Mathematical physics refers to 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".
Scope
There are several distinct branches of mathematical
physics, and these roughly correspond to particular
historical periods.
Classical mechanics
The rigorous, abstract and advanced re-formulation
of Newtonian mechanics adopting the Lagrangian
mechanics and the Hamiltonian mechanics even
in the presence of constraints. Both formulations
are embodied in the so-called 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.
Partial differential equations
The theory of partial differential equations
are perhaps most closely associated with mathematical
physics. These were developed intensively
from the second half of the eighteenth century
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 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 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.
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 to use heuristic, intuitive, and
approximate arguments. Such arguments are
not considered rigorous by mathematicians.
Arguably, rigorous mathematical physics is
closer to mathematics, and theoretical physics
is closer to physics. This is reflected institutionally:
mathematical physicists are often members
of the mathematics department.
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 already be solved. However, they can sometimes
show that the previous solution was incomplete,
incorrect, or simply, too naive. Issues about
attempts to infer the second law of thermodynamics
from statistical mechanics are examples. Other
examples concerns all the subtleties involved
with synchronisation procedures in special
and general relativity
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. Not
quite radical, Copernicus merely sought to
simplify astronomy and achieve orbits of more
perfect circles, stated by Aristotelian physics
to be the intrinsic motion of Aristotle's
fifth element—the quintessence or universal
essence known in Greek as aither 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, however,
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, upon his telescopic observations,
supported heliocentrism. Having introducing
experimentation, Galileo then refuted geocentric
cosmology by refuting Aristotelian physics
itself. Galilei's 1638 book Discourse on Two
New Sciences established 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 Galilean invariance, also called
Galilean relativity, for any object experiencing
inertia, there is empirical justification
of knowing only its being 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 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.
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 Galilean invariance/relativity
was merely implicit in Newton's theory of
motion. Having ostensibly reduced Keplerian
celestial laws of motion as well as Galilean
terrestrial laws of motion to a unifying force,
Newton achieved great mathematic rigor if
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 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 handle 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.
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
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]
did a lot of major work: Stokes was a leader
in optics and fluid dynamics; Kelvin made
substantial discoveries in thermodynamics;
Hamilton did notable work on analytical mechanics
finding out 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 the so-called canonical
transformations. The German Hermann von Helmholtz
[1821–1894] is greatly contributed to 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, prominent was the 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. Physicists' mathematical process
to translate the positions in one reference
frame to predictions of positions in another
reference frame, all plotted on Cartesian
coordinates, had been the Galilean transformation,
which was newly replaced with 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. Hypotheses at
the aether thus kept Maxwell's electromagnetic
field aligned with the principle 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 special theory of relativity, newly
explaining both the electromagnetic field's
invariance and Galilean invariance by discarding
all hypotheses at aether, including aether
itself. Refuting the framework of Newton's
theory—absolute space and absolute time—special
relativity states relative space and relative
time, whereby length contracts and time dilates
along the travel pathway of an object experiencing
kinetic energy.
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 to great
elegance in general theory of relativity,
extending invariance to all reference frames—whether
perceived as inertial or as accelerated—and
thanked 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.
