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quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the Wigner–Weyl transform or Weyl–Wigner transform (after
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
and
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
) is the invertible mapping between functions in the quantum phase space formulation and
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
operators in the Schrödinger picture. Often the mapping from functions on phase space to operators is called the Weyl transform or Weyl quantization, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized ''classical'' phase space functions to operators, a procedure known as ''Weyl quantization''. It is now understood that Weyl quantization does not satisfy all the properties one would require for consistent quantization and therefore sometimes yields unphysical answers. On the other hand, some of the nice properties described below suggest that if one seeks a single consistent procedure mapping functions on the classical phase space to operators, the Weyl quantization is the best option: a sort of
normal coordinates In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a torsion tensor, symmetric affine connection are a local coordinate system in a neighborhood (mathematics), neighborhood of ''p'' obtained by ...
of such maps. ( Groenewold's theorem asserts that no such map can have all the ideal properties one would desire.) Regardless, the Weyl–Wigner transform is a well-defined integral transform between the phase-space and operator representations, and yields insight into the workings of quantum mechanics. Most importantly, the Wigner quasi-probability distribution is the Wigner transform of the quantum
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
, and, conversely, the density matrix is the Weyl transform of the Wigner function. In contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation within quantum mechanics; it need not connect "classical" with "quantum" quantities. For example, the phase-space function may depend explicitly on the reduced Planck constant ''ħ'', as it does in some familiar cases involving angular momentum. This invertible representation change then allows one to express quantum mechanics in phase space, as was appreciated in the 1940s by Hilbrand J. Groenewold and José Enrique Moyal. In more generality, Weyl quantization is studied in cases where the phase space is a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
, or possibly a Poisson manifold. Related structures include the Poisson–Lie groups and Kac–Moody algebras.


Definition of the Weyl quantization of a general observable

The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be , and let be a function defined everywhere on phase space. In what follows, we fix operators ''P'' and ''Q'' satisfying the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p ...
s, such as the usual position and momentum operators in the Schrödinger representation. We assume that the exponentiated operators e^ and e^ constitute an irreducible representation of the Weyl relations, so that the Stone–von Neumann theorem (guaranteeing uniqueness of the canonical commutation relations) holds.


Basic formula

The Weyl transform (or Weyl quantization) of the function is given by the following operator in Hilbert space, Throughout, ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. It is instructive to perform the and integrals in the above formula first, which has the effect of computing the ordinary Fourier transform \tilde of the function , while leaving the operator e^. In that case, the Weyl transform can be written as :\Phi = \frac\iint\tilde(a,b)e^\,da\,db. We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function f(p,q), but then when applying the Fourier inversion formula, we substitute the quantum operators P and Q for the original classical variables and , thus obtaining a "quantum version of ." A less symmetric form, but handy for applications, is the following, : \Phi \frac\iint \!\!\!\iint\!\! dq\, dp\, d\tilde \, d\tilde \ e^~ f(q,p) ~ , \tilde\rangle\langle \tilde, .


In the position representation

The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator, : \langle x, \Phi , y \rangle = \int_^\infty ~e^~ f\left(,p\right) .


Inverse map

The inverse of the above Weyl map is the Wigner map (or Wigner transform), which was introduced by Eugene Wigner, which takes the operator back to the original phase-space kernel function , For example, the Wigner map of the oscillator thermal distribution operator \exp (-\beta (P^2+Q^2)/2) is : \exp_\star \left (-\beta (p^2+q^2)/2 \right )= \left ( \cosh\left(\frac\right)\right ) ^ \exp\left ( \frac \tanh\left(\frac\right) (p^2+q^2)/2\right ) . If one replaces \Phi /math> in the above expression with an arbitrary operator, the resulting function may depend on the reduced Planck constant , and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below. In turn, the Weyl map of the Wigner map is summarized by ''Groenewold's formula'', : \Phi = h \iint \,da\,db ~e^ \operatorname ( e^ \Phi).


Weyl quantization of polynomial observables

While the above formulas give a nice understanding of the Weyl quantization of a very general observable on phase space, they are not very convenient for computing on simple observables, such as those that are polynomials in q and p. In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators Q and P. 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 , which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.


Properties


Weyl quantization of polynomials

The action of the Weyl quantization on polynomial functions of q and p is completely determined by the following symmetric formula: : (aq+bp)^n\longmapsto (aQ+bP)^n for all complex numbers a and b. From this formula, it is not hard to show that the Weyl quantization on a function of the form q^k p^l gives the average of all possible orderings of k factors of Q and l factors of P:\prod_^N \xi_ ~~ \longmapsto ~~ \frac \sum_ \prod_^N \Xi_ where \xi_j = q_j, \xi_ = p_j, and S_N is the set of permutations on ''N'' elements. For example, we have : 6 p^2 q^2 ~~ \longmapsto ~~ P^2 Q^2 + Q^2 P^2 + PQPQ+PQ^2P+QPQP+QP^2Q. While this result is conceptually natural, it is not convenient for computations when k and l are large. In such cases, we can use instead McCoy's formulaMcCoy, Neal (1932). "On the Function in Quantum Mechanics which Corresponds to a Given Function in Classical Mechanics", ''Proc Nat Acad Sci USA'' 19 674
online
.
: p^m q^n ~~ \longmapsto ~~ \sum_^ Q^r P^m Q^=\sum_^ P^s Q^P^. This expression gives an apparently different answer for the case of p^2 q^2 from the totally symmetric expression above. There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator. (The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of p^2q^2 in terms of the operators P^2Q^2, QP^2Q, and Q^2P^2 and verify the first expression in McCoy's formula with m=n=2.) It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side. (An exact correspondence is impossible, in light of Groenewold's theorem.) For example, Moyal showed the : Theorem: If f(q,p) is a polynomial of degree at most 2 and g(q,p) is an arbitrary polynomial, then we have \Phi(\)=\frac Phi(f),\Phi(g)/math>.


Weyl quantization of general functions

* If is a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
, then its Weyl-map image is
self-adjoint In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...
. * If is an element of Schwartz space, then is trace-class. * More generally, is a densely defined unbounded operator. * The map is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).


See also

*
Canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p ...
* Deformation quantization *
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b' ...
* Moyal bracket * Weyl algebra *
Functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
* Pseudo-differential operator * Wigner quasi-probability distribution * Stone–von Neumann theorem * Phase space formulation of quantum mechanics * Kontsevich quantization formula * Gabor–Wigner transform * Oscillator representation


References

*


Further reading

* (Sections I to IV of this article provide an overview over the ''Wigner–Weyl transform'', the Wigner quasiprobability distribution, the phase space formulation of quantum mechanics and the example of the
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.) * * Terence Tao's 201
notes on Weyl ordering
{{DEFAULTSORT:Wigner-Weyl transform Mathematical quantization Mathematical physics Foundational quantum physics Concepts in physics