Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in. The  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, 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 and the  even in the presence of constraints. Both formulations are embodied in. 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. These approaches and ideas can be, and in fact have been, extended to other areas of physics as, , and. Moreover, they have provided several examples and basic ideas in (e.g. the theory of  and several notions in ).

Partial differential equations
The theory of s (and the related areas of, , , and ) are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the 18th century (by, for example,, , and ) until the 1930s. Physical applications of these developments include, , , , , , , , and.

Quantum theory
The theory of (and, later, ) developed almost concurrently with the mathematical fields of, the  of ,  and more broadly,. Nonrelativistic quantum mechanics includes operators, and it has connections to. theory is another subspecialty.

Relativity and quantum relativistic theories
The and  theories of relativity require a rather different type of mathematics. This was, which played an important role in both quantum field theory and. This was, however, gradually supplemented by and   in the mathematical description of  as well as quantum field theory phenomena. In this area both and  are important nowadays.

Statistical mechanics
forms a separate field, which includes the theory of s. It relies upon the (or its quantum version) and it is closely related with the more mathematical  and some parts of. There are increasing interactions between, in particular statistical physics.

Usage
The usage of the term "mathematical physics" is sometimes. Certain parts of mathematics that initially arose from the development of are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, s and are generally viewed as purely mathematical disciplines, whereas s and  belong to mathematical physics. 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 s within a mathematically framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of pure and. Although related to, 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, which often requires theoretical physicists (and mathematical physicists in the more general sense) to use , , 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. 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 ( and ).

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 parallel each other in many ways. The mathematical study of quantum mechanics, quantum field theory, and quantum statistical mechanics has motivated results in. The attempt to construct a rigorous quantum field theory has also brought about progress in fields such as. Use of and  plays an important role in.

Before Newton
The roots of mathematical physics can be traced back to the likes of in Greece,  in Egypt,  in, and  in.

In the first decade of the 16th century, amateur astronomer proposed, and published a treatise on it in 1543. He retained the idea of s, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According to, the circle was the perfect form of motion, and was the intrinsic motion of Aristotle's —the quintessence or universal essence known in Greek as  for the English pure air—that was the pure substance beyond the , and thus was celestial entities' pure composition. The German [1571–1630], 's assistant, modified Copernican orbits to s, formalized in the equations of Kepler's.

An enthusiastic atomist, 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 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. By the Galilean as well as the principle of, 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.

adopted Galilean principles and developed a complete system of heliocentric cosmology, anchored on the principle of vortex motion,, whose widespread acceptance brought the demise of Aristotelian physics. Descartes sought to formalize mathematical reasoning in science, and developed for geometrically plotting locations in 3D space and marking their progressions along the flow of time.

was the first to use mathematical formulas to describe the laws of physics, and for that reason Huygens is regarded as the first and the founder of mathematical physics.

Newtonian and post Newtonian
(1642–1727) developed new mathematics, including and several  such as  to solve problems in physics. Newton's theory of motion, published in 1687, modeled three Galilean laws of motion along with Newton's on a framework of —hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming, 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 (1700–1782) made contributions to, and s. The Swiss  (1707–1783) did special work in , dynamics, fluid dynamics, and other areas. Also notable was the Italian-born Frenchman, (1736–1813) for work in : he formulated ) and variational methods. A major contribution to the formulation of Analytical Dynamics called  was also made by the Irish physicist, astronomer and mathematician,  (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  (1768 – 1830) introduced the notion of  to solve the, giving rise to a new approach to solving partial differential equations by means of.

Into the early 19th century, the French (1749–1827) made paramount contributions to mathematical, , and. (1781–1840) worked in and potential theory. In Germany, (1777–1855) made key contributions to the theoretical foundations of, , , and. In England, (1793-1841) published  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 (1629–1695) developed the wave theory of light, published in 1690. By 1804, '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, was accepted. modeled hypothetical behavior of the aether. introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four. Initially, optics was found consequent of Maxwell's field. Later, radiation and then today's known were found also consequent of this electromagnetic field.

