This year at Sapienza University of Rome I followed Theory of Fundamental Interactions, a course built around a single question: what do you get if you insist on special relativity and quantum mechanics at the same time, and refuse to assume anything else?
The answer, developed over thirty-two lectures, turns out to be most of the Standard Model — not postulated, but forced. The route is Wigner’s: classify the unitary irreducible representations of the Poincaré group, discover that “particle” simply means “irrep”, and then watch fields, spin–statistics, gauge invariance, gravity and spontaneous symmetry breaking appear one after another as consistency conditions rather than as inputs.
What I’m Sharing Here
These are typed versions of the handwritten lecture notes, one document per lecture. Each one is:
- a faithful transcription of the notes as they were given, with the results that were highlighted in the original preserved in coloured boxes;
- extended with my own additions in blue boxes — motivations, the algebra that gets skipped at the board, terminology, and the occasional correction;
- accompanied by redrawn figures and page markers
[p. N]that point back to where each original page begins.
The point of typing them up was selfish: you cannot transcribe a derivation you do not follow. Every blue box marks a place where I had to stop and work something out. If they are useful to anyone else, so much the better.
How the Course Is Built
The thirty-two lectures fall into three movements:
- Lectures 1–12 — from symmetry to particles and fields. The Euclidean group as a warm-up, then the Poincaré group: massive and massless irreps, Wigner’s little group, helicity, and the passage from one-particle states to quantum fields, with microcausality doing the real work.
- Lectures 13–21 — interactions, and gravity as a field theory. The Lagrangian formalism and Noether’s theorem, minimal coupling and scalar QED, then the massive and massless spin-2 field, the graviton propagator, Newton’s law from graviton exchange, and why linearized gravity cannot be the end of the story.
- Lectures 22–32 — gauge theory and symmetry breaking. Yang–Mills, why a naive gauge-boson mass fails, spontaneous symmetry breaking and the Goldstone theorem, the CCWZ construction, the Higgs mechanism, and finally the electroweak spectrum and the fermionic sector.
A Note on Collaboration
These are student notes, so treat them as a work in progress. If you spot a typo in a formula, a sign that has gone missing, or somewhere I have misunderstood the argument in one of my own blue boxes, please tell me — the blue boxes are mine and the mistakes in them are mine too. Science is a collective effort, and I would much rather be corrected than quoted.
Connect with me on Email or X if you have any questions!
| Lecture | Topic | Material |
|---|---|---|
| 1 | The Euclidean group, its Lie algebra, and unitary irreps of SE(2) | |
| 2 | Completing the irreps of SE(2); the Poincaré group, its structure and Lie algebra | |
| 3 | The Poincaré algebra in covariant form; the massive case and Wigner’s little group | |
| 4 | Spin, SU(2) vs SO(3), the Pauli–Lubanski vector, and the Wigner rotation | |
| 5 | Unitarity of the massive irreps, the spin–polarization basis, and the massless case: helicity | |
| 6 | Massless irreps completed: the Wigner phase, parity, and the vacuum; from particles to fields | |
| 7 | Relativistic causality and quantum fields: microcausality and the particle–field connection | |
| 8 | The (j₁, j₂) representations of the Lorentz group, and the Klein–Gordon equation | |
| 9 | The massive vector field: polarization vectors, the polarization sum, and spin–statistics | |
| 10 | Completing the massive vector field, and the massless case: the photon field | |
| 11 | The obstruction, gauge invariance, and the graviton field | |
| 12 | Internal symmetries and the reason for antiparticles | |
| 13 | From free to interacting theories | |
| 14 | The massive spin-1 field and its massless limit | |
| 15 | The minimal coupling prescription | |
| 16 | Gravity as a QFT | |
| 16a | Addendum: the Weinberg soft theorem for photons | |
| 17 | Linearized gravity and the Weinberg soft theorem | |
| 18 | The graviton propagator and forces from particle exchange | |
| 19 | Newton’s law from graviton exchange, and why gravity must be non-linear | |
| 20–21 | Interactions between matter and gravity, and gravity as an effective field theory | |
| 22 | Non-abelian gauge theories | |
| 23 | Semi-simple gauge groups, the matter sector, and the failure of massive Yang–Mills | |
| 24 | Spontaneous symmetry breaking | |
| 25 | Goldstone bosons: field redefinitions, effective field theory, and the Goldstone theorem | |
| 26 | The Goldstone theorem proved, and two notable breaking patterns | |
| 27 | The coset space, the Goldstone matrix, and the transformation of Goldstone bosons | |
| 28 | Coset representatives, and the CCWZ prescription | |
| 29 | The two-derivative CCWZ Lagrangian, and the Higgs mechanism (abelian case) | |
| 30 | The non-abelian Higgs mechanism, and a Standard Model primer | |
| 31 | The electroweak spectrum: W±, Z, the photon, and the bosonic interactions | |
| 32 | The fermionic sector, the generation of mass, and the electroweak phase transition |