Weyl Quantization from geometric quantization

Abstract

In [1] a nice looking formula is conjectured for a deformed product of functions on a symplectic manifold in case it concerns a hermitian symmetric space of non‐compact type. We derive such a formula for simply connected symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2‐dimensional symplectic manifolds R2, H2, and S2. For R2 we obtain the well known Moyal‐Weyl product. The other cases show that the original idea in [1] should be interpreted with care.