Soluzioni classiche per la stringa relativistica

We will develop the dynamics of an elementary string in a relativistic non-quantistic theory: the study of the elementary string’s motion is the first step in the formulation of the string theory. We will begin by writing the action of a free elementary string, presenting it in two different formulations (Nambu-Goto and Polyakov): both are consistent with Special Relativity, but the first one is simpler to obtain while the second one is easier to handle. Using Polyakov action and the variational principle, we will derive the equations of motion for the string in a generally curved space-time and the constraints that the motion must undergo. We will also introduce Noether’s theorem as a tool with the purpose of extracting conserved quantities along the solutions of the equations of motion. In the final section we will explore the particular case of a folded string rotating in an Anti de Sitter space. This example is interesting because of the AdS/CFT correspondence: this theory introduce a duality between the motion of a string in this curved space-time and an important conformal field theory. It has been proved that some calculations that are very complicated for the CFT become much simpler in the string theory, when the related entities have been identified.