# 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 (Green function formalism, path integrals) 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

## Course contents summary

Quantum Field Theory will be the main subject, whose methods will be emphasized for their power and generality. Lorentz and Poincare' groups representations will be presented togheter with brief references to group theory. Green function theory will be presented in different contexts (non-relativistic and relativistic quantum mechanics; classical and quantum field theory). Quantum field theory will be deducted both in canonical and in path-integral formulation, stressing the analogies of the latter with statistical mechanics. Renormalization will be heuristically motivated in the framework of the relationships between path integral and statistical mechanics before renormalization theory will be approached in the form of renormalized pertubation theory.

## 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.

- Path integral for non relativistic quantum mechanics; relationship with classical minimal action principle. Deduction of Schroedinger equation, introduction of Hilbert space and canonical commutation rules. Sources and generating functional for quantum mechanics path integral.

- The Green function formalism for non relativistic quantum mechanics: propagator. Prescriptions for singularity, causality and scattering theory; iterative solution and Born approximation.

- The Green function formalism for classical field theory. Prescription for singularities, advanced and retarded solutions; applications to moving point charge.

- Covariance properties of Dirac equation, charge coniugation and CPT. The Green function formalism for Dirac equation; prescription for singularities from Dirac hole theory. Iterative solution for the propagator and matrix S computation. Applications to Mott cross section and electron-proton scattering.

- 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.

- Spectral assumptions on interacting theory Hilbert space. 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 Feynam graphs.

- Construction of scalar field theory path integral, with a lattice regulator and euclidean metric. Analogies with statistical mechanics and introducion of correlation functions. Computation of free field generating functional. Introduction of an interacting term in the action and Feynman graphs. Equivalence with canonical formulation of quantum field theory: continuum limit of the lattice regulated propagator and analytical continuation in time; transfer matrix and deduction of canonical commutation relations. Heuristic arguments in support of a non trivial continuum limit and relationships with phase transitions; need for a renormalization program.

- 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.

## Recommended readings

There many excellent books on quantum field theory, and it is fairly easyto find them in a library. None of them will be taken as the onlyreference. 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

A. Zee, "Quantum Field Theory in a Nutshell", Princeton University 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.