View source for Mathematical physics

{{Short description|Branch of applied mathematics}}
{{Math topics TOC}}
{{TopicTOC-Physics}}

[[File:StationaryStatesAnimation.gif|300px|thumb|right|An example of mathematical physics: solutions of the [[Schrödinger equation]] <!--in [[quantum mechanics]]--> for [[quantum harmonic oscillator]]s&nbsp;(left) with their [[Probability amplitude|amplitude]]s (right).]]

'''Mathematical physics''' is the development of [[mathematics|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".<ref>Definition from the ''Journal of Mathematical Physics''. {{cite web |url=http://jmp.aip.org/jmp/staff.jsp |title=Archived copy |access-date=2006-10-03 |url-status=dead |archive-url=https://web.archive.org/web/20061003233339/http://jmp.aip.org/jmp/staff.jsp |archive-date=2006-10-03 }}</ref> An alternative definition would also include those mathematics that are inspired by physics, known as [[physical mathematics]].<ref>{{Cite web |title=Physical mathematics and the future |url=https://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf |access-date=2022-05-09 |website=www.physics.rutgers.edu}}</ref>

== Scope ==
There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world.

=== Classical mechanics ===
{{Main|Lagrangian mechanics|Hamiltonian mechanics}}
Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of [[Lagrangian mechanics]] and [[Hamiltonian mechanics]] (including both approaches in the presence of constraints). Both formulations are embodied in [[analytical mechanics]] and lead to an understanding of the deep interplay between the notions of [[symmetry (physics)|symmetry]] and [[conservation law|conserved quantities]] during the dynamical evolution of mechanical systems, as embodied within the most elementary formulation of [[Noether's theorem]]. These approaches and ideas have been extended to other areas of physics, such as [[statistical mechanics]], [[continuum mechanics]], [[classical field theory]], and [[quantum field theory]]. Moreover, they have provided multiple examples and ideas in [[differential geometry]] (e.g., several notions in [[symplectic geometry]] and [[vector bundle]]s).

=== Partial differential equations ===
{{Main|Partial differential equations}}
Within mathematics proper, the theory of [[partial differential equation]], [[variational calculus]], [[Fourier analysis]], [[potential theory]], and [[vector analysis]] are perhaps most closely associated with mathematical physics. These fields were developed intensively from the second half of the 18th century (by, for example, [[Jean le Rond d'Alembert|D'Alembert]], [[Leonhard Euler|Euler]], and [[Joseph-Louis Lagrange|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 ===
{{Main|Quantum mechanics}}
The theory of [[atomic spectra]] (and, later, [[quantum mechanics]]) developed almost concurrently with some parts of the mathematical fields of [[linear algebra]], the [[spectral theory]] of [[Operator (mathematics)|operators]], [[operator algebra]]s and, more broadly, [[functional analysis]]. Nonrelativistic quantum mechanics includes [[Schrödinger]] operators, and it has connections to [[atomic, molecular, and optical physics|atomic and molecular physics]]. [[Quantum information]] theory is another subspecialty.

=== Relativity and quantum relativistic theories ===
{{main|Theory of relativity|Quantum field theory}}
The [[Special relativity|special]] and [[general relativity|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 [[physical cosmology|cosmological]] as well as [[quantum field theory]] phenomena. In the mathematical description of these physical areas, some concepts in [[homological algebra]] and [[category theory]]<ref>{{cite web|title=quantum field theory|url=https://ncatlab.org/nlab/show/quantum+field+theory|website=nLab}}</ref> are also important.

=== Statistical mechanics ===
{{main|Statistical mechanics}}
[[Statistical mechanics]] forms a separate field, which includes the theory of [[phase transition]]s. 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.

== Usage ==
[[File:Mathematical Physics and other sciences v1.png|thumb|Relationship between mathematics and physics]] 
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 equation]]s and [[symplectic geometry]] are generally viewed as purely mathematical disciplines, whereas [[dynamical system]]s 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".<ref>[[John Herapath]] (1847) [https://catalog.hathitrust.org/Record/011557061?type%5B%5D=author&lookfor%5B%5D=John%20Herapath&ft=ft 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]]</ref>

=== Mathematical vs. theoretical physics ===
The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or [[thought experiment]]s within a mathematically [[mathematical rigour|rigorous]] framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of some mathematical aspect and theoretical physics aspect. Although related to [[theoretical physics]],<ref>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.</ref> mathematical physics in this sense emphasizes the mathematical rigour of the similar 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]], [[Intuition (knowledge)|intuitive]], or approximate arguments.<ref>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.</ref> Such arguments are not considered rigorous by mathematicians.

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 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.{{Citation needed|date=July 2023}} 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 not only developed physics but also has influenced developments of some mathematical areas. 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 algebra]]s. The attempt to construct a rigorous mathematical formulation of [[quantum field theory]] has also brought about some progress in fields such as [[representation theory]].

