Author: Carolina Saggiomo

Academic Team

Course Content

Type: Optional – PEC Track – 1st Trimester

This course offers a comprehensive introduction to the theory and practice of quantum information and quantum computation, exploring their theoretical foundations, applications, and physical implementations. It addresses the concept of quantum entanglement both as an essential resource for quantum computing and communication, and as a tool for analyzing many-particle systems.

The syllabus ranges from classical computation models (Turing machine, circuit model, and logic gates) and thermodynamic principles of information (Landauer’s principle, Szilard engine) to key quantum phenomena such as the no-cloning theorem, quantum teleportation, and dense coding. Fundamental quantum algorithms are studied (including Deutsch, Grover, Quantum Fourier Transform, Phase Estimation, and Shor) along with universal quantum computation.

The course delves into quantum information theory (density matrix, Schmidt decomposition, generalized measurements, Shannon and Schumacher theorems) and techniques for characterizing and measuring entanglement. It also examines decoherence, quantum noise, noisy channels, and strategies for quantum error correction and fault tolerance.

A core component of the training is hands-on experience: students will use online quantum computing platforms such as IBM Quantum Experience and work with Qiskit to design, simulate, and execute simple quantum circuits.

Reading List

• M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information. Cambridge University Press, Cambridge, UK (2000).
• J. Preskill, Notes on Quantum Computation. N.D. Mermin, “Quantum computer science: an introduction”. Cambridge University Press, Cambridge, UK (2007).

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Course Content

Type: Optional – PEC Track – 3rd Trimester

This course provides a comprehensive overview of the current state of experimental particle physics, introducing students to the key experiments that led to the development of the Standard Model.

The first block covers the fundamentals and techniques of modern High Energy Physics experiments, including particle accelerators, detectors, and methods for reconstructing particle properties. It also reviews experimental tests of the Standard Model, such as measurements in QCD, QED, and electroweak theory, CKM matrix determinations, and searches for phenomena like the Higgs boson and new particles.

The second block focuses on hands-on learning through supervised projects using real data from the ATLAS and CMS experiments at the LHC. Students will practice programming, data analysis, and visualization, develop strategies for signal selection, control background noise, and evaluate statistical significance.

Reading List

• Robert N. Cahn, G. Goldhaber; The Experimental Foundations of Particle Physics; Cambridge University Press (1993).
• W.R. Leo; Techniques for Nuclear and Particle-Physics Experiments; Springer-Verlag Berlin (1987).
• Andrew Pickering; Constructing Quarks; The University of Chicago Press (1984).
• A. Ferrer, E. Ros; Fisica de partículas y de astropartículas; Publicaciones Universidad de Valencia (2005)

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Type: Optional– PEC Track – 3rd Trimester

This course explores the key unresolved questions in the Standard Model of Fundamental Interactions and the theoretical approaches proposed to address them. Students will examine topics such as chiral symmetry and light quarks, anomalies and their phenomenological impact, the hierarchy and flavour problems, neutrino physics, and the strong CP problem.

The syllabus covers:

• Symmetries of the Standard Model and the nature of fermion masses (Dirac and Majorana).
• Anomalies, B–L conservation, and chiral symmetry.
• Effective field theories and the hierarchy problem.
• Flavour physics: CKM and PMNS matrices, CP violation, the GIM mechanism, and matter–antimatter asymmetry.
• Neutrino oscillations, experimental results, and the Seesaw mechanism.
• The strong CP problem, electric dipole moments, and the axion.

Through lectures and discussions, students will gain a deeper understanding of the conceptual and experimental challenges that drive research beyond the Standard Model, equipping them with the knowledge base needed to engage with cutting-edge developments in particle physics.

Reading List

• Gauge Theory of Elementary Particle Physics, T.-P. Cheng & L.-F. Li (Oxford University Press)

• Dynamics of the Standard Model, J. Donoghue, E. Golowich & B. Holstein (Cambridge University Press)

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Type: Optional– PEC Track – 2nd Trimester

This advanced Quantum Field Theory (QFT) course focuses on the path integral formulation as a unifying framework to describe and compute physical processes in high-energy physics. The course builds a deep understanding of renormalization in QFT, with a strong emphasis on gauge theories—both abelian (QED) and non-abelian (QCD). Students will develop technical skills to derive Feynman rules from the path integral, calculate loop corrections, and analyze the behavior of couplings via Renormalization Group (RG) equations.

