- Docente: Roberto Soldati
- Credits: 6
- SSD: FIS/02
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Physics (cod. 8025)
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 main
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.
THE 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. The zero point energy and the cosmological
constant puzzle. Normal modes decomposition of the free real scalar
field. Quantization of the free real scalar field. Creation and
destruction operators. 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 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 QUANTIZATION 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.
Readings/Bibliography
See the up-to-dated bibliography
included in the lecture notes available on-line with title:
First Semester Course, Introduction To Quantum Field Theory, A
Primer For Basic Education.
Teaching methods
Front teaching.
Assessment methods
Written test.
Teaching tools
Extended Notes for the whole Course are available on-line, as well as the text of the written exams with solutions.
Title: First Semester Course,
Introduction To Quantum Field Theory, A Primer For Basic
Education.
Lectures are delivered at the blackboad or by making use of slides.
Links to further information
Office hours
See the website of Roberto Soldati