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A Holomorphic perspective of Strict Deformation Quantization

Michael Heins

TL;DR

A Holomorphic perspective of Strict Deformation Quantization develops a holomorphic analytic framework for deformation quantization, moving beyond formal power series in the Planck constant. It introduces strict deformation quantization by replacing the formal parameter with a complex deformation parameter and demanding convergence and continuity on suitable topologies, including Fréchet spaces. The work then extends this holomorphic viewpoint to infinite-dimensional settings, using complexifications of manifolds, Cauchy estimates, and entire functions on vector spaces to analyze star products and their tensorial structures. Key contributions include a general analytic formulation of star products, a systematic treatment of holomorphy in Gâteaux and Fréchet senses, and the embedding of tensor algebras as bounded entire holomorphic functions on strong duals, with illustrative examples in constant Poisson structures and cotangent bundles of Lie groups. The results provide tools to control convergence, holomorphic dependence on $ihbar$ and the algebraic structure of observables in quantum systems.

Abstract

We provide and discuss complex analytic methods for overcoming the formal character of formal deformation quantization. This is a necessity for returning to physically meaningful statements, and accounts for the fact that the formal parameter $\hbar$ carries the interpretation of Planck's constant. As formal star products are given by a formal power series, this naturally leads into the realm of holomorphic functions and analytic continuation, both in finite and infinite dimensions. We propose a general notion of strict deformation quantization and investigate how one can use established results from complex analysis to think about the resulting objects. Within the main body of the text, the outlined program is then put into practice for strict deformation quantizations of constant Poisson structures on locally convex vector spaces and the strict deformation quantization of canonical mechanics on the cotangent bundle of a Lie group. Numerous auxiliary results, many of which are well-known yet remarkable in their own right, are provided throughout.

A Holomorphic perspective of Strict Deformation Quantization

TL;DR

A Holomorphic perspective of Strict Deformation Quantization develops a holomorphic analytic framework for deformation quantization, moving beyond formal power series in the Planck constant. It introduces strict deformation quantization by replacing the formal parameter with a complex deformation parameter and demanding convergence and continuity on suitable topologies, including Fréchet spaces. The work then extends this holomorphic viewpoint to infinite-dimensional settings, using complexifications of manifolds, Cauchy estimates, and entire functions on vector spaces to analyze star products and their tensorial structures. Key contributions include a general analytic formulation of star products, a systematic treatment of holomorphy in Gâteaux and Fréchet senses, and the embedding of tensor algebras as bounded entire holomorphic functions on strong duals, with illustrative examples in constant Poisson structures and cotangent bundles of Lie groups. The results provide tools to control convergence, holomorphic dependence on and the algebraic structure of observables in quantum systems.

Abstract

We provide and discuss complex analytic methods for overcoming the formal character of formal deformation quantization. This is a necessity for returning to physically meaningful statements, and accounts for the fact that the formal parameter carries the interpretation of Planck's constant. As formal star products are given by a formal power series, this naturally leads into the realm of holomorphic functions and analytic continuation, both in finite and infinite dimensions. We propose a general notion of strict deformation quantization and investigate how one can use established results from complex analysis to think about the resulting objects. Within the main body of the text, the outlined program is then put into practice for strict deformation quantizations of constant Poisson structures on locally convex vector spaces and the strict deformation quantization of canonical mechanics on the cotangent bundle of a Lie group. Numerous auxiliary results, many of which are well-known yet remarkable in their own right, are provided throughout.

Paper Structure

This paper contains 21 sections, 158 theorems, 976 equations, 3 figures.

Table of Contents

  1. Deformation Quantization
  2. Formal Deformation Quantization
  3. Strict Deformation Quantization
  4. Holomorphic Deformation Quantization
  5. Entire Functions on Vector Spaces
  6. Polynomials
  7. Gâteaux Holomorphic Functions
  8. Fréchet Holomorphic Functions
  9. Uniform Convergence on Bounded Sets
  10. Symmetric Tensors as Polynomials
  11. A Collection of Examples
  12. Hilbert Tensor Product
  13. Entire Functions on Lie groups
  14. Standard Ordered Quantization
  15. Lie-Taylor Expansions and Entire Functions
  16. ...and 6 more sections

Key Result

theorem 1

Let $M$ be a Poisson manifold.That is, a smooth manifold $M$ such that its algebra of smooth functions $\Cinfty(M)$ carries the structure of a Poisson algebra. Then there exists a formal star product quantizing $(\Cinfty(M), \{\argument, \argument\})$.

Figures (3)

  • Figure 1: A schematic picture of the domain $\Omega \subseteq \widehat{\field{C}}^2$
  • Figure 2: Schematic picture of $V_g$
  • Figure 3: Schematic picture of $V_g \cap V_h$

Theorems & Definitions (325)

  • definition 1: Formal star product, bayen.et.al:1978a
  • theorem 1: Kontsevich's Formality Theorem, kontsevich:2003a
  • proof : Sketchy sketch
  • definition 2: Star product
  • lemma 1: Cauchy estimates
  • lemma 2
  • proof
  • proposition 1: Polarization identity, dineen:1999a
  • proof
  • corollary 1
  • ...and 315 more