Simetria conforme em teoria de campos

Abstract

This work presents an introduction to conformal symmetry, emphasizing their main properties and their role in field theory. After providing a motivation and the general definition of conformal transformations, we begin our analysis with conformal isometries in their infinitesimal formulation, generated by conformal Killing vectors, and systematically reconstruct the structure of the conformal group from the corresponding Killing equation. Next, after determining the representations of the conformal group, we apply Noether’s theorem to conformal symmetries and derive the conditions under which a Lagrangian field theory is conformally invariant. We then extend the analysis to curved spacetime, clarifying the distinction between conformal isometries and Weyl transformations—an issue that often leads to confusion and misunderstandings. To this end, we gauge the global Weyl symmetry and examine the relationship between conformal symmetry in flat spacetime and Weyl invariance. Weyl geometry is then reviewed in contrast to the pseudo-riemannian structure of General Relativity. Finally, we conclude with some remarks on the broader implications of conformal transformations and indicate possible directions for further study.