A significant part of the course is devoted to the spontaneous breaking of symmetries—both global and local—and the Higgs mechanism, providing the theoretical foundation for the mass generation of gauge bosons in the Standard Model. By the end of the course, students will be equipped with the mathematical and conceptual tools needed for cutting-edge research in theoretical particle physics and related fields.

Reading List

• T. Banks, Modern Quantum Field Theory: A Concise Introduction, Cambridge University Press.
• M. Peskin & D. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley.
• M. Srednicki, Quantum Field Theory, Cambridge University Press.
• S. Weinberg, The Quantum Theory of Fields, Vols. I & II, Cambridge University Press.
• A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press.
• J. Collins, Renormalization, Cambridge Monographs on Mathematical Physics.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press.

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Course Content

Type: Optional– PEC Track – 1st Trimester
This course provides a solid foundation in the key mathematical tools used in modern theoretical physics, with a focus on differential geometry, group theory, and probability & statistics. Students will not only learn the theory but also apply these methods to real physical problems and data analysis. Emphasis is placed on mathematical reasoning, problem-solving, and collaborative work.

Main Topics

• Differential Geometry – Manifolds, tensor algebra, vector bundles, connections, curvature, Riemannian geometry, and basics of manifold topology.
• Group Theory – Lie groups and algebras, representation theory, classification of algebras, and applications to symmetries in physics (Poincaré, Lorentz, internal symmetries).
• Probability & Statistics – Probability distributions, estimators, hypothesis testing, Bayesian statistics, and Monte Carlo simulations.

Reading List

Group Theory:
1) H. Georgi: “Lie Algebras in Particle Physics”,
2) H.F. Jones: “Groups, Representations and Physics”
3) J.F. Cornwell: “Group Theory in Physcis”, Vols. 1,2

Differential Geometry:
1) Nash, Sen: “Topology and Geometry for Physicists”
2) M. Nakahara: “Geometry, Topology and Physics”

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Course Content

Type: Compulsory– PEC Track – 2nd Trimester

This course offers a comprehensive introduction to the Standard Model (SM), the fundamental theoretical framework describing electromagnetic, weak, and strong interactions. Students will explore the structure, symmetries, and phenomenology of the SM, along with its experimental verification.

Main Topics

• Structure and symmetries of the Standard Model.
• Quantum Electrodynamics (QED) refresher.
• Quantum Chromodynamics (QCD) and strong interactions.
• Electroweak theory and unification.
• Higgs mechanism and mass generation.
• Flavour physics, mixing, and CP violation.
• Experimental tests and validation of the SM.

Skills Acquired

• Understand the theoretical foundations of the SM and its construction.
• Describe the role of symmetries and conservation laws in particle physics.
• Explain how mass arises via the Higgs mechanism.
• Connect theoretical predictions with experimental results.

Reading List

• T.-P. Cheng & L.-F. Li, Gauge Theory of Elementary Particle Physics.
• I. Aitchison & A. Hey, Gauge Theories in Particle Physics.

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Academic Team

Course Content

Type: Compulsory– PEC Track – 1st Trimester

This course introduces Quantum Field Theory (QFT) in the operator formalism, providing the essential tools to describe interactions among elementary particles. Beginning with the quantization of scalar and spinor fields, students progress to Quantum Electrodynamics (QED), symmetry principles, and an introduction to non-abelian gauge theories. The course blends conceptual foundations with practical computational techniques, including Feynman diagrams, scattering amplitudes, and renormalization.

Main Topics

• Foundations of QFT – Motivation, connection between quantum mechanics and special relativity, spin–statistics, locality, and causality.
• Scalar Fields – Canonical quantization, propagators, vacuum energy, Casimir effect, and interactions in scalar theories (λϕ⁴, Yukawa).
• Spinor Fields – Dirac and Weyl equations, chirality, Majorana fermions, fermion quantization, and Feynman rules for fermions.
• Quantum Electrodynamics (QED) – Gauge invariance, quantization of the electromagnetic field, QED Feynman rules, key scattering processes (Bhabha, Compton), and loop corrections.
• Renormalization – Power counting, divergences, dimensional regularization, and electron self-energy.
• Non-Abelian Gauge Theories – Lagrangians for gauge fields and fermions, gauge invariance, running couplings, and asymptotic freedom.