== Prominent mathematical physicists ==

=== Before Newton ===
There is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples include [[Euclid]] (''Optics''), [[Archimedes]] (''On the Equilibrium of Planes'', ''On Floating Bodies''), and [[Ptolemy]] (''Optics'', ''Harmonics'').<ref>{{Cite journal|last=Pellegrin|first=P.|title=Physics|journal=Greek Thought: A Guide to Classical Knowledge|year=2000|editor-last=Brunschwig|editor-first=J.|pages=433–451|editor-last2=Lloyd|editor-first2=G. E. R.}}</ref><ref>{{Cite journal|last=Berggren|first=J. L.|date=2008|title=The Archimedes codex|url=https://www.ams.org/notices/200808/tx080800943p.pdf|journal=Notices of the AMS|volume=55|issue=8|pages=943–947}}</ref> Later, [[Science in the medieval Islamic world|Islamic]] and [[Byzantine science|Byzantine]] scholars built on these works, and these ultimately were reintroduced or became available to the West in the [[Renaissance of the 12th century|12th century]] and during the [[History of science in the Renaissance|Renaissance]].

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 [[Ptolemy|Ptolemaic]] idea of [[epicycle]]s, 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 [[classical elements|fifth element]]—the quintessence or universal essence known in Greek as ''[[Aether (classical element)|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 ''[[ellipse]]s'', 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".<ref>Peter Machamer [http://plato.stanford.edu/archives/spr2010/entries/galileo "Galileo Galilei"]—sec 1 "Brief biography", in Zalta EN, ed, ''The Stanford Encyclopedia of Philosophy'', Spring 2010 edn</ref> His 1632 book, about his telescopic observations, supported heliocentrism.<ref name=Flew1984p129>Antony G Flew, ''Dictionary of Philosophy'', rev 2nd edn (New York: St Martin's Press, 1984), p [https://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA129 129]</ref> Having made use of experimentation, Galileo then refuted geocentric [[cosmology]] by refuting Aristotelian physics itself. Galileo's 1638 book ''Discourse on Two New Sciences'' established the law of equal free fall as well as the principles of inertial motion, two central concepts of what today is known as [[classical mechanics]].<ref name=Flew1984p129/> 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]] developed a complete system of heliocentric cosmology anchored on the principle of vortex motion, [[Cartesian physics]], whose widespread acceptance helped bring the demise of Aristotelian physics. Descartes used mathematical reasoning as a model for science, and developed [[analytic geometry]], which in time allowed the plotting of locations in 3D space ([[Cartesian coordinate system|Cartesian coordinates]]) and marking their progressions along the flow of time.<ref>Antony G Flew, ''Dictionary of Philosophy'', rev 2nd edn (New York: St Martin's Press, 1984), p [https://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA89&dq=mathematical+reasoning 89]</ref>

[[Christiaan Huygens]], a talented mathematician and physicist and older contemporary of Newton, was the first to successfully idealize a physical problem by a set of mathematical parameters in ''[[Horologium Oscillatorium|Horologium Oscillatorum]]'' (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon in ''[[Treatise on Light|Traité de la Lumière]]'' (1690). He is thus considered a forerunner of [[theoretical physics]] and one of the founders of modern mathematical physics.<ref>Dijksterhuis, F. J. (2008). Stevin, Huygens and the Dutch republic. ''Nieuw archief voor wiskunde, 5,'' pp. 100–107. https://research.utwente.nl/files/6673130/Dijksterhuis_naw5-2008-09-2-100.pdf</ref><ref>Andreessen, C.D. (2005) ''Huygens: The Man Behind the Principle''. Cambridge University Press: 6</ref>

=== Newtonian physics and post Newtonian ===

The prevailing framework for science in the 16th and early 17th centuries was one borrowed from [[Greek mathematics|Ancient Greek mathematics]], where geometrical shapes formed the building blocks to describe and think about space, and time was often thought as a separate entity. With the introduction of algebra into geometry, and with it the idea of a coordinate system, time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries. 

By the middle of the 17th century, important concepts such as the [[fundamental theorem of calculus]] (proved in 1668 by Scottish mathematician [[James Gregory (mathematician)|James Gregory]]) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematician [[Pierre de Fermat]]) were already known before Leibniz and Newton.<ref name=geometriae>{{cite book| last=Gregory | first=James | title=Geometriae Pars Universalis | url=https://archive.org/details/gregory_universalis | publisher= Patavii: typis heredum Pauli Frambotti | year=1668 | location=[[Museo Galileo]] }}</ref> [[Isaac Newton]] (1642–1727) developed [[calculus]] (although [[Gottfried Wilhelm Leibniz]] developed similar concepts outside the context of physics) and [[Newton's method]] to solve problems in mathematics and physics. He was extremely successful in his application of [[calculus]] and other methods to the study of motion. Newton's theory of motion, culminating in his ''[[Philosophiæ Naturalis Principia Mathematica]]'' (''Mathematical Principles of Natural Philosophy'') 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.<ref>{{citation|contribution=The Mathematical Principles of Natural Philosophy|title=Encyclopædia Britannica|place=London|contribution-url=https://www.britannica.com/EBchecked/topic/369153/The-Mathematical-Principles-of-Natural-Philosophy}}</ref> 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.<ref name=Lakatos1980>Imre Lakatos, auth, Worrall J & Currie G, eds, ''The Methodology of Scientific Research Programmes: Volume 1: Philosophical Papers'' (Cambridge: Cambridge University Press, 1980), pp [https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA213 213–214], [https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA220 220]</ref>

