09520 - Theory of Fields

Learning outcomes

at the end of this course, the student will possess the main knowledges concerning the physical principles at the ground of the relativistic quantum field theory, of the mathematical methods of the analytic and algebraic type which stand below the models describing the quantized scalar, spinor and vector field, massive and massless, of the spatial-temporal and internal simmetries which specify the dynamics of such models. The student will become acquainted with perturbation thery for interacting quantum fields, collision theory, radiative corrections and the basic principles beyond the Standard Model of electroweak and strong interactions.

Course contents

ELEMENTS OF THE GROUP THEORY.

Symmetry groups. Group representations. Equivalence, reducibility and irreducibility of group representations. Unitary representations. Decomposition theorem. Continuous groups. Lie groups. The rotations group. Infinitesimal generators and the structure constants. Fundamental theorems on the Lie algebras. Canonical coordinates and exponential representations. Baker-Campbell-Hausdorff formula. Examples : the Lie groups SU(2) and SO(3), exponential representations and topological properties. Unitary irreducible finite dimensional representations of the Lie group SU(2). The Lorentz group. Connected components and proper orthochronus subgroup. Examples. Exponential representation and Lie algebra. Finite dimensional irreducible representations of the Lorentz group. Simple and semisimple Lie groups and Lie algebras. Cartan-Killing metric. Casimir operators and Dynkin indices. The Poincaré group. Casimir operators of mass and spin. Unitary irreducible representations of the Poincaré group : the concept of elementary particle.

CLASSICAL RELATIVISTIC WAVE FIELDS.

Definitions and basic properties. Local, total and differential infinitesimal field variations. Scalar and pseudoscalar wave fields. Vector and arbitrary rank tensor fields. Weyl spinorial fields. Spatial inversion. Dirac spinor fields. Dirac matrices and the Clifford algebra. Invariants from the Weyl and Dirac spinors. Spin tensor for the Weyl and Dirac wave fields. Charge conjugation. Self-conjugated Majorana spinors. The Action functional. The Euler-Lagrange field equations. The Noether theorem. Conserved charges. Examples : energy-momentum density tensor, total angular momentum and spin angular momentum densities, current densities for internal symmetries.

QUANTIZATION OF THE KLEIN-GORDON FIELD.

Real scalar wave field : Lagrangian density, energy-momentum, Hamiltonian and field equations. Free real scalar field. The Klein-Gordon equation. Normal modes decomposition of the free real scalar field. Quantization of the free real scalar field. Creation and destruction operators. The zero point energy and the cosmological constant puzzle. Normal products of field operators. The Fock space of states and the Bose-Einstein statistics. Manifestly Lorentz covariant particle states. Unitary Poincaré transformations for the quantized real scalar field. Special distributions : the Pauli-Jordan commutator and the Feynman propagator. Wick rotation and the Euclidean formulation for the real scalar field theory. The generating functional for the Green's functions of the real scalar free field. Symanzik functional differential equation for the generating functional of the real scalar free field and its explicit solution. Functional integration. Vacuum to vacuum transition amplitude and the determinant of a differential operator. Zeta function regularisation for the determinant of a differential operator.

THE QUANTIZATION OF THE DIRAC FIELD.

The Dirac equation. Covariance of the Dirac equation. Plane wave solutions of the Dirac equation. Normal modes decomposition of the Dirac field. Properties of the spin states : orthonormality and closure relations. Projectors on the spin states. Explicit realization of the spin states. Noether currents for the Dirac field : energy-momentum, helicity and electric current density. Quantization of the Dirac field : canonical anticommutation relations. The generators of the space-time translations. Fock space and the Fermi-Dirac statistics. Observables for the quantized Dirac field : energy-momentum, helicity and electric charge. Covariance of the quantized Dirac field : unitary representation of the Poincarè group. Discrete symmetries : charge conjugation, parity and time reversal (CPT). Special distributions : anticommutator at arbitrary times and the Feynman propagator. The Euclidean formulation and the spinor Euclidean Action. The fermion generating functional. Symanzik equation for fermuions. Integration over Grassmann variables. Functional integration with fermions.

THE VECTOR FIELD.

General covariant gauges. Normal modes decomposition of the massive vector field. Normal modes decomposition of the vector gauge potential. Covariant canonical quantization of the massive vector field. Covariant canonical quantization of the vector gauge potential. Fock space with indefinite metric. Subsidiary condition and observables.

THE FEYNMAN RULES.

Connected Green's functions. Self-interacting real scalar field. Yukawa theory. Fermion determinants. Non-relativistic Yukawa potential. Quantum electrodynamics (QED). Euclidean field theories.

THE SCATTERING MATRIX.

The S-matrix in non-relativistic quantum mechanics. Green's functions and the S-matrix in perturbation theory. Lehmann-Symanzik-Zimmermann reduction formulae. Cross section. Scattering amplitude. Luminosity. Quasi-elestic scattering. Electron-positron into muon-antimuon pairs. Evaluation of R : the color hypothesis and the asymptotic freedom. Electron-muon collisions. Annihilation of an electron-positron pair into a photon pair.

RADIATIVE CORRECTIONS.

Evaluation of the simplest divergent Feynman integrals. Ultraviolet cut-off regularisation. Dimensional regularisation. The photon self-energy tensor in QED. Vacuum polarization effects. Effective Action. Divergences of the Feynman diagrams. Power counting criterion. Renormalization. Renormalization group equations.

Readings/Bibliography

see the up-to-dated bibliography included in the lecture notes available on-line.

Teaching methods

front lectures, written and oral exams.

Assessment methods

five or six written exams for academic year, oral exams by appointment.

Teaching tools

the notes of the whole course are available on-line, as well as the text of the written exams with solutions.

Lectures are delivered at the blackboad or by making use of transparencies.

Links to further information

http://www.robertosoldati.com

Office hours

See the website of Roberto Soldati