Reading List

• An Introduction to Quantum Field Theory, M.E. Peskin and D.V. Schroeder. Addison-Wesley Pub. Co. (1995).
• The Quantum Theory of Fields, Vols. I and II, S. Weinberg. Cambridge Univ Press (1995). Curso 2018-2019.
• Quantum Field Theory in a Nutshell, A. Zee. Princeton University (2003).
• Quantum Field Theory, M. Srednicki. Cambridge Univ Press (2007).
• Modern Quantum Field Theory, T. Banks. Cambridge Univ Press (2008).
• Quantum Field Theory, C. Itzykson and J.B. Zuber. McGraw Hill (1980).
• Field Theory: a Modern Primer, P. Ramond. Benjamin (1981).

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Content Course

Type: Optional– PEC&AC Track – 3rd Trimester

This course covers advanced mathematical and theoretical tools for modern gravitational physics. It combines in-depth study of differential geometry, Hamiltonian formulations of General Relativity, and black hole thermodynamics, with an introduction to extensions of Einstein’s theory.

Main Topics

• Advanced differential geometry: differential forms, Lie groups and algebras, Yang–Mills fields, spinors in curved space, first-order formalism of General Relativity, and elements of supergravity.
• Hamiltonian formalism of GR: ADM mass, time, and conserved quantities.
• Conserved charges in gauge and gravitational theories: Komar integrals, Abbott–Deser approach, Witten’s positive mass theorem.
• Black hole thermodynamics: laws of black hole mechanics, cosmic censorship, singularity theorems, horizon topology, and no-hair results.
• Extensions of GR: scalar–tensor theories, f(R)f(R)f(R) gravity, Lovelock theories, topological terms, and 3D gravity.

Reading List

• Spacetime and Geometry, Carroll, Cambridge University Press (2019)
• Gravitation; Misner, Thorne, Wheeler, and Freeman (1970)
• General Relativity; Wald, The University Chicago Press (1984)
• Gravitation and Cosmology; Weinberg, Addison Wesley (1978)
• Gravity: Newtonian, Post Newtonian, Relativistic; Poisson and Will, Cambridge University Press (2014)
• The large scale structure of spacetime; Hawking and Ellis, Cambridge University Press (1973)

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Course Content

Type: Optional– PEC&AC Track – 3rd Trimester

Module A. Introduction. The Homogeneous Universe
The standard Cosmological Model. Cosmological parameters.
Problems of the Big Bang model.
The Inflationary solution.

Module B: The Non-Homogeneous Universe: Perturbations in the Matter Distribution
Linear theory of perturbation evolution in Friedmann universes.
Non-linear perturbation theory: Zeldovich approximation and N-body simulations.

Module C: Background Radiation
Propagation of a radiation field in an inhomogeneous Friedmann Universe.
Origin of anisotropies in the Background Radiation: Sachs-Wolfe effect, integrated Sachs-Wolfe effect.
Acoustic oscillations on the last scattering surface. Adiabatic and isocurvature modes.
Recent observations and implications for structure formation models. Cosmological parameters.
Sunyaev-Zeldovich effect: hot gas in clusters and secondary anisotropies in the background radiation.

Module D: Large-Scale Structure
Observations of matter distribution.
Statistical methods in Cosmology.
Measurement of the correlation function and power spectrum from galaxy catalogs.
Luminosity functions and mass functions. Comparison between simulations and observations.
Baryon acoustic oscillations in the galaxy distribution. Spectroscopic and photometric surveys.
Determination of cosmological parameters from large-scale structure.

Module E: Gravitational Lenses
History. Basic concepts.
Equations of gravitational lensing effect.
Strong and weak lenses.
Applications in Cosmology.