In the 18th century, the Swiss [[Daniel Bernoulli]] (1700–1782) made contributions to [[fluid dynamics]], and [[vibrating string]]s. The Swiss [[Leonhard Euler]] (1707–1783) did special work in [[Calculus of variations|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, following mathematicians in France, Germany and England had contributed to mathematical physics. The French [[Pierre-Simon Laplace]] (1749–1827) made paramount contributions to mathematical [[astronomy]], [[potential 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 (mathematician)|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 (scientist)|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. The English physicist [[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{{clarify|date=January 2018}} Maxwell's field. Later, radiation and then today's known [[electromagnetic spectrum]] were found also consequent of{{clarify|date=January 2018}} 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 [[William Thomson, 1st Baron Kelvin|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 mathematician [[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, [[fluid]]s, 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.

=== Relativistic ===

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{{clarify|date=January 2018}} 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 [[luminiferous aether|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.

Austrian theoretical physicist and philosopher [[Ernst Mach]] criticized Newton's postulated absolute space.  Mathematician [[Henri Poincaré|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.<ref name=Lakatos1980/> 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 time|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.

Cartesian coordinates arbitrarily used rectilinear coordinates. Gauss, inspired by Descartes' work, introduced the curved geometry, replacing rectilinear axis by curved ones. Gauss also introduced another key tool of modern physics, the curvature. Gauss's work was limited to two dimensions. Extending it to three or more dimensions introduced a lot of complexity, with the need of the (not yet invented) tensors. It was Riemman the one in charge to extend curved geometry to N dimensions. In 1908, Einstein's former mathematics professor [[Hermann Minkowski]], applied the curved geometry construction to model 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. <ref>Minkowski, Hermann (1908–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift, 10: 75–88. Actually the union of space and time was implicit in Descartes's work first, with space and time being represented as axis of coordinates, and in Lorentz transformation later, but its physical interpretation was still hidden to common sense.
</ref> Einstein initially called this "superfluous learnedness", but later used [[Minkowski spacetime]] with great elegance in his [[general theory of relativity]],<ref>Salmon WC & Wolters G, eds, ''Logic, Language, and the Structure of Scientific Theories'' (Pittsburgh: University of Pittsburgh Press, 1994), p [https://books.google.com/books?id=Z9K8llQufcMC&pg=PA125&dq=superfluous+learnedness+Einstein+Minkowski+general+relativity 125]</ref> 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 [[Euclidean vector|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]]. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of [[spacetime]] and free falling particles with mass move along a geodesic curve in the spacetime" ([[Riemannian geometry]] already existed before the 1850s, by mathematicians [[Carl Friedrich Gauss]] and [[Bernhard Riemann]] in search for intrinsic geometry and non-Euclidean geometry.), 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–energy equivalence|mass equivalence]] locally "curving" the geometry of the four, unified dimensions of space and time.)

=== Quantum ===
Another revolutionary development of the 20th century was [[Quantum mechanics|quantum theory]], which emerged from the seminal contributions of [[Max Planck]] (1856–1947) (on [[Planck's law|black-body radiation]]) and Einstein's work on the [[photoelectric effect]].  In 1912, mathematician [[Henri Poincaré]] published ''Sur la théorie des quanta''.<ref name=McCormmach>
{{cite journal
  | last =McCormmach
  | first =Russell
  | title = Henri Poincaré and the Quantum Theory
  | journal = Isis
  | volume = 58
  | issue = 1
  | pages = 37–55
  | date = Spring 1967
  | doi =10.1086/350182
  | s2cid =120934561
}}</ref><ref name=Irons>
{{cite journal
  | last =Irons
  | first =F. E.
  | title = Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms
  | journal =  American Journal of Physics
  | volume = 69
  | issue = 8
  | pages = 879–84
  | date = August 2001
  | doi =10.1119/1.1356056
  |bibcode = 2001AmJPh..69..879I
}}</ref> He introduced the first non-naïve definition of quantization in this paper. The development of early quantum physics 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), [[Louis de Broglie]] (1892–1987), [[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 operator]]s on an infinite-dimensional vector space. That is called [[Hilbert space]] (introduced by mathematicians [[David Hilbert]] (1862–1943), [[Erhard Schmidt]] (1876–1959) and [[Frigyes Riesz]] (1880–1956) in search of generalization of Euclidean space and study of integral equations), 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]] (introduced by [[David Hilbert]] who investigated [[quadratic forms]] with infinitely many variables. Many years later, it had been revealed that his spectral theory is associated with the spectrum of the hydrogen atom. He was surprised by this application.) 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 contributors to mathematical physics in the 20th century ===
Prominent contributors to the 20th century's mathematical physics include (ordered by birth date):

