# Mathematical physics

#### Background Information

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**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 theoretical physics|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 [1852-1928], 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.