Reading List

• Modern Cosmology – S. Dodelson. Academic Press (2003).
  The Cosmic Microwave Background – R. Durrer. Cambridge University Press (2008).
• The Early Universe – E.W. Kolb, M.S. Turner. Addison-Wesley (1990).
• The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure – D.H. Lyth, A. Liddle. Cambridge U.P. (2009).
• Statistics of the Galaxy Distribution – V. J. Martínez and E. Saar. Chapman & Hall (2002).
• Structure Formation in the Universe – T. Padmanabhan. Cambridge U. P. (1993).
• Cosmological Physics – J.A. Peacock. Cambridge University Press (1998).
• Large Scale Structure of the Universe – P.J.E. Peebles. Princeton U.P. (1980).
• Cosmology – S. Weinberg. Oxford U.P. (2008).
• Extragalactic Astronomy and Cosmology – P. Schneider. Springer-Verlag (2006).
• Dark Energy – Yun Wang. Wiley-VCH (2010).
• Practical Statistics for Astronomers – J.V. Wall and C.R. Jenkins. Cambridge Univ. Press.
• Measuring our Universe from Galaxy Redshift Surveys, O. Lahav and Y. Suto (2004)
• Lectures on Gravitational Lensing, Ramesh Narayan & Matthias Bartelmann (1996), astro-ph/9606001
• Gravitational Lensing in Astronomy, Joachim Wambsganss (1998), Living Reviews,
• Cosmological Parameters from Observations of Galaxy Clusters, S. Allen (2011)
• The evolution of the large-scale structure of the universe: beyond the linear regime, F. Bernardeau (2013)
• Dark Energy and CMB (S. Dodelson et al., 2013)
• Distance Probes of Dark Energy (Kim, Padmanabhan et al., 2013)
• Cosmological Perturbation theory (classic paper by Ma and Bertschinger)
• Planck cosmological results
• Supernova Ia cosmology
• Dark energy review

All the course information is available on its website.

Teaching Plan Information

Course Schedule

Academic Team

Course Content

Type: Optional– PEC&AC Track – 3rd Trimester

This course has been taught in the postgraduate programmes at Madrid Autónoma University, Durham University, the University of Valencia (2018), and The Max Planck Institute München (2009). It has also been taught (in different formats) in a number of postgraduate schools: Taller de Altas Energías (TAE 2014, 2015, 2017, 2018, 2019, 2021), 1st International School on Particle Physics and Cosmology (UIMP 2019), Higgs Centre School of Theoretical Physics (2016), STFC HEP School (2015, 2016, 2017).

Some notes for the course (pdf).
Exercises (pdf).
Extra exercise (pdf).

1.- Introduction

In this session, we present the observational evidence that points towards the existence of dark matter. We introduce the concept of dark matter halo and think about its properties (slides).

2.- Cosmology 101

A brief reminder of Early Universe Cosmology. Emphasis is put on how to compute the abundance of a given species in equilibrium. The Boltzmann equation that describes the evolution of the number density is derived (slides).

3.- Dark Matter Production (Freeze-out)

We apply the Boltzmann equation to the case of a non-relativistic DM particle that “freezes-out”, and define WIMPs (slides).

4.- Dark Matter Production (WIMP models and Freeze-in)

We study some specific particle realisations of WIMPs (concentrating on simplified models) and study special cases, such as resonant annihilation and co-annihilations. We then introduce a new paradigm, where dark matter with very small couplings “freezes-in” (slides).

5- Axions

We review the misalignment mechanism for the production of axions. The cosmological implications are reviewed. We then comment on axion detection.

6- Direct Detection

We show the basics of direct dark matter detection. We derive the equation for the observed detection rate and comment on uncertainties associated to nuclear physics and astrophysical parameters of the dark matter halo (slides).

EXERCISES

Exercises for the course (solutions available upon request) (pdf). 

Extra exercise (pdf)

Reading List

Group Theory:
1) H. Georgi: “Lie Algebras in Particle Physics”,
2) H.F. Jones: “Groups, Representations and Physics”
3) J.F. Cornwell: “Group Theory in Physcis”, Vols. 1,2

Differential Geometry:
1) Nash, Sen: “Topology and Geometry for Physicists”
2) M. Nakahara: “Geometry, Topology and Physics”

All the course information is available on its website.

Teaching Plan Information

Course Schedule