Abstract
Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.
Highlights
A Historical OverviewMore than one century after the publication by Fourier of his “Théorie analytique de la chaleur” [1,2], the Fourier transform revealed its tremendous importance at the advent of quantum mechanics with the setting of its specific formalism, especially with the seminal contributions of Weyl (1927) [3] on phase space symmetry, and Wigner (1932) [4] on phase space distribution
In the continuation of the procedure exposed in the previous sections, we investigate special cases of affine covariant integral quantization that leads to remarkable properties
Through the above specifications of covariant integral quantization, in their Wigner-Weyl-like restrictions, to two basic cases, the euclidean plane with its translational symmetry on one hand, the open half-plane with its affine symmetry on the other hand, we have provided an illustration of the crucial role of the Fourier transform, which is needed at each step of the calculations
Summary
More than one century after the publication by Fourier of his “Théorie analytique de la chaleur” [1,2], the Fourier transform revealed its tremendous importance at the advent of quantum mechanics with the setting of its specific formalism, especially with the seminal contributions of Weyl (1927) [3] on phase space symmetry, and Wigner (1932) [4] on phase space distribution The phase space they were concerned with is essentially the Euclidean plane R2 = {(q, p) , q, p, ∈ R}, q (mathematicians prefer to use x) for position and p for momentum. We observe that the Fourier transform lies at the heart of the above interplay of Weyl and Wigner approaches Please note that both the maps (46) and (5) allow one to set up a quantum mechanics in phase space, as was developed at a larger extent in the 1940s by Groenewold [7] and. Detailed proofs of two of our results are given in Appendix A
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