The English physicist [1842–1919] worked on. The Irishmen (1805–1865),  (1819–1903) and  (1824–1907) produced several major works: Stokes was a leader in  and fluid dynamics; Kelvin made substantial discoveries in ; Hamilton did notable work on, discovering a new and powerful approach nowadays known as. Very relevant contributions to this approach are due to his German colleague (1804–1851) in particular referring to. The German (1821–1894) made substantial contributions in the fields of, waves, s, and sound. In the United States, the pioneering work of (1839–1903) became the basis for. Fundamental theoretical results in this area were achieved by the German (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 within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of the, physicists inferred that motion within the aether resulted in , shifting the electromagnetic field, explaining the observer's missing speed relative to it. The 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, but this process was replaced by , modeled by the Dutch  [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. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all, while Newton's theory of motion was spared.

In the 19th century, 's contributions to, or geometry on curved surfaces, laid the groundwork for the subsequent development of by  (1826–1866). Austrian theoretical physicist and philosopher criticized Newton's postulated absolute space. Mathematician (1854–1912) questioned even absolute time. In 1905, published a devastating criticism of the foundation of Newton's theory of motion. Also in 1905, (1879–1955) published his, 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——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 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 with great elegance in his, 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, and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's of hypothetical gravitational force—an instant —with a gravitational field. The gravitational field is Minkowski spacetime itself, the 4D of Einstein aether modeled on a  that "curves" geometrically, according to the, 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 locally "curving" the geometry of the four, unified dimensions of space and time.)

Quantum
Another revolutionary development of the 20th century was, which emerged from the seminal contributions of (1856–1947) (on ) and Einstein's work on the. This was, at first, followed by a heuristic framework devised by (1868–1951) and  (1885–1962), but this was soon replaced by the  developed by  (1882–1970),  (1901–1976),  (1902–1984),  (1887–1961),  (1894–1974), and  (1900–1958). This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of s on an infinite dimensional vector space. That is called, introduced in its elementary form by (1862–1943) and  (1880-1956), and rigorously defined within the axiomatic modern version by  in his celebrated book , where he built up a relevant part of modern functional analysis on Hilbert spaces, the  in particular. Paul Dirac used algebraic constructions to produce a relativistic model for the, predicting its and the existence of its antiparticle, the.

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; since the page can be edited by anyone, sometimes less deserved mentions can pop up in the list) include, ordered by birth date, [1824–1907],  [1850–1925],  [1854–1912],  [1862–1943],  [1868–1951],  [1873–1950],  [1879–1955],  [1882–1970],  [1884-1944],  [1885–1955],  [1894-1974],  [1894–1964],  (1897–1995),  [1900–1958],  [1902–1984],  [1902–1995],  [1903-1987],  [1903-1976],  [1903–1957],  [1906–1979],  [1907–1981],  [1909–1992],  [1910-1995],  [1914–1984],  [1918–1994],  [1918–1988],  [1918–1998],  [1920–1995],  [1922–2013],  [1922– ],  [1922–2016],  [1923– ],  [1925–2014],  [1926–1996],  [1928–1999],  [1929–2019],  [1930– ],  [1931– ],  [1932– ],  [1932– ],  [1933– ],  [1934–2017],  [1935– ],  [1935– ],  [1937–2010],  [1937– ],  [1939– ],  [1940– ],  [1940– ],  [1941- ],  [1941- ],  [1942–2018],  [1942–2010],  [1945– ],  [1946– ],  [1947– ],  [1948– ],  [1951– ],  [1951?– ],  [1956-] and  [1968– ].

Generic works

 * (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.)
 * (This is a reprint of the original (1953) edition of this title.)
 * (This tome was reprinted in 1985.)
 * (This tome was reprinted in 1985.)

Textbooks for undergraduate studies

 * (pbk.)
 * (set : pbk.)
 * (set : pbk.)
 * (set : pbk.)
 * (set : pbk.)
 * (set : pbk.)
 * (set : pbk.)
 * (set : pbk.)