# QUANTUM FIELD THEORY I

## Learning outcomes of the course unit

The purpose of the course is to give the student the theoretical concepts, based on quantum mechanics and special relativity theory, that are needed to describe relativistic particles and their interactions. The aims of the course are that, upon completion of the course, the student should have acquired the following knowledge and skills:

Learning skills: The student understands the importance of formulating theories in a Lorentz invariant way and how this manifests itself for different kinds of fields. The student understands how scalar, Dirac and electromagnetic fields are quantized and can use these to calculate conserved quantities such as energy and momentum. The student understands what a propagator is and how its properties are related to causality. The student understands the basic notion of perturbationtheory and the meaning of asymptotic states as well as the definitions of cross section. The student masters the perturbative expansion of correlation functions and how these calculations can be simplified using Feynman diagrams. The student masters the Feynman rules for simple theories and understands how they can be derived from the Lagrange density. The student can make simple calculations of processes at tree level such as electron-positron scattering and Compton scattering as well as relating different processes using crossing relations. The student has a basic understanding of how the theory can be reformulated in a consistent way in order to include processes with higher order radiative corrections.

Making judgements: students have to demonstrate that they improved their critical abilities on the different quantization procedures fo the electromagnetic field, that they can discuss and comment critically on the possible interactions that can produce consistent quantum filed theories.

Communication skills: students have to demonstrate that they can effectively expose the topics described above. In particular, they must be able to introduce these topics in a clear and accessible way, not only for a specialist in the field, but also for a physicist with a different background.

## Prerequisites

Quantum Mechanics, Classical electrodynamics, special relativity.

## Course contents summary

Quantum Field Theory will be the main subject. After an introduction to classical field theory, and the role of symmetries in field theory, the fundamental concepts of quantum field theory, including the quantization of the free Klein-Gordon and Dirac fields and the derivation of the Feynman propagator, will be derived. The Gupta Bleuler quantization of the electromagnetic field will also be considered. Interactions are introduced and a systematic procedure to calculate scattering amplitudes using Feynman diagrams will be derived. We will compute explicit tree-level scattering amplitudes of QED process. The one corrections, loop divergences of the QDE will be computed and the need of a renormalization procedure will be introduced.

## Course contents

- Brief review of classical mechanics; minimal action principle. Lagrange equations. Deduction of field Lagrange equations in the case of coupled oscillators in the limit of infinite degrees of freedom. Classical field theory: brief review of electromagnetism in Lagrangian formalism.

- Invariance in classical field theory: Noether theorem and its applications to Lorentz transformations. Brief review of Lie groups and algebras. Internal symmetries and relative conserved currents. Poincare' group and its generators.

- Canonical quantization of scalar field theory, relativistic invariance and fields transformation properties. Internal symmetries. Solution for free field, Fock space, normal ordering. Complex field and conserved charge. Wave packets and particle interpretation. Propagator, temporal ordering; the propagator as a Green function and relative singularities prescription.

- Dirac equation: covariance, charge coniugation and CPT. The Dirac field; Lagrangian for the free Dirac field and interacting with an external field;

Canonical quantization of Dirac field theory, solution for free field, Fock space. Normal and temporal ordering for fermionic fields; the propagator of the Dirac field.

- Canonical quantization of Maxwell field theory, Gupta Bluer condition for the Fock space. Propagator of the Maxwell field.

- Interacting scalar theory. Asymptotic IN and OUT fields. LSZ reduction formulas. Green functions for interacting theory and solution in interaction representation. Wick theorem and Feynman graphs in configuration and momentum space. Vacuum graphs and disconnected graphs cancellation. Loop counting. S Matrix and cross sections. Charged scalar field and its Feynman graphs. Feynman rules for the QED theory.

- Tree level scattering amplitudes in QED; cross sections;

- One loop structure of QED: g-2, renormalization of the electric charge; Ward identity

## Recommended readings

There many excellent books on quantum field theory. None of them will be taken as the only reference. A useful list is the following:

C. Itzykson, C. Zuber, "Quantun field theory", McGraw-Hill

M. Peskin, D. Schroeder, "An Introduction to quantum filed theory",

Addison Welsey

G. Sterman, "An Introduction to quantum filed theory", CambridgeUniversity Press

Notes will be provided by the lecturer when needed.

For a few subjects it is useful to consult

J. Bjorken, S. Drell, "Relativistic Quantum Mechanics", Mcgraw-Hill

## Teaching methods

We will have both frontal lectures and problem solving sessions. The contents of the latter are to be regarded as a distinguished part of the knowledge the student is supposed to gain. Students will be directly involved in the solution of problems.

## Assessment methods and criteria

At the end of the semester, each student will be assigned a problem to solve. Discussing the solution will be the starting point for the oral examination; a correct solution is a prerequisite for passing the exam.

## Other informations

--