Hilbert Space Quantization

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

西岡, 道夫

By introducing a complex phase space into classical mechanical system, we present three models each of which bu means of geometric quantization naturally introduces the Bargmann Hilbert space as Hilbert space to represent the corresponding quantum …

Rukmini Dey

In this paper, we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is as follows: we embed the smooth manifold of real dimension [Formula: see text] into [Formula: see text] 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 [Formula: see text]. In the Odzijewicz-type quantization, one has to consider a tensor product of the geometric quantization line bundle with holomorphic [Formula: see text]-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space [Formula: see text]. 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 [Formula: see text] and we induce the quantization on the smooth compact embedded manifold from [Formula: see text].

A. B. Govorkov

H. S. Sharatchandra

Quantization of $R^2$ and $S^1 \times S^1$ phase spaces are explicitly carried out tweaking the techniques of geometric quantization. Crucial is a combined use of left and right invariant vector fields. Canonical bases, operators and their actions are explicitly presented. Arguments of Dirac and also Wu and Yang for monopoles are applied for obtaining the quantization of the phase space area. Equivalence of states in the infinite dimensional prequantum Hilbert space resulting in a physical Hilbert space of dimension equal to the phase space area in units of the Planck constant is demonstrated. These techniques can be applied to any manifold with a symplectic structure.