Objectives of Programme's Learning Outcomes

• Skill 6: Have acquired professional skills in relation to the objective defining the degree.
• Have gained expertise and specialised knowledge in a field of mathematics in order to enter fully into the world of research.
• Demonstrate intuition and creativity to tackle new mathematical problems.
• Expose high-level mathematical results to a specialised audience.
• Have integrated and elaborate mathematical knowledge.
• Mobilise the Bachelor's course in mathematics to address complex issues and have profound mathematical expertise to complement the knowledge developed in the Bachelor's course.
• Use prior knowledge to independently learn high-level mathematics.
• Research mathematical literature in an efficient and relevant way.
• Read research articles in at least one discipline of mathematics.
• Carry out major projects.
• Work in teams and, in particular, communicate effectively and with respect for others.
• Carry out major projects.
• Work in teams and, in particular, communicate effectively and with respect for others.
• Communicate clearly.
• Communicate the results of mathematical or related fields, both orally and in writing, by adapting to the public.
• make a structured and reasoned presentation of the content and principles underlying a piece of work, mobilised skills and the conclusions it leads to.
• Have sufficient knowledge of English for basic scientific communication.
• Adapt to different contexts.
• Have developed a high degree of independence to acquire additional knowledge and new skills to evolve in different contexts.
• Critically reflect on the impact of mathematics and the implications of projects to which they contribute.
• Demonstrate thoroughness, independence, creativity, intellectual honesty, and ethical values.

Learning Outcomes of UE

By the end of the course, the student should know the very basics of Relativistic Quantum Field Theory in flat spacetime. In particular, he/she should be able to identify the relevance of the Poincaré and its unitary irreducible representations for the classification of linear relativistic wave equations in flat spacetime. The student should be able to derive the various fundamental relativistic field equations (Klein-Gordon, Dirac, Maxwell, Fronsdal) from a corresponding variational principle. He/she should be able to apply Noether's theorem for relativistic field theories in order to construct conserved current densities.  The student should know our to canonically quantize a free field of arbitrary spin. In particular, how to deal with gauge invariance for the quantization of Maxwell's vector field. He/she should be able to compute the scattering S matrix in time-dependent perturbation theory and explicitly compute some Feynman diagrams at tree level for simple scattering processes in Quantum Electrodynamics.

UE Content: description and pedagogical relevance

Lorentz and  Poincaré groups. Classification of the unitary irreducible representations of the Poincaré group. Variational principles in Relativistic Field Theory: Klein-Gordon, Dirac, Maxwell, Fierz-Pauli and Fronsdal field equations. Gauge invariances and rigid symmetries. Relativistic Hydrogen atom. Noether theorem in Field Theory. Canonical quantization of free fields of spin less than two. Method of Dirac for constained systems. Propagators, Wick theorem. Time-dependent perturbation theory for the scattering S matrix. Reduction formula. Feynmann rules for quantum electrodynamics.

Prior Experience

Classical electrodynamics, analytical mechanics, quantum mechanics, group theory, special relativity and complex analysis.