Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "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 is a branch of applied mathematics, but deals with physical problems.

An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).


There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods.

Classical mechanics

The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics. It leads, for instance, to discover the deep interplay of the notion of symmetry and that of conserved quantities during the dynamical evolution, stated within the most elementary formulation of Noether's theorem. These approaches and ideas can be, and in fact have been, extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory and quantum field theory. Moreover, they have provided several examples and basic ideas in differential geometry (e.g. the theory of vector bundles and several notions in symplectic geometry).

Partial differential equations

The theory of partial differential equations (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 18th century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.

Quantum theory

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and more broadly, functional analysis. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic and molecular physics. Quantum information theory is another subspecialty.

Relativity and quantum relativistic theories

The special and general theories of relativity require a rather different type of mathematics. This was group theory, which played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the mathematical description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays.

Statistical mechanics

Statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics (or its quantum version) and it is closely related with the more mathematical ergodic theory and some parts of probability theory. There are increasing interactions between combinatorics and physics, in particular statistical physics.


Mathematical Physics and other sciences

The usage of the term "mathematical physics" is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics. John Herapath used the term for the title of his 1847 text on "mathematical principles of natural philosophy"; the scope at that time being "the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature".[2]

Mathematical vs. theoretical physics

The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of pure mathematics and physics. Although related to theoretical physics,[3] 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.[4] Such arguments are not considered rigorous by mathematicians, but that is changing over time.

Such mathematical physicists primarily expand and elucidate physical theories. Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to be already solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incomplete, incorrect, or simply too naïve. Issues about attempts to infer the second law of thermodynamics from statistical mechanics are examples. Other examples concern the subtleties involved with synchronisation procedures in special and general relativity (Sagnac effect and Einstein synchronisation).

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 mechanics, quantum field theory, and quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has also brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory.

Prominent mathematical physicists

Before Newton

The roots of mathematical physics can be traced back to the likes of Archimedes in Greece, Ptolemy in Egypt, Alhazen in Iraq, and Al-Biruni in Persia.

In the first decade of the 16th century, amateur astronomer Nicolaus Copernicus proposed heliocentrism, and published a treatise on it in 1543. He retained the Ptolemaic idea of epicycles, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According to Aristotelian physics, the circle was the perfect form of motion, and was the intrinsic motion of Aristotle's fifth element—the quintessence or universal essence known in Greek as aether for the English pure air—that was the pure substance beyond the sublunary sphere, and thus was celestial entities' pure composition. The German Johannes Kepler [1571–1630], Tycho Brahe's assistant, modified Copernican orbits to ellipses, formalized in the equations of Kepler's laws of planetary motion.

An enthusiastic atomist, Galileo Galilei in his 1623 book The Assayer asserted that the "book of nature" is written in mathematics.[5] His 1632 book, about his telescopic observations, supported heliocentrism.[6] Having introduced experimentation, Galileo then refuted geocentric cosmology by refuting Aristotelian physics itself. Galilei's 1638 book Discourse on Two New Sciences established the law of equal free fall as well as the principles of inertial motion, founding the central concepts of what would become today's classical mechanics.[6] By the Galilean law of inertia as well as the principle of Galilean invariance, also called Galilean relativity, for any object experiencing inertia, there is empirical justification for knowing only that it is at relative rest or relative motion—rest or motion with respect to another object.

René Descartes adopted Galilean principles and developed a complete system of heliocentric cosmology, anchored on the principle of vortex motion, Cartesian physics, whose widespread acceptance brought the demise of Aristotelian physics. Descartes sought to formalize mathematical reasoning in science, and developed Cartesian coordinates for geometrically plotting locations in 3D space and marking their progressions along the flow of time.[7]

Christiaan Huygens was the first to use mathematical formulas to describe the laws of physics, and for that reason Huygens is regarded as the first theoretical physicist and the founder of mathematical physics.[8][9]

