When Poisson and Moyal Brackets are equal? (original) (raw)
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An operator approach to Poisson brackets
Annals of Physics, 1976
The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.
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Communications in Mathematical Physics, 2011
In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.
In sections of the literature it is assumed, following early comments by Dirac, that a desirable quantization operation should preserve the Possion bracket (rather than merely agree in the limit h → 0). A celebrated mathematical theorem establishes that this is impossible; hence a consistent quantization is often considered to be unattainable. Here we question whether the premise of exact correspondence is sensible from a physical viewpoint. In particular, we give an exact quantum mechanical version of classical dynamics in which the commutator of any two quantized operators is proportional to the Poisson bracket of the corresponding functions (in a manner which subtly evades the aforementioned theorem). We then relate this novel dynamical system to the standard quantum theory, and identify the bracket structure for the case of symmetric ordering. Our conclusion is that a consistent quantization should not be expected to preserve the Poisson bracket.
Bracket Products for Weyl—Heisenberg Frames
Advances in Gabor Analysis, 2003
We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions (g n ) into a sequence (e n ) with the property that (E mb e n ) m,n∈Z is orthonormal in L 2 (R). Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions g ∈ L 2 (R) and ab = 1 so that the family (E mb T na g) is complete in L 2 (R). One consequence of this is that for functions g supported on a half-line [α, ∞) (in particular, for compactly supported g), (g, 1, 1) is complete if and only if sup 0≤t<a |g(t − n)| = 0 a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any g ∈ L 2 (R), A ≤ n |g(t − na)| 2 ≤ B is equivalent to (E m/a g) being a Riesz basic sequence.
A Poisson bracket on multisymplectic phase space
Reports on Mathematical Physics, 2001
A new Poisson bracket for Hamiltonian forms on the full multisymplectic phase space is defined. At least for forms of degree n − 1, where n is the dimension of space-time, Jacobi's identity is fulfilled.
Banach Journal of Mathematical Analysis, 2011
The relevance of modulation spaces for deformation quantization, Landau-Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that Q s (R 2d ) is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces M p vs (R 2d ) are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class Q s (R 2d ) is the modulation space M 2 vs (R 2d ). The main result gives estimates of the singular values of pseudodifferential operators with symbols in M p vs (R 2d ) for the standard Weyl calculus and for the Landau-Weyl calculus.
Poisson Brackets on Lie Algebras
1995
Classi cation of a Poisson bivectors on some Lie algebras 3.1 Poisson bivectors on a compact Lie algebra 3.2 Poisson bivectors and closed 2-forms on elementary (K ahler) Lie algebras 4 Poisson bivectors subordinated to a closed 2-form and groups acting simply transitively on orbits 4.1 Relations between Poisson bivectors and closed 2-forms on a Lie algebra 4.2 Poisson bivectors subordinate d to a closed 2-form 4.3 Case of exact 2-form. Frobenius Lie algebras 4.4 Case of semi-simple Lie algebra. Problem of classi cation of almost simply transitive Lie subgroups of G 4.5 Iwasawa decomposition for SO(p; 2) and the canonical symplectic form on the Iwasawa subalgebra 5 Poisson bivectors associated to a Hermitian symmetric domains for the function X(;) taking the values in G G, where G is a Lie algebra. In order to de ne the quantity X 12 (1 ; 2), following BD 1982], we x an associative algebra A with unit, which contains G and the linear maps ' 12 ; ' 23 and ' 13 , so that ' 12 : G G ! A A A; ' 12 (a b) = a b 1 (0:2) and analogously for maps ' 23 and ' 13. Note that if X(;) is a solution of eq. (0.1) and '(u) is a function with values in G, thenX(;) = ('() '())X(;) is also a solution of (0.1) and we will consider the solutions X andX as equivalent. Let us introduce the following de nition. De nition 0.1. The function X(;) is invariant relative to g 2 Aut G, if (g g)X(;) = X(;): The set of all such g forms the group that is called the invariance group of X(;). The function X(;) is said to be invariant with respect to h 2 G if h 2 1 + 1 h; X(;)] = 0; i.e. if it is invariant relative to expfadhg for any t. Note that if X(;) is a solution of (0.1), which is invariant relative to the subalgebra
5 Extended Weyl Calculus and Application to the
2015
We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.