Canonical Quantum Gravity

Fundamentals and Recent Developments

This book aims to present a pedagogical and self-consistent treatment of the canonical approach to Quantum Gravity, starting from its original formulation to the most recent developments in the field.

We start with an innovative and enlightening introduction to the formalism and concepts on which General Relativity has been built, giving all the information necessary in the later analysis. A brief sketch of the Standard Cosmological Model describing the Universe evolution is also given alongside the analysis of the inflationary mechanism. After deepening the fundamental properties of constrained dynamic systems, the Lagrangian approach to the Einsteinian Theory is presented in some detail, underlining the parallelism with non-Abelian gauge theories. Then, the basic concepts of the canonical approach to Quantum Mechanics are provided, focusing on all those formulations which are relevant for the Canonical Quantum Gravity problem. The Hamiltonian formulation of General Relativity and its constrained structure is then analyzed by comparing different formulations. The resulting quantum dynamics, described by the Wheeler–DeWitt equation, is fully discussed in order to outline its merits and limits. Afterwards, the reformulation of Canonical Quantum Gravity in terms of the Ashtekar–Barbero–Immirzi variables is faced by a detailed discussion of the resulting Loop Quantum Gravity Theory. Finally, we provide a consistent picture of canonical Quantum Cosmology by facing the main features of the Wheeler–DeWitt equation for the homogeneous Bianchi models and then by a detailed treatment of Loop Quantum Cosmology, including very recent developments.

Contents:

• Introduction to General Relativity
• Elements of Cosmology
• Constrained Hamiltonian Systems
• Lagrangian Formulations
• Quantization Methods
• Quantum Geometrodynamics
• Gravity as a Gauge Theory
• Loop Quantum Gravity
• Quantum Cosmology

Readership: Researchers in theoretical physics, quantum physics, general relativity and astrophysics.
Key Features:

• An independent and sound analysis touching many crucial points in the attempt toward a consistent Canonical Quantum Gravity theory
• A pedagogical and exhaustive review of fundamental concepts of Quantum Mechanics and General Relativity
• A solid introduction to Hamiltonian constrained systems with a detailed analysis of the notion of canonical transformation in this class of systems with some concrete examples giving useful insight
• A self-consistent derivation of the Loop Quantum Gravity Approach, outlining the main progresses and the shortcomings of this theory