Minggu, 18 Desember 2011

Mathematically Rigorous Physics



The term 'mathematical' physics is also sometimes used in a special sense, to denote 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. This also has an institutional side: Many mathematical physicists are members of mathematics departments.


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 four 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.
  4. quantum information theory
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.

Minggu, 11 Desember 2011

Mathematical Physics



Mathematical physics refers to development of mathematical methods for application to problems in physics.

Scope of the subject


The Journal of Mathematical Physics defines this area 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.". There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equations (and the related areas of variational calculusFourier analysispotential theory, and vector analysis) are perhaps most closely associated with mathematical physics.

These were developed intensively from the second half of the eighteenth century (by, for example, D'AlembertEuler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamicscelestial mechanicselasticity theoryacoustics,thermodynamicselectricitymagnetism, and aerodynamics.

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, thespectral theory of operators, and more broadly, functional analysis. These constitute the mathematical basis of another branch of mathematical physics.

The special and general theories of relativity require a rather different type of mathematics. This was group theory: and it played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology in the mathematical description of cosmological as well as quantum field theory phenomena.

Statistical mechanics forms a separate field, which 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.

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 are. For example, ordinary differential equationsand symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.




Mathematical Methods for Physicists: A concise introduction



Author: TAI L. CHOW


was born and raised in China. He received a BS degree in physics from the National Taiwan University, a Masters degree in physics from Case Western Reserve University, and a PhD in physics from the University of Rochester. Since 1969, Dr Chow has been in the Department of Physics at California State University, Stanislaus, and served as department chairman for 17 years, until 1992. He served as Visiting Professor of Physics at University of California (at Davis and Berkeley) during his sabbatical years. He also worked as Summer Faculty Research Fellow at Stanford University and at NASA. Dr Chow has published more than 35 articles in physics journals and is the author of two textbooks and a solutions manual.

Sumber:

1. Wikipedia
2. Google book

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