# QUANTUM FIELD THEORY I

## Learning outcomes of the course unit

Quantum field theory is the general framework for our current understanding of elementary particle physics. It is also a much more general language and provides a very useful formalism for solving a number of problems in a variety of subjects. The course aims at both preparing the ground for the study of high energy physics and at providing tools and methods which are useful in a broader area. After attending the course students are supposed to master a few tools whose capabilities and power they should recognize in a number of problems. Both students involvment in solving problems in front of their collegues and the presentation of the solution they will provide to the final problem they will be assigned are intended as a training of their communication skills (they should be able to argue in public).

## Prerequisites

Quantum Mechanics, Classical electrodynamics

## Course contents summary

Quantum Field Theory will be the main subject. Lorentz and Poincare' groups representations will be presented together with brief references to group theory. After an introduction to classical field theory, and the role of symmetries in field theory, the fundamental concepts of quantum field theory, including the quantisation of the free Klein-Gordon and Dirac fields and the derivation of the Feynman propagator, will be derived. Interactions are introduced and a systematic procedure to calculate scattering amplitudes using Feynman diagrams is derived. We will also compute some explicit tree-level scattering amplitudes in a number of simple examples. The one loop divergences of the scalar phi^4 theory and of the Yukawa theory will be computed and the need of a renormalization procedure is 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 electromagntism in lagrangian formalism.

- Invariance in classical field theory: Noether theorem and its applications to Lorentz trasnformations. 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 packects 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.

- 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 cancelation. Loop counting. S Matrix and cross sections. Charged scalar field and its Feynman graphs. Feynman rules for the Yukawa theory.

- Superficial degree of divergence of a Feynman graph. Renormalizable, super-renormalizable and non renormalizable theories. Dimensional regularization. Renormalized perturbation theory and renormalization conditions. One loop structure of scalar field theory with a quartic interaction and for the Yukawa theory.

## 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. Thecontents of the latter are to be regarded as a distinguished part ofthe 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.