Editing Mathematical physics (section)

== Usage ==
[[File:Mathematical Physics and other sciences v1.png|thumb|Relationship between mathematics and 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, in fact, 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. [[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".<ref>[[John Herapath]] (1847) [https://catalog.hathitrust.org/Record/011557061?type%5B%5D=author&lookfor%5B%5D=John%20Herapath&ft=ft 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]]</ref>

=== Mathematical vs. theoretical physics ===
The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or [[thought experiment]]s within a mathematically [[mathematical rigour|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]],<ref>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.</ref> 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]], [[Intuition (knowledge)|intuitive]], or approximate arguments.<ref>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.</ref> 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.{{Citation needed|date=July 2023}} 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 algebra]]s. The attempt to construct a rigorous mathematical formulation of [[quantum field theory]] has also brought about some progress in fields such as [[representation theory]].