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A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT

Albert Much

Abstract

We provide a deformation quantization, in the sense of Rieffel, for \textit{all} globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e.\ a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).

A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT

Abstract

We provide a deformation quantization, in the sense of Rieffel, for \textit{all} globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e.\ a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).

Paper Structure

This paper contains 21 sections, 24 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Embedding Formalism
  4. Moving in the Embedded Spacetime
  5. Poisson Structure
  6. A Generalized Rieffel Deformation
  7. Oscillatory Integrals
  8. The Deformed Product
  9. Deformed Product for Two Different Points
  10. Quantum Field Theory in Globally Hyperbolic Spacetimes
  11. The Quantized Scalar Field
  12. States and GNS Theorem
  13. The Microlocal Spectrum Condition and Hadamard
  14. Deformation of QFT in Globally Hyperbolic Spacetimes
  15. Representation of the Deformation
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let $(\mathcal{M},g)$ be a Lorentzian manifold. The following assertions are equivalent:$\,$

Theorems & Definitions (65)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5: Orthogonal Projector
  • Lemma 2.6
  • Proposition 2.7
  • Definition 2.8
  • Corollary 2.9
  • Definition 2.10
  • ...and 55 more