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Lectures 
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 loop
 quantum
 gravity


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Lectures(on(loop(quantum(gravity(Rodolfo(Gambini(1) Why quantize gravity? 2) General relativity 3) Hamiltonian treatment of constraint systems 4) Totally constrained systems and the issue of time. 5) Quantization of constrained systems 6) Canonical analysis of general relativity 7) Canonical analysis in terms of Ashtekar variables 8) Loop representation for general relativity 9) Spin networks and quantum geometry 10) The issue of the dynamics. 11) Applications: loop quantum cosmology, black hole entropy, and potentially observable effects. 12) Conclusions1) Why quantize gravity? Quantum mechanics and general relativity have given us a profound understanding of the physical world, including scales ranging from the atomic to the cosmological. Quantum mechanics describes nuclear and atomic physics, condensed matter, semiconductors, superconductors, lasers, superfluids and led to important technological developments, for instance, in modern electronics. General relativity leads to relativistic astrophysics, cosmology and the GPS technology. These two theories have nevertheless destroyed the coherent vision of the world given by classical mechanics and non-relativistic theories. General relativity is local, deterministic and continuum, whereas quantum mechanics is probabilistic, non-local and discrete. In spite of their empirical success, GR and QM offer a schizophrenic understanding of the physical world. General relativity has taught us that space-time is a dynamical entity just like any physical object. Quantum mechanics has taught us that physical objects are composed of quanta and have states that can be superposition of different behaviors.With the exception of classical mechanics, all current theories of physics are incomplete and contain inconsistencies. They are all valid to describe phenomena at certain scales and in certain regimes but they display inconsistencies when applied outside their range of validity. Electromagnetism: The energy and the mass of a point charge are infinite. The self-interaction of a charge with its own field is ill-defined, yielding “runaway” solutions. The treatment of the charged point particle is clearly incomplete. The quantum description eliminates some infinities, for instance avoiding the collapse of the electrons into the nucleus. But even in Quantum Field Theories a) Divergent vacuum energy <0|H|0>=∞. b) Distributional field operators These observations lead us to expect that at high energies and small scales the universe should behave as composed of quanta of space-time. How is one to describe such objects?c) Ill defined interactive theories We only have rigorous theories in dimensions less than four or highly symmetric theories as N=4 supersymmetry. d) Physical quantities as scattering cross sections are infinite when all radiative corrections are taken into account, The divergences in G may be reabsorbed redefining the constants and the fields λ,m,φ, so G results well defined. The series, however, for many physically Interesting cases are divergent. Renormalization may be considered as a short-cut which allow us to compute physical quantities without worrying about what is going on at extremely short distances. We are ignoring any possible space-time microstructure.One also has infinities in general relativity. A generic space-time containing matter will develop singularities in its evolution (Hawking and Penrose singularity theorems). At a singularity (big bang, black holes) the curvature diverges and matter acquires pathological behaviors. More generally, a space-time is singular if it contains at least one incomplete geodesic. The geometric description of space-time breaks down at the singularities and only quantum considerations could solve these pathologies. Summarizing: all known theories of modern physics are partial. Inconsistencies appear when we attempt to apply them beyond their realm of validity. Only quantum gravity could be complete. It will be relevant at scales when inconsistencies and infinities arise (big bang, black holes singularities, ultra high energy, black hole evaporation).The problem of unifying quantum mechanics and general relativity is quite complex. Both theories are radically different. Quantum mechanics in its most developed form, quantum field theory, uses a background space-time in which the notion of particles makes sense. This preferred structure is incompatible with general relativity where space-time is dynamical. The properties of continuity and differentiability of space-time are essential in general relativity. But in quantum mechanics a quantized space-time is possibly discrete. We lack experimental evidence of phenomena that are dominated by quantum gravity effects, a theory that becomes relevant in regimes highly difficult to access. A lot of physicists, motivated by the last observation, have been led to ignore quantum gravity. But ignoring a problem does not make it go away. We can state that quantum gravity effects are going to be very small, but we do not know how to prove that they actually are (“How do you know the effects of a theory you do not know are small” A. Salam).The search for consistency: Searching for consistency in physics has been the source of great discoveries. Maxwell theory + classical mechanics -> Special Relativity Special Relativity+ quantum mechanics -> antiparticles, quantum field theory Special Relativity+ Newtonian gravity -> General Relativity In all cases progress resulted from taking seriously both points of view and constructing a better synthesis. Two main approaches: Canonical quantization and path integral quantization of general relativity-> Loop quantum gravity. Unification of gravity with other interactions -> string theory. The existence of more than one approach reflects the state of the art. We still do not have a theory that is completely satisfactory.General relativity: Riemannian geometry, a brief review. Einstein noticed that non inertial systems of reference are locally equivalent to systems in a gravitational field and therefore a theory of gravity will be generally covariant. General relativity is a theory of gravity but instead of describing the latter as a force, it describes it as a deformation of space-time. The geometrical properties in a given coordinate system are given by the metric tensor: Let us recall the properties of a Riemannian geometry in a metric manifold without torsion. The


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