Weyl quantization of functions on $\mathbb R^{2d}$ motivates the extension of harmonic analysis to quantum harmonic analysis. Klaas Landsman has generalized Weyl quantization to an arbitrary Riemannian manifold $M$ (in the place of $\mathbb R^d$), where functions on $T^*M$ (the cotangent bundle of $M$) become operators on $L^2(M)$. This is the natural generalization of Weyl quantization from the point of view of physics.
In this talk, we will see that many of the correspondences that underlie quantum harmonic analysis have analogues in the setting where $\mathbb R^d$ is replaced by a Riemannian manifold $M$. The proper formulation of these correspondences makes contact with a number of fascinating modern mathematical constructions and frameworks, such as Connes' tangent groupoid, continuous fields of C$^*$-algebras, Kirillov's orbit method and infinite dimensional Lie groups. Moreover, this framework leads us to a novel notion of Fourier transform for functions on $T^*M$, as well as a Fourier-Wigner transform for operators on $L^2(M)$—a potential starting point for a theory of quantum harmonic analysis on Riemmanian manifolds.