QUANTUM FIELD THEORY COURSE OVERVIEW

### Objectives

An important objective of this course is to enable the students to calculate lifetimes and cross sections at an early stage. This course has been planned to take full advantage of the student preparation of elemenatry scattering theory in quantum mechanics, time dependent perturbation theory, Fermi Golden rule.

### Prerequisites

Following topics should have been covered in quantum mechanics course elemenatry

- Introduction to Scattering theory in quantum mechanics,
- Time dependent perturbation theory,
- Fermi Golden rule.

The course will be covered in nine parts. The syllabus for different parts is as follows.

### Part-I Time Evolution in Quantum Theory

Pictures in Quantum Theory. Time dependent perturbation theory in .

interaction picture. Transition to continuum. Scattering, Fermi Golden Rule

Computation of cross sections.

Prerequisites: Introduction to Schrodinger Wave Mechanics; Cross

section in classical and quantum mechanics.

Assignments: Computing unequal time commutation relations; Normal

ordering;

### Part-II Second Quantization of Non-relativistic Equation

Schrodinger equation as classical field. Action principle and canonical

qunatization. Schrodinger field as a collection of harmonic oscillators.

Number and raising and lowering operators. Complete set of commuting

operators.Hilbert space of states of quantized Schrodinger field.

Prerequisites: Lagrangian and Hamiltonian formalism of classical

mechanics; Poisson brackets

Assignments:Cross section for Rutherford scattering in second quantized

theory. Scattering from a finite range potential and determination of nuclear

size.

### Part-III Second Quantization of Klein Gordon equation.

Review of free particle solutions of Klein Gordon; Second Quantization

Prerequisites: Introduction to relativistic quantum mechanics.Klein

Gordon equation and its free particle solutions.

Asssignments: Life time of \(K\) meson decays

\(K^+ \to \pi^+ \ \pi^0\) and \(K_0 \to \pi^+\ \pi^0\);

Implications for change in isospin.

Computation of cross section for \(\pi^+ \pi^-\) scattering.

### Part-IV Quantization of Electromagnetic Field

Classical electromagnetic field. Physical Degrees of freedom. Electromagnetic

field as assembly of harmonic oscillators. Quantization of electromagnetic

field using harmonic oscillator representation.

Prerequisites: Maxwell's equations, scalar and vector potential;

Gauge transformations and gauge invariance; Coulomb gauge.

Assignments: Scattering of light from non-relativistic electrons.

Rayleigh and Thosmson scattering. Dipole transitions in atomic physics.

### Part-V Second Quantization of Dirac Field

Free Dirac field; Properties of free particle solutions; Second quantization;

Use of anticommutators; Spin statistics Theorem;

Prerequisites:Dirac equation; Properties of Free particle solutions.

Assignments: Hyperon decay \(\Lambda \to p \pi^0\);

Polarization and parity violation in Hyperon decays. Compare V-A and

scalar pseudoscalar interactions.

Branching ratio of pion decay modes \(\pi \to \mu \nu\) to \(\pi\to e\nu\)

using

(i) V-A interaction and (ii) scalar-pseudoscalar interaction.

Part-VI Lowest order processes Coulomb scattering for nonrelativistic case, Coulomb Scattering for Klein

Gordon equation and Dirac equation. Mott scattering.

### Part-VI Lorentz Invariance and Spin

Inhomogenous Lorentz group; Lie Algebra and Infinitesimal generators;

Commutation relations of generators.Pauli Lubanski Operator. Lorentz

transformations of scalar, Dirac and vector fields. Energy momentum and

angular momentum tensors for scalar, Dirac and vector fields.

Prerequisites: Special theory of relativity. Lorentz transformations.

Spin in quantum mechanics.

Assignments: Properties of time like,light like and space like vectors;

General properties of Lorentz transformations.

Deriving commutators of Lorentz generators using group property; Using

Noether's theroem toobtain expressions for energy momentum and angular

momentum.

### Part-VII Discrete symmetries, and CPT theorem

Partity Charge conjugation and time reversal for scalar, vector and Dirac

fields. CPT theorem.

Time reversal in quantum mechanics of particles.

Time reversal classical and quantum fields.

Prerequisites: Time reversal invariance in Newtonian

mechanics. Electromagentic fields and Charged particles.

Assignments: Verifying CPT properties of Bilinear covariants;

Principle of detailed balance. Applications determination of spin of pion.

### Part-VIII Interacting fields; Wick's Theorem, Feynman diagrams

Fields in interaction; Gauge invariance, Lorentz invariant phenomenological

interaction Lagrangian. Beta decay of neutron as an example.

Normal ordering; S matrix, Wick's theorem; FeynmanmDiagram;

Assignments: Understanding processes allowed by a

phenomenlogical Lagrangian; Expectation value of commutator for unequal times.

Feynman propagator for scalar field; Exercises on applying Wicks theorem.

### Part- IX Higher order corrections

Loop diagrams in higher orders; Appearance of divergences;

Anomalous magnetic moment of electron; Lamb Shift

Assignments: Evaluation of simple loop integrals using Feynman

parametric form. Filling in details of anomalous magnetic moment

and Lamb shift calculation

The course study material made available here is based on Quantum Field Theory Course taught at IIT Bhubanswar duirng Autumn Semesters of 2016 and 2017. For more information visit Course Site URL

### COURSE SITE

All study material will also be made available on the following site.

http://physics-lessons.globalcloudhost.com

You must visit these sites regularly. Important announcements will be put up

there and activities will be assigned there