand Comp.

Textbooks for undergraduate studies

 * Sir Harold Jeffreys and Bertha Swirles (Lady Jeffreys), Methods of Mathematical Physics, third revised edition (Cambridge University Press, 1956 &mdash; reprinted 1999). ISBN 0-521-66402-0, ISBN 978-0-521-66402-8.
 * Eugene Butkov, Mathematical Physics. Addison Wesley, 1968.
 * Ivar Stakgold, Boundary Value Problems of Mathematical Physics. Vols. I and II. Macmillan, 1970.
 * Mary L. Boas, Mathematical Methods in the Physical Sciences. John Wiley & Sons, 3 ed., 2005.
 * George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists. Academic Press, 1995.
 * Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 2e, Addison-Wesley, 1970. ISBN 0-8053-7002-1

Other specialised subareas

 * Jamil Aslam and Faheem Hussain Mathematical Physics, Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006, World Scientific, Singapore, 2007. ISBN 978-981-270-591-4
 * P. Szekeres, A Course in Modern Mathematical Physics: Groups, Hilbert Space and differential geometry. Cambridge University Press, 2004.
 * J. Baez, Gauge Fields, Knots, and Gravity. World Scientific, 1994.
 * A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.
 * A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.
 * R. Geroch, Mathematical Physics. University of Chicago Press, 1985.