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] An
alternative definition would also include those
mathematics that are inspired by physics (also known
as physical mathematics).[2]

Scope
There are several distinct branches of mathematical
physics, and these roughly correspond to particular
An example of mathematical physics: solutions of
historical periods. Schrödinger's equation for quantum harmonic
oscillators (left) with their amplitudes (right).
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 and lead to understanding the deep interplay of the notions of
symmetry and conserved quantities during the dynamical evolution, as embodied within the most
elementary formulation of Noether's theorem. These approaches and ideas 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 ideas in differential
geometry (e.g. several notions in symplectic geometry and vector bundle).

Partial differential equations

Following mathematics: the theory of partial differential equation, 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
some parts of 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 the mathematical
description of these physical areas, some concepts in homological algebra and category theory[3] are
also important.

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.

Usage
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";
Relationship between mathematics
the scope at that time being "the causes of heat, gaseous elasticity,
and physics
gravitation, and other great phenomena of nature".[4]

Mathematical vs. theoretical physics

The term "mathematical physics" is sometimes used to denote research aimed at studying and solving
problems in 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
some mathematical aspect and physics theoretical aspect. Although related to theoretical physics,[5]
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, intuitive, or approximate arguments.[6] 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. 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 algebras. 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).[7][8] Later, Islamic and Byzantine scholars built on these works, and these
ultimately were reintroduced or became available to the West in the 12th century and during the
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 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".[9] His 1632 book, about his telescopic observations, supported
heliocentrism.[10] Having introduced 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, founding the central concepts of
what would become today's classical mechanics.[10] 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 famously 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.[11]

An older contemporary of Newton, Christiaan Huygens, was the first to idealize a physical problem by
a set of parameters and the first to fully mathematize a mechanistic explanation of unobservable
physical phenomena, and for these reasons Huygens is considered the first theoretical physicist and
one of the founders of modern mathematical physics.[12][13]

Newtonian and post Newtonian

In this era, important concepts in calculus such as the fundamental theorem of calculus (proved in
1668 by Scottish mathematician James Gregory[14]) 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. Isaac Newton (1642–1727) developed some concepts in calculus
(although Gottfried Wilhelm Leibniz developed similar concepts outside the context of physics) and
Newton's method to solve problems in physics. He was extremely successful in his application of
calculus to the theory of motion. Newton's theory of motion, shown in his Mathematical Principles of
Natural Philosophy, published in 1687,[15] 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.[16]

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, 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 (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. 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 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 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, 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.

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
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.

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.[16]
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 mathematics 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.[17] Einstein
initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance
in his general theory of relativity,[18] 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. 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 equivalence locally "curving" the geometry of the four, unified
dimensions of space and time.)

Quantum

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. In 1912, a mathematician Henri Poincare published Sur la théorie des
quanta.[19][20] 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), 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 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 (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],
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 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], Ryogo Kubo [1920–1995], Arthur Strong Wightman [1922–
2013], Chen-Ning Yang [1922– ], Rudolf Haag [1922–2016], Freeman John Dyson [1923–2020],
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– ], Yakir Aharonov [1932– ], Sheldon Glashow [1932– ], Steven Weinberg [1933–2021],
Ludvig Dmitrievich Faddeev [1934–2017], David Ruelle [1935– ], Yakov Grigorevich Sinai [1935– ],
Vladimir Igorevich Arnold [1937–2010], Arthur Michael 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], John Michael Kosterlitz
[1943-], Jerrold Eldon Marsden [1942–2010], Michael C. Reed [1942– ], Israel Michael Sigal [1945–
], Alexander Markovich Polyakov [1945– ], Barry Simon [1946– ], Herbert Spohn [1946– ], John
Lawrence Cardy [1947– ], Giorgio Parisi [1948– ], Edward Witten [1951– ], F. Duncan Haldane
[1951-], Ashoke Sen [1956-] and Juan Martín Maldacena [1968– ].