Newtonian and post Newtonian

Isaac Newton (1642–1727) developed new mathematics, including calculus and several numerical methods such as Newton's method to solve problems in physics. Newton's theory of motion, published in 1687, modeled three Galilean laws of motion along with Newton's law of universal gravitation on a framework of absolute space—hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming absolute time, supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space. The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced the Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity.[10]

In the 18th century, the Swiss Daniel Bernoulli (1700–1782) made contributions to fluid dynamics, and vibrating strings. The Swiss Leonhard Euler (1707–1783) did special work in variational calculus, dynamics, fluid dynamics, and other areas. Also notable was the Italian-born Frenchman, Joseph-Louis Lagrange (1736–1813) for work in analytical mechanics: he formulated Lagrangian mechanics) and variational methods. A major contribution to the formulation of Analytical Dynamics called Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician, William Rowan Hamilton (1805-1865). Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist Joseph Fourier (1768 – 1830) introduced the notion of Fourier series to solve the heat equation, giving rise to a new approach to solving partial differential equations by means of integral transforms.

Into the early 19th century, the French Pierre-Simon Laplace (1749–1827) made paramount contributions to mathematical astronomy, potential theory, and probability theory. Siméon Denis Poisson (1781–1840) worked in analytical mechanics and potential theory. In Germany, Carl Friedrich Gauss (1777–1855) made key contributions to the theoretical foundations of electricity, magnetism, mechanics, and fluid dynamics. In England, George Green (1793-1841) published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828, which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism.

A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch Christiaan Huygens (1629–1695) developed the wave theory of light, published in 1690. By 1804, Thomas Young's double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the luminiferous aether, was accepted. Jean-Augustin Fresnel modeled hypothetical behavior of the aether. Michael Faraday introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish James Clerk Maxwell (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four Maxwell's equations. Initially, optics was found consequent of Maxwell's field. Later, radiation and then today's known electromagnetic spectrum were found also consequent of this electromagnetic field.

The English physicist Lord Rayleigh [1842–1919] worked on sound. The Irishmen William Rowan Hamilton (1805–1865), George Gabriel Stokes (1819–1903) and Lord Kelvin (1824–1907) produced several major works: Stokes was a leader in optics and fluid dynamics; Kelvin made substantial discoveries in thermodynamics; Hamilton did notable work on analytical mechanics, discovering a new and powerful approach nowadays known as Hamiltonian mechanics. Very relevant contributions to this approach are due to his German colleague Carl Gustav Jacobi (1804–1851) in particular referring to canonical transformations. The German Hermann von Helmholtz (1821–1894) made substantial contributions in the fields of electromagnetism, waves, fluids, and sound. In the United States, the pioneering work of Josiah Willard Gibbs (1839–1903) became the basis for statistical mechanics. Fundamental theoretical results in this area were achieved by the German Ludwig Boltzmann (1844-1906). Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics.


By the 1880s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field. Thus, although the observer's speed was continually lost relative to the electromagnetic field, it was preserved relative to other objects in the electromagnetic field. And yet no violation of Galilean invariance within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of the aether, physicists inferred that motion within the aether resulted in aether drift, shifting the electromagnetic field, explaining the observer's missing speed relative to it. The Galilean transformation had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on Cartesian coordinates, but this process was replaced by Lorentz transformation, modeled by the Dutch Hendrik Lorentz [1853–1928].

In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion into the aether prompted aether's shortening, too, as modeled in the Lorentz contraction. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all inertial frames of reference, while Newton's theory of motion was spared.

In the 19th century, Gauss's contributions to non-Euclidean geometry, or geometry on curved surfaces, laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann (1826–1866). Austrian theoretical physicist and philosopher Ernst Mach criticized Newton's postulated absolute space. Mathematician Jules-Henri Poincaré (1854–1912) questioned even absolute time. In 1905, Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion.[10] Also in 1905, Albert Einstein (1879–1955) published his special theory of relativity, newly explaining both the electromagnetic field's invariance and Galilean invariance by discarding all hypotheses concerning aether, including the existence of aether itself. Refuting the framework of Newton's theory—absolute space and absolute time—special relativity refers to relative space and relative time, whereby length contracts and time dilates along the travel pathway of an object.

