A Brief Description on Mathematical Physics

About the Study

Mathematical physics is the study of mathematical methods and their application to physics problems. The Journal of Mathematical Physics defines the topic as "the application of mathematics to physics problems and the creation of mathematical methods suitable for such applications and the formulation of physical theories". Mathematical physics is a large academic field differentiated solely by the incorporation of some mathematical and theoretical aspects of physics. Although mathematical physics is related to theoretical physics, it emphasizes mathematical rigour similar to that found in mathematics.

Partial differential equations

Following mathematics, mathematical physics is most closely related with partial differential equation theory, variational calculus, Fourier analysis, potential theory, and vector analysis. From the second part of the 18th century until the 1930s, these were vigorously developed (for example, by D'Alembert, Euler, and Lagrange). Hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics are some of the physical applications of these findings.

Quantum theory

Some portions of the mathematical sciences of linear algebra, spectral theory of operators, operator algebras, and, more broadly, functional analysis emerged roughly simultaneously with the theory of atomic spectra (and, later, quantum mechanics). Schrödinger operators are part of nonrelativistic quantum mechanics, which has linkages to atomic and molecular physics. Another specialization is quantum information theory.

Relativity and quantum relativistic theories

The special and general theories of relativity both necessitate a unique sort of math. This was group theory, which was crucial in quantum field theory as well as differential geometry. Topology and functional analysis, on the other hand, eventually complemented this in the mathematical explanation of cosmological and quantum field theory phenomena. Some homological algebra and category theory notions are also significant in the mathematical representation of these physical fields.

Statistical mechanics

Statistical mechanics is a distinct field that encompasses phase transition theory. It is closely related to the more mathematical ergodic theory and other areas of probability theory, and it is based on Hamiltonian mechanics (or its quantum variant). Combinatorics and physics, particularly statistical physics, are becoming increasingly intertwined.

Mathematical vs. Theoretical physics

The phrase "mathematical physics" is occasionally used to refer to research that uses a mathematically rigorous framework to examine and solve physics problems or thought experiments. In this respect, mathematical physics is a large academic field differentiated solely by the incorporation of some mathematical and theoretical aspects of physics. Although mathematical physics is related to theoretical physics, it emphasises mathematical rigour similar to that found in mathematics.

Theoretical physics, on the other hand, stresses the connections between observations and experimental physics, which frequently necessitates the employment of heuristic, intuitive, and imprecise arguments by theoretical physicists (and mathematical physicists in general). Mathematicians do not regard such arguments to be rigorous.

Physical theories are primarily expanded and elucidated by such mathematical scientists. Because of the high level of mathematical rigour required, these researchers frequently work on problems that theoretical physicists consider to be solved. They can, however, occasionally reveal that the preceding answer was either partial, wrong, or simply too naive. Examples include attempts to deduce the second law of thermodynamics from statistical mechanics. Other examples include the complexities of special and general relativity synchronisation processes (Sagnac effect and Einstein synchronisation).

The effort to set physical theories on a theoretically sound foundation has influenced the development of several mathematical domains as well as physics. In many ways, the development of quantum mechanics and some areas of functional analysis, for example, parallel each other. Results in operator algebras have been prompted by the mathematical study of quantum mechanics, quantum field theory, and quantum statistical mechanics. The effort to develop a formal mathematical formulation of quantum field theory has aided research in topics such as representation theory.