Stone–von Neumann theorem (guaranteeing uniqueness of the canonical commutation relations) holds. The 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. It is instructive to perform the and integrals in the above formula first, which has the effect of computing the ordinary Fourier transform of the function , while leaving the operator . In that case, the Weyl transform can be written as . We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function , but then when applying the Fourier inversion formula, we substitute the quantum operators and for the original classical variables and , thus obtaining a "quantum version of ." A less symmetric form, but handy for applications, is the following, In the position representation The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator, The inverse map The inverse of the above Weyl map is the Wigner map, which takes the operator back to the original phase-space kernel function , For example, the Wigner map of the oscillator thermal distribution operator is If one replaces in the above expression with an arbitrary operator, the resulting function may depend on Planck's 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, The 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 and . In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators and . 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 and is completely determined by the following symmetric formula: for all complex numbers and . From this formula, it is not hard to show that