In 1908, Einstein's former professor Hermann Minkowski modeled 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time. Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance in his general theory of relativity,[11] extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with Gaussian coordinates, and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's vector of hypothetical gravitational force—an instant action at a distance—with a gravitational field. The gravitational field is Minkowski spacetime itself, the 4D topology of Einstein aether modeled on a Lorentzian manifold that "curves" geometrically, according to the Riemann curvature tensor, in the vicinity of either mass or energy. (Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.)


Another revolutionary development of the 20th century was quantum theory, which emerged from the seminal contributions of Max Planck (1856–1947) (on black body radiation) and Einstein's work on the 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), Satyendra Nath Bose (1894–1974), 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. That is called Hilbert space, introduced in its elementary form by David Hilbert (1862–1943) and Frigyes Riesz (1880-1956), and rigorously defined within the axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics, where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory in particular. 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.

List of prominent mathematical physicists in the 20th century

Prominent contributors to the 20th century's mathematical physics (although the list contains some typically theoretical, not mathematical, physicists and leaves many contributors out; please also note that since the page can be edited by anyone, sometimes less deserved mentions can pop up in the list) include, ordered by birth date, William Thomson (Lord Kelvin) [1824–1907], Oliver Heaviside [1850–1925], Jules Henri Poincaré [1854–1912] , David Hilbert [1862–1943], Arnold Sommerfeld [1868–1951], Constantin Carathéodory [1873–1950], Albert Einstein [1879–1955], Max Born [1882–1970], George David Birkhoff [1884-1944], Hermann Weyl [1885–1955], Satyendra Nath Bose [1894-1974], Norbert Wiener [1894–1964], Wolfgang Pauli [1900–1958], Paul Dirac [1902–1984], Eugene Wigner [1902–1995], Andrey Kolmogorov [1903-1987], Lars Onsager [1903-1976], John von Neumann [1903–1957], Sin-Itiro Tomonaga [1906–1979], Hideki Yukawa [1907–1981], Nikolay Nikolayevich Bogolyubov [1909–1992], Subrahmanyan Chandrasekhar [1910-1995], Mark Kac [1914–1984], Julian Schwinger [1918–1994], Richard Phillips Feynman [1918–1988], Irving Ezra Segal [1918–1998], Arthur Strong Wightman [1922–2013], Chen-Ning Yang [1922– ], Rudolf Haag [1922–2016], Freeman Dyson [1923– ], Martin Gutzwiller [1925–2014], Abdus Salam [1926–1996], Jürgen Moser [1928–1999], Michael Francis Atiyah [1929–2019], Joel Louis Lebowitz [1930– ], Roger Penrose [1931– ], Elliott Hershel Lieb [1932– ], Sheldon Lee Glashow [1932– ], Steven Weinberg [1933– ], Ludvig D. Faddeev [1934–2017], David Ruelle [1935– ], Yakov Grigorevich Sinai [1935– ], Vladimir Igorevich Arnold [1937–2010], Arthur Jaffe [1937– ], Roman Wladimir Jackiw [1939– ], Leonard Susskind [1940– ], Rodney James Baxter [1940– ], Michael Victor Berry [1941- ], Giovanni Gallavotti [1941- ], Stephen William Hawking [1942–2018], Jerrold Eldon Marsden [1942–2010], Alexander Markovich Polyakov [1945– ], Gerardus 't Hooft [1946– ], John L. Cardy [1947– ], Giorgio Parisi [1948– ], Edward Witten [1951– ], Herbert Spohn [1951?– ], Ashoke Sen [1956-] and Juan Martín Maldacena [1968– ].

