The main goal of the course is twofold: 1) on one side, to develop a different approach to Quantum Mechanics and Quantum Field Theory based on the Path Integral approach and 2) on the other, to understand and be proficient in the renormalization of a theory. This is a fundamental requisite to arrive to any physical result involving loop diagrams. Besides understanding the concept and renormalization procedure we will focus on its interaction with symmetries and we will conclude by establishing the renormalization group equations.

Understand the bases of advanced topics selected at the frontier of high energy physics, astrophysics and cosmology and apply them consistently.

- Apply the mechanisms of renormalisation systematically.
- Understand the foundations of functional formalism in quantum field theory.

- Functional Methods 1.1 Path Integral in Quantum Mechanics. 1.2 Functional Quantization and Path Integral in Quantum Field Theory 1.3 Symmetries in the functional formalism language
- Renormalization Theory 2.1 Ultraviolet Divergences, conceptual meaning. 2.2 Classification of theories according to their renormalization properties 2.2 Renormalized perturbation theory
- Renormalization and symmetry 3.1 Spontaneous Symmetry Breaking and linear sigma model: how they should be renormalized.
- Aspects of non abelian gauge theories
- Renormalization Group Equations

It is recommended to have followed the course of introduction to Quantum Field Theory of the Master, or at least basic courses on Quantum Field Theory during the undergrate courses.

- M. E. Peskin and D. V. Schroeder, An introduction to Quantum Field Theory, Westview Press.
- R. J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press.
- S. Pokorski, Gauge Field Theories, Cambridge Monographs on Mathematical Physics.