Lecturer. Massimiliano Gubinelli (University of Oxford).
This course is part of the Probability and Stochastic Analysis Trimester Programme at SISSA.
In these lectures I will review some mathematical problems associated to the definition and construction of quantum field theories. Mainly from the probabilistic viewpoint but also paying some attention to the quantum aspects. The goal is to give a wide perspective on the open problems and opportunities for research, in the era of large language models.
Lecture 1 (20 October 2026): Relativistic quantum fields and imaginary time. Wightman axioms and quantum fields as operator-valued distributions; the two-point function; positivity of energy and continuation to imaginary time; the Feynman–Kac representation, lattice spin systems and the Osterwalder–Schrader axioms. [slides (PDF)] [script (PDF)]
The following plans are provisional and may be adjusted as the course progresses.
Lecture 2: The Gaussian free field. Construction on the torus and Euclidean space; reflection positivity and the free relativistic field; Wick products; Gaussian integration by parts and the Dyson–Schwinger equation; parabolic and elliptic stochastic quantisation.
Lecture 3: Stochastic observables, the Polchinski flow and forward–backward equations. Decomposition of the field by resolution and stochastic observables; effective potentials and the Polchinski equation; forward–backward stochastic differential equations; the Wetterich equation and the variational structure of the flow. Further topics, time permitting: products of observables, perturbation theory and counterterms, and Dyson–Schwinger equations at every scale.
All dates are in 2026; times are local to SISSA, Trieste.
20 October, 15:00–17:00, Room 134.
22 October, 11:00–13:00, Room 133.
30 October, 11:00–13:00, Room 136.
4 November, 14:00–16:00, Room 133.
5 November, 11:00–13:00, Room 136.
6 November, 14:00–16:00, Room 136.