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Formal Deformation quantization as a Fréchet algebra

Qin Li

Abstract

We define a Fréchet topology on the space $C^\infty(X)[[\hbar]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^\infty(X)[[\hbar]]$ making it a formal deformation quantization of $X$, we will show that the quantum product $\star$ is jointly continuous, and making it a Fréchet algebra. We will show a quantum Weierstrass theorem which says quantum polynomials are locally dense in all formal smooth functions. We will also show that the canonical trace of any formal deformation quantization is continuous under this Fréchet topology.

Formal Deformation quantization as a Fréchet algebra

Abstract

We define a Fréchet topology on the space of formal smooth functions on a symplectic manifold , by constructing a sequence of semi-norms on it. For any star product on making it a formal deformation quantization of , we will show that the quantum product is jointly continuous, and making it a Fréchet algebra. We will show a quantum Weierstrass theorem which says quantum polynomials are locally dense in all formal smooth functions. We will also show that the canonical trace of any formal deformation quantization is continuous under this Fréchet topology.

Paper Structure

This paper contains 15 sections, 16 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Deformation quantization as a Fréchet algebra
  3. Fréchet spaces
  4. Fréchet algebra
  5. Example: smooth functions on manifolds
  6. Formal deformation quantization as a Fréchet algebra
  7. Fedosov quantization and Quantum Weierstrass Approximation Theorem
  8. Fedosov's approach to deformation quantization
  9. Quantum Darboux Theorem and Quantum Weierstrass Approximation
  10. Some computation in Fedosov quantization
  11. Continuity of the trace map
  12. Trace as the correlation functions
  13. The geometry of BV bundle and quantization
  14. A graphical description of equation \ref{['equation: RG-flow-quantum-observable']}
  15. The proof of Proposition \ref{['proposition: integrand-locality']}

Key Result

Theorem 1.1

(Theorem theorem: quantum-Weierstrass) For any Darboux chart $U\subset X$, the space of quantizable functions are dense in all formal smooth functions with respect to the Fréchet topology. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 34 more