See also
International Association of Mathematical Physics
Notable publications in mathematical physics
List of mathematical physics journals
Gauge theory (mathematics)
Relationship between mathematics and physics
Theoretical, computational and philosophical physics

Notes
1. Definition from the Journal of Mathematical Physics. "Archived copy" (https://web.archive.org/web/
20061003233339/http://jmp.aip.org/jmp/staff.jsp). Archived from the original (http://jmp.aip.org/jm
p/staff.jsp) on 2006-10-03. Retrieved 2006-10-03.
2. "Physical mathematics and the future" (https://www.physics.rutgers.edu/~gmoore/PhysicalMathem
aticsAndFuture.pdf) (PDF). www.physics.rutgers.edu. Retrieved 2022-05-09.
3. "quantum field theory" (https://ncatlab.org/nlab/show/quantum+field+theory). nLab.
4. 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 (https://catalog.hathitrust.org/Record/011557061?type%5B%5D=author&lookfor%5B%5D=
John%20Herapath&ft=ft), Whittaker and company via HathiTrust
 5. 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.
6. 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.
7. Pellegrin, P. (2000). Brunschwig, J.; Lloyd, G. E. R. (eds.). Physics. Greek Thought: A Guide to
Classical Knowledge. pp. 433–451.
8. Berggren, J. L. (2008). "The Archimedes codex" (https://www.ams.org/notices/200808/tx08080094
3p.pdf) (PDF). Notices of the AMS. 55 (8): 943–947.
9. Peter Machamer "Galileo Galilei" (http://plato.stanford.edu/archives/spr2010/entries/galileo)—sec
1 "Brief biography", in Zalta EN, ed, The Stanford Encyclopedia of Philosophy, Spring 2010 edn
10. Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 129
(https://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA129)
11. Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 89 (h
ttps://books.google.com/books?id=MmJHVU9Rv3YC&pg=PA89&dq=mathematical+reasoning)
12. 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
13. Andreessen, C.D. (2005) Huygens: The Man Behind the Principle. Cambridge University Press: 6
14. Gregory, James (1668). Geometriae Pars Universalis (https://archive.org/details/gregory_universa
lis). Museo Galileo: Patavii: typis heredum Pauli Frambotti.
15. "The Mathematical Principles of Natural Philosophy" (https://www.britannica.com/EBchecked/topi
c/369153/The-Mathematical-Principles-of-Natural-Philosophy), Encyclopædia Britannica, London
16. 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 (https://books.google.com/books?id=RRniFBI8Gi4C&pg=PA213), 220 (https://books.
google.com/books?id=RRniFBI8Gi4C&pg=PA220)
17. Minkowski, Hermann (1908–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift,
10: 75–88
18. Salmon WC & Wolters G, eds, Logic, Language, and the Structure of Scientific Theories
(Pittsburgh: University of Pittsburgh Press, 1994), p 125 (https://books.google.com/books?id=Z9K
8llQufcMC&pg=PA125&dq=superfluous+learnedness+Einstein+Minkowski+general+relativity)
19. McCormmach, Russell (Spring 1967). "Henri Poincaré and the Quantum Theory". Isis. 58 (1): 37–
55. doi:10.1086/350182 (https://doi.org/10.1086%2F350182). S2CID 120934561 (https://api.sema
nticscholar.org/CorpusID:120934561).
20. Irons, F. E. (August 2001). "Poincaré's 1911–12 proof of quantum discontinuity interpreted as
applying to atoms". American Journal of Physics. 69 (8): 879–84. Bibcode:2001AmJPh..69..879I
(https://ui.adsabs.harvard.edu/abs/2001AmJPh..69..879I). doi:10.1119/1.1356056 (https://doi.org/
10.1119%2F1.1356056).

References
Zaslow, Eric (2005), Physmatics, arXiv:physics/0506153 (https://arxiv.org/abs/physics/0506153),
 Bibcode:2005physics...6153Z (https://ui.adsabs.harvard.edu/abs/2005physics...6153Z)

Further reading

Generic works
Allen, Jont (2020), An Invitation to Mathematical Physics and its History, Springer, ISBN 978-3-
030-53758-6
Courant, Richard; Hilbert, David (1989), Methods of Mathematical Physics, Vol 1–2, Interscience
Publishers
Françoise, Jean P.; Naber, Gregory L.; Tsun, Tsou S. (2006), Encyclopedia of Mathematical
Physics, Elsevier, ISBN 978-0-1251-2660-1
Joos, Georg; Freeman, Ira M. (1987), Theoretical Physics (3rd ed.), Dover Publications, ISBN 0-
486-65227-0
Kato, Tosio (1995), Perturbation Theory for Linear Operators (2nd ed.), Springer-Verlag, ISBN 3-
540-58661-X
Margenau, Henry; Murphy, George M. (2009), The Mathematics of Physics and Chemistry
(2nd ed.), Young Press, ISBN 978-1444627473
Masani, Pesi R. (1976–1986), Norbert Wiener: Collected Works with Commentaries, Vol 1–4, The
MIT Press
Morse, Philip M.; Feshbach, Herman (1999), Methods of Theoretical Physics, Vol 1–2, McGraw
Hill, ISBN 0-07-043316-X
Thirring, Walter E. (1978–1983), A Course in Mathematical Physics, Vol 1–4, Springer-Verlag
Tikhomirov, Vladimir M. (1991–1993), Selected Works of A. N. Kolmogorov, Vol 1–3, Kluwer
Academic Publishers
Titchmarsh, Edward C. (1985), The Theory of Functions (2nd ed.), Oxford University Press

