Lecturer. Massimiliano Gubinelli (University of Oxford).
Joint work. The results presented in Lectures 6–7 are joint work with Sarah-Jean Meyer, obtained recently with the use of AI.
Acknowledgements. M. Gubinelli is partly supported by UK Research and Innovation via the grant “StochFields” EP/Z534328/1 and by the Simons Collaboration on Probabilistic Paths to QFT. S.-J. Meyer is supported by the grant “StochFields” EP/Z534328/1.
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. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 7 November 2026.
Lecture 2 (22 October 2026): The Gaussian free field. Construction on the torus and Euclidean space; reflection positivity and the free relativistic field; Nelson's Markov property and Symanzik's Brownian-path representation; Wick products; Gaussian integration by parts and the Dyson–Schwinger equation. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 7 November 2026.
Lecture 3 (30 October 2026): Stochastic quantisation and stochastic observables. Parabolic and elliptic stochastic quantisation of the Gaussian free field; decomposition of the field by resolution; stochastic observables, with examples from Brownian motion and the free field; Wick powers in every dimension, products of observables and the operator product expansion. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 7 November 2026.
Lecture 4 (4 November 2026): Forward–backward equations. Renormalisation, optimal control and filtering as three ways to generate the interacting measure; force coordinates; the value function as a generator, the Legendre transform and the Wetterich equation. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 9 November 2026.
Lecture 5 (5 November 2026): Observables and field equations. Observables of the interacting theory; from the Minkowski to the Euclidean field equation, and Nelson's Euclidean noise; the field equation at every resolution, with a sub-Gaussian noise; Dyson–Schwinger equations, also in pathwise form; infinite volume. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 9 November 2026.
Lecture 6 (6 November 2026): The FBSDE in force coordinates, two renormalisations and the large-field problem. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 9 November 2026.
Lecture 7 (9 November 2026): A priori estimates, UV stability and the limiting FBSDE: an explicit density for the measure and a renormalised variational formula. Notes: [PDF] [HTML] [TeXmacs]. Slides: [PDF] [HTML] [TeXmacs]. Revised 9 November 2026.
Supplement to Lectures 6–7: Detailed proofs of what the lectures cite or condense: the coefficient construction, the large-field ledger, consistency of the coefficients, and older routes. Notes: [PDF] [HTML] [TeXmacs]. Revised 9 November 2026.
Materials revised 9 November 2026. The TeXmacs sources of all notes and
slides, together with the style packages they use, are also available as
one archive [zip]. In the HTML slides, use the arrow keys
or click to navigate; press o for an overview.
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, 11:00–13:00, Room 133.
9 November, 11:00–13:00, Room 133.