* [[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] (1824–1907)
* [[Oliver Heaviside]] (1850–1925)
* [[Henri Poincaré|Jules Henri Poincaré]] (1854–1912)
* [[David Hilbert]] (1862–1943)
* [[Arnold Sommerfeld]] (1868–1951)
* [[Constantin Carathéodory]] (1873–1950)
* [[Albert Einstein]] (1879–1955)
* [[Emmy Noether]] (1882–1935)
* [[Max Born]] (1882–1970)
* [[George David Birkhoff]] (1884–1944)
* [[Hermann Weyl]] (1885–1955)
* [[Satyendra Nath Bose]] (1894–1974)
* [[Louis de Broglie]] (1892–1987)
* [[Norbert Wiener]] (1894–1964)
* [[John Lighton Synge]] (1897–1995)
* [[Mário Schenberg]] (1914–1990)
* [[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 Bogolyubov|Nikolay Nikolayevich Bogolyubov]] (1909–1992)
* [[Subrahmanyan Chandrasekhar]] (1910–1995)
* [[Mark Kac]] (1914–1984)
* [[Julian Schwinger]] (1918–1994)
* [[Richard Feynman|Richard Phillips Feynman]] (1918–1988)
* [[Irving Ezra Segal]] (1918–1998)
* [[Ryogo Kubo]] (1920–1995)
* [[Arthur Strong Wightman]] (1922–2013)
* [[Chen-Ning Yang]] (1922–)
* [[Rudolf Haag]] (1922–2016)
* [[Freeman Dyson|Freeman John Dyson]] (1923–2020)
* [[Martin Gutzwiller]] (1925–2014)
* [[Abdus Salam]] (1926–1996)
* [[Jürgen Moser]] (1928–1999)
* [[Michael Atiyah|Michael Francis Atiyah]] (1929–2019)
* [[Joel Lebowitz|Joel Louis Lebowitz]] (1930–)
* [[Roger Penrose]] (1931–)
* [[Elliott H. Lieb|Elliott Hershel Lieb]] (1932–)
* [[Yakir Aharonov]] (1932–)
* [[Sheldon Glashow]] (1932–)
* [[Steven Weinberg]] (1933–2021)
* [[Ludvig Faddeev|Ludvig Dmitrievich Faddeev]] (1934–2017)
* [[David Ruelle]] (1935–)
* [[Yakov Sinai|Yakov Grigorevich Sinai]] (1935–)
* [[Vladimir Arnold|Vladimir Igorevich Arnold]] (1937–2010)
* [[Arthur Jaffe|Arthur Michael Jaffe]] (1937–)
* [[Roman Jackiw|Roman Wladimir Jackiw]] (1939–)
* [[Leonard Susskind]] (1940–)
* [[Rodney Baxter|Rodney James Baxter]] (1940–)
* [[Michael Berry (physicist)|Michael Victor Berry]] (1941–)
* [[Giovanni Gallavotti]] (1941–)
* [[Stephen Hawking|Stephen William Hawking]] (1942–2018)
* [[Jerrold E. Marsden|Jerrold Eldon Marsden]] (1942–2010)
* [[Michael C. Reed]] (1942–)
* [[John Michael Kosterlitz]] (1943–)
* [[Israel Michael Sigal]] (1945–)
* [[Alexander Markovich Polyakov]] (1945–)
* [[Barry Simon]] (1946–)
* [[Herbert Spohn]] (1946–)
* [[John Cardy|John Lawrence Cardy]] (1947–)
* [[Giorgio Parisi]] (1948-)
* [[Abhay Ashtekar]] (1949-)
* [[Edward Witten]] (1951–)
* [[Frederick Duncan Michael Haldane|F. Duncan Haldane]] (1951–)
* [[Ashoke Sen]] (1956–)
* [[Juan Martín Maldacena]] (1968–)

== See also ==
* [[International Association of Mathematical Physics]]
* [[List of publications in physics#Mathematical physics|Notable publications in mathematical physics]]
* [[List of mathematical physics journals]]
* [[Gauge theory (mathematics)]]
* [[Relationship between mathematics and physics]]
* [[Theoretical physics|Theoretical]], [[computational physics|computational]] and [[philosophy of physics|philosophical physics]]

== Notes ==
{{Reflist|40em}}

== References ==
*{{Citation |last=Zaslow |first=Eric |author-link=Eric Zaslow|year=2005 |title=Physmatics |arxiv=physics/0506153|bibcode = 2005physics...6153Z }}

