Mathematical physics
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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 physical theories."<ref>Definition from the ''[[Journal of Mathematical Physics]]''.<sup>[http://jmp.aip.org/jmp/staff.jsp]</sup></ref>
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<ref>Zaslow E.,Physmatics</ref>, 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 equation]]s (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 [[Joseph-Louis Lagrange|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 relativity|special]] and [[general relativity|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 [[physical cosmology|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 equation]]s and [[symplectic geometry]] are generally viewed as purely ''mathematical'' disciplines, whereas [[dynamical system]]s 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 string|vibrating strings]])'', and, more especially, [[Leonhard Euler]] [1707-1783], (for his work in ''[[Calculus of variations|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 [[Leonhard Euler|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 [[Maxwell's equations|equations of electromagnetism]], and his countryman, [[William Thomson, 1st Baron Kelvin|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'', ''[[fluid|fluids]]'', and ''[[sound]].'' In the U.S.A., the pioneering work of [[Willard Gibbs|Josiah Willard Gibbs]] [1839-1903] became the basis for ''[[statistical mechanics]].'' Together, these men laid the foundations of [[electromagnetism|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]] [1853-1928], with important insights from [[Henri Poincaré|Jules-Henri Poincaré]] [1854-1912], but which were brought to full clarity by [[Albert Einstein]] [1879-1955]. [[Albert Einstein|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.
[[Albert Einstein|Einstein]]'s [[special relativity]] replaced the [[Galilean transformations]] of space and time with [[Lorentz transformations]] in four dimensional [[Minkowski space]]-time. His [[general relativity|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 [[Isaac Newton|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 [[Planck's law|black body radiation]]) and [[Albert Einstein|Einstein]]'s work on the [[Photoelectric effect|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 Schrödinger]] [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 [[mathematical rigour|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) [[quantum mechanics|nonrelativistic quantum mechanics]] ([[Schrödinger]] [[operators]]), including the connections to [[atomic, molecular, and optical physics|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.
== Notes ==
<references />
== References ==
{{Citation |last=Zalsow |first=Eric |date=2005 |title=Physmatics |url=http://arxiv.org/abs/physics/0506153 }}
== Bibliographical references ==
===The classics===
:*{{citation |first1 = Ralph |last1 = Abraham |author1-link = Ralph Abraham |first2 = Jerrold E. |last2 = Marsden |author2-link = Jerrold E. Marsden |title = 'Foundations of mechanics: a mathematical exposition of classical mechanics with an introduction to the qualitative theory of dynamical systems' |edition = 2nd |place = Providence, [RI.] |publisher = AMS Chelsea Pub. |year = 2008 |isbn = 9780821844380}}
:*{{citation |first1 = Vladimir I. |last1 = Arnold |author-link = Vladimir Arnold |first2 = K. |last2 = Vogtmann |first3 = A. (tr.) |last3 = Weinstein |title = 'Mathematical methods of classical mechanics / [Matematicheskie metody klassicheskoĭ mekhaniki]' |edition = 2nd |place = New York, [NY.] |publisher = Springer-Verlag |year = 1997 |isbn = 0-387-96890-3}}
:*{{citation |first1 = Richard |last1 = Courant |author1-link = Richard Courant |first2 = David |last2 = Hilbert |author2-link = David Hilbert |title = 'Methods of mathematical physics / [Methoden der mathematischen Physik]' |place = New York, [NY.] |publisher = Interscience Publishers |year = 1989}}
:*{{citation |first1 = James |last1 = Glimm |author1-link = James Glimm |first2 = Arthur |last2 = Jaffe |author2-link = Arthur Jaffe |title = 'Quantum physics: a functional integral point of view' |edition = 2nd |place = New York, [NY.] |publisher = Springer-Verlag |year = 1987 |isbn = 0-387-96477-0}} (pbk.)
