The main purpose of this course is to learn the basic concepts and techniques behind the theory of quantum fields, with aplications to elementary particle physics, in particular Quantum Electrodynamics.

- Apply the main principles to specific areas such as particle physics, astrophysics of stars, planets and galaxies, cosmology and physics beyond the Standard Model.
- Formulate and tackle problems, both open and more defined, identifying the most relevant principles and using approaches where necessary to reach a solution, which should be presented with an explanation of the suppositions and approaches.
- Use acquired knowledge as a basis for originality in the application of ideas, often in a research context. Use critical reasoning, analytical capacity and the correct technical language and formulate logical arguments.
- Use mathematics to describe the physical world, select the appropriate equations, construct adequate models, interpret mathematical results and make critical comparisons with experimentation and observation.

- Analyze the concept of renormalization and apply it in to electromagnetic processes.
- Apply the language of Feynman diagrams in quantum field theory .
- Apply quantum field theory electromagnetic processes.
- Calculate cross sections of electromagnetic processes .
- Understand the basics of quantum field theory.

- Introduction

1.1 Motivation

1.2 Elements of classical field theory

1.2.1 Functional calculus

1.2.2 Lagrangian and Hamiltonian. Euler-Lagrange equations

1.3 Natural units - Quantization of free fields

2.1 Non-relativistic fields. Bosons and Fermions. Number operator and statistics

2.2 Field of Klein-Gordon real. Propagators and causality

2.3 Continuous symmetries. Noether theorem: currents and energy-momentum tensor

2.4. Discrete symmetries: C,P,T

2.5 Field of Klein-Gordon complex. Charge symmetry

2.6 Dirac field. Propagators, symmetries, spin: helicity and quirality

2.7 Electromagnetic field - Interaction

3.1 Cross section and S matrix

3.2 Interaction picture and S matrix

3.3 Wick theorem

3.4 First computation at tree level: $\lambda \phi^4$

3.5 Feynman diagrams

3.6 Decays - QED

4.1 Quantization of QED

4.2 S-matrix to O(e^2)

4.3 Compton scattering at tree level. Feynman diagrams and computational techniques: traces, spin, ...

4.4 About gauge invariance. Example of Ward identity

4.5 Generalized Feynman rules and for QED

4.6 Other elementary QED processes at tree level: e^+e^- -> \mu^+\mu^-, ...

4.7 Radiative transitions of Hydrogen - Beyond tree level. Introduction

5.1 Infinities and dimensional regularization

5.2 Vacuum polarization

5.3 Renormalization of the electric charge

5.4 Optical theorem - Beyond perturbation theory

6.1 LSZ formalism and crossing symmetry (examples)

It is recommended to have followed the course Introduction to the Physics of the Cosmos.

- D. Lurie. Particles and Fields
- M. Peskin and D. Schroeder. An introduction to Quantum Field Theory L.H. Ryder. Quantum Field Theory
- S. Weinberg. The Quantum Theory of Fields
- C. Itzykson and J. Zuber. Quantum Field Theory