- CGM Seminars 2007 (University of Rome “Sapienza”)

An intruduction Wilson Loop (21 June 07)
Speaker: Francesco Cianfrani

Abstract: The emergence of Wilson loops in the strong coupling limit of lattice gauge theories is outlined, stressing how this formulation easily accomplished the issue of gauge invariance. Then, the appealing features of a quantization of gravity in terms of loops are sketched, in particular with respect to the appearance of non canonical commutation relations and of a numerable basis in the Hilbert space.

Mixmaster chaos from the Loop Quantum Cosmology point of view (15 June 07)
Speaker: Marco Valerio Battisti

Abstract: The chaotic behavior of the Bianchi IX model, in the dynamics toward the classical singularity, is investigated in the Loop Quantum Cosmology framework. Starting from an isotropic settings, we review the key points that brings to a non-chaotic dynamics for such a model, as soon as the quantum effects become important. We also point out the problems and ambiguities of such a framework, in particular in the formulation of the Hamiltonian constraint.

Quantization of theories with constraints - Part I (15 May 07)
Speaker: Simone Zonetti
-- Proceeding (pdf) --

Abstract: Gauge theories, when treated with an hamiltonian formalism, behave as constrained theories, where conditions between the canonical variables hold. At first two kinds of constraints can be recognized: primary and secondary constraints. The former appearing with Lagrange multipliers, the latter arising from the consistency conditions, i.e. time indipendence, of the others. This distinction, however, is not essential, and the more usefull classification based on the Poisson algebra is adopted. This way it is possible to discard the Poisson bracktes and adopt the Dirac ones, constructed with 2nd class constraint algebra matrix {φ_a,φ_b}, and embed these constraints in the inner structure of the theory. Now one gets a theory with at most 1st class constraints, that generate the gauge transformations.

 

Quantization of theories with constraints - Part II (15 May 07)
Speaker: Francesco Cianfrani
-- Proceeding (pdf) --

Abstract: The geometrical interpretation of first- and second-class constraints in the phase space is outlined, with the aim to demonstrate the different reduction of degrees of freedom they produce. Furthermore, the quantization of such constrained system is analyzed, which provide us with a demonstration of how the Fadeev-Popov determinant arises in the path-integral formulation.

 

 

BRST symmetries (20 April 07)
Speaker: Michele Castellana
-- Proceeding (pdf) --

Abstract: After the experimental observation of neutral-currents processes in 1973, the requirement for a proof of renormalizability of non-abelian gauge theories predicting the existence of such processes became an essential point in quantum field theory. The discovery of BRST symmetry for the Yang - Mills action made the electroweak model predicting these processes a consistent theoretical framework. As a matter of fact, this underlying symmetry of the gauge fixed action allowed to show it to posses all the required term to make it renormalizable. BRST symmetry showed to apply to a really wide class of systems of physical interest, and can be easily generalized to a generic system which possessing some basic “gauge symmetry”. To this end, we recall that in the literature there exist different formulations for the BRST formalism, with substantial differences from each other. On one side there exists a formulation of BRST symmetry for constrained systems based on canonical quantization methods which is widely diffused and on the other hand there is another approach to derive BRST symmetry based entirely on path integral methods and is applicable to systems with infinite degrees of freedom avoiding those inconsistencies proper of canonical quantization methods we discussed above. In this paper we will follow the latter derivation.

 

 

An introduction to spin foams (30 March 07)
Speaker: Orchidea Maria Lecian
-- Proceeding (pdf) --

Abstract: Spin foams will be introduced from a geometrical and field-theoretical point of view [1]. Starting from the definition of spin networks as a generalization of Wilson loops and drawing the analogy with the concept of “plaquettes” will allow one to outline the possibility of recognizing spin-network states as basis for the functionals of the connection.
Spin foams, defined as branched surfaces, will accomplish the dual transformation that leads to a physically equivalent description of lattice gauge theory. Particular attention will be paid to the description of physical observables in terms of the path-integral formulation within this formalism, and to the mathematical meaning of these operations [2].
A spin-foam model for Yang-Mills theories will follow, and a background-independent spin-foam model for quantum gravity will be obtained by slightly modifying the duality map.
The relation between spin-network states and the geometry of spacetime will be further investigated [3]. In particular, covariant quantum gravity will be approached considering spin-network states as states of the gravitational field. Equivalence classes for spin foams will be established, and the interpretation of spin foams as quantum histories will be proposed.
References:
[1] F. Conrady, “Geometric spin foams, Yang-Mills theory and background-independent models” [gr-qc/0504059] and the references therein.
[2] See, for example, P. Caressa [link to the site].
[3] D. Oriti, “Spacetime geometry from algebra: spin foams models for non-perturbative quantum gravity” [gr-qc/0106091] and the references therein.