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Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance

Marcus Gaul, Carlo Rovelli

Abstract

This series of lectures gives a simple and self-contained introduction to the non-perturbative and background independent loop approach of canonical quantum gravity. The Hilbert space of kinematical quantum states is constructed and a complete basis of spin network states is introduced. An application of the formalism is provided by the spectral analysis of the area operator, which is the quantum analogue of the classical area function. This leads to one of the key results of loop quantum gravity: the derivation of the discreteness of the geometry and the computation of the quanta of area. Finally, an outlock on a possible covariant formulation of the theory is given leading to a "sum over histories" approach, denoted as spin foam model. Throughout the whole lecture great significance is attached to conceptual and interpretational issues. In particular, special emphasis is given to the role played by the diffeomorphism group and the notion of observability in general relativity.

Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance

Abstract

This series of lectures gives a simple and self-contained introduction to the non-perturbative and background independent loop approach of canonical quantum gravity. The Hilbert space of kinematical quantum states is constructed and a complete basis of spin network states is introduced. An application of the formalism is provided by the spectral analysis of the area operator, which is the quantum analogue of the classical area function. This leads to one of the key results of loop quantum gravity: the derivation of the discreteness of the geometry and the computation of the quanta of area. Finally, an outlock on a possible covariant formulation of the theory is given leading to a "sum over histories" approach, denoted as spin foam model. Throughout the whole lecture great significance is attached to conceptual and interpretational issues. In particular, special emphasis is given to the role played by the diffeomorphism group and the notion of observability in general relativity.

Paper Structure

This paper contains 27 sections, 120 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Basic Formalism of Loop Quantum Gravity
  3. A brief Outline of the Connection Formalism
  4. Basic Definitions
  5. The Construction of a Hilbert Space $\mathcal{H}$
  6. Holonomies.
  7. Graphs.
  8. Cylindrical Functions.
  9. A Basis in the Hilbert Space
  10. The $SU(2)$ Gauge Constraint
  11. Operators on $\mathcal{H}$
  12. An $SU(2)$ Gauge Invariant Operator.
  13. Quantization of the Area
  14. The Physical Contents of Quantum Gravity and the Meaning of Diffeomorphism Invariance
  15. Passive and Active Diffeomorphism Invariance
  16. ...and 12 more sections

Figures (18)

  • Figure 1: A simple example of a graph.
  • Figure 2: A step by step construction of the physical Hilbert space.
  • Figure 3: Examples of $n=1,2,3$-valent nodes.
  • Figure 4: A simple spin network with two trivalent nodes.
  • Figure 5: The decomposition of the spin network state (\ref{['example']}) into loop states.
  • ...and 13 more figures