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Deformation quantization on the cotangent bundle of a Lie group

Ziemowit Domański

Abstract

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

Deformation quantization on the cotangent bundle of a Lie group

Abstract

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

Paper Structure

This paper contains 26 sections, 18 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie groups basics
  4. Poisson structure on $M$
  5. Measures and integration
  6. Baker-Campbell-Hausdorff product
  7. Notation
  8. Spaces $\mathcal{F}(M)$ and $\mathcal{L}(M)$
  9. Deformation quantization of the classical system
  10. Star-product on $\mathcal{F}(M)$
  11. Extension of the $\star$-product to the Hilbert space $\mathcal{L}(M)$
  12. $C^*$-algebra of observables and states
  13. Extension of the $\star$-product to an algebra of distributions
  14. Time evolution
  15. Operator representation
  16. ...and 11 more sections

Key Result

Lemma 1

Let $f,g \in \mathcal{F}(M)$ and $\mathsf{f},\mathsf{g} \in C^\infty_c(G \times G)$ be the corresponding integral kernels. Then,

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 21 more