Mathematical physics
Encyclopedia
Mathematical physics refers to development of mathematical methods for application to problems in physics
. The Journal of Mathematical Physics
defines this area 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.".
s (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 eighteenth century (by, for example, D'Alembert
, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics
, elasticity theory, acoustics
, thermodynamics
, electricity
, magnetism
, and aerodynamics
.
The theory of atomic spectra (and, later, quantum mechanics
) developed almost concurrently with the mathematical fields of linear algebra
, the spectral theory
of operators, and more broadly, functional analysis
. These constitute the mathematical basis of another branch of mathematical physics.
The special
and general
theories of relativity require a rather different type of mathematics
. This was group theory
: and it played an important role in both quantum field theory
and differential geometry. This was, however, gradually supplemented by topology
in the mathematical description of cosmological
as well as quantum field theory phenomena.
Statistical mechanics
forms a separate field, which 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.
The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics
are not 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.
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. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics.
Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.
The field has concentrated in three main areas:
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 statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory
. Use of geometry
and topology
plays an important role in string theory
.
and mathematician
, Isaac Newton
[1642–1727], developed a wealth of new mathematics (for example, calculus
and several numerical methods [e.g. Newton's method
]) to solve problems in physics
. Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens [1629–1695] (famous for suggesting the wave theory of light), and the German Johannes Kepler
[1571–1630] (Tycho Brahe
's assistant, and discoverer of the equations for planetary motion/orbit).
In the eighteenth century, two of the innovators of mathematical physics were Swiss: Daniel Bernoulli
[1700–1782] (for contributions to fluid dynamics
, and vibrating string
s), and, more especially, Leonhard Euler
[1707–1783], (for his work in variational calculus
, dynamics, fluid dynamics, and many other things). Another notable contributor was the Italian-born Frenchman, Joseph-Louis Lagrange [1736–1813] (for his work in mechanics
and variational methods).
In the late eighteenth and early nineteenth centuries, important French figures were Pierre-Simon Laplace
[1749–1827] (in mathematical astronomy
, potential theory
, and mechanics) and Siméon Denis Poisson
[1781–1840] (who also worked in mechanics and potential theory). In Germany
, both Carl Friedrich Gauss
[1777–1855] (in magnetism
) and Carl Gustav Jacobi [1804–1851] (in the areas of dynamics and canonical transformations) made key contributions to the theoretical foundations of electricity
, magnetism, mechanics, and fluid dynamics.
Gauss
's contributions to non-Euclidean geometry
laid the groundwork for the subsequent development of Riemannian geometry
by Bernhard Riemann
[1826–1866]. As we shall see later, this work is at the heart of general relativity
.
The nineteenth century also saw the Scot, James Clerk Maxwell
[1831–1879], win renown for his four equations of electromagnetism
, and his countryman, Lord Kelvin
[1824–1907] make substantial discoveries in thermodynamics
. Among the English physics community, Lord Rayleigh [1842–1919] worked on sound
; and George Gabriel Stokes
[1819–1903] was a leader in optics
and fluid dynamics; while the Irishman William Rowan Hamilton
[1805–1865] was noted for his work in dynamics. The German Hermann von Helmholtz
[1821–1894] is best remembered for his work in the areas of electromagnetism
, waves, fluid
s, and sound. In the U.S.A., the pioneering work of Josiah Willard Gibbs [1839–1903] became the basis for statistical mechanics
. Together, these men laid the foundations of electromagnetic theory, fluid dynamics and statistical mechanics.
The late nineteenth and the early twentieth centuries saw the birth of special relativity
. This had been anticipated in the works of the Dutchman, Hendrik Lorentz
[1853–1928], with important insights from Jules-Henri Poincaré
[1854–1912], but which were brought to full clarity by Albert Einstein
[1879–1955]. Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity. This was based on the non-Euclidean geometry created by Gauss and Riemann in the previous century.
Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations in four dimensional Minkowski space
-time. His general theory of relativity replaced the flat Euclidean geometry
with that of a Riemannian manifold
, whose curvature is determined by the distribution of gravitational matter. This replaced Newton's vector gravitational force by the Riemann curvature tensor
.
