Canonical Quantum Gravity: Fundamentals and Recent Developments

Book Preface

The request of a coherent quantization for the gravitational field dynamics emerges as a natural consequence of Einstein’s Equations: the energymomentum of a field is source of the spacetime curvature and therefore its microscopic quantum features must be reflected onto microgravity effects. The use of expectation values as sources is well-grounded as a first approximation only, and does not fulfill the requirements of a fundamental theory.

However, as is well-known, the achievement of a consistent Quantum Gravity theory remains a complete open task, due to a plethora of different subtleties, which are here summarized in the two main categories: i) General Relativity is a background-independent theory and therefore any analogy with the non-Abelian gauge formalisms must deal with the concept of a dynamic metric field; ii) the implementation in the gravitational sector of standard prescriptions, associated with the quantum mechanics paradigms, appears as a formal procedure, whose real physical content is still elusive.

There exists a qualitative consensus on the idea that a convincing solution to the Quantum Gravity problem will not arise before General Relativity and Quantum Mechanics are both deeply revised in view of a converging picture. Indeed, over the last fifteen years, the three most promising approaches to Quantum Gravity (i.e. String Theories, Loop Quantum Gravity and Non-commutative Geometries) revealed common features in defining a “lattice” nature for the microphysics of spacetime. This consideration makes clear that, up to the best of our present understanding, the main effort to improve fundamental formalisms must be the introduction of a “cut-off” physics, able to replace the notion of a spacetime continuum with a consistent discrete scenario. The correctness of such a statement will probably be regarded as the main success of the end of the last century reached in Theoretical Physics. A proper task for the present century is now to constrain the morphology of such a discrete microstructure of spacetime, to get, at least, a phenomenological description for Quantum Gravity effects. From a theoretical point of view, an important aim would consist of a unified picture containing common features of the present approaches, but synthesized into a more powerful mathematical language. On the one hand, it would be important to recognize non-commutative properties in the loop representation of spacetime; on the other hand, transporting the background independence of the “spin networks” into the interaction framework characterizing String Theories would constitute a relevant progress. The viability of these two goals is an intuitive perspective, but it contrasts with the rigidity of the corresponding formalisms, which confirms the request for more general investigation tools.

Contents

1. Introduction to General Relativity

2. Elements of Cosmology

3. Constrained Hamiltonian Systems

4. Lagrangian Formulations

5. Quantization Methods

6. Quantum Geometrodynamics

7. Gravity as a Gauge Theory

8. Loop Quantum Gravity

9. Quantum Cosmology