Broken symmetries in field theory

Abstract

The thesis discusses the role of symmetries in Quantum Field Theory. Quantum Field Theory is the mathematical framework to describe the physics of elementary particles. A symmetry here means a transformation under which the model at hand is invariant. Three types of symmetry are distinguished: 1. Internal symmetries, 2. Space-time symmetries and 3. Supersymmetries. Supersymmetry is discussed in detail in the first chapter, where emphasis is put on attempts to formulate this symmetrie on a space-time lattice, with the ultimate goal to contribute to the study of computer simulations of this symmetry. Main conclusion here is that two types of mathematical framework to approach this issue are shown to be mathematically inconsistent. The quantum mechanical breaking due regularization and renormalization effects of conformal symmetries (which are a type of space-time symmetries) is discussed in chapters 2 and 3. In chapter 2 in the context of the Non-Linear Schroedinger model, in chapter 3 in the context of the Jackiw-Pi model. Main conclusion here is that the conformal symmetries of both models break quantum mechanically. However, in case of the Jackiw-Pi model these symmetries survive under a special condition which also allows vortex solutions.