== Further reading ==

=== Generic works ===
*{{citation |first1 = Jont |last1 = Allen |title = An Invitation to Mathematical Physics and its History |publisher = Springer |year = 2020 |bibcode = 2020imph.book.....A |isbn = 978-3-030-53758-6}}
*{{citation |first1 = Richard |last1 = Courant |author1-link = Richard Courant |first2 = David |last2 = Hilbert |author2-link = David Hilbert |title = [[Methods of Mathematical Physics]] |others=Vol 1–2 |publisher = Interscience Publishers |year = 1989|bibcode = 1989mmp..book.....C }}
*{{citation |first1 = Jean P. |last1 = Françoise  |first2 = Gregory L. |last2 = Naber |first3 = Tsou S. |last3 = Tsun |title = Encyclopedia of Mathematical Physics |publisher = Elsevier |year = 2006 |isbn = 978-0-1251-2660-1}}
* {{citation |author1=Joos, Georg |author1-link = Georg Joos |author2=Freeman, Ira M. | title=Theoretical Physics |edition = 3rd | publisher=Dover Publications | year=1987 | isbn=0-486-65227-0}}
*{{citation |first = Tosio |last = Kato  |author-link=Tosio Kato|title =  Perturbation Theory for Linear Operators |edition = 2nd |publisher =  Springer-Verlag |year = 1995 |isbn = 3-540-58661-X}}
*{{citation |first1 = Henry |last1  = Margenau |author-link = Henry Margenau |first2 = George M. |last2 = Murphy |title =  The Mathematics of Physics and Chemistry |edition = 2nd |publisher = Young Press |year = 2009 |isbn = 978-1444627473}}
*{{citation |first = Pesi R. |last = Masani |title = [[Norbert Wiener]]: Collected Works with Commentaries |others=Vol 1–4 |publisher =  The MIT Press |year = 1976–1986}}
*{{citation |first1 = Philip M. |last1 = Morse |author1-link = Philip M. Morse |first2 = Herman |last2 = Feshbach |author2-link = Herman Feshbach  |title =  Methods of Theoretical Physics |others=Vol 1–2 |publisher = McGraw Hill |year = 1999 |isbn = 0-07-043316-X}}
*{{citation |first1 = Walter E. |last1 = Thirring |author-link = Walter Thirring |title = A Course in Mathematical Physics |others=Vol 1–4  |publisher =  Springer-Verlag |year = 1978–1983}}
*{{citation |first1 = Vladimir M. |last1 = Tikhomirov |title = Selected Works of [[Andrey Kolmogorov|A. N. Kolmogorov]] |others=Vol 1–3 |publisher = Kluwer Academic Publishers |year = 1991–1993}}
*{{citation |first = Edward C. |last = Titchmarsh |author-link = E. C. Titchmarsh |title = The Theory of Functions  |edition = 2nd |publisher = Oxford University Press |year = 1985}}

===Textbooks for undergraduate studies===
*{{citation |first1 = George B. |last1 = Arfken |author1-link = George B. Arfken |first2 = Hans J. |last2 = Weber |first3 = Frank E. |last3 = Harris |title =  Mathematical Methods for Physicists: A Comprehensive Guide  |edition = 7th |publisher =  Academic Press |year = 2013 |isbn = 978-0-12-384654-9}}, ([https://archive.org/details/Mathematical_Methods_for_Physicists Mathematical Methods for Physicists], [https://archive.org/details/SolutionMathematicalMethodForPhysics7 Solutions for ''Mathematical Methods for Physicists (7th ed.)''], archive.org)
*{{citation |first1 = Selçuk Ş. |last1 = Bayın |title =  Mathematical Methods in Science and Engineering  |edition = 2nd |publisher =  Wiley |year = 2018 |isbn = 9781119425397}}
*{{citation |first = Mary L. |last = Boas |author1-link = Mary L. Boas |title = [[Mathematical Methods in the Physical Sciences]]  |edition = 3rd |publisher =  Wiley |year = 2006 |isbn = 978-0-471-19826-0}}
*{{citation |first = Eugene |last = Butkov |title =  Mathematical Physics  |publisher = Addison-Wesley |year = 1968}}
*Hassani, Sadri (2009), ''Mathematical Methods for Students of Physics and Related Fields'', (2nd ed.), New York, Springer, eISBN 978-0-387-09504-2
*{{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 |publisher = Cambridge University Press |year = 1956}}
*{{citation |first = Adam |last = Marsh |title =  Mathematics for Physics: An Illustrated Handbook  |journal = Contemporary Physics |publisher = World Scientific |year = 2018 |volume = 59 |issue = 3 |page = 329 |doi = 10.1080/00107514.2018.1501430 |bibcode = 2018ConPh..59..329N |isbn = 978-981-3233-91-1}}
*{{citation |first1 =  Jon |last1 = Mathews  |author1-link = Jon Mathews |first2 =  Robert L. |last2 = Walker |title =   Mathematical Methods of Physics |edition = 2nd |publisher = W. A. Benjamin |year = 1970  |bibcode = 1970mmp..book.....M |isbn =  0-8053-7002-1}}
* {{citation | author=Menzel, Donald H. |author1-link = Donald Howard Menzel | title=Mathematical Physics | publisher=Dover Publications | year=1961 | isbn=0-486-60056-4}}
*{{citation |first1 = Ken F. |last1 = Riley |author1-link = Ken Riley (physicist) |first2 = Michael P. |last2 = Hobson |first3 = Stephen J. |last3 = Bence |title =  Mathematical Methods for Physics and Engineering  |edition = 3rd |publisher =  Cambridge University Press |year = 2006 |isbn = 978-0-521-86153-3}}
*{{citation |first = Ivar |last = Stakgold |author1-link = Ivar Stakgold |title =  Boundary Value Problems of Mathematical Physics |others=Vol 1-2. |publisher =  Society for Industrial and Applied Mathematics |year = 2000 |isbn = 0-89871-456-7}}
*{{citation |first = Steven P. |last = Starkovich |title =  The Structures of Mathematical Physics: An Introduction |publisher =  Springer |year = 2021 |bibcode = 2021smpa.book.....S |isbn = 978-3-030-73448-0}}

