D1. Knowledge and understanding: Students should become familiar with the basic principles of quantum field theory.
D2. Ability to apply knowledge and understanding: Students should be able to calculate the main physical quantities (energy, charge, momentum...) as a function of the creators and annihilation operators of particles; they will also have to be able to compute the simplest transition amplitudes starting from Feynman's rules.
D3. Making judgements: Students will be able to discern which of the contents learned are relevant to various physical problems, from many-body systems to particle physics.
D4. Communication skills: Students must be able to communicate the contents learned with the appropriate terminology, in particular learning the language and technical terms of perturbative field theory.
D5. Learning skills: At the end of the course, students should have developed the know-how necessary to solve basic quantum field theory problems, and to tackle more advanced topics.

Knowledge in Quantum Mechanics (bachelor level)

The following topics will be covered.
1. Introduction to classical field theory
2. Non relativistic QFT (second quantized Schroedinger equation)
3. Second quantized Klein-Gordon equation (neutral and charged field)
4. Dirac equation, Dirac matrices and associated notation
5. Second quantization of the Dirac equation
6. Second quantization of the electromagnetic field
7. Scattering theory
8. Iteracting fields, perturbative expansion, Wick theorem, Feynman diagrams.
9. Computation of a Feynman diagram to first perturbative order

Greiner, Relativistic Quantum Mechanics, Springer.
Greiner and Reinhardt, Field Quantization, Springer.
Peskin and Schroeder, An Introduction to Quantum Field Theory, Perseus Books.
Mandl and Shaw, Quantum Field Theory, 2nd ed., Wiley.

- Basic review of relativistic notation. Relativistic generalization of the Schrödinger equation based on the correspondence principle: the Klein-Gordon equation. Plane wave solution, relativistic dispersion relation. Problem with negative energies. Problem with negative probabilities in the Klein Gordon equation.
- Dirac equation: Introduction and Clifford Algebra. Dirac matrices. Pauli's fundamental theorem. Equation of continuity for the Dirac equation.
- Eigenstates and eigenvalues of the Dirac free particle Hamiltonian. Positive and negative energy solutions.
- Dirac spinors and their properties (orthogonality and completeness). Exercises.
- Helicity and spin.
- The Dirac sea and its implications.
- Non-relativistic limit of the Dirac equation coupled to the electromagnetic field.
- Lorentz invariance of the Dirac equation. Spinors and transformation matrices. Free particle solutions of the Dirac equation using Lorentz transformations.
- Bilinear of Dirac matrices.
- Overview of classical single particle physics and canonical quantization.
- Functional derivatives: brief introduction. Review of Lagrangian and Hamiltonian formalism for classical fields.
- Noether's theorem (without proof). Applications: invariance under translations and internal symmetries.
- Second bosonic quantization of the Schroedinger field: Lagrangian, Hamiltonian, commutation brackets; differences with the first quantization. Equations of motion. Expansion in creation and destruction operators, their commutators. States of single and multiple particles. Representation in energy and position. Calculation of matrix elements.
- Second Fermionic quantization of the Schroedinger field.
- Second quantization of the neutral scalar field: Lagrangian, Hamiltonian formalism, field expansion into creators and annihilators. Energy, momentum and number operators and divergences. Normal ordering for the Klein Gordon field.
- Quantization of the charged scalar field. Invariance under phase transformation and conserved charge.
- Micro-causality for the Klein Gordon field. Violation of micro-causality if the field is quantized as a fermionic field.
- Feynman propagator for the Klein Gordon field.
- Dirac's Lagrangian and Hamiltonian. Second quantization of the Dirac field. Expansion of the quantized field in plane waves. Problem with negative energies. Solving the problem of negative energies. Particle and antiparticle states. Hamiltonian, renormalized charge and momentum operator. Feynman propagator of the Fermi field. Micro-causality for the Dirac field (outline).
- Relativistic formulation of electromagnetism: fields, potentials, equations, gauge invariance, gauge choice (Lorentz), free solutions, polarization vectors (definition, orthogonality properties and completeness).
- Quantization of the electromagnetic field: Gupta-Bleuler method. Interacting fields, general introduction. Picture of Heisenberg, Schroedinger and interaction.
- Dyson series expansion. S-matrix. Wick's theorem for transforming temporal-ordered products into normal-ordered products.
- Introduction to QED.
- Perturbative expansion of first order QED. Conclusion: all processes are either null or kinematically forbidden.
- Perturbative expansion of second order QED. Review of all physical processes.
- Second order electron-electron and electron-photon collision.

Blackboard lessons and exercises

Video recordings of lessons and exercises via MS Teams Teaching material available on Moodle: https://moodle2.units.it

The exam consists of written exercises on the topics covered during the lectures. The duration of the exam will be two hours. After the written part is assessed, a discussion of the exam will follow. The exam may be held in Italian or English, at the student's choice