## Quantum Geometry and Quantum Gravity

*Fernando Barbero (IEM-CSIC, Madrid)*

### Course description

Loop quantum gravity (LQG) is one of the leading approaches to the quantization of General Relativity. Some of its most salient features are the background independence of the formalism, its non-perturbative character, and the special role played by gauge connections as the main variables used to describe the gravitational field. Quantum geometry deals with the definition of quantum operators representing lengths, areas, and volumes. This is the formalism best suited to the description of background-free theories of connections and is at the core of LQG. The purpose of this minicourse is to give an extended introduction to LQG with special emphasis on its geometrical aspects. The fact that we will be looking for a framework suitable to the study of quantum gravity will require the introduction of appropriate Hilbert spaces. As a consequence a very interesting interplay between geometric and functional-analytic issues will take place. The minicourse will be divided in three main parts: In the first I will present the Ashtekar formalism for general relativity, the second will deal with the introduction of the Ashtekar-Lewandowski Hilbert space where the quantization will take place, and the third will be devoted to quantum geometry and, in particular, to the introduction and discussion of the relevant operators for areas and volumes. I will conclude with a brief description of the main successes of the framework, open problems, and future directions.

### Background materials

#### Books on quantum mechanics

Comments: The amount of material on the subject is overwhelming. Most of the founding fathers have written classic books that can be easily found as references in any text book. We are just providing three titles that we feel appropriate for the participants in the meeting.

F. Strocchi. *"An Introduction to the Mathematical Structure of
Quantum Mechanics. A Short Course for Mathematicians".* Advanced
Series in Mathematical Physics, Vol. 27, World Scientific (2005).

Strocchi's book offers an excellent and short presentation of the mathematical aspects of QM using the C*-algebraic structure of the algebra of observables that defines a physical system. This C*-algebraic formulation, not yet standard in QM textbooks, has played a crucial role for the formulation of Loop Quantum Gravity. In this approach the usual description of states and observables as Hilbert space vectors and operators is derived from the GNS and Gelfand-Naimark Theorems. Schrodinger QM follows from the von Neumann uniqueness theorem about the representations of the Weyl algebra that codifies the Heisenberg uncertainty relations. The book deals also with the problem of quantum dynamics. This is related to the self-adjointness of the differential operator describing the Hamiltonian and can be solved, in many cases, by the Rellich-Kato theorems.

C. J. Isham. *"Lectures on Quantum Theory: Mathematical and
Structural Foundations"*. World Scientific (1995).

This is an elegant and clear introduction of QM that discusses in detail the fundamental conceptual problems of quantum theory, their differences with respect classical physics, and the new mathematical framework employed in quantum mechanics.

A. Galindo, P. Pascual. *"Quantum Mechanics I & II"*. Springer (1991).

This is a comprehensive, deep, mathematically rigorous (some physicists would say unforgiving), and elegant classical university text.

#### Books on quantum gravity

Comments: This short list contains a series of books mainly devoted to loop quantum gravity (the approach where quantum geometry is used). They contain introductions that review, to some extent, other approaches to the subject. We have included a book on 2+1 gravity with a more general scope and one on strings and gravity for those people who want to read something about the prevailing (orthodox?) view on quantum gravity.

C. Rovelli, *Quantum Gravity*, Cambridge University Press (2004).

A. Ashtekar, *Non Perturbative Canonical Gravity*, World Scientific (1991).

R. Gambini, J. Pullin, *Loops, Knots, Gauge Theories and Quantum Gravity*, Cambridge University Press (2000).

S. Carlip, *Quantum Gravity in 2+1 Dimensions*, Cambridge University Press (2003).

T. Ortin, *Gravity and Strings*, Cambridge University Press (2004).

#### Introductory papers and reviews on loop quantum gravity and quantum geometry

Comments: We are giving a short list of recent review articles from leading experts in loop quantum gravity and quantum geometry. Some of them cover the basic formalism whereas others discuss applications, in particular to quantum cosmology. A complete (and reasonably up to date) list of references on the subject is also given.

A. Ashtekar, J. Lewandowski, *Background Independent Quantum Gravity:
A Status Report*, ref: Class.Quant.Grav. 21 (2004) R53, gr-qc/0404018.

A. Corichi, *Loop Quantum Geometry: A primer*, J.Phys.Conf.Ser. 24
(2005) 1-22, gr-qc/0507038

M. Bojowald, *Loop Quantum Cosmology*, gr-qc/0601085

T. Thiemann, *Loop Quantum Gravity: An Inside View*, hep-th/0608210

A. Corichi, A. Hauser, Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity, gr-qc/0509039

#### Links

Comments: These are links to Departments or personal home pages of (some) people working in this area. Other links can be found there.