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Quantization is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. It is a procedure for constructing quantum mechanics from classical mechanics. A generalization involving infinite degrees of freedom is field quantization, as in the "quantization of the electromagnetic field", referring to photons as field "quanta". This procedure is basic to theories of atomic physics, chemistry, particle physics, nuclear physics, condensed matter physics, and quantum optics.
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.
In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian.
David Louis Goodstein was an American physicist and educator. From 1988 to 2007 he served as Vice-provost of the California Institute of Technology (Caltech), where he was also a professor of physics and applied physics, as well as the Frank J. Gilloon Distinguished Teaching and Service Professor.
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In mathematics, the Moyal product is an example of a phase-space star product. It is an associative, non-commutative product, ★, on the functions on , equipped with its Poisson bracket. It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra.
George Frederick Chapline Jr. is an American theoretical physicist, based at the Lawrence Livermore National Laboratory. His most recent interests have mainly been in quantum information theory, condensed matter, and quantum gravity. In 2003 he received the Computing Anticipatory Systems award for a new interpretation of quantum mechanics based on the similarity of quantum mechanics and Helmholtz machines. He was awarded the E. O. Lawrence Award in 1982 by the United States Department of Energy for leading the team that first demonstrated a working X-ray laser.
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José Enrique Moyal was an Australian mathematician and mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields.
The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possible Lorentz invariant quantization of string theory in 4-dimensional Minkowski space-time.
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
Usually non-critical string theory is considered in frames of the approach proposed by Polyakov. The other approach has been developed in. It represents a universal method to maintain explicit Lorentz invariance in any quantum relativistic theory. On an example of Nambu-Goto string theory in 4-dimensional Minkowski space-time the idea can be demonstrated as follows:
The Husimi Q representation, introduced by Kôdi Husimi in 1940, is a quasiprobability distribution commonly used in quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It is used in the field of quantum optics and particularly for tomographic purposes. It is also applied in the study of quantum effects in superconductors.
Cosmas K. Zachos is a theoretical physicist. He was educated in physics at Princeton University, and did graduate work in theoretical physics at the California Institute of Technology under the supervision of John Henry Schwarz.
Hilbrand Johannes "Hip" Groenewold (1910–1996) was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization.
The phase-space formulation is a formulation of quantum mechanics that places the position and momentum variables on equal footing in phase space. The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution and operator multiplication is replaced by a star product.
David B. Fairlie is a British mathematician and theoretical physicist, Professor Emeritus at the University of Durham (UK).
References
- ↑ Professor Curtright's physics publications are available on the INSPIRE Database and the GoogleCite database .
- ↑ Curtright, T. (1977). "Conformal spinor current anomalies". Physics Letters B. 71 (1): 185–188. Bibcode:1977PhLB...71..185C. doi:10.1016/0370-2693(77)90773-0.
- ↑ Curtright, T.; Thorn, C. (1982). "Conformally Invariant Quantization of the Liouville Theory". Physical Review Letters. 48 (19): 1309. Bibcode:1982PhRvL..48.1309C. doi:10.1103/PhysRevLett.48.1309.
- ↑ http://worldcat.org/identities/lccn-n92082148/ [ bare URL ]
- ↑ Cosmas K. Zachos, David B. Fairlie, and Thomas L. Curtright, Quantum Mechanics in Phase Space, (World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .
- ↑ Thomas L Curtright, David B Fairlie, Cosmas K Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, (World Scientific, Singapore, 2014) ISBN 9789814520430