Textbooks for undergraduate studies

Arfken, George B.; Weber, Hans J.; Harris, Frank E. (2013), Mathematical Methods for Physicists:
A Comprehensive Guide (7th ed.), Academic Press, ISBN 978-0-12-384654-9, (Mathematical
Methods for Physicists (https://archive.org/details/Mathematical_Methods_for_Physicists),
Solutions for Mathematical Methods for Physicists (7th ed.) (https://archive.org/details/SolutionMat
hematicalMethodForPhysics7), archive.org)
Bayın, Selçuk Ş. (2018), Mathematical Methods in Science and Engineering (2nd ed.), Wiley,
ISBN 9781119425397
Boas, Mary L. (2006), Mathematical Methods in the Physical Sciences (3rd ed.), Wiley, ISBN 978-
0-471-19826-0
Butkov, Eugene (1968), Mathematical Physics, Addison-Wesley
Hassani, Sadri (2009), Mathematical Methods for Students of Physics and Related Fields, (2nd
ed.), New York, Springer, eISBN 978-0-387-09504-2
Jeffreys, Harold; Swirles Jeffreys, Bertha (1956), Methods of Mathematical Physics (3rd ed.),
Cambridge University Press
Marsh, Adam (2018), Mathematics for Physics: An Illustrated Handbook, World Scientific,
ISBN 978-981-3233-91-1
Mathews, Jon; Walker, Robert L. (1970), Mathematical Methods of Physics (2nd ed.), W. A.
Benjamin, ISBN 0-8053-7002-1
 Menzel, Donald H. (1961), Mathematical Physics, Dover Publications, ISBN 0-486-60056-4
Riley, Ken F.; Hobson, Michael P.; Bence, Stephen J. (2006), Mathematical Methods for Physics
and Engineering (3rd ed.), Cambridge University Press, ISBN 978-0-521-86153-3
Stakgold, Ivar (2000), Boundary Value Problems of Mathematical Physics, Vol 1-2., Society for
Industrial and Applied Mathematics, ISBN 0-89871-456-7
Starkovich, Steven P. (2021), The Structures of Mathematical Physics: An Introduction, Springer,
ISBN 978-3-030-73448-0

Textbooks for graduate studies

Blanchard, Philippe; Brüning, Erwin (2015), Mathematical Methods in Physics: Distributions,
Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics (2nd ed.),
Springer, ISBN 978-3-319-14044-5
Cahill, Kevin (2019), Physical Mathematics (2nd ed.), Cambridge University Press, ISBN 978-1-
108-47003-2
Geroch, Robert (1985), Mathematical Physics, University of Chicago Press, ISBN 0-226-28862-5
Hassani, Sadri (2013), Mathematical Physics: A Modern Introduction to its Foundations (2nd ed.),
Springer-Verlag, ISBN 978-3-319-01194-3
Marathe, Kishore (2010), Topics in Physical Mathematics, Springer-Verlag, ISBN 978-1-84882-
938-1
Milstein, Grigori N.; Tretyakov, Michael V. (2021), Stochastic Numerics for Mathematical Physics
(2nd ed.), Springer, ISBN 978-3-030-82039-8
Reed, Michael C.; Simon, Barry (1972–1981), Methods of Modern Mathematical Physics, Vol 1-4,
Academic Press
Richtmyer, Robert D. (1978–1981), Principles of Advanced Mathematical Physics, Vol 1-2.,
Springer-Verlag
Rudolph, Gerd; Schmidt, Matthias (2013–2017), Differential Geometry and Mathematical Physics,
Vol 1-2, Springer
Serov, Valery (2017), Fourier Series, Fourier Transform and Their Applications to Mathematical
Physics, Springer, ISBN 978-3-319-65261-0
Simon, Barry (2015), A Comprehensive Course in Analysis, Vol 1-5, American Mathematical
Society
Stakgold, Ivar; Holst, Michael (2011), Green's Functions and Boundary Value Problems (3rd ed.),
Wiley, ISBN 978-0-470-60970-5
Stone, Michael; Goldbart, Paul (2009), Mathematics for Physics: A Guided Tour for Graduate
Students, Cambridge University Press, ISBN 978-0-521-85403-0
Szekeres, Peter (2004), A Course in Modern Mathematical Physics: Groups, Hilbert Space and
Differential Geometry, Cambridge University Press, ISBN 978-0-521-53645-5
Taylor, Michael E. (2011), Partial Differential Equations, Vol 1-3 (2nd ed.), Springer.
Whittaker, Edmund T.; Watson, George N. (1950), 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 (4th ed.), Cambridge University Press

