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Special Features of the Upper Half-Plane in Semiclassical Theory

Hirotaka Wakuta

Abstract

We study conditions that the semiquantised Riemannian metric $g_Q$ and Levi-Civita connection $\nabla_Q$ have the same form as classical one in semiclassical theory. Concrete examples of semiclassical theory have been computed by Majid et al. for upper half-planes, hemispheres and complex projective spaces. However, only in the example of the upper half-planes the semiquantised metric and connection have the same form as classical one. In this study, we first investigate the conditions that the Riemannian metric has the same form as classical one when its components are replaced by general functions. Next, based on these conditions we compute the generalised Ricci form and investigate the conditions that a quantised connection is a quantum Levi-Civita connection. Finally, we semiquantise the upper half-plane with Poincaré metric generalised by a parameter $t$. This paper is based on my Master thesis.

Special Features of the Upper Half-Plane in Semiclassical Theory

Abstract

We study conditions that the semiquantised Riemannian metric and Levi-Civita connection have the same form as classical one in semiclassical theory. Concrete examples of semiclassical theory have been computed by Majid et al. for upper half-planes, hemispheres and complex projective spaces. However, only in the example of the upper half-planes the semiquantised metric and connection have the same form as classical one. In this study, we first investigate the conditions that the Riemannian metric has the same form as classical one when its components are replaced by general functions. Next, based on these conditions we compute the generalised Ricci form and investigate the conditions that a quantised connection is a quantum Levi-Civita connection. Finally, we semiquantise the upper half-plane with Poincaré metric generalised by a parameter . This paper is based on my Master thesis.
Paper Structure (54 sections, 15 theorems, 197 equations)

This paper contains 54 sections, 15 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Classical differential geometry
  3. Fundamentals of vector bundles
  4. Definitions and examples
  5. Connections on a vector bundle
  6. Riemannian manifolds
  7. Affine connection
  8. Curvature and torsion
  9. Definition of Riemannian manifolds
  10. In local coordinates
  11. Riemannian geometry with differential forms
  12. Symplectic manifolds
  13. Definitions and examples
  14. Hamiltonian vector fields
  15. Poisson manifolds
  16. ...and 39 more sections

Key Result

Lemma 2.4

$\Gamma(E)$ is a module over $C^{\infty}(M)$.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 51 more