Mathematical physics is the development of mathematical methods for use in physics and their applications.Definition from the Journal of Mathematical Physics. A broader definition would include the development of mathematical ideas inspired by physics, known as physical mathematics.
Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world.
Classical mechanics
Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead to an understanding of the deep interplay between the notions of symmetry and conserved quantities during the dynamical evolution of mechanical systems, as embodied within the most elementary formulation of Noether's theorem. These approaches and ideas have been extended to other areas of physics, such as statistical mechanics, continuum mechanics, classical field theory, and quantum field theory. Moreover, they have provided multiple examples and ideas in differential geometry (e.g., several notions in symplectic geometry and vector bundles).
Partial differential equations
Within mathematics proper, the theory of partial differential equation, variational calculus, Fourier analysis, potential theory, and vector analysis are perhaps most closely associated with mathematical physics. These fields were developed intensively from the second half of the 18th century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.
Quantum theory
The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with some parts of the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and, more broadly, functional analysis. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic and molecular physics. Quantum information theory is another subspecialty.
Relativity and quantum relativistic theories
The special and general theories of relativity require a rather different type of mathematics. This was group theory, which played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the mathematical description of cosmological as well as quantum field theory phenomena. In the mathematical description of these physical areas, some concepts in homological algebra and category theory are also important.
Statistical mechanics
Statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics (or its quantum version) and it is closely related with the more mathematical ergodic theory and some parts of probability theory. There are increasing interactions between combinatorics and physics, in particular statistical physics.
Usage
The usage of the term "mathematical physics" is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics. John Herapath used the term for the title of his 1847 text on "mathematical principles of natural philosophy", the scope at that time being "the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature".John Herapath (1847) Mathematical Physics; or, the Mathematical Principles of Natural Philosophy, the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature, Whittaker and company via HathiTrust
Mathematical vs. theoretical physics The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or thought experiments within a mathematically rigorous framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of some mathematical aspect and theoretical physics aspect. Although related to theoretical physics,Quote: " ... a negative definition of the theorist refers to his inability to make physical experiments, while a positive one... implies his encyclopaedic knowledge of physics combined with possessing enough mathematical armament. Depending on the ratio of these two components, the theorist may be nearer either to the experimentalist or to the mathematician. In the latter case, he is usually considered as a specialist in mathematical physics.", Ya. Frenkel, as related in A.T. Filippov, The Versatile Soliton, pg 131. Birkhauser, 2000. mathematical physics in this sense emphasizes the mathematical rigour of the similar type as found in mathematics.
On the other hand, theoretical physics emphasizes the links to observations and experimental physics, which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, or approximate arguments.Quote: "Physical theory is something like a suit sewed for Nature. Good theory is like a good suit. ... Thus the theorist is like a tailor." Ya. Frenkel, as related in Filippov (2000), pg 131. Such arguments are not considered rigorous by mathematicians.
Such mathematical physicists primarily expand and elucidate physical theories. Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to be already solved. However, they can sometimes show that the previous solution was incomplete, incorrect, or simply too naïve. Issues about attempts to infer the second law of thermodynamics from statistical mechanics are examples. Other examples concern the subtleties involved with synchronisation procedures in special and general relativity (Sagnac effect and Einstein synchronisation).
The effort to put physical theories on a mathematically rigorous footing not only developed physics but also has influenced developments of some mathematical areas. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum mechanics, quantum field theory, and quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous mathematical formulation of quantum field theory has also brought about some progress in fields such as representation theory.
Prominent mathematical physicists
Before Newton There is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples include Euclid (Optics), Archimedes (On the Equilibrium of Planes, On Floating Bodies), and Ptolemy (Optics, Harmonics). Later, Islamic and Byzantine scholars built on these works, and these ultimately were reintroduced or became available to the West in the 12th century and during the Renaissance.
In the first decade of the 16th century, amateur astronomer Nicolaus Copernicus proposed heliocentrism, and published a treatise on it in 1543. He retained the Ptolemaic idea of epicycles, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According to Aristotelian physics, the circle was the perfect form of motion, and was the intrinsic motion of Aristotle's fifth element—the quintessence or universal essence known in Greek as aether for the English pure air—that was the pure substance beyond the sublunary sphere, and thus was celestial entities' pure composition. The German Johannes Kepler [1571–1630], Tycho Brahe's assistant, modified Copernican orbits to ellipses, formalized in the equations of Kepler's laws of planetary motion.
An enthusiastic atomist, Galileo Galilei in his 1623 book The Assayer asserted that the "book of nature is written in mathematics".Peter Machamer "Galileo Galilei"—sec 1 "Brief biography", in Zalta EN, ed, The Stanford Encyclopedia of Philosophy, Spring 2010 edn His 1632 book, about his telescopic observations, supported heliocentrism.Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 129 Having made use of experimentation, Galileo then refuted geocentric cosmology by refuting Aristotelian physics itself. Galileo's 1638 book Discourse on Two New Sciences established the law of equal free fall as well as the principles of inertial motion, two central concepts of what today is known as 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 developed a complete system of heliocentric cosmology anchored on the principle of vortex motion, Cartesian physics, whose widespread acceptance helped bring the demise of Aristotelian physics. Descartes used mathematical reasoning as a model for science, and developed analytic geometry, which in time allowed the plotting of locations in 3D space (Cartesian coordinates) and marking their progressions along the flow of time.Antony G Flew, Dictionary of Philosophy, rev 2nd edn (New York: St Martin's Press, 1984), p 89
Christiaan Huygens, a talented mathematician and physicist and older contemporary of Newton, was the first to successfully idealize a physical problem by a set of mathematical parameters in Horologium Oscillatorum (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon in Traité de la Lumière (1690). He is thus considered a forerunner of theoretical physics and one of the founders of modern mathematical physics.Dijksterhuis, F. J. (2008). Stevin, Huygens and the Dutch republic. Nieuw archief voor wiskunde, 5, pp. 100–107. https://research.utwente.nl/files/6673130/Dijksterhuis_naw5-2008-09-2-100.pdfAndreessen, C.D. (2005) Huygens: The Man Behind the Principle. Cambridge University Press: 6
Newtonian and post-Newtonian physics
The prevailing framework for science in the 16th and early 17th centuries was one borrowed from ancient Greek mathematics, where geometrical shapes formed the building blocks to describe and think about space, and time was often thought of as a separate entity. With the introduction of algebra into geometry, and with it the idea of a coordinate system, time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and is used in further mathematical frameworks developed in subsequent centuries.