===Textbooks for graduate studies===
*{{citation |first1 = Philippe |last1 = Blanchard |author1-link = Philippe Blanchard |first2 = Erwin |last2 = Brüning  |title =  Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics |edition = 2nd |publisher =  Springer |year = 2015 |bibcode = 2015mmpd.book.....B |isbn = 978-3-319-14044-5 }}
*{{citation |first = Kevin |last = Cahill |title =  Physical Mathematics |edition = 2nd |publisher =  Cambridge University Press |year = 2019 |isbn = 978-1-108-47003-2 }}
*{{citation |first = Robert |last = Geroch |author-link = Robert Geroch |title =  Mathematical Physics |publisher =  University of Chicago Press |year = 1985 |isbn = 0-226-28862-5}}
*{{citation |first = Sadri |last = Hassani  |title =  Mathematical Physics: A Modern Introduction to its Foundations |edition = 2nd |publisher =  Springer-Verlag |year = 2013 |bibcode = 2013mpmi.book.....H |isbn = 978-3-319-01194-3 }}
*{{citation |first = Kishore |last = Marathe  |title =  Topics in Physical Mathematics |publisher =  Springer-Verlag |year = 2010 |isbn = 978-1-84882-938-1 }}
*{{citation |first1 = Grigori N. |last1 = Milstein |first2 = Michael V. |last2 = Tretyakov |title =  Stochastic Numerics for Mathematical Physics |edition = 2nd |publisher = Springer |year = 2021 |isbn = 978-3-030-82039-8 |author1-link = Grigori Milstein }}
*{{citation |author-link=Michael C. Reed |first1=Michael C. |last1=Reed |author-link2=Barry Simon |first2=Barry |last2=Simon |title=Methods of Modern Mathematical Physics |others=Vol 1-4 |publisher=Academic Press |year=1972–1981}}
*{{citation |author-link=Robert D. Richtmyer |first1=Robert D. |last1=Richtmyer |title=Principles of Advanced Mathematical Physics |others=Vol 1-2. |publisher=Springer-Verlag |year=1978–1981}}
*{{citation |first1 = Gerd |last1 = Rudolph |first2 = Matthias |last2 = Schmidt |title =  Differential Geometry and Mathematical Physics |others=Vol 1-2 |publisher =  Springer |year = 2013–2017}}
*{{citation |first1=Valery |last1=Serov |title=Fourier Series, Fourier Transform and Their Applications to Mathematical Physics |publisher =  Springer |year=2017 |isbn = 978-3-319-65261-0}}
*{{citation |author-link=Barry Simon |first1=Barry |last1=Simon |title =  A Comprehensive Course in Analysis |others=Vol 1-5 |publisher =  American Mathematical Society |year = 2015}}
*{{citation |first1 = Ivar |last1 = Stakgold |author1-link = Ivar Stakgold |first2 = Michael |last2 = Holst |title =  Green's Functions and Boundary Value Problems |edition = 3rd |publisher =  Wiley |year = 2011 |isbn = 978-0-470-60970-5}}
*{{citation |first1 = Michael |last1 = Stone |first2 = Paul |last2 = Goldbart |author2-link=Paul Goldbart |title =  Mathematics for Physics: A Guided Tour for Graduate Students |journal = Physics Today |publisher =  Cambridge University Press |year = 2009 |volume = 62 |issue = 10 |page = 57 |doi = 10.1063/1.3248483 |bibcode = 2009PhT....62j..57S |isbn = 978-0-521-85403-0}}
*{{citation |first = Peter |last = Szekeres  |title =  A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry |publisher =  Cambridge University Press |year = 2004 |isbn = 978-0-521-53645-5}}
*{{citation |first = Michael E. |last = Taylor  |author-link=Michael E. Taylor |title =  Partial Differential Equations |edition = 2nd |others=Vol 1-3 |publisher =  Springer. |year = 2011 }}
*{{citation |first1 = Edmund T. |last1 = Whittaker |author1-link = E. T. Whittaker  |first2 = George N. |last2 = Watson  |author2-link = G. N. Watson |title = [[Whittaker and Watson|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]] |date = 1950 |edition = 4th |publisher = Cambridge University Press}}

