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RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold

Rukmini Dey

TL;DR

This work develops Berezin-type and Odzijewicz-type quantizations for arbitrary compact smooth manifolds by embedding the manifold into $\mathbb{C}P^n$ and pulling back the associated coherent-state, reproducing-kernel Hilbert-space structures. It first constructs explicit quantizations on $\mathbb{C}P^n$ and then induces corresponding operators and symbols on the embedded manifold $X$, establishing a consistent correspondence principle in both frameworks. A key technical contribution is solving a Monge-Ampère equation to realize the Odzijewicz construction on $\mathbb{C}P^n$ and then transporting it to $X$ via the pullback of coherent states, thereby enabling a path-integral interpretation of transition amplitudes through holomorphic line bundles with connections. The approach yields an embedding-dependent, geometrically grounded quantization scheme suitable for systems lacking a natural symplectic action, with potential links to higher-dimensional quantum Hall physics and quantum chaos phenomena.

Abstract

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 ${\mathbb C}P^n$ 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 ${\mathbb C}P^n$. 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 ${\mathbb C}P^n$. 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 ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$.

RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold

TL;DR

This work develops Berezin-type and Odzijewicz-type quantizations for arbitrary compact smooth manifolds by embedding the manifold into and pulling back the associated coherent-state, reproducing-kernel Hilbert-space structures. It first constructs explicit quantizations on and then induces corresponding operators and symbols on the embedded manifold , establishing a consistent correspondence principle in both frameworks. A key technical contribution is solving a Monge-Ampère equation to realize the Odzijewicz construction on and then transporting it to via the pullback of coherent states, thereby enabling a path-integral interpretation of transition amplitudes through holomorphic line bundles with connections. The approach yields an embedding-dependent, geometrically grounded quantization scheme suitable for systems lacking a natural symplectic action, with potential links to higher-dimensional quantum Hall physics and quantum chaos phenomena.

Abstract

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 into 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 . In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic -forms. In the Berezin case, the operators that are quantized are those induced from the ambient space . 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 and we induce the quantization on the smooth compact embedded manifold from .
Paper Structure (10 sections, 10 theorems, 36 equations)

This paper contains 10 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Review of Geometric Quantization and Coherent states on ${\mathbb C}P^n$
  3. A reproducing kernel Hilbert space on a compact smooth manifolds and coherent states
  4. Induced Berezin-type quantization on compact smooth manifolds
  5. Review of Berezin quantizaion on ${\mathbb C}P^n$:
  6. Induced operators on $X$ and correspondence principle
  7. Odzijewicz-type quantization on compact smooth manifolds
  8. Review of Odzijewicz quantization on ${\mathbb C}P^n$
  9. Induced Odzijewicz-type quantization on compact smooth manifolds
  10. Acknowledgments

Key Result

Proposition 2.1

Reproducing kernel property. If $\Psi$ is any other section, then $\left< \psi_{\mu}, \Psi \right> = \Psi(\mu)$. In particular, $\left< \psi_{\mu}, \psi_{\nu} \right> = \psi_{\nu}(\mu)$.

Theorems & Definitions (18)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Theorem 4.2: Berezin
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 8 more