Coherent states and geometric quantization

Abstract

Based on the concept of generalized coherent states, a theory of mechanical systems is formulated in a way which naturally exhibits the mutual relation of classical and quantum aspects of physical phenomena.

Quantization Methods: A Guide for Physicists and Analysts

“…(More precisely, Kontsevich's formula is an expansion of a certain Feynman integral at a saddle point, see Cattaneo and Felder [60].) Connections between Feynman path integrals, coherent states, and the Berezin quantization are discussed in Kochetov and Yarunin [152], Odzijewicz [189], Horowski, Kryszen and Odzijewicz [130], Klauder [149], Chapter V in Berezin and Shubin [33], Marinov [168], Charles [61], and Bodmann [41]. For a discussion of Feynman path integrals in the context of geometric quantization, see Gawedzki [98], Wiegmann [258], and Chapter 9 in the book of Woodhouse [263].…”

“…Thus, in some sense, the quantizers ∆ y above and their associated reproducing kernel Hilbert spaces may be regarded as generalizations of the coherent states to the situation when there is no group action present. For more information on coherent states and their applications in quantization, see for instance Klauder [149], Odzijewicz [189], Unterberger [251], Ali and Goldin [10], Antoine and Ali [6], Ali [4], Bartlett, Rowe and Repka [220], and the survey by Ali, Antoine, Gazeau and Mueller [7], as well as the recent book [8], and the references therein. An interesting characterization of the cut locus of a compact homogeneous Kähler manifold in terms of orthogonality of coherent states has recently been given by Berceanu [30].…”

“…Later on this connection was examined in detail in a series of papers by Cahen [55] and Cahen, Gutt and Rawnsley [56] (parts I and II of [56] deal with compact manifolds, part III with the unit disc, and part IV with homogeneous spaces). See also Nishioka [187] and Odzijewicz [189].…”

Bergman Spaces and Kernels for Holomorphic Vector Bundles

“…Recently reproducing kernels of Bergman type have been used to define the quantization of classical states in holomorphic models of quantum field theory (see [11], [7] or [8]). Earlier they have been appeard in studies of wave and Dirac equations (see [2]).…”

“…spaces of sections of trivial bundles over the open domains in C™. In holomorphic models of quantum field theory however ( [11] and [8]) the holomorphic sections of non-trivial line bundles over the Káhler manifold have been appeard and reproducing kernels for such spaces have been used. The purpose of this paper is to introduce reproducing kernels in the case of an arbitrary holomorphic vector bundle and to prove basic properties of these kernels.…”