A comic with four panels, each depicting a pair of scientists in front of a blackboard scribbled with equations and diagrams. The first panel depicts a pair of young theoretical physicists, in 1968, joyously discussing physics in front of a board with Feynman diagrams. The second panel depicts a similar scene in 1968 with a pair of young mathematicians, in front of a board with advanced mathematics. The next two panels depict the same people and same boards, but in 1998. Both pairs of people are now old, and the mathematicians stand in front of the board with physics, perplexed, as perplexed as the physicists in front of the board with mathematics.

The Mathematics of Quantum Field Theory

a learning seminar

by

John Huerta and Gonçalo Oliveira

Practical information

What is this?

This semester (Spring 2024), John Huerta and Gonçalo Oliveira are running a learning seminar on the mathematics of quantum field theory. We started on March 18th, and will continue into July.

The seminar consists of two tracks:

Gonçalo is discussing constructive quantum field theory, leading up to the construction of interacting scalar field theories in two-dimensions.
John is discussing the Batalin–Vilkovisky (BV) formalism, leading up to the perturbative quantization of Yang–Mills theory in 4-dimensions.

Where and when?

We normally meet on Mondays, 2 PM - 4 PM, and Wednesdays, 1 PM - 3 PM, in room P3 of the mathematics building at Instituto Superior Técnico. However, we often change our schedule on a whim. If you are local to Lisbon and would like to be part of this, please email me.

The BV Track

Notes

You can see our in-progress notes here: (Draft version! Use with care!!)

Notes for the BV track by Rui Peixoto and Björn Gohla, from talks by John Huerta unless noted otherwise.

Upcoming talks

The following is a tentative schedule of upcoming speakers in the BV track and their topics. The dates are subject to change, which will be announced by email.

~~20 May: John Huerta on errata, Rui Peixoto on the finite-dimensional quantum BV formalism (part two).~~
~~27 May: Manuel Araújo on finite-dimensional Feynman diagrams.~~
~~3 June: Manuel Araújo on finite-dimensional Feynman diagrams (part two).~~
~~5 June: Roger Picken on the Wilsonian definition of a scalar QFT.~~
10 June: Holiday, no seminar.
~~12 June: Leander Stecker on elliptic complexes, gauge fixing operators, and the propagator.~~
~~17 June: Leander Stecker on elliptic complexes, gauge fixing operators, and the propagator (part two).~~
24 June - 3 July: No seminar.
8 July: John Huerta on the definition of BV quantization.
15 July: Diogo Freire de Andrade on the simplicial set of BV quantizations.
17 July: Björn Gohla on obstructions to BV quantization.
22 July: John Huerta on renormalizability.
29 July: John Huerta on the BV quantization of Yang–Mills theory.

Older talks

Please refer to the notes!

References

Kevin Costello, Renormalisation and the Batalin–Vilkovisky formalism, arXiv:0706.1533.
Kevin Costello, Renormalization and effective field theory, American Mathematical Society, Providence, 2011. Draft version.
Kevin Costello and Owen Gwilliam, Factorization algebras and quantum field theory, vol 2, Cambridge University Press, Cambridge, 2021. Draft version.
Chris Elliott, Brian Williams, Philsang Yoo, Asymptotic freedom in the BV formalism, J. Geom. Phys. 123 (2018), 246–283. arXiv:1702.05973.
Owen Gwilliam, Factorization algebras and free field theories, PhD thesis, Department of Mathematics, Northwestern University, 2012.
Pavel Mnev, Quantum field theory: Batalin-Vilkovisky formalism and its applications, American Mathematical Society, Providence, 2019. (Based on lecture notes available as arXiv:1707.08096.)

The Constructive Track

Please see this page, maintained by Gonçalo.