2024/05/05 by Dey, Rukmini
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2405.02838
In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension n into \mathbb CPn and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of \mathbb CPn. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic n-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space \mathbb CPn. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on \mathbb CPn and we induce the quantization on the smooth compact embedded manifold from \mathbb CPn.