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

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 include David Hilbert and almost all of the original founders of quantum mechanics. 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 compensation 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. Quantum mechanics cannot be understood without a good knowledge of mathematics. It is not surprising, then, that its developed version under the name of quantum field theory is one of the most abstract, mathematically-based areas of the physical sciences, dealing with algebraic structures such as Lie Algebras - a topic of which ordinary physicists are often ignorant. Among the most relevant areas of contemporary mathematics in mathematically rigorous physics research are functional analysis and probability theory. Other subjects researched by rigorous mathematical physicists include operator algebras, geometric algebra, noncommutative geometry, string theory, group theory, random fields etc.


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

External links

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