:*{{citation |first = Rudolf |last = Haag |author-link = Rudolf Haag |title = 'Local quantum physics: fields, particles, algebras' |edition = 2nd rev. & enl. |place = Berlin, [Germany] ; New York, [NY.] |publisher = Springer-Verlag |year = 1996 |isbn = 3-540-61049-9}} (softcover)
:*{{citation |first1 = Stephen W. |last1 = Hawking |author1-link = Stephen Hawking |first2 = George F. R. |last2 = Ellis |title = 'The large scale structure of space-time' |place = Cambridge, [England]
|publisher = Cambridge University Press |year = 1973 |isbn = 0-521-20016-4}}
:*{{citation |first = Tosio |last = Kato |title = 'Perturbation theory for linear operators' |edition = 2nd repr. |place = Berlin, [Germany] |publisher = Springer-Verlag |year = 1995 |isbn = 3-540-58661-X}}
:*''This is a'' reprint'' of the second (1980) edition of this title.''
:*{{citation |first1 = Henry |last1 = Margenau |author-link = Henry Margenau |first2 = George Moseley |last2 = Murphy |title = 'The mathematics of physics and chemistry' |edition = 2nd repr. |place = Huntington, [NY.] |publisher = R. E. Krieger Pub. Co. |year = 1976 |isbn = 0-882-75423-8}}
:*''This is a reprint of the'' 1956 ''second edition.''
:*{{citation |first1 = Philip McCord |last1 = Morse |author1-link = Philip M. Morse |first2 = Herman
|last2 = Feshbach |author2-link = Herman Feshbach |title = 'Methods of theoretical physics' |edition = repr. |place = Boston, {Mass.] |publisher = McGraw Hill |year = 1999 |isbn = 0-070-43316-X}}
:*''This is a'' reprint'' of the original (1953) edition of this title.''
:*{{citation |first1 = John |last1 = von Neumann |author-link = John von Neumann |first2 = Robert T. (tr.) |last2 = Beyer |title = 'Mathematical foundations of quantum mechanics' |place = Princeton, [NJ.] |publisher = Princeton University Press |year = 1955}}
:*{{citation |first1 = Michael C. |last1 = Reed |first2 = Barry |last2 = Simon |author2-link = Barry Simon |title = 'Methods of modern mathematical physics (4 vol.)' |place = New York. {NY.] |publisher = Academic Press |year = 1972-1977 |isbn = 0-125-85001-8}}
:*{{citation |first = Edward Charles |last = Titchmarsh |author-link = E. C. Titchmarsh |title = 'The theory of functions' |edition = 2nd |place = London, [England] |publisher = Oxford University Press |year = 1939}}
:*''This tome was reprinted in 1985.
:*{{citation |first1 = Walter E. |last1 = Thirring |author-link = Walter Thirring |first2 = Evans M. (tr.) |last2 = Harrell |title = 'A course in mathematical physics / [Lehrbuch der mathematischen Physik] (4 vol.)' |place = New York, [NY.] |publisher = Springer-Verlag |year = 1978-1983}}
:*{{citation |first1 = Hermann |last1 = Weyl |author1-link = Hermann Weyl |first2 = H. P. (tr.) |last2 = Robertson |title = 'The theory of groups and quantum mechanics / [Gruppentheorie und Quantenmechanik]' |place = London, [England] |publisher = Methuen & Co. |year = 1931}}
:*{{citation |first1 = Edmund Taylor |last1 = Whittaker |author1-link = E. T. Whittaker |first2 = George Neville |last2 = Watson |author2-link = G. N. Watson |title = 'A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions' |edition = 1st AMS |place = New York, [NY.] |publisher = AMS Press |year = 1979 |isbn = 0-404-14736-4}}
===Textbooks for undergraduate studies===
:*{{citation |first1 = George B. |last1 = Arfken |first2 = Hans J. |last2 = Weber |title = 'Mathematical methods for physicists' |edition = 4th |place = San Diego, [CA.] |publisher = Academic Press |year = 1995 |isbn = 0-120-59816-7}} (pbk.)