See also


  1. ^ Definition from the Journal of Mathematical Physics. "Archived copy". Archived from the original on 2006-10-03. Retrieved 2006-10-03.CS1 maint: Archived copy as title (link)
  2. ^ John Herapath (1847) Mathematical Physics; or, the Mathematical Principles of Natural Philosophy, the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature, Whittaker and company via HathiTrust
  3. ^ Quote: " ... a negative definition of the theorist refers to his inability to make physical experiments, while a positive one.. implies his encyclopaedic knowledge of physics combined with possessing enough mathematical armament. Depending on the ratio of these two components, the theorist may be nearer either to the experimentalist or to the mathematician. In the latter case, he is usually considered as a specialist in mathematical physics.", Ya. Frenkel, as related in A.T. Filippov, The Versatile Soliton, pg 131. Birkhauser, 2000.
  4. ^ Quote: "Physical theory is something like a suit sewed for Nature. Good theory is like a good suit. ... Thus the theorist is like a tailor." Ya. Frenkel, as related in Filippov (2000), pg 131.
  5. ^ Peter Machamer "Galileo Galilei"—sec 1 "Brief biography", in Zalta EN, ed, The Stanford Encyclopedia of Philosophy, Spring 2010 edn
  6. ^ a b Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 129
  7. ^ Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 89
  8. ^ Dijksterhuis E.J (1950) De mechanisering van het wereldbeeld. Meulenhoff, Amsterdam.
  9. ^ Andreessen, C.D. (2005) Huygens: The Man Behind the Principle. Cambridge University Press: 6
  10. ^ a b Imre Lakatos, auth, Worrall J & Currie G, eds, The Methodology of Scientific Research Programmes: Volume 1: Philosophical Papers (Cambridge: Cambridge University Press, 1980), pp 213–214, 220
  11. ^ Salmon WC & Wolters G, eds, Logic, Language, and the Structure of Scientific Theories (Pittsburgh: University of Pittsburgh Press, 1994), p 125


Further reading

Generic works

Textbooks for undergraduate studies

  • Arfken, George B.; Weber, Hans J. (1995), Mathematical methods for physicists (4th ed.), San Diego: Academic Press, ISBN 0-12-059816-7 (pbk.)
  • Boas, Mary L. (2006), Mathematical Methods in the Physical Sciences (3rd ed.), Hoboken: John Wiley & Sons, ISBN 978-0-471-19826-0
  • Butkov, Eugene (1968), Mathematical physics, Reading: Addison-Wesley
  • Jeffreys, Harold; Swirles Jeffreys, Bertha (1956), Methods of mathematical physics (3rd rev. ed.), Cambridge, [England]: Cambridge University Press
  • Kusse, Bruce R. (2006), Mathematical Physics: Applied Mathematics for Scientists and Engineers (2nd ed.), Germany: Wiley-VCH, ISBN 3-527-40672-7
  • Joos, Georg; Freeman, Ira M. (1987), Theoretical Physics, Dover Publications, ISBN 0-486-65227-0
  • Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1
  • Menzel, Donald Howard (1961), Mathematical Physics, Dover Publications, ISBN 0-486-60056-4
  • Stakgold, Ivar (c. 2000), Boundary value problems of mathematical physics (2 vol.), Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-456-7 (set : pbk.)

Textbooks for graduate studies

  • Hassani, Sadri (1999), Mathematical Physics: A Modern Introduction to Its Foundations, Berlin, [Germany]: Springer-Verlag, ISBN 0-387-98579-4
  • Reed, M.; Simon, B. (1972–1977). Methods of Mathematical Physics. Vol 1-4. Academic Press.
  • Teschl, G. (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 978-0-8218-4660-5.
  • Moretti, V. (2018). Spectral Theory and Quantum Mechanics; Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation 2nd Edition. Berlin, Milan: Springer. ISBN 978-3-319-70705-1.
  • Landsman, K. (2017). Foundations of Quantum Theory. Berlin, Milan: Springer. ISBN 978-3-319-51776-6.

