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".
It is a branch of applied mathematics, but
deals with physical problems.
== Scope ==
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.
== 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"; the scope at that
time being
"the causes of heat, gaseous elasticity, gravitation,
and other great phenomena of nature".
=== 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, mathematical
physics in this sense emphasizes the mathematical
rigour of the same type as found in mathematics.
On the other hand, theoretical physics emphasizes
the links to observations and experimental
physics, which often requires theoretical
physicists (and mathematical physicists in
the more general sense) to use heuristic,
intuitive, and approximate arguments.
Such arguments are not considered rigorous
by mathematicians, 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.
His 1632 book, about his telescopic observations,
supported heliocentrism.
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.
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.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.
=== 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.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.
=== 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.
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.
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, 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.)
=== 
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.
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 ==
International Association of Mathematical
Physics
Notable publications in mathematical physics
List of mathematical physics journals
== Notes
