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 technique was first described by Hermann Weyl in 1927.

Example

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The following demonstrates Weyl quantization on a simple, two-dimensional Euclidean phase space. Let the cordinates 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

$\backslash widehat\{f\}(a,b)\; =\; \backslash frac\; \{1\}\{(2\backslash pi)^2\}$

\int \int f(q,p) e^{-i(aq+bp)} dq\, dp

The associated Weyl operator is

$\backslash Phi(f)\; =\; \backslash int\backslash int\backslash widehat\{f\}(a,b)$

\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:

$*=PQ-QP=-i\backslash hbar\backslash ,$

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

$aQ+bP-i\backslash hbar\; z$

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

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

is an element of the Heisenberg group. Given some particular group representation $\backslash Phi$ of the Heisenberg group, the quantity

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

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

Properties

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

$*=PQ-QP=-i\backslash hbar\backslash ,\; \backslash operatorname\{Id\}\_\backslash mathcal\{H\}$

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 $\backslash Phi(f)$ is self-adjoint.

• If f is an element of Schwartz space, then $\backslash Phi(f)$ is trace-class.

• More generally, $\backslash Phi(f)$ 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).

Deformation quantization

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In the context of the above example, the Moyal product introduced by Groenewold in 1946, $*\_h$ of a pair of functions in $f\_1,f\_2\; \backslash in\; C^\backslash infty(\backslash mathbb\{R\}^2)$ is given by

$\backslash Phi(f\_1\; *\_h\; f\_2)\; =\; \backslash Phi(f\_1)\backslash Phi(f\_2)\backslash ,$

The product is not commutative in general, but goes over to the ordinary commutative product of functions in the limit of $h\backslash to\; 0$. As such, the Moyal product is said to define a deformation of the commutative algebra of $C^\backslash infty(\backslash mathbb\{R\}^2)$. Insofar as the algebra of functions on a space determines the geometry of that space, the study of the Moyal product leads to the study of non-commutative geometry.

For the example above, the Moyal product may be written in terms of the Poisson bracket as

$f\_1\; *\_h\; f\_2\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\; \{1\}\{n!\}\; \backslash left(\backslash frac\{-i\backslash hbar\}\{2\}\; \backslash right)^n\; \backslash Pi^n(f\_1,\; f\_2)$

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

$\backslash Pi^0(f\_1,f\_2)=f\_1f\_2$

and

$\backslash Pi^1(f\_1,f\_2)=\backslash \{f\_1,f\_2\backslash \}=$

\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 $\backslash \{f\_1,f\_2\backslash \}$ is the Poisson bracket and, more generally,

$\backslash Pi^n(f\_1,f\_2)=\; (-1)^n$

\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 $\{n\; \backslash choose\; k\}$ is the binomial coefficient.

Generalizations

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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.

Historical References

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• H.Weyl, "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1-46.
• 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.
• H.J. Groenewold, "On the Principles of elementary quantum mechanics",Physica,12 (1946) pp. 405-460.

Mathematical quantization