Mathematical physics

Mathematical physics is a scientific discipline aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although mathematical physics and theoretical physics are related, these two notions are often distinguished. Mathematical physics emphasizes the mathematical rigor of the same type as found in mathematics while theoretical physics emphasizes the links to observations and experimental physics which often requires the theoretical physicists to use heuristic, intuitive, and approximate arguments. Arguably, mathematical physics is closer to mathematics, and theoretical physics is closer to physics. Some compensation for the fact that mathematicians tend to call mathematical physicists 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.

Because of the required rigor, mathematical physicists often deal with questions that theoretical physicists have considered to be solved. However, the mathematical physicists 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 mathematical physics research are functional analysis and probability theory. Other subjects researched by mathematical physicists include operator algebras, geometric algebra, noncommutative geometry, string theory, group theory, random fields etc.

People such as David Hilbert, James Clerk Maxwell, and almost all of the original founders of quantum mechanics were mathematical physicists. Other mathematical physicists include Joseph-Louis Lagrange, Jules-Henri Poincaré, Lord Kelvin, Satyendra Nath Bose, and Josiah Willard Gibbs.

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, 2004.
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.

Internal links

Important publications in Mathematical physics

External links

• Communications in Mathematical Physics  (http://link.springer.de/link/service/journals/00220/index.htm)
• Journal of Mathematical Physics  (http://jmp.aip.org/)
• Mathematical Physics Electronic Journal  (http://www.ma.utexas.edu/mpej/MPEJ.html)
• International Association of Mathematical Physics (http://www.iamp.org/)
• Erwin Schrödinger International Institute for Mathematical Physics  (http://www.esi.ac.at)
• Linear Mathematical Physics Equations: Exact Solutions (http://eqworld.ipmnet.ru/en/solutions/lpde.htm) - from EqWorld
• Mathematical Physics Equations: Index (http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-mphe.htm) - from EqWorld
• Nonlinear Mathematical Physics Equations: Exact Solutions (http://eqworld.ipmnet.ru/en/solutions/npde.htm) - from EqWorld
• Nonlinear Mathematical Physics Equations: Methods (http://eqworld.ipmnet.ru/en/methods/meth-pde.htm) - from EqWorldde:Mathematische Physik

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