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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 $n$ into $ℂP^n$ 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 $ℂP^n$. 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 $ℂP^n$. 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 $ℂP^n$ and we induce the quantization on the smooth compact embedded manifold from $ℂP^n$.

This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball as the sum of the Kähler two-form on ℂⁿ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a hermitian linear Hamiltonian in the generators of the Jacobi group are described by a matrix Riccati equation on and a linear first-order differential equation in z ∈ ℂⁿ, with coefficients depending also on . H_n denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on . When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane , where denotes the Siegel upper half plane.

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain as the sum of the Kähler two-form on ℂ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a linear Hamiltonian in the generators of the Jacobi group is described by a Riccati equation on and a linear first-order differential equation in z ∈ ℂ, where H₁ denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space , where denotes the Siegel upper half-plane.

A boson-fermion correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic anti-normal Berezin-Sudarshan symbols is rigorously constructed.