===Specialized texts in classical physics===
*{{citation |first1 = Ralph |last1 = Abraham |author1-link = Ralph Abraham (mathematician) |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 |publisher = AMS Chelsea Publishing |year = 2008 |isbn = 978-0-8218-4438-0}}
*{{citation |first = John A. |last = Adam |title = Rays, Waves, and Scattering: Topics in Classical Mathematical Physics |publisher = Princeton University Press. |year = 2017 |isbn = 978-0-691-14837-3}}
* {{citation |first1 = Vladimir I. |last1 = Arnold |author-link = Vladimir Arnold |title = [[Mathematical Methods of Classical Mechanics]] |edition = 2nd |publisher = Springer-Verlag |year = 1997 |isbn = 0-387-96890-3 }}
*{{citation |first1 = Frederick |last1 = Bloom |title =  Mathematical Problems of Classical Nonlinear Electromagnetic Theory |publisher = CRC Press |year = 1993 |isbn = 0-582-21021-6}}
*{{citation |first1 = Franck |last1 = Boyer |first2 = Pierre |last2 = Fabrie |title =  Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models |publisher =  Springer |year = 2013 |isbn = 978-1-4614-5974-3}}
*{{citation |first1 = David |last1 = Colton |first2 = Rainer |last2 = Kress |title =  Integral Equation Methods in Scattering Theory |publisher =  Society for Industrial and Applied Mathematics |year = 2013 |isbn = 978-1-611973-15-0}}
*{{citation |first1 = Philippe G. |last1 = Ciarlet |author1-link = Philippe G. Ciarlet |title =  Mathematical Elasticity |others=Vol 1–3 |publisher =  Elsevier |year = 1988–2000}}
*{{citation |first = Giovanni P. |last = Galdi |title =  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems |edition = 2nd |publisher =  Springer |year = 2011 |isbn = 978-0-387-09619-3}}
*{{citation |first1 = George W. |last1 = Hanson |first2 = Alexander B. |last2 = Yakovlev |title =  Operator Theory for Electromagnetics: An Introduction |publisher = Springer |year = 2002 |isbn = 978-1-4419-2934-1}}
*{{citation |first1 = Andreas |last1 = Kirsch |first2 = Frank |last2 = Hettlich |title =  The Mathematical Theory of Time-Harmonic Maxwell's Equations: Expansion-, Integral-, and Variational Methods |publisher = Springer |year = 2015 |bibcode = 2015mttm.book.....K |isbn = 978-3-319-11085-1}}
*{{citation |first1 = Andreas |last1 = Knauf |title =  Mathematical Physics: Classical Mechanics |publisher = Springer |year = 2018 |bibcode = 2018mpcm.book.....K |isbn = 978-3-662-55772-3}}
*{{citation |first1 = Kurt |last1 = Lechner |title =  Classical Electrodynamics: A Modern Perspective |publisher = Springer |year = 2018 |isbn = 978-3-319-91808-2}}
*{{citation |first1 = Jerrold E. |last1 = Marsden |author1-link = Jerrold E. Marsden |first2 = Tudor S. |last2 = Ratiu |author2-link = Tudor Ratiu |title =  Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |edition = 2nd |publisher = Springer |year = 1999 |isbn = 978-1-4419-3143-6}}
*{{citation |first1 = Claus |last1 = Müller |title =  Foundations of the Mathematical Theory of Electromagnetic Waves |publisher = Springer-Verlag |year = 1969 |isbn = 978-3-662-11775-0}}
*{{citation |first = Alexander G. |last = Ramm |author-link = Alexander Ramm |title =  Scattering by Obstacles and Potentials |publisher = World Scientific |year = 2018 |isbn = 9789813220966}}
*{{citation |first1 = Gary F. |last1 = Roach |first2 = Ioannis G. |last2 = Stratis |first3 = Athanasios N. |last3 = Yannacopoulos |title =  Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics |publisher = Princeton University Press |year = 2012 |bibcode = 2012mads.book.....R |isbn = 978-0-691-14217-3}}