Other specialised subareas

  • Baez, John C.; Muniain, Javier P. (1994), Gauge fields, knots, and gravity, Singapore ; River Edge: World Scientific, ISBN 981-02-2034-0 (pbk.)
  • Geroch, Robert (1985), Mathematical physics, Chicago: University of Chicago Press, ISBN 0-226-28862-5 (pbk.)
  • Polyanin, Andrei D. (2002), Handbook of linear partial differential equations for engineers and scientists, Boca Raton: Chapman & Hall / CRC Press, ISBN 1-58488-299-9
  • Polyanin, Alexei D.; Zaitsev, Valentin F. (2004), Handbook of nonlinear partial differential equations, Boca Raton: Chapman & Hall / CRC Press, ISBN 1-58488-355-3
  • Szekeres, Peter (2004), A course in modern mathematical physics: groups, Hilbert space and differential geometry, Cambridge; New York: Cambridge University Press, ISBN 0-521-53645-6 (pbk.)
  • Yndurain, Francisco J (2006), Theoretical and Mathematical Physics. The Theory of Quark and Gluon Interactions, Berlin: Springer, ISBN 978-3642069741 (pbk.)

External links

Advances in Theoretical and Mathematical Physics

Advances in Theoretical and Mathematical Physics (ATMP) is a peer-reviewed, mathematics journal, published by International Press.

Established in 1997, the journal publishes articles on theoretical physics and mathematics.

The current managing editor is Charles Doran (University of Alberta).

Andrei Okounkov

Andrei Yuryevich Okounkov (Russian: Андре́й Ю́рьевич Окунько́в, Andrej Okun'kov) (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at Columbia University and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."

Communications in Mathematical Physics

Communications in Mathematical Physics is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, statistical mechanics and quantum field theory, and in operator algebras, quantum information and relativity. The current editor-in-chief is Horng-Tzer Yau.

Articles from 1965 to 1997 are available in electronic form free of charge, via Project Euclid, a non-profit organization initiated by Cornell University Library. This portion of the journal is provided through the Electronic Mathematical Archiving Network Initiative (EMANI) to support the long-term electronic preservation of mathematical publications.

Dannie Heineman Prize for Mathematical Physics

Dannie Heineman Prize for Mathematical Physics is an award given each year since 1959 jointly by the American Physical Society and American Institute of Physics. It is established by the Heineman Foundation in honour of Dannie Heineman. As of 2010, the prize consists of US$ 10,000 and a certificate citing the contributions made by the recipient plus travel expenses to attend the meeting at which the prize is bestowed.

Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a closed-form expression for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Edward Witten

Edward Witten (born August 26, 1951) is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey.

Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics.

In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity.

Elliott H. Lieb

Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.

In particular, his scientific works pertain to: the quantum and classical many-body problem, the stability of matter, atomic structure, the theory of magnetism, and the Hubbard model.

He is a prolific author in mathematics and physics with over 300 publications. He received his B.S. in physics from MIT (1953) and his Ph.D. in mathematical physics from the University of Birmingham in England (1956). Lieb was a (1956–1957) Fulbright Fellow at Kyoto University, Japan and for some time worked as the Staff Theoretical Physicist for IBM.

He has been a professor at Princeton since 1975, following a leave from his professorship at MIT. Lieb has been awarded several prizes in mathematics and physics, including the 1978 Heineman Prize for Mathematical Physics of the American Physical Society and the American Institute of Physics (1978), the Max Planck Medal of the German Physical Society (1992), the Boltzmann medal of the International Union of Pure and Applied Physics (1998), the Schock Prize (2001), and the Henri Poincaré Prize of the International Association of Mathematical Physics (2003). He is a member of the U.S. National Academy of Sciences and has twice served (1982–1984 and 1997–1999) as the President of the International Association of Mathematical Physics. Lieb was awarded the Austrian Decoration for Science and Art in 2002. In 2012 he became a fellow of the American Mathematical Society and in 2013 a Foreign Member of the Royal Society.He is married to fellow Princeton professor Christiane Fellbaum.

Eric Weinstein

Eric Ross Weinstein (born October 1965) is an American mathematician, economist, and the managing director of Thiel Capital, Peter Thiel's investment firm. He also writes on investments, capitalism, science, and mathematics.

