Master's Courses

General Relativity I.pdf

General Relativity I

Given by Prof. Roberto Casadio, the course General Relativity I includes the following topics in order.

Introduction: Old covariant formalism, differentiable manifolds, vector spaces, integral curves, one-forms, tensors and tensor fields, lengths and angles, metric tensor.
Symmetries: Active and passive diffeomorphisms, Lie dragging, Lie derivatives, isometries, Frobenius' theorem.
Integrals and Forms: p-forms, volume forms, differential forms.
Covariant Derivatives: Parallel transport, connections, geodesics and geodesic equation, Riemann tensor, Ricci tensor, Ricci scalar, Einstein tensor.
General Theory of Relativity: Equivalence principles, energy-momentum tensor, Einstein field equations, Einstein-Hilbert action.
Linearised Einstein Gravity: Weak field limit and Newtonian regime, de Donder gauge, gravitational waves, quadrupole formula.
Schwarzschild Spacetime: Schwarzschild solution, radial geodesics, gravitational redshift, infalling probes, orbits, star solutions.
Cosmological Models: Cosmological principle, FLRW metric, Friedmann equations.

Quantum Field Theory I

Given by Prof. Michele Cicoli, the course Quantum Field Theory I includes the following topics in order:

Introduction: Necessity of QFT, a mechanical model of fields, quantisation of elastic springs.
Spacetime symmetries: Finite and infinite representations of SO(1,3), spinor representations, Dirac and Weyl bases.
Classical Field Theory: Nöether theorem, symmetries, energy-momentum and Belinfante-Rosenfeld tensors.
Klein-Gordon Theory: Canonical quantisation of scalar fields, Klein-Gordon spectrum, Heisenberg picture, correlators.
Dirac Theory: Spinors and their canonical quantisation, Dirac equation, helicity, symmetries of Dirac theory, propagators.
Spin-1 Particles: Maxwell Lagrangian, gauge theory of electromagnetism, quantisation of the electromagnetic waves, the physical Hilbert space and Gupta-Bleuler condition.

Quantum Field Theory I.pdf

Relativistic Quantum Mechanics and Path Integrals.pdf

Relativistic Quantum Mechanics and Path Integrals

Given by Prof. Fiorenzo Bastianelli, the course Relativistic Quantum Mechanics and Path Integrals includes the following topics in order:

Relativistic Quantum Mechanics

Introduction to Group Theory: Vector spaces, group axioms, representations of groups.
Lie Algebras: Generators and algebras, special relativity, finite representations of the Lorentz group, unitary representations of the Poincaré group.
Klein-Gordon Equation: Mode expansions of scalar fields, Yukawa potential, Green functions and propagators, Klein-Gordon action, conserved charges.
Dirac Equation: Spinors and Clifford algebra, non-relativistic limit of the Dirac equation, covariance of the Dirac equation, spinor identities, discrete symmetries, Dirac action.
Particles of Higher Spin: Pauli-Fierz equations, Proca equation, Maxwell theory, spin-2 particles.

Path Integrals:

Introduction: Transition amplitudes, path Integrals in phase and configuration space, Wick rotation.
Correlation Functions: Correlators, generating functionals, Gaussian theories, harmonic oscillators.
Perturbative Expansions: Third and fourth order expansions, Dyson formula, Feynman diagrams.

Statistical Mechanics

Given by Prof. Elisa Ercolessi, the course Statistical Mechanics includes the following topics in order:

Review of Thermodynamics: Laws of thermodynamics, fundamental equations, thermodynamic potentials.
Review of Classical Mechanics: Phase space, Hamiltonian formalism, ensembles and probabilities.
Ensembles: Microcanonical ensemble, Boltzmann's formula, canonical ensemble, partition function, equipartition theorem, grandcanonical ensemble, virial expansion.
Phase Transitions: Clasius-Clapeyron equation, first and second order phase transitions, theorems of Lee and Young.
Ising Model: Magnetisation, spontaneous and explicit symmetry breaking, mean field approximation, critical exponents, correlators.
Review of Quantum Mechanics: States, density matrix, identical particles, second quantisation, quantum ensembles.
Quantum Gases: Quantum partition functions, Bose-Einstein and Fermi-Dirac distributions, semi-classical limit, free Fermi gas, Bose gas, Bose-Einstein condensation.