Specialized texts in classical physics

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

Specialized texts in modern physics

Baez, John C.; Muniain, Javier P. (1994), Gauge Fields, Knots, and Gravity, World Scientific,
ISBN 981-02-2034-0
Blank, Jiří; Exner, Pavel; Havlíček, Miloslav (2008), Hilbert Space Operators in Quantum Physics
(2nd ed.), Springer, ISBN 978-1-4020-8869-8
Engel, Eberhard; Dreizler, Reiner M. (2011), Density Functional Theory: An Advanced Course,
Springer-Verlag, ISBN 978-3-642-14089-1
Glimm, James; Jaffe, Arthur (1987), Quantum Physics: A Functional Integral Point of View
(2nd ed.), Springer-Verlag, ISBN 0-387-96477-0
Haag, Rudolf (1996), Local Quantum Physics: Fields, Particles, Algebras (2nd ed.), Springer-
Verlag, ISBN 3-540-61049-9
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Springer, ISBN 978-1-4614-7115-8
Hamilton, Mark J. D. (2017), Mathematical Gauge Theory: With Applications to the Standard
Model of Particle Physics, Springer, ISBN 978-3-319-68438-3
Hawking, Stephen W.; Ellis, George F. R. (1973), The Large Scale Structure of Space-Time,
Cambridge University Press, ISBN 0-521-20016-4
 Jackiw, Roman (1995), Diverse Topics in Theoretical and Mathematical Physics, World Scientific,
ISBN 9810216963
Landsman, Klaas (2017), Foundations of Quantum Theory: From Classical Concepts to Operator
Algebras, Springer, ISBN 978-3-319-51776-6
Moretti, Valter (2018), Spectral Theory and Quantum Mechanics: Mathematical Foundations of
Quantum Theories, Symmetries and Introduction to the Algebraic Formulation (2nd ed.), Springer,
ISBN 978-3-319-70705-1
Robert, Didier; Combescure, Monique (2021), Coherent States and Applications in Mathematical
Physics (2nd ed.), Springer, ISBN 978-3-030-70844-3
Tasaki, Hal (2020), Physics and mathematics of quantum many-body systems (https://www.worldc
at.org/oclc/1154567924), Springer, ISBN 978-3-030-41265-4, OCLC 1154567924 (https://www.wo
rldcat.org/oclc/1154567924)
Teschl, Gerald (2009), Mathematical Methods in Quantum Mechanics: With Applications to
Schrödinger Operators (https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/), American
Mathematical Society, ISBN 978-0-8218-4660-5
Thirring, Walter E. (2002), Quantum Mathematical Physics: Atoms, Molecules and Large Systems
(2nd ed.), Springer-Verlag, ISBN 978-3-642-07711-1
von Neumann, John (2018), Mathematical Foundations of Quantum Mechanics, Princeton
University Press, ISBN 978-0-691-17856-1
Weyl, Hermann (2014), The Theory of Groups and Quantum Mechanics, Martino Fine Books,
ISBN 978-1614275800
Ynduráin, Francisco J. (2006), The Theory of Quark and Gluon Interactions (4th ed.), Springer,
ISBN 978-3642069741
Zeidler, Eberhard (2006–2011), Quantum Field Theory: A Bridge Between Mathematicians and
Physicists, Vol 1-3, Springer

External links
Media related to Mathematical physics at Wikimedia Commons

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