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] For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This is a list of mathematics-based methods, by Wikipedia page. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ...

It can be seen as underpinning both theoretical physics and computational physics. Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ...

Scope of the subject

There are several quite distinct branches of mathematical physics, which roughly correspond to different historical periods. The theory of partial differential equations and related areas of variational calculus, Fourier analysis, potential theory and vector analysis are perhaps most closely associated with mathematical physics, and developed intensively from the second half of 18th century (D'Alembert, Euler, Lagrange) until the 1930s. Their physical applications include hydrodynamics, celestial mechanics, elasticity theory, acoustics, thermodynamics, electricity and magnetism, aerodynamics. The theory of atomic spectra and later quantum mechanics developed almost concurrently with the mathematical fields of linear algebra, spectral theory of operators, and more broadly, functional analysis, which constitute the mathematical side of another branch of mathematical physics. Special relativity and general relativity require rather different type of mathematics, represented by group theory (also playing an important role in quantum theory) and differential geometry. They were gradually supplemented by topology, as it gained prominence in mathematical description of cosmological as well as quantum field theory phenomena. Statistical mechanics forms a separate field, which is closely related with more mathematical ergodic theory and some parts of probability theory. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... Potential theory may be defined as the study of harmonic functions. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean Le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Lagrange may mean: Joseph Louis Lagrange, mathematician and mathematical physicist A Lagrange point in physics and astronomy The Lagrange_Multiplier mathematical technique Places in the United States: Lagrange, Georgia Lagrange, Indiana Lagrange, Maine Lagrange, New York (three places): Lagrange, Dutchess County Lagrange, Orange County Lagrange, Wyoming County Lagrange, Ohio Lagrange, Virginia... Hydrodynamics is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Solid mechanics (also known as the theory of elasticity) is a branch of physics, which governs the response of solid material to applied stress (e. ... Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... For the Daft Punk song, see Aerodynamic (song). ... Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of spectra, that is, the dependence of physical quantities on frequency. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... This article is about the physics subject. ... Quantum field theory (QFT) is the quantum theory of fields. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...

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 would be included. For example, ordinary differential equations and symplectic geometry are generally viewed as mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... The Lorenz attractor is an example of a non-linear dynamical system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

Prominent mathematical physicists

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, George Gabriel Stokes, 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. 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 Carl Friedrich Gauss, Jules-Henri Poincaré, Richard Feynman, Roger Penrose, and Satyendra Nath Bose. Carl Friedrich Gauss is largely considered to be one of the three greatest mathematicians of all time. His influence in mathematical physics is largely felt by his developing of the mathematical field non-Euclidean Geometry, which Albert Einstein's General Theory of Relativity as well as our understanding of the event horizon in black holes rely so heavily on. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... For other uses, see Calculus (disambiguation). ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance — eponymously named Maxwells equations — including an important modification (extension) of the Ampères... For other persons named William Thomson, see William Thomson (disambiguation). ... Sir George Gabriel Stokes, 1st Baronet (13 August 1819–1 February 1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics (including Stokes theorem). ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... A bar magnet. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... “Einstein” redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... This article is in need of attention. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... This article is about the physicist. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Satyendra Nath Bose Bengali: ) (January 1, 1894 – February 4, 1974) was an Indian physicist, specializing in mathematical physics. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... “Einstein” redirects here. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... For the science fiction film, see Event Horizon (film). ... This article is about the astronomical body. ...

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. 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. For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ... Look up Framework in Wiktionary, the free dictionary. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Look up Rigour in Wiktionary, the free dictionary. ... Experimental physics is the part of physics that deals with experiments and observations pertaining to natural/physical phenomena, as opposed to theoretical physics. ... Look up Heuristic in Wiktionary, the free dictionary. ... Intuition has many meanings across many cultures, including: quick and ready insight seemingly independent of previous experiences and empirical knowledge immediate apprehension or cognition knowledge or conviction gained by intuition the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference. ...

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. In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ...

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 field theory (QFT) is the quantum theory of fields. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ... This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ...

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. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... For other uses, see Geometry (disambiguation). ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point...

Notes

1. ^ Definition from the Journal of Mathematical Physics.^[1]

The Journal of Mathematical Physics is a peer-reviewed journal published by the American Institute of Physics devoted to the publication of papers in mathematical physics. ...

