Field Theory

In the meeting of this year, there were a total of 165 contributions in the form of panel and oral sections, besides two plenary and three parallel talks. As occurred in the previous meetings, these talks refer to recent and/or signi cative subjects in the Field Theory. At this year, one talk was on the Anti-de Sitter (AdS){ Conformal Field Theory (CFT) correspondence, two on nonconcommutative theories, one on application of eld theory in condensed matter and one on derivative expansions and nite temperature. I nd also important to mention that there were 14 contributions from the Cosmology and Gravitation sector which also concerns to eld theory. The quantum and semiclassical aspects of gravitation as well as quantum elds in classical and curved background are subjects whose interest has been increasing year after year. It might be tedious and uninteresting to make an analysis of each contribution (or group of them) isolated from the general view of where this work is in the context of the Field Theory as a whole. The main reason is that one of the purposes of this talk is to let people of other areas know what has been done in Field Theory. In my opinion, another important point would be to situate the works within the development of the corresponding research groups in Brazil in order to have an idea of what has been done in each group during the last years. Concerning this last part, we refer to the careful analysis of Prof. Marcelo Gomes in the summary talk of the last meeting. I am going to concentrate on the rst part. We may say that the success of the Field Theory starts from the quantization of the electromagnetic and fermion elds, describing the electromagnetic interaction (QED), where there is a fantastic agreement with experiments. QED is a gauge theory whose symmetry group is the U(1). In the year 1954, Yang and Mills proposed an extension for this gauge theory in order to include non-Abelian elds. This was achieved by considering more general groups SU(N), where the number of gauge elds is N 1. Even though considered a very nice theoretical idea, the rst successful application of the Yang-Mills elds just occurred almost fteen years later (1968), in the consistent description of the weak interaction in a uni ed theory also involving QED. The corresponding symmetry group is the SU(2)L U(1)Y (where \L" means \left doublets" and \Y" refers to the hypercharge) and its development was due to Glashow, Salam and Weinberg. It is opportune to mention that the experimental observation of weak gauge elds took fteen years more, after the construction of powerful accelerators. Later on, the strong interaction was also formulated as a Yang-Mills theory, whose symmetry group is the SU(3) and the basic ingredients are not protons and neutrons, as in the old Nuclear Physics, but quarks and gluons. Nowadays, the gauge theory whose symmetry group is SU(3) SU(2)L U(1)Y , also known as standard model, correctly explains all the events we know involving weak, electromagnetic and strong interactions. The standard model and problems related with con nement, vacuum QCD etc. are always an interesting area of research in quantum eld theory [1, 2, 3, 4]. We also mention that problems related to the Casimir E ect have increased of interest in recent years [5,6,7,8,9,10,11,12]. The path integral formalism in Field Theory has a very high resemblance with the partition function in Statistical Physics. The intersection of these initially distinct subjects leads to a fruitful line of research because the knowledge of each one could be used into the other. The so called Quantum Field Theory at Finite Temperature, or Thermal Field Theory, has always been an interesting research subject