Wigner–Weyl transform

The Weyl quantization is a method in quantum mechanics to systematically map a reversible quantum mechanical Hermitian operator to a classical distribution in phase space. Therefore it is also called the phase space quantization.

The quantization of these underlying fundamental correspondence map of phase space functions on operators in Hilbert space is called a Weyl transformation. It was first described in 1927 by Hermann Weyl.

In contrast to Weyl's original intention to find a consistent quantization scheme, this figure is only a representation change. You must not combine classical and quantum mechanical parameters: the phase-space distribution may also depend on the Planck constant h. In some known cases that involve an angular momentum that is so.

The converse of this Weyl transform is the Wigner function. It forms from the Hilbert space into the phase space representation. This reversible change of representation allows to express quantum mechanics in phase space, as proposed in the 1940s by Groenewold and Moyal.

Example

The following are the Weyl transformation on the 2- dimensional Euclidean phase space is presented. The coordinates of the phase space are (q, p); Further, f is a function defined everywhere in phase space. The Weyl transform of f is given by the following operator in the Hilbert space (mostly analogous to the delta function ):

Now the P and Q operators as generators of a Lie algebra, the Heisenberg algebra are taken:

It is the reduced Planck's constant. A generic element of a Heisenberg algebra can be written as

The exponential of an element of a Lie algebra is then an element of the corresponding Lie group:

An element of the Heisenberg group. Given a specific group representation Φ of the Heisenberg group, denoted

The element of the corresponding representation of the group element g

The inverse of the above Weylfunktion is the Wigner function, which the operator Φ back to the phase space function f brings:

In general, the function f depends on the Planck constant h and can describe quantum mechanical processes well, provided it is properly assembled with the star product listed below.

For example, the Wigner a quantum mechanical operator for a square angular momentum (L ²) is not identical to the conventional operator, but additionally contains the term corresponding to the non-zero angular momentum of the ground state of the Bohr orbit.

Properties

The typical representation of a Heisenberg group is by the generators of its Lie algebra: A pair of self-adjoint ( Hermitian ) on a Hilbert space, so that their commutator, a central element of the group, the identity element in the Hilbert space results ( the canonical commutation relation )

The Hilbert space can as a set of square integrable functions on the real number line ( plane waves ) or a more limited quantity, such as the Schwartz space will be accepted. Depending on the involved area, followed by several properties:

  • If f is a real-valued function, then the image of the Weyl function Φ [ f] is self - adjoint.
  • If f is an element of the Schwartz space, then Φ [ f] is a trace class operator.
  • More generally, Φ [ f] unlimited densely defined operator.
  • For the standard representation of the Heisenberg group on the square integrable functions, corresponding to the Φ [ f] one-to-one the Schwartz space ( as a subspace of square integrable functions) function.

Generalizations

The Weyl quantization is studied in greater generality in cases where the phase space is a symplectic manifold, or possibly a Poisson manifold. Related structures are, for example, Poisson - Lie groups and the Kac -Moody algebras.

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