Constructive Euclidean Quantum Field Theory
Reading Seminar WS 2026
Constructive Euclidean Quantum Field Theory
Reading Seminar WS 2026
News: The reading seminar will start on Thursday 22.10.2026
Description: Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, they can be described by Gibbs measures associated with Euclidean action functionals over spaces of distributions. In this seminar, we discuss the rigorous construction of such theories, in particular we will see the construction of the phi^4_d theory.
Audience: Master or PhD students of Mathematics as well as motivated Bachelor (third year) students
Language: English
Time and place: Thursdays 10:15-12:00, Online via Zoom. You can find here the Zoom link
Important: If you are interested in participating to this reading seminar please write me an email at cristina.caraci@math.lmu.de
Prerequisites: Measure-theoretic probability (Gaussian measures, weak convergence) and functional analysis (L^p and Sobolev spaces, Fourier series, distributions). Familiarity with Brownian motion and Itô calculus is helpful; Besov spaces and the necessary tools from stochastic analysis will be introduced during the seminar. No background in physics or quantum field theory is required.
(Tentative) Schedule:
22.10.2026 Introduction and distribution of the material
05.11.2026 Construction of the Gaussian free field. Wick ordering and hypercontractivity.
19.11.2026 Construction of the phi^4_2 measure.
03.12.2026 Renormalization in three dimensions and the variational formula.
17.12.2026 Construction of the phi^4_3 measure I: the renormalized variational problem and ultraviolet stability.
07.01.2027 Construction of the phi^4_3 measure II
21.01.2027 (tbc) Singularity of the phi^4_3 measure with respect to the free field.
04.02.2027 The phi^4_3 theory on R^3 and the Osterwalder–Schrader axioms.
References:
B. Simon. The P(φ)₂ Euclidean (Quantum) Field Theory. Princeton University Press, 1974.
J. Glimm and A. Jaffe. Quantum Physics: A Functional Integral Point of View. 2nd ed., Springer, 1987.
E. Nelson. A quartic interaction in two dimensions. In: R. Goodman and I. Segal (eds.), Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA, 1966, pp. 69–73.
E. Nelson. The free Markoff field. J. Funct. Anal. 12 (1973), 211–227.
K. Osterwalder and R. Schrader. Axioms for Euclidean Green’s functions. Comm. Math. Phys. 31 (1973), 83–112; II, Comm. Math. Phys. 42 (1975), 281–305.
J.-C. Mourrat, H. Weber, and W. Xu. Construction of Φ⁴₃ diagrams for pedestrians. In: P. Gonçalves and A. J. Soares (eds.), From Particle Systems to Partial Differential Equations, Springer Proc. Math. Stat. 209, Springer, 2017, pp. 1–46.
M. Gubinelli and N. Perkowski. Lectures on singular stochastic PDEs. Ensaios Matemáticos 29 (2015), 1–89. arXiv:1502.00157.
M. Boué and P. Dupuis. A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 (1998), 1641–1659.
N. Barashkov and M. Gubinelli. A variational method for Φ⁴₃. Duke Math. J. 169 (2020), 3339–3415.
N. Barashkov and M. Gubinelli. The Φ⁴₃ measure via Girsanov’s theorem. Electron. J. Probab. 26 (2021), paper no. 81, 1–29.
M. Gubinelli and M. Hofmanová. A PDE construction of the Euclidean Φ⁴₃ quantum field theory. Comm. Math. Phys. 384 (2021), 1–75.
Other references:
Lecture notes from the course "Gaussian spaces and field theory"
Lecture notes by Antti Knowles "Bosons and Gaussian fields"
Lecture notes by Martin Hairer "Advance Stochastic analysis"
Janson, Gaussian Hilbert Spaces;