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Geometric Quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. Origins One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field 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". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry and conserved quantities during the dynamical evoluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertram Kostant
Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945. He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups. Career in mathematics After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coadjoint Orbit
In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G, which again may be complicated, while the orbits are relatively tractable. Formal definition Let G be a Lie group and \mathfrak be its Lie algebra. Let \mathrm : G \rightarrow \mathrm(\mathfrak) denote the adjoint re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-autonomous Mechanics
Non-autonomous mechanics describe non- relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle Q\to \mathbb R over the time axis \mathbb R coordinated by (t,q^i). This bundle is trivial, but its different trivializations Q=\mathbb R\times M correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection \Gamma on Q\to\mathbb R which takes a form \Gamma^i =0 with respect to this trivialization. The corresponding covariant differential (q^i_t-\Gamma^i)\partial_i determines the relative velocity with respect to a reference frame \Gamma. As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamilton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superintegrable Hamiltonian System
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2n-dimensional symplectic manifold for which the following conditions hold: (i) There exist k>n independent integrals F_i of motion. Their level surfaces (invariant submanifolds) form a fibered manifold F:Z\to N=F(Z) over a connected open subset N\subset\mathbb R^k. (ii) There exist smooth real functions s_ on N such that the Poisson bracket of integrals of motion reads \= s_\circ F. (iii) The matrix function s_ is of constant corank m=2n-k on N. If k=n, this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows. Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold F is a fiber bundle in tori T^m. There exists a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal–Bargmann Space
In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integrability condition: :\, F\, ^2 := \pi^ \int_ , F(z), ^2 \exp(-, z, ^2)\,dz < \infty, where here ''dz'' denotes the 2''n''-dimensional Lebesgue measure on It is a Hilbert space with respect to the associated inner product: : The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see and . Basic information about the material in this section may be found in and . Segal worked from the beginning in the infinite-dimensional setting; see and Section 10 of for more information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase-space Formulation
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 (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner. The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Manifold
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalently, \ defines a Lie algebra structure on the vector space (M) of smooth functions on M such that X_:= \: (M) \to (M) is a vector field for each smooth function f (making (M) into a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