:*{{citation |first = Mary L. |last = Boas |title = 'Mathematical methods in the physical sciences' |edition = 3rd |place = Hoboken, [NJ.] |publisher = John Wiley & Sons |year = 2006 |isbn = 9780471198260}}
:*{{citation |first = Eugene |last = Butkov |title = 'Mathematical physics' |place = Reading, [Mass.] |publisher = Addison-Wesley |year = 1968}}
:*{{citation |first1 = Harold |last1 = Jeffreys |author1-link = Harold Jeffreys |first2 = Bertha |last2 = Swirles Jeffreys |author2-link = Bertha Swirles |title = 'Methods of mathematical physics' |edition = 3rd rev. |place = Cambridge, [England] |publisher = Cambridge University Press |year = 1956}}
:*{{citation |first1 = Jon |last1 = Mathews |first2 = Robert L. |last2 = Walker |title = 'Mathematical methods of physics' |edition = 2nd |place = New York, [NY.] |publisher = W. A. Benjamin |year = 1970 |isbn = 0-8053-7002-1}}
:*{{citation |first = Ivar |last = Stakgold |title = 'Boundary value problems of mathematical physics (2 vol.)' |place = Philadelphia, [PA.] |publisher = Society for Industrial and Applied Mathematics |year = c.2000 |isbn = 0-898-71456-7}} (set : pbk.)
===Other specialised subareas===
:*{{citation |first1 = Jamil |last1 = Aslam |first2 = Faheem |last2 = Hussain |title = 'Mathematical physics' Proceedings of the 12th Regional Conference], [[Islamabad]], [[Pakistan]], 27 March - 1 April [[2006]] |url = http://www.worldscibooks.com/physics/6405.html |place = Singapore |publisher = World Scientific |year = 2007 |isbn = 978-981-270-591-4}}
:*{{citation |first1 = John C. |last1 = Baez |author-link = John Baez |first2 = Javier P. |last2 = Muniain |title = 'Gauge fields, knots, and gravity' |place = Singapore ; River Edge, [NJ.] |publisher = World Scientific |year = 1994 |isbn = 9-810-22034-0}} (pbk.)
:*{{citation |first = Robert |last = Geroch |title = 'Mathematical physics' |place = Chicago, [IL.]
|publisher = University of Chicago Press |year = 1985 |isbn = 0-226-28862-5}} (pbk.)
:*{{citation |first = Andrei D. |last = Polyanin |title = 'Handbook of linear partial differential equations for engineers and scientists' |place = Boca Raton, [FL.] |publisher = Chapman & Hall / CRC Press |year = 2002 |isbn = 1-584-88299-9}}
:*{{citation |first1 = Alexei D. |last1 = Polyanin |first2 = Valentin F. |last2 = Zaitsev |title = 'Handbook of nonlinear partial differential equations' |place = Boca Raton, [FL.] |publisher = Chapman & Hall / CRC Press |year = 2004 |isbn = 1-584-88355-3}}
:*{{citation |first = Peter |last = Szekeres |title = 'A course in modern mathematical physics: groups, Hilbert space and differential geometry' |place = Cambridge, [England] ; New York, [NY.] |publisher = Cambridge University Press |year = 2004 |isbn = 0-521-53645-6}} (pbk.)
== See also ==
* [[List of publications in physics#Mathematical physics|Important publications in Mathematical Physics]]
* [[Theoretical physics]]
== External links ==
* [http://link.springer.de/link/service/journals/00220/index.htm Communications in Mathematical Physics ]
* [http://jmp.aip.org/ Journal of Mathematical Physics ]
* [http://www.ma.utexas.edu/mpej/MPEJ.html Mathematical Physics Electronic Journal ]
* [http://www.iamp.org/ International Association of Mathematical Physics]
* [http://www.esi.ac.at Erwin Schrödinger International Institute for Mathematical Physics ]
* [http://eqworld.ipmnet.ru/en/solutions/lpde.htm ''Linear Mathematical Physics Equations: Exact Solutions''] - from EqWorld
* [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-mphe.htm ''Mathematical Physics Equations: Index''] - from EqWorld
* [http://eqworld.ipmnet.ru/en/solutions/npde.htm ''Nonlinear Mathematical Physics Equations: Exact Solutions''] - from EqWorld
* [http://eqworld.ipmnet.ru/en/methods/meth-pde.htm ''Nonlinear Mathematical Physics Equations: Methods''] - from EqWorld
{{Mathematics-footer}}
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