Bibliographical references

The Classics

• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge University Press, 1927.
• E. C. Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, 1939 (reprinted 1985).
• John von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955.
• Richard Courant and David Hilbert, Methods of Mathematical Physics. Vols. I and II. John Wiley & Sons, 1989.
• Hermann Weyl, The Theory of Groups and Quantum Mechanics. 1931.
• Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics. Parts I and II. McGraw Hill, 1953.
• Tosio Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1995.
• Barry Simon and Michael Reed, Methods of Modern Mathematical Physics. Vol. I: Functional Analysis, Academic Press, 1972; Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975; Vol. III: Scattering Theory, Academic Press, 1978; Vol. IV: Analysis of Operators, Academic Press, 1977.
• Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, 1996.
• James Glimm and Arthur Jaffe, Quantum Physics: A Functional Integral Point of View. Springer-Verlag, 1987.
• Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, 1975.
• Vladimir I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, 1997.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qualitative Theory of Dynamical Systems. Addison Wesley, 1994.
• Walter Thirring, A Course in Mathematical Physics I-IV. Springer-Verlag, 1998.
• Henry Margenau and George M. Murphy, The Mathematics of Physics and Chemistry. Van Nostrand Comp.

Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ... (George) Neville Watson (31 January 1886 - 2 February 1965) was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. ... Edward Charles (Ted) Titchmarsh (born 1 June 1899 in Newbury died 18 January 1963 at Oxford) was a leading British mathematician. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Full name : Philip McCord Morse Founding ORSA President (1952) B.S. Physics, 1926, Case Institute; Ph. ... Institute Professor Emeritus Herman Feshbach of Cambridge, a renowned nuclear physicist and champion of equal opportunity at MIT and around the world, died December 22 2000 of congestive heart failure at Youville Hospital in Cambridge. ... Katos conjecture is a mathematical problem named after mathematician Tosio Kato, of University of California at Berkeley. ... Barry Simon (born 16 April 1946) is an eminent American mathematical physicist and the IBM Professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics. ... Rudolf Haag (* 1922 in Tübingen, Germany) is a German physicist. ... James Glimm is an American mathematical physicist. ... Arthur Jaffe is an American mathematical physicist and a professor at Harvard University. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... Vladimir Igorevich Arnold (Russian: Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... Ralph H. Abraham (born July 4, 1936) is an American mathematician. ... Jerrold E. Marsden. ... Walter Thirring (born 1927-04-29) is an Austrian physicist after whom the Thirring model in quantum field theory is named. ... Henry Margenau (1901 - February 8, 1997) was a German-U.S. physicist, historian and philosopher of science, and Christian writer. ...

Textbooks for undergraduate studies

• Sir Harold Jeffreys and Bertha Swirles (Lady Jeffreys), Methods of Mathematical Physics, third revised edition (Cambridge University Press, 1956 — reprinted 1999). ISBN 0-521-66402-0, ISBN 978-0-521-66402-8.
• Eugene Butkov, Mathematical Physics. Addison Wesley, 1968.
• Ivar Stakgold, Boundary Value Problems of Mathematical Physics. Vols. I and II. Macmillan, 1970.
• Mary L. Boas, Mathematical Methods in the Physical Sciences. John Wiley & Sons, 3 ed., 2005.
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists. Academic Press, 1995.
• Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 2e, Addison-Wesley, 1970. ISBN 0-8053-7002-1

Sir Harold Jeffreys (22 April 1891 – 18 March 1989) was a mathematician, statistician, geophysicist, and astronomer. ... Bertha Swirles (Lady Jeffreys), (22 May 1903 - 18 December 1999) carried out research on quantum theory, particularly in its early days. ...

Other specialised subareas

• Jamil Aslam and Faheem Hussain Mathematical Physics, Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006, World Scientific, Singapore, 2007. ISBN 978-981-270-591-4
• P. Szekeres, A Course in Modern Mathematical Physics: Groups, Hilbert Space and differential geometry. Cambridge University Press, 2004.
• J. Baez, Gauge Fields, Knots, and Gravity. World Scientific, 1994.
• A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.
• R. Geroch, Mathematical Physics. University of Chicago Press, 1985.

Islamabad (Urdu: اسلام آباد) is the capital city of Pakistan, and is located in the Potohar Plateau in the northwest of the country. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... John Carlos Baez (b. ...

See also

• Important publications in Mathematical Physics
• Theoretical physics

This is a list of important publications in physics, organized by field. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ...

External links

• 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
• Mathematical Physics Equations: Index - from EqWorld
• Nonlinear Mathematical Physics Equations: Exact Solutions - from EqWorld
• Nonlinear Mathematical Physics Equations: Methods - from EqWorld