Mathematical physics
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.
The great 17th century mathematician and physicist
Isaac Newton developed a wealth of new mathematics, in an informal way, to solve problems in physics, including
calculus and several
numerical methods (most notably
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 the mathematician
David Hilbert who devised the theory of
Hilbert spaces for
integral equations which would 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.
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.
* Definition from the
Journal of Mathematical Physics [1].
* 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, 1955.
*
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.
*
Important publications in Mathematical physics*
Theoretical physics*
Communications in Mathematical Physics *
Journal of Mathematical Physics *
Mathematical Physics Electronic Journal *
International Association of Mathematical Physics*
Erwin Schrödinger International Institute for Mathematical Physics *
Linear Mathematical Physics Equations: Exact Solutions - from EqWorld
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Mathematical Physics Equations: Index - from EqWorld
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Nonlinear Mathematical Physics Equations: Exact Solutions - from EqWorld
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Nonlinear Mathematical Physics Equations: Methods - from EqWorld