Erwin Schrödinger International Institute for Mathematical Physics

The Erwin Schrödinger International Institute for Mathematics and Physics (ESI) is a research institute located in Vienna, Austria, whose aim is to stimulate cross fertilization between mathematics and physics. Named after Austrian physicist Erwin Schrödinger, it is a part of the University of Vienna and receives its basic funding from the Austrian Federal Ministry of Science and Research.

Gaussian q-distribution

In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel, is a q-analogue of the Gaussian or normal distribution.

The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.

Greg Moore (physicist)

Gregory Winthrop Moore (born 1961) is an American theoretical physicist who specializes in mathematical physics and string theory. Moore is a professor in the Physics and Astronomy Department of Rutgers University and a member of the University's High Energy Theory group. He is married to Karin Rabe, and son of Arthur Cotton Moore.

Moore's research has focused on: D-branes on Calabi–Yau manifolds and BPS state counting; relations to Borcherds products, automorphic forms, black-hole entropy, and wall-crossing; applications of the theory of automorphic forms to conformal field theory, string compactification, black hole entropy counting, and the AdS/CFT correspondence; potential relation between string theory and number theory; effective low energy supergravity theories in string compactification and the computation of nonperturbative stringy effects in effective supergravities; topological field theories, and applications to invariants of manifolds; string cosmology and string field theory.

Moore won a 2007 Essays on Gravitation Award from the Gravity Research Foundation for his essay, joint with Frederik Denef, How Many Black Holes Fit on the Head of a Pin? In 2012 he became a fellow of the American Mathematical Society.Moore won the 2014 Dannie Heineman Prize for Mathematical Physics "For eminent contributions to mathematical physics with a wide influence in many fields, ranging from string theory to supersymmetric gauge theory, conformal field theory, condensed matter physics and four-manifold theory." In 2015, he was jointly awarded the 2015 Dirac Medal by ICTP.Moore was a member of the Advisory Board for Springer's Encyclopedia of Mathematical Physics.He received an AB in physics from Princeton University in 1982 and a PhD in the same subject from Harvard University in 1985.

International Association of Mathematical Physics

The International Association of Mathematical Physics (IAMP) was founded in 1976 to promote research in mathematical physics. It brings together research mathematicians and theoretical physicists, including students. The association's ordinary members are individual researchers, although associate membership is available to organizations and companies. The IAMP is governed by an executive committee elected by the ordinary members.

The association sponsors the International Congress on Mathematical Physics (ICMP), which takes place every three years, and it also supports smaller conferences and workshops. There is a quarterly news bulletin.

IAMP currently awards two kinds of research prizes in mathematical physics at its triannual meetings, the Henri Poincaré Prize (created in 1997) and the Early Career Award (created in 2009).

John C. Baez

John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He is known for his work on spin foams in loop quantum gravity. For some time, his research had focused on applications of higher categories to physics and other things.Baez is also known to science fans as the author of This Week's Finds in Mathematical Physics, an irregular column on the internet featuring mathematical exposition and criticism. He started This Week's Finds in 1993 for the Usenet community, and it now has a following in its new form, the blog "Azimuth". This Week's Finds anticipated the concept of a personal weblog. Additionally, Baez is known on the World Wide Web as the author of the crackpot index.

Journal of Mathematical Physics

The Journal of Mathematical Physics is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen. Its 2017 Impact Factor is 1.165

Lists of physics equations

In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Physics is derived of formulae only.

Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.

Quantization (physics)

In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. (It is a procedure for constructing a quantum field theory starting from a classical field theory.) This is a generalization of the procedure for building quantum mechanics from classical mechanics. One also speaks of field quantization, as in the "quantization of the electromagnetic field", where one refers to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics.

Robert Geroch

Robert Geroch (born 1 June 1942 in Akron, Ohio) is an American theoretical physicist and professor at the University of Chicago. He has worked prominently on general relativity and mathematical physics and has promoted the use of category theory in mathematics and physics. He was the Ph.D. supervisor for Abhay Ashtekar and Gary Horowitz. He also proved an important theorem in spin geometry.

Statistical ensemble (mathematical physics)

In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1902, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system.A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.

Number theory
Probability theory
Decision sciences
Other applications
See also

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