What is Mathematical Physics?

Information about Mathematical Physics

Mathematical physics is the scientific discipline concerned with "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."[1]

It can be seen as underpinning both theoretical physics and computational physics.

Scope of the subject

There are several quite distinct branches of mathematical physics, which roughly correspond to different historical periods. The theory of partial differential equations and related areas of variational calculus, Fourier analysis, potential theory and vector analysis are perhaps most closely associated with mathematical physics, and developed intensively from the second half of 18th century (D'Alembert, Euler, Lagrange) until the 1930s. Their physical applications include hydrodynamics, celestial mechanics, elasticity theory, acoustics, thermodynamics, electricity and magnetism, aerodynamics. The theory of atomic spectra and later quantum mechanics developed almost concurrently with the mathematical fields of linear algebra, spectral theory of operators, and more broadly, functional analysis, which constitute the mathematical side of another branch of mathematical physics. Special relativity and general relativity require rather different type of mathematics, represented by group theory (also playing an important role in quantum theory) and differential geometry. They were gradually supplemented by topology, as it gained prominence in mathematical description of cosmological as well as quantum field theory phenomena. Statistical mechanics forms a separate field, which is closely related with more mathematical ergodic theory and some parts of probability theory.

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 would be included. For example, ordinary differential equations and symplectic geometry are generally viewed as mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.

Prominent mathematical physicists

The great 17th century mathematician and physicist Isaac Newton developed a wealth of new mathematics, in an informal way, to solve problems in physics, including calculus and several numerical methods (most notably Newton's method). James Clerk Maxwell, Lord Kelvin,George Gabriel Stokes, William Rowan Hamilton, and J. Willard Gibbs were mathematical physicists who had a profound impact on 19th century science. Revolutionary mathematical physicists at the turn of the 20th century included the mathematician David Hilbert who devised the theory of Hilbert spaces for integral equations which would find a major application in quantum mechanics. Paul Dirac used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron. Albert Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations, and his general relativity replaced the flat geometry of the large scale universe by that of a Riemannian manifold, whose curvature replaced Newton's gravitational force. Other prominent mathematical physicists include Carl Friedrich Gauss, Jules-Henri Poincaré, Richard Feynman, Roger Penrose, and Satyendra Nath Bose. Carl Friedrich Gauss is largely considered to be one of the three greatest mathematicians of all time. His influence in mathematical physics is largely felt by his developing of the mathematical field non-Euclidean Geometry, which Albert Einstein's General Theory of Relativity as well as our understanding of the event horizon in black holes rely so heavily on.

Mathematically rigorous physics

The term 'mathematical' physics is also sometimes used in a special sense, to distinguish 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 mathematics 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: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.

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. The above are just a few examples. An examination of the current research literature would undoubtedly give other such instances.

Notes

1. ^ Definition from the Journal of Mathematical Physics.[1]

Bibliographical references

The Classics

  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge University Press, 1927.
  • E. C. Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, 1939 (reprinted 1985).
  • John von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955.
  • Richard Courant and David Hilbert, Methods of Mathematical Physics. Vols. I and II. John Wiley & Sons, 1989.
  • Hermann Weyl, The Theory of Groups and Quantum Mechanics. 1931.
  • Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics. Parts I and II. McGraw Hill, 1959.
  • Tosio Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1995.
  • Barry Simon and Michael Reed, Methods of Modern Mathematical Physics. Vol. I: Functional Analysis, Academic Press, 1972; Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975; Vol. III: Scattering Theory, Academic Press, 1978; Vol. IV: Analysis of Operators, Academic Press, 1977.
  • Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, 1996.
  • James Glimm and Arthur Jaffe, Quantum Physics: A Functional Integral Point of View. Springer-Verlag, 1987.
  • Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, 1975.
  • Vladimir I. Arnold, Mathematical Methods of Classical Mechanics. Springer-Verlag, 1997.
  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qitative Theory of Dynamical Systems. Addison Wesley, 1994.
  • Walter Thirring, A Course in Mathematical Physics I-IV. Springer-Verlag, 1998.
  • Henry Margenau and George M. Murphy, The Mathematics of Physics and Chemistry. Van Nostrand Comp.

Textbooks for undergraduate studies

  • Sir Harold Jeffreys and Bertha Swirles (Lady Jeffreys), Methods of Mathematical Physics, third revised edition (Cambridge University Press, 1956 — reprinted 1999). ISBN 0-521-66402-0, ISBN 978-0-521-66402-8.
  • Eugene Butkov, Mathematical Physics. Addison Wesley, 1968.
  • Ivar Stakgold, Boundary Value Problems of Mathematical Physics. Vols. I and II. Macmillan, 1970.
  • Mary L. Boas, Mathematical Methods in the Physical Sciences. John Wiley & Sons, 3 ed., 2005.
  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists. Academic Press, 1995.
  • Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 2e, Addison-Wesley, 1970. ISBN 0-8053-7002-1

Other specialised subareas

  • Jamil Aslam and Faheem Hussain Mathematical Physics, Proceedings of the 12th Regional Conference, Islamabad, Pakistan, 27 March - 1 April 2006, World Scientific, Singapore, 2007. ISBN 978-981-270-591-4
  • P. Szekeres, A Course in Modern Mathematical Physics: Groups, Hilbert Space and differential geometry. Cambridge University Press, 2004.
  • J. Baez, Gauge Fields, Knots, and Gravity. World Scientific, 1994.
  • A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.
  • R. Geroch, Mathematical Physics. University of Chicago Press, 1985.

See also

External links

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Major fields of mathematics
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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This is a list of mathematics-based methods, by Wikipedia page.

See also list of graphical methods.
  • Adams' method (differential equations)
  • Akra-Bazzi method (asymptotic analysis)
  • Condorcet method (voting systems)
  • Coombs' method (voting systems)

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Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. Its central core is mathematical physics 1, though other conceptual techniques are also used.
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Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. Its central core is mathematical physics 1, though other conceptual techniques are also used.
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Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. It is often regarded as a subdiscipline of theoretical physics but some consider it an intermediate branch between
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In mathematics, a partial differential equation (PDE) is a type of differential equation, i. e. a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables.
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Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives.
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Fourier analysis, named after Joseph Fourier's introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. their frequencies) that can be recombined to obtain the original function.
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Potential theory may be defined as the study of harmonic functions.

Definition and comments

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied
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Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
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Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707(1707--)
Basel, Switzerland
Died September 18 [O.S.
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See also Lagrange and La Grange


LaGrange may refer to:

Places

United States

  • LaGrange, Arkansas (the United States Postal Service spells this town as La Grange)

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Hydrodynamics, also known as liquid-dynamics in limited academic circles, (literally, "water motion") is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. However, this distinction from fluid dynamics as a whole is not always fully observed.
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Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data.
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elastica theory is a theory of mechanics of solid materials developed by Euler that allows for very large scale elastic deflections of structures. Euler (1744) and Jakob Bernoulli developed the theory for elastic lines and studied buckling.
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Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering.
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Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on
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Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles.
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For the Daft Punk song, see .


Aerodynamics (shaping of objects that affect the flow of air or gas) is a branch of fluid dynamics concerned with the study of forces generated on a body in a flow.
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Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. Spectrometry is the measurement of these interactions and an instrument which performs such measurements is a spectrometer or
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
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In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in
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Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier
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special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies". Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and
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General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16.[1] It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational
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Group theory is the mathematical study of symmetry, as embodied in the structures known as groups. These are sets with a closed binary operation satisfying the following three properties:
  1. The operation must be associative.
  2. There must be an identity element.

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In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf.
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Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
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