Weyl Quantization - Example

Example

The following illustrates the Weyl transformation on the simplest, 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 Weyl transform of the function f is given by the following operator in Hilbert space, broadly analogous to a Dirac delta function,

\Phi = \frac{1}{(2\pi)^2}\iint\!\!\! \iint f(q,p) \left(e^{i(a(Q-q) +b(P-p))}\right) \text{d}q\, \text{d}p\, \text{d}a\, \text{d}b.

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

where ħ is the reduced Planck constant. A general element of the Heisenberg algebra may thus be written as aQ+bP+c .

The exponential map of this element of the Lie algebra is then an element of the corresponding Lie group,

the Heisenberg group. Given some particular group representation Φ of the Heisenberg group, the operator

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

This Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,

The inverse of the above Weyl map is the Wigner map, which takes the operator Φ back to the original phase-space kernel function f,

In general, the resulting function f depends on Planck's constant ħ, and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below.

For example, the Wigner map of the quantum angular-momentum-squared operator L2 is not just the classical angular momentum squared, but it further contains an offset term − 3ħ2/2, which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.

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