S4F3 - Graduate Seminar on Applied Probability – WS2021

Nonperturbative Renormalization

Massimiliano Gubinelli 

Preliminary meeting: January 28nd, 16. Via Zoom.

Seminar will be held from April 7th to April 15th 2021 (see below details, mail me for connection parameters).

The idea of this seminar is to give a full worked out example of the use of rigorous non-perturbative renormalization group equations to study the scaling limit of certain models of mathematical physics. In particular we will base the core of the seminar on the paper

Giuliani, Alessandro, Vieri Mastropietro, and Slava Rychkov. ‘Gentle Introduction to Rigorous Renormalization Group: A Worked Fermionic Example’. ArXiv:2008.0436 2020. http://arxiv.org/abs/2008.04361

taking some additional material from the book

Mastropietro, Vieri. Non-Perturbative Renormalization. Hackensack, NJ: World Scientific Publishing Co Pte Ltd, 2008.

While the setting of Fermionic theory in the Euclidean formalism is not very familiar to mathematicians, it has the great advantage that perturbation theory for Fermions is very often convergent. In this way one can obtain truly non-perturbative results for models which do not have closed-form solutions and one can make simpler many arguments which are quite challenging (if not not yet fully understood) in Bosonic theories.

Material

• Notes from the organization seminar (pdf)
• The history of the paper (link, skim down in the "What's new" section)
• A seminar by S. Rychkov on the paper (link)
• Useful lecture notes on the Grassmann formalism in RG (link) "Fermionic Functional Integrals and the Renormalization Group" by Feldman, Knörrer and Trubowitz.

Topics of the various talks and speakers

• 7/4 10h00 - Definition of the model, Berezin integrals, definition of the renormalization step and the integrating out map [Eq. (5.2) / Appendix B ] - Sebastian Schmidt (slides)
• 8/4 10h00 - Various representation for the fermionic expectations (Appendix D / Book) - Massimiliano Gubinelli (slides)
• 8/4 15h00 - Finite volume representation and infinite volume limit (Appendix H)  Chunqiu Song (slides)
• 9/4 10h00 - Renomalization map in the trimmed representation and fixed point equation (Sect 5.4, 5.5, Appendix C) Introduce the Banach space ℬ for effective actions - Margherita Disertori (slides)
• 9/4 15h00 - Norm bounds (Sect 5.6 / Appendix E) (needs Appendix D). Control of ℛ_γ:ℬ→ℬ  - Francesco de Vecchi (slides)
• 12/4 10h00 - Construction of the fixed point (Section 6) - Luca Fresta (slides)
• 13/4 10h00 - Proof of the key lemma (Section 7) - Mattia Turra (slides)
• 15/4 10h00 - The RG in classical PDE theory -- Lucio Galeati (slides)

 

Complementary literature

• Giovanni Jona-Lasinio, ‘Renormalization Group and Probability Theory', Physics Reports 352, no. 4–6 (October 2001): 439–58, https://doi.org/10.1016/S0370-1573(01)00042-4.
• Kenneth G. Wilson, ‘The Renormalization Group and Critical Phenomena', Reviews of Modern Physics 55, no. 3 (1983): 583–600, https://doi.org/10.1103/RevModPhys.55.583.
• P.K.Mitter:The Exact Renormalization Group, Encyclopedia in Mathematical Physics, Elsevier 2006, http://arXiv:math-ph/0505008
•  Bertrand Delamotte, ‘An Introduction to the Nonperturbative Renormalization Group', ArXiv:Cond-Mat/0702365, 15 February 2007, http://arxiv.org/abs/cond-mat/0702365.
• Giovanni Gallavotti, ‘Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods', Reviews of Modern Physics 57, no. 2 (1 April 1985): 471–562, https://doi.org/10.1103/RevModPhys.57.471.
• Manfred Salmhofer, Renormalization: An Introduction, 1st Corrected ed. 1999, Corr. 2nd printing 2007 edition (Berlin; New York: Springer, 2007).
• Joseph Polchinski, ‘Renormalization and Effective Lagrangians', Nuclear Physics B 231, no. 2 (January 1984): 269–95, https://doi.org/10.1016/0550-3213(84)90287-6.
• David C. Brydges, Roberto Fernández, Functional Integrals and Their Applications, 1993.