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Mathematical physics

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Mathematical physics is the scientific discipline concerned with "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"1.

It can be seen as underpinning both theoretical physics and computational physics.

Prominent mathematical physicists

The great 17th century physicist Isaac Newton developed a wealth of new mathematics, in an informal way, to solve problems in physics, including a form of calculus and numerical methods such as Newton's method. James Clerk Maxwell, Lord Kelvin, William Rowan Hamilton, and J. Willard Gibbs were mathematical physicists who had a profound impact on 19th century science. Revolutionary mathematical physicists at the turn of the 20th century included David Hilbert who devised the theory of Hilbert spaces for Integral equations, to find a major application in quantum mechanics. The "very mathematical" Paul Dirac used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron. Albert Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations, and his general relativity replaced the flat geometry of the large scale universe by that of a Riemannian manifold, whose curvature replaced Newton's gravitational force. Other prominent mathematical physicists include Jules-Henri Poincaré and Satyendra Nath Bose.

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 while 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. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics.

Some recompense for the fact that mathematicians tend to call researchers in this area physicists and that physicists tend to call them mathematicians is provided by the breadth of physical subject matter and beauty of various unexpected interconnections in the mathematical structure of rather distinct physical situations.

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 such a mathematically rigorous footing has inspired many mathematical developements. 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 mathematical fields such as representation theory. Use of geometry and topology plays an important role in string theory. The above are just a few examples. A examination of the currently research literature would undoubtedly give other such instances.

Notes

Bibliographical references

  • P. Szekeres, A Course in Modern Mathematical Physics: Groups, Hilbert Space and differential geometry. Cambridge University Press, 2004.
  • J. von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1996.
  • J. Baez, Gauge Fields, Knots, and Gravity. World Scientific, 1994.
  • R. Geroch, Mathematical Physics. University of Chicago Press, 1985.
  • R. Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, 1996.
  • J. Glimm & A. Jaffe, Quantum Physics: A Functional Integral Point of View. Springer-Verlag, 1987.
  • A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
  • George B. Arfken & Hans J. Weber, Mathematical Methods for Physicists, Academic Press; 4th edition, 1995.

See also

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