Statistical Mechanics.pdf

Introduction to String Theory.pdf

Introduction to String Theory

Given by Prof. Ling Lin and Prof. Michele Cicoli, the course is divided into two modules and includes the following topics in order.

Ling Lin

Classical Relativistic Strings: Nambu-Goto and Polyakov action, classical solutions to Polyakov string, boundaries of open strings, the Witt algebra.
Quantum String: Old covariant quantisation, Virasoro algebra, bosonic string theory, closed strings in spacetime, lightcone quantisation, string spectrum.
String Scatterings: Diagrams, vertex operators, state-operator correspondence, moduli space, 1-loop amplitudes.
Strings in Background Fields: Spacetime perspective, non-linear sigma model, D-branes, DBI action.
Superstrings: Supersymmetry, superstring mode expansion, quantum superstring, superstring spectrum, GSO projection, Type II string theories.

Michele Cicoli

String Phenomenology: Compactifications, cosmological moduli problem, moduli stabilisation in heterotic theories, Type II compactifications, dualities, Type IIB string phenomenology, 10D Type IIB supergravity

Statistical Field Theory

Given by Prof. Lorenzo Piroli and Prof. Francesco Ravanini, the course is divided into two modules and includes the following topics in order.

Statistical Field Theory (Lorenzo Piroli)

Review of Statistical Mechanics: Postulate of equal a priori probabilities, microcanonical and canonical ensembles, quantum statistical mechanics.
Phase Transitions: Critical exponents, mean field theory, exact solution of Ising model in 1D, Peierl's argument.
Landau-Ginzburg Theory: Mesoscopic degrees of freedom, free energy, scaling hypothesis.
Renormalisation Group: RG techniques, fixed points, conceptual and formal frameworks, Gaussian models.

Conformal Field Theory (Francesco Ravanini)

QFT and Statistical Mechanics: Wick rotation, classical dimensions and relevance of operators.
Conformal Symmetry: Conformal transformations, Noether current, algebra of local fields.
Conformal Group in D=2: Witt algebra, conformal transformations of primary fields, operator product expansions, conformal generators, conformal Ward identities.
Virasoro Algebra: Virasoro generators, central extension, global conformal subalgebra.
Radial Quantisation: State-operator correspondence, action of Virasoro generators, Verma module.
Minimal Models: Gram matrix, Kac determinant, Kac table, fusion rules, topology of models.

Statistical Field Theory.pdf

Quantum Field Theory II.pdf

Quantum Field Theory II

Given by Prof.Tiziano Peraro, the course includes the following topics in order.

Introduction and Review of Quantum Field Theory
Interacting Fields: S-matrix, cross sections and decay rates, Green functions and the LSZ reduction formula.
Feynman Diagrams: Dyson series, Wick theorem, Feynman rules in momentum space, applications of Feynman diagrams, crossing symmetry.
QED Scattering: Unpolarised cross sections, external photons, QED as an Abelian gauge theory, Born approximation.
Path Integrals: Path integrals in interacting theory, quantisation of QED, symmetries with path integrals.
Loop Amplitudes: Regularisation and renormalisation, renormalisation schemes, one-loop integrals in dimensional regularisation, renormalisation with counterterms, minimal subtraction schemes, systematics of renormalisation, renormalisability from dimensional analysis, non-renormalisable theories.
QED at One-Loop Level: On-shell renormalisation for QED, vacuum polarisation, anomalous magnetic moment of the electron.

Group Theory

Given by Prof. Roberto Zucchini and Prof. Ling Lin, the course is divided into two modules and contains the following subjects:

Module 1 (Roberto Zucchini)

Groups in Quantum Mechanics: Quantum physics, dynamical symmetry groups, unitary representations.
Lie Groups: Unitary representations of Lie groups, dynamical Lie symmetries.
Vector Geometry of R(3): Direct and wedge products, Pauli matrix formalism, rotation group O(3), Lorentz group.

Module 2 (Ling Lin)

Lie Manifolds: Topological structures, Lie algebras, Lie algebras of matrix groups, the exponential map.
Representation Theory: Adjoint representation, reducibility of representations.
Roots and Weights: Killing forms and Cartan subalgebras, root decomposition, geometry of roots, weights and representations.
Note: This module also contains applications of representation theory including Clebsch-Gordon decomposition, classification of Lie groups, Gell-Mann Eightfold Way, Standard Model and Beyond Standard Model theories. These are not included in my notes.

Group Theory for Physics.pdf

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