===Specialized texts in modern physics===
*{{citation |first1 = John C. |last1 = Baez |author-link = John Baez |first2 = Javier P. |last2 = Muniain |title =  Gauge Fields, Knots, and Gravity  |publisher =  World Scientific |year = 1994 |isbn = 981-02-2034-0}}
*{{citation |first1 = Jiří |last1 = Blank |first2 = Pavel |last2 = Exner |author2-link = Pavel Exner |first3 = Miloslav |last3 = Havlíček |title =  Hilbert Space Operators in Quantum Physics |edition = 2nd |publisher =  Springer |year = 2008 |bibcode = 2008hsoq.book.....B |isbn = 978-1-4020-8869-8 }}
*{{citation |first1 = Eberhard |last1 = Engel |first2 = Reiner M. |last2 = Dreizler |title =  Density Functional Theory: An Advanced Course  |publisher =  Springer-Verlag |year = 2011 |isbn = 978-3-642-14089-1}}
*{{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 |publisher =  Springer-Verlag |year = 1987 |isbn = 0-387-96477-0}}
*{{citation |first = Rudolf |last = Haag |author-link = Rudolf Haag |title =  Local Quantum Physics: Fields, Particles, Algebras |edition = 2nd |publisher = Springer-Verlag |year = 1996 |isbn = 3-540-61049-9}}
*{{citation |first = Brian C. |last = Hall |title =  Quantum Theory for Mathematicians |publisher = Springer |year = 2013 |bibcode = 2013qtm..book.....H |isbn = 978-1-4614-7115-8}}
*{{citation |first=Mark J. D. |last=Hamilton |title=Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics |publisher=Springer |year=2017 |bibcode=2017mgta.book.....H |isbn=978-3-319-68438-3}}
*{{citation |first1 = Stephen W. |last1 = Hawking |author1-link = Stephen Hawking |first2 = George F. R. |last2 = Ellis |author2-link = George F. R. Ellis |title = [[The Large Scale Structure of Space-Time]] |publisher = Cambridge University Press |year = 1973 |isbn = 0-521-20016-4}}
*{{citation |first=Roman |last=Jackiw |author-link =Roman Jackiw |title=Diverse Topics in Theoretical and Mathematical Physics |publisher=World Scientific |year=1995 |isbn=9810216963}}
*{{citation |first=Klaas |last=Landsman |title=Foundations of Quantum Theory: From Classical Concepts to Operator Algebras |publisher=Springer |year=2017 |bibcode=2017fqtf.book.....L |isbn=978-3-319-51776-6}}
*{{citation |first=Valter |last=Moretti |title=Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation |series=Unitext |url=https://link.springer.com/book/10.1007/978-3-319-70706-8 |edition=2nd |publisher=Springer |year=2017 |volume=110 |doi=10.1007/978-3-319-70706-8 |bibcode=2017stqm.book.....M |isbn=978-3-319-70705-1|s2cid=125121522 |url-access=subscription }}
*{{citation |first1=Didier |author2-link=Monique Combescure |last1=Robert |first2=Monique |last2=Combescure |title=Coherent States and Applications in Mathematical Physics |edition = 2nd |publisher=Springer |year=2021 |bibcode=2021csam.book.....R |isbn=978-3-030-70844-3}}
*{{citation |last=Tasaki |first=Hal |url=https://www.worldcat.org/oclc/1154567924 |pages= |title=Physics and mathematics of quantum many-body systems |date=2020 |publisher=Springer |isbn=978-3-030-41265-4 |location= |oclc=1154567924}}
*{{citation |author-link=Gerald Teschl |first=Gerald |last=Teschl |title=Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators |publisher=American Mathematical Society |year=2009 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/|isbn=978-0-8218-4660-5}}
*{{citation |first1 = Walter E. |last1 = Thirring |author-link = Walter Thirring |title = Quantum Mathematical Physics: Atoms, Molecules and Large Systems |edition = 2nd |publisher =  Springer-Verlag |year = 2002 |bibcode = 2002qmpa.book.....T |isbn = 978-3-642-07711-1}}
*{{citation |first1 = John |last1 = von Neumann |author-link = John von Neumann |title = [[Mathematical Foundations of Quantum Mechanics]] |publisher = Princeton University Press |year = 2018 | isbn = 978-0-691-17856-1}}
*{{citation |first1 = Hermann |last1 = Weyl |author1-link = Hermann Weyl |title =  The Theory of Groups and Quantum Mechanics |publisher =  Martino Fine Books |year = 2014 | isbn = 978-1614275800}}
*{{citation |first = Francisco J. |last = Ynduráin |author-link=Francisco José Ynduráin |title = The Theory of Quark and Gluon Interactions |edition = 4th |publisher = Springer  |year = 2006 |bibcode = 2006tqgi.book.....Y |isbn = 978-3642069741}}
*{{citation |first = Eberhard |last = Zeidler  |author-link=Eberhard Zeidler (mathematician) |title = Quantum Field Theory: A Bridge Between Mathematicians and Physicists |others=Vol 1-3 |publisher = Springer  |year = 2006–2011}}

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{{DEFAULTSORT:Mathematical Physics}}
[[Category:Mathematical physics| ]]