Deformation 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 invertibly. The crucial map from phase-space functions to Hilbert-space operators underlying the method is called the Weyl transformation, (not to be confused with another definition of the Weyl transformation), and was first detailed by Hermann Weyl[1] in 1927.

Formally, however, in some contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation, and need 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.

The inverse of this Weyl transformation is the Wigner map, which reverts from Hilbert space to the phase-space representation, (cf. the Wigner quasi-probability distribution, which is the Wigner map of the quantum density matrix). This invertible representation change then allows expressing quantum mechanics in phase space, as was appreciated in the 1940s by Groenewold [2] and Moyal [3] in the 1940s.

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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 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 then

\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:

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

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.

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

f(x,p)= 2 \int_{-\infty}^{\infty}dy~e^{2ipy/\hbar}~ \langle x-y| \Phi |x+y \rangle .

In general, the resulting function f depends on Planck's constant \hbar, and may well describe quantum mechanical processes, provided it is properly composed through the star product, below [4].

For example, the Wigner map of the quantum angular-momentum-squared operator is not just the classical angular momentum squared, but it further contains a term -3\hbar^2 /2 , which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.

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 be 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.
  • 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).

Deformation quantization

In the context of the above flat phase-space example, the star product (Moyal product, actually 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

Π0(f1,f2) = f1f2

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 {f1,f2} 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 yields the Moyal bracket, the proper quantum deformation of the Poisson bracket, and the phase-space isomorph of the quantum commutator in the more usual Hilbert-space formulation of quantum mechanics.

There results a complete phase-space representation of quantum mechanics, completely equivalent to the Hilbert-space operator representation, with star multiplications paralleling operator multiplications isomorphically [5].

Expectation values in phase-space quantization are obtained isomorphically to tracing operator observables Φ with the density matrix in Hilbert space: they are obtained by phase-space integrals of observables such as the above f with the Wigner quasi-probability distribution effectively serving as a measure.

Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), the above Weyl map makes it easy to recognize quantum mechanics as a deformation (generalization) of classical mechanics, with deformation parameter \hbar /S. (Other familiar deformations in physics involve the deformation of classical Newtonian into relativistic mechanics, with deformation parameter v/c; or the deformation of Newtonian gravity into General Relativity, with deformation parameter Scwarzschild-radius/characteristic-dimension.)

Classical expressions, observables, and operations (such as Poisson brackets) are modified by h-dependent quantum corrections, and the conventional commutative multiplication utilized in classical mechanics is generalized to the noncommutative star-multiplication characterizing quantum mechanics and underlying its uncertainty principle.


Generalizations

In more generality, Weyl quantization is studied in cases where the phase space is a symplectic manifold, or possibly a Poisson manifold. Related structures include the Poisson-Lie groups and Kac-Moody algebras.


See also

References

  1. ^ H.Weyl , "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1-46.
  2. ^ H.J. Groenewold, "On the Principles of elementary quantum mechanics",Physica,12 (1946) pp. 405-460.
  3. ^ J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99-124.
  4. ^ R. Kubo, "Wigner Representation of Quantum Operators and Its Applications to Electrons in a Magnetic Field", Jou. Phys. Soc. Japan,19 (1964) pp. 2127-2139.
  5. ^ C. Zachos, D. Fairlie, and T. Curtright, "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .
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