Another revolutionary development of the twentieth century has been 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], 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 (Hilbert space
, introduced by David Hilbert
[1862–1943]). Paul Dirac, for example, used algebraic constructions to produce a relativistic model for the electron
, predicting its magnetic moment
and the existence of its antiparticle, the positron
.
Later important contributors to twentieth century mathematical physics include Satyendra Nath Bose
[1894–1974], Julian Schwinger
[1918–1994], Sin-Itiro Tomonaga
[1906–1979], Richard Feynman
[1918–1988], Freeman Dyson
[1923– ], Hideki Yukawa
[1907–1981], Roger Penrose
[1931– ], Stephen Hawking
[1942– ], Edward Witten
[1951– ] and Rudolf Haag
[1922– ]
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. The Journal of Mathematical Physics
Journal of Mathematical Physics
The Journal of Mathematical Physics is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The...
defines this area 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 of the subject
There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (and the related areas of variational calculus, Fourier analysis, potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, and vector analysis) are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...
, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, elasticity theory, acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, electricity
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...
, magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
, and aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
.
The theory of atomic spectra (and, later, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
) developed almost concurrently with the mathematical fields of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
of operators, and more broadly, functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. These constitute the mathematical basis of another branch of mathematical physics.
The special
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and general
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
theories of relativity require a rather different type of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. This was group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
: and it played an important role in both quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and differential geometry. This was, however, gradually supplemented by topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
in the mathematical description of cosmological
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
as well as quantum field theory phenomena.
Statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
forms a separate field, which is closely related with the more mathematical ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
and some parts of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
.
There are increasing interactions between combinatorics and physics
Combinatorics and physics
Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics."Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory."."Physical combinatorics...
, in particular statistical physics.
The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
are not considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s and Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
belong to mathematical physics.
Mathematically rigorous physics
The term 'mathematical' physics is also sometimes used in a special sense, to denote research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Although related to theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, 'mathematical' physics in this sense emphasizes the mathematical rigour
Rigour
Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
of the same type as found in mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics
Experimental physics
Within the field of physics, experimental physics is the category of disciplines and sub-disciplines concerned with the observation of physical phenomena in order to gather data about the universe...
which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
, intuitive
Intuition (knowledge)
Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...
, 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.
Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.
The field has concentrated in three main areas:
- quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, especially the precise construction of models; - statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, especially the theory of phase transitions; and - nonrelativistic quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
(Schrödinger operators), including the connections to atomic and molecular physicsAtomic, molecular, and optical physicsAtomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of the...
.
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
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
. Use of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
plays an important role in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
Prominent mathematical physicists
The seventeenth century English physicistPhysicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
[1642–1727], developed a wealth of new mathematics (for example, calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and several numerical methods [e.g. Newton's method
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
]) to solve problems in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens [1629–1695] (famous for suggesting the wave theory of light), and the German Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
[1571–1630] (Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
's assistant, and discoverer of the equations for planetary motion/orbit).
In the eighteenth century, two of the innovators of mathematical physics were Swiss: Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
[1700–1782] (for contributions to fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, and vibrating string
Vibrating string
A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency in most cases is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note....
s), and, more especially, Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
[1707–1783], (for his work in variational calculus
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, dynamics, fluid dynamics, and many other things). Another notable contributor was the Italian-born Frenchman, Joseph-Louis Lagrange [1736–1813] (for his work in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
and variational methods).
In the late eighteenth and early nineteenth centuries, important French figures were Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
[1749–1827] (in mathematical astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, and mechanics) and Siméon Denis Poisson
Siméon Denis Poisson
Siméon Denis Poisson , was a French mathematician, geometer, and physicist. He however, was the final leading opponent of the wave theory of light as a member of the elite l'Académie française, but was proven wrong by Augustin-Jean Fresnel.-Biography:...
[1781–1840] (who also worked in mechanics and potential theory). In Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
, both Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
[1777–1855] (in magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
) and Carl Gustav Jacobi [1804–1851] (in the areas of dynamics and canonical transformations) made key contributions to the theoretical foundations of electricity
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...