Quantum
Another revolutionary development of the twentieth
century has been quantum theory, which emerged
from the seminal contributions of Max Planck
[1856–1947] 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 the so-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 on 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 important mathematical physicists
in the 20th century
Prominent contributors to the 20th century's
mathematical physics include Arnold Sommerfeld
[1868–1951], Albert Einstein [1879–1955],
Max Born [1882–1970], Niels Bohr [1885–1962],
Hermann Weyl [1885–1955], Satyendra Nath
Bose [1894–1974], Wolfgang Pauli [1900–1958],
Werner Heisenberg [1901–1976], Paul Dirac
[1902–1984], Eugene Wigner [1902–1995],
John von Neumann [1903–1957], Sin-Itiro
Tomonaga [1906–1979], Hideki Yukawa [1907–1981],
Lev Landau [1908-1968], Nikolay Bogolyubov
[1909–1992], Mark Kac [1914–1984], Julian
Schwinger [1918–1994], Richard Feynman [1918–1988],
Arthur Strong Wightman [1922–2013], Chen-Ning
Yang [1922– ], Rudolf Haag [1922– ], Freeman
Dyson [1923– ], Martin Gutzwiller [1925–2014],
Abdus Salam [1926–1996], Jürgen Moser [1928–1999],
Peter Higgs [1929– ], Michael Atiyah [1929–
], Joel Lebowitz [1930– ], Roger Penrose
[1931– ], Elliott H. Lieb [1932– ], Sheldon
Lee Glashow [1932– ], Steven Weinberg [1933–
], Ludvig D. Faddeev [1934– ], David Ruelle
[1935– ], Yakov G. Sinai [1935– ], Vladimir
Arnold [1937–2010], Arthur Jaffe [1937–
], Roman Jackiw [1939– ], Leonard Susskind
[1940– ], Rodney J. Baxter [1940– ], Stephen
Hawking [1942– ], Alexander M. Polyakov
[1945– ], Barry Simon [1946– ], John L.
Cardy [1947– ], Edward Witten [1951– ], and
Juan M. Maldacena [1968– ].
See also
International Association of Mathematical
Physics
Notable publications in mathematical physics
Notes
References
Zalsow, Eric, Physmatics, arXiv:physics/0506153,
Bibcode:2005physics...6153Z 
Further reading
The 
Classics
Abraham, Ralph; Marsden, Jerrold E., 'Foundations
of mechanics: a mathematical exposition of
classical mechanics with an introduction to
the qualitative theory of dynamical systems',
Providence, [RI.]: AMS Chelsea Pub., ISBN 978-0-8218-4438-0 
Arnold, Vladimir I.; Vogtmann, K.; Weinstein,
A., 'Mathematical methods of classical mechanics
/ [Matematicheskie metody klassicheskoĭ mekhaniki]',
New York, [NY.]: Springer-Verlag, ISBN 0-387-96890-3 
Courant, Richard; Hilbert, David, Methods
of mathematical physics, New York, [NY.]:
Interscience Publishers 
Glimm, James; Jaffe, Arthur, 'Quantum physics:
a functional integral point of view', New
York, [NY.]: Springer-Verlag, ISBN 0-387-96477-0 
Haag, Rudolf, 'Local quantum physics: fields,
particles, algebras', Berlin, [Germany] ;
New York, [NY.]: Springer-Verlag, ISBN 3-540-61049-9 
Hawking, Stephen W.; Ellis, George F. R.,
'The large scale structure of space-time',
Cambridge, [England]: Cambridge University
Press, ISBN 0-521-20016-4 
Kato, Tosio, 'Perturbation theory for linear
operators', Berlin, [Germany]: Springer-Verlag,
ISBN 3-540-58661-X  edition of this title.)
Margenau, Henry; Murphy, George Moseley, 'The
mathematics of physics and chemistry', Huntington,
[NY.]: R. E. Krieger Pub. Co., ISBN 0-88275-423-8 
Morse, Philip McCord; Feshbach, Herman, 'Methods
of theoretical physics', Boston, [Mass.]:
McGraw Hill, ISBN 0-07-043316-X  edition
of this title.)
von Neumann, John; Beyer, Robert T., 'Mathematical
foundations of quantum mechanics', Princeton,
[NJ.]: Princeton University Press 
Reed, Michael C.; Simon, Barry, Methods of
modern mathematical physics 4, New York City:
Academic Press, ISBN 0-12-585001-8 
Titchmarsh, Edward Charles, 'The theory of
functions', London, [England]: Oxford University
Press 
Thirring, Walter E.; Harrell, Evans M., 'A
course in mathematical physics / [Lehrbuch
der mathematischen Physik]', New York, [NY.]:
Springer-Verlag 
Weyl, Hermann; Robertson, H. P., 'The theory
of groups and quantum mechanics / [Gruppentheorie
und Quantenmechanik]', London, [England]:
Methuen & Co. 
