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In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. 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.
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.
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.
The Wigner quasiprobability distribution is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space.
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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.
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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 Feynman Lectures on Physics is a physics textbook based on a great number of lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the California Institute of Technology (Caltech), during 1961–1963. The book's co-authors are Feynman, Robert B. Leighton, and Matthew Sands.
The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum representations. 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/
- ↑ 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