, magnetism, mechanics, and fluid dynamics.
Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...
's contributions to non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
laid the groundwork for the subsequent development of Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
by Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
[1826–1866]. As we shall see later, this work is at the heart of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
The nineteenth century also saw the Scot, James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
[1831–1879], win renown for his four equations of electromagnetism
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, and his countryman, Lord Kelvin
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
[1824–1907] make substantial discoveries in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
. Among the English physics community, Lord Rayleigh [1842–1919] worked on sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
; and George Gabriel Stokes
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
[1819–1903] was a leader in optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
and fluid dynamics; while the Irishman William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
[1805–1865] was noted for his work in dynamics. The German Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
[1821–1894] is best remembered for his work in the areas of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, waves, fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s, and sound. In the U.S.A., the pioneering work of Josiah Willard Gibbs [1839–1903] became the basis for statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. Together, these men laid the foundations of electromagnetic theory, fluid dynamics and statistical mechanics.
The late nineteenth and the early twentieth centuries saw the birth of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. This had been anticipated in the works of the Dutchman, Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
[1853–1928], with important insights from Jules-Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
[1854–1912], but which were brought to full clarity by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
[1879–1955]. Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity. This was based on the non-Euclidean geometry created by Gauss and Riemann in the previous century.
Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations in four dimensional Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
-time. His general theory of relativity replaced the flat Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
with that of a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
, whose curvature is determined by the distribution of gravitational matter. This replaced Newton's vector gravitational force by the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
.
Another revolutionary development of the twentieth century has been quantum theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, which emerged from the seminal contributions of Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
[1856–1947] (on black body radiation) and Einstein's work on the photoelectric effect
Photoelectric effect
In the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...
. This was, at first, followed by a heuristic framework devised by Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
[1868–1951] and Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
[1885–1962], but this was soon replaced by the quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
developed by Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
[1882–1970], Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
[1901–1976], Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
[1902–1984], Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
[1887–1961], and Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
[1900–1958]. This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operator
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s on an infinite dimensional vector space (Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, introduced by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
[1862–1943]). Paul Dirac, for example, used algebraic constructions to produce a relativistic model for the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, predicting its magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
and the existence of its antiparticle, the positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
.
Later important contributors to twentieth century mathematical physics include Satyendra Nath Bose
Satyendra Nath Bose
Satyendra Nath Bose FRS was an Indian mathematician and physicist noted for his collaboration with Albert Einstein in developing a theory regarding the gaslike qualities of electromagnetic radiation. He is best known for his work on quantum mechanics in the early 1920s, providing the foundation...
[1894–1974], Julian Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
[1918–1994], Sin-Itiro Tomonaga
Sin-Itiro Tomonaga
was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.-Biography:...
[1906–1979], Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
[1918–1988], Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
[1923– ], Hideki Yukawa
Hideki Yukawa
né , was a Japanese theoretical physicist and the first Japanese Nobel laureate.-Biography:Yukawa was born in Tokyo and grew up in Kyoto. In 1929, after receiving his degree from Kyoto Imperial University, he stayed on as a lecturer for four years. After graduation, he was interested in...
[1907–1981], Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
[1931– ], Stephen Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
[1942– ], Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
[1951– ] and Rudolf Haag
Rudolf Haag
Rudolf Haag is a German physicist. He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms...
[1922– ]
See also
- Important publications in Mathematical Physics
- International Association of Mathematical PhysicsInternational Association of Mathematical PhysicsThe International Association of Mathematical Physics was founded in 1976 to promote research in mathematical physics. It brings together research mathematicians and theoretical physicists, including students. The association's ordinary members are individual researchers, although associate...
- Theoretical physicsTheoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
The Classics
(pbk.) (softcover) (This is a reprint of the second (1980) edition of this title.) (This is a reprint of the 1956 second edition.) (This is a reprint of the original (1953) edition of this title.) (This tome was reprinted in 1985.)Textbooks for undergraduate studies
(pbk.)- (set : pbk.)
Textbooks for graduate studies
- Reed,Michael; Simon,Barry Methods of Modern Mathematical Physics (Academic Press).[Four volumes]