Weyl quantization

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In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian operator with a "classical" distribution in phase space. The crucial map from phase-space functions to Hilbert space operators underlying the method was first described by Hermann Weyl in 1927. Formally, however, this map merely amounts to a change of representation, and needs not connect "classical" with "quantum" quantities--the starting phase-space distribution may well depend on Planck's constant $\hbar$ , and in some familiar cases involving angular momentum it does.

1 Example
2 Properties
3 Deformation quantization
4 Generalizations
5 References

[edit] Example

The following demonstrates Weyl quantization on a simple, two-dimensional Euclidean phase space. Let the coordinates on phase space be $(q, p)$ and let f be a function defined everywhere on phase space. The Fourier transform of f is given by

$\widehat{f}(a,b) = \frac {1}{(2\pi)^2} \int \int f(q,p) e^{-i(aq+bp)} dq\, dp$

The associated Weyl map operator in Hilbert space is

$\Phi(f) = \int\int\widehat{f}(a,b) \Phi\left(e^{i(aQ+bP)}\right) da\, db$

Here, P and Q are taken to be the generators of a Lie algebra, the Heisenberg algebra:

$[P,Q]=PQ-QP=-i\hbar\,$

where $\hbar$ is the reduced Planck constant. A general element of the Heisenberg algebra may thus be written as

$aQ+bP-i\hbar z$

The exponential map of an element of a Lie algebra is then an element of the corresponding Lie group. Thus,

$g=e^{aQ+bP-i\hbar z}$

is an element of the Heisenberg group. Given some particular group representation $Φ$ of the Heisenberg group, the quantity

$\Phi\left( e^{aQ+bP-i\hbar z} \right)$

denotes the element of the representation corresponding to the group element g.

[edit] Properties

Typically, the standard quantum mechanical representation of the Heisenberg group is as a pair of self-adjoint (Hermitian) operators on some Hilbert space $\mathcal{H}$ , such that the their commutator is the identity on the Hilbert space:

$[P,Q]=PQ-QP=-i\hbar\, \operatorname{Id}_\mathcal{H}$

The Hilbert space may taken to be the set of square integrable functions on the real number line (the plane waves), or a more bounded set, such as Schwartz space. Depending on the space, various results follow:

If f is a real-valued function, then $Φ(f)$ is self-adjoint.

If f is an element of Schwartz space, then $Φ(f)$ is trace-class.

More generally, $Φ(f)$ is a densely defined unbounded operator.

For the standard representation of the Heisenberg group by square integrable functions, the map Φ is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).

[edit] Deformation quantization

In the context of the above flat phase space example, the star product (Moyal product, introduced by Groenewold in 1946) $* h$ of a pair of functions in $f_1,f_2 \in C^\infty(\mathbb{R}^2)$ is specified by

$\Phi(f_1 *_h f_2) = \Phi(f_1)\Phi(f_2)\,$

The star product is not commutative in general, but goes over to the ordinary commutative product of functions in the limit of $h\to 0$ . As such, it is said to define a deformation of the commutative algebra of $C^\infty(\mathbb{R}^2)$ .

Insofar as the algebra of functions on a space determines the geometry of that space, the study of the star product leads to the study of a non-commutative geometry deformation of that space.

For the Weyl-map example above, the star product may be written in terms of the Poisson bracket as

$f_1 *_h f_2 = \sum_{n=0}^\infty \frac {1}{n!} \left(\frac{i\hbar}{2} \right)^n \Pi^n(f_1, f_2)$

Here, Π is an operator defined such that its powers are

Π 0 (f 1, f 2) = f 1 f 2

and

$\Pi^1(f_1,f_2)=\{f_1,f_2\}= \frac{\partial f_1}{\partial q} \frac{\partial f_2}{\partial p} - \frac{\partial f_1}{\partial p} \frac{\partial f_2}{\partial q}$

where ${f 1, f 2}$ is the Poisson bracket and, more generally,

$\Pi^n(f_1,f_2)= \sum_{k=0}^n (-1)^k {n \choose k} \left( \frac{\partial^k }{\partial p^k} \frac{\partial^{n-k}}{\partial q^{n-k}} f_1 \right) \times \left( \frac{\partial^{n-k} }{\partial p^{n-k}} \frac{\partial^k}{\partial q^k} f_2 \right)$

where ${n \choose k}$ is the binomial coefficient. Antisymmetrization of this star product leads to the Moyal bracket.

[edit] Generalizations

More generally, Weyl quantization is studied in the case where the phase space is a symplectic manifold or possibly a Poisson manifold. Structures include the Poisson-Lie groups and Kac-Moody algebras.

[edit] References

H.Weyl, "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1-46.
H.J. Groenewold, "On the Principles of elementary quantum mechanics",Physica,12 (1946) pp. 405-460.
J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99-124.
F.Bayen, M.Flato, C.Fronsdal, A.Lichnerowicz and D.Sternheimer, "Deformation theory and quantization I, II",Ann. Phys. (N.Y.),111 (1978) pp. 61-110,111-151.
C. Zachos, D. Fairlie, and T. Curtright, "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005).