Whittaker, Edmund Taylor; Watson, George Neville,
'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',
Cambridge: Cambridge University Press, ISBN 978-0-521-58807-2 
Textbooks for undergraduate studies
Arfken, George B.; Weber, Hans J., 'Mathematical
methods for physicists', San Diego, [CA.]:
Academic Press, ISBN 0-12-059816-7 
Boas, Mary L., 'Mathematical Methods in the
Physical Sciences', Hoboken, [NJ.]: John Wiley
& Sons, ISBN 978-0-471-19826-0 
Butkov, Eugene, 'Mathematical physics', Reading,
[Mass.]: Addison-Wesley 
Jeffreys, Harold; Swirles Jeffreys, Bertha,
'Methods of mathematical physics', Cambridge,
[England]: Cambridge University Press 
Kusse, Bruce R., 'Mathematical Physics: Applied
Mathematics for Scientists and Engineers',
[Germany]: Wiley-VCH, ISBN 3-527-40672-7 
Joos, Georg; Freeman, Ira M., Theoretical
Physics, Dover Publications, ISBN 0-486-65227-0 
Mathews, Jon; Walker, Robert L., 'Mathematical
methods of physics', New York, [NY.]: W. A.
Benjamin, ISBN 0-8053-7002-1 
Menzel, Donald Howard, Mathematical Physics,
Dover Publications, ISBN 0-486-60056-4 
Stakgold, Ivar, 'Boundary value problems of
mathematical physics', Philadelphia, [PA.]:
Society for Industrial and Applied Mathematics,
ISBN 0-89871-456-7 
Textbooks for graduate studies
Hassani, Sadri, 'Mathematical Physics: A Modern
Introduction to Its Foundations', Berlin,
[Germany]: Springer-Verlag, ISBN 0-387-98579-4 
Reed, M.; Simon, B.. Methods of Mathematical
Physics. Vol 1-4. Academic Press. 
Teschl, G.. Mathematical Methods in Quantum
Mechanics; With Applications to Schrödinger
Operators. Providence: American Mathematical
Society. ISBN 978-0-8218-4660-5. 
Moretti, V.. Spectral Theory and Quantum Mechanics;
With an Introduction to the Algebraic Formulation.
Berlin, Milan: Springer. ISBN 978-88-470-2834-0. 
Other specialised subareas
Aslam, Jamil; Hussain, Faheem, 'Mathematical
physics' Proceedings of the 12th Regional
Conference, Islamabad, Pakistan, 27 March
– 1 April 2006, Singapore: World Scientific,
ISBN 978-981-270-591-4 
Baez, John C.; Muniain, Javier P., 'Gauge
fields, knots, and gravity', Singapore ;
River Edge, [NJ.]: World Scientific, ISBN 981-02-2034-0 
Geroch, Robert, 'Mathematical physics', Chicago,
[IL.]: University of Chicago Press, ISBN 0-226-28862-5 
Polyanin, Andrei D., 'Handbook of linear partial
differential equations for engineers and scientists',
Boca Raton, [FL.]: Chapman & Hall / CRC Press,
ISBN 1-58488-299-9 
Polyanin, Alexei D.; Zaitsev, Valentin F.,
'Handbook of nonlinear partial differential
equations', Boca Raton, [FL.]: Chapman & Hall
/ CRC Press, ISBN 1-58488-355-3 
Szekeres, Peter, 'A course in modern mathematical
physics: groups, Hilbert space and differential
geometry', Cambridge, [England]; New York,
[NY.]: Cambridge University Press, ISBN 0-521-53645-6 
Yndurain, Francisco J, 'Theoretical and Mathematical
Physics. The Theory of Quark and Gluon Interactions',
Berlin, [Germany]: Springer, ISBN 978-3642069741 
