lecture notes on Lagrangian Field Theory

Below are my lecture notes for courses on Lagrangian Field Theory that I have taught at the University of Bonn and the Max Planck Institute for Mathematics. I make them available here due to popular demand. They are currently in the state of being continuously rewritten and revised. To allow referencing them, I will keep all the versions posted here. There may be major changes from version to version, so please make sure to always check the newest version.

The purpose of the notes is to give an account of the basic notions, structure, and results of Lagrangian Field Theory using a rigorous and reasonably modern mathematical framework. The target readership are mathematicians or mathematical physicists who want to work rigorously.

On the one hand, I have tried to avoid statements that are only “morally” true or under unrealistic assumptions. The prime example is the action principle, which states that the solutions of the Euler-Lagrange equations are the critical points of the action. In this “mythological form” (Graeme Segal), it requires that the spacetime is compact, which excludes time in classical mechanics and Minkowski space in Maxwell theory. Other oversimplifying assumptions are that the lagrangian is of first jet order or that the fields are sections of an affine bundle, which exclude general relativity. In fact, here is what I think should be the 12th commandment in Lagrangian Field Theory:

  • Thou shalt not make any assumptions that exclude general relativity.

See Section 1.5.2 for more assumptions that violate this principle.

On the other hand, I have tried to steer clear of gratuitous generalizations that bend classical field theory to the will of mathematicians, rather than describe the theory as it is. I use the following mathematical formalism: diffeology for the differential geometry of the spaces of fields; pro-categories for infinite jet manifolds and, dually, ind-categories for forms on them; varational cohomology for the action principle and Noether’s theorems; multisymplectic geometry and homotopical algebra for hamiltonian symmetries. Together, this is a good framework for a rigorous formulation of Lagrangian Field Theory that is general enough to include Einstein gravity and its diffeomorphism symmetry. At the same time, it should be a good starting point for more general approaches.

These lecture notes are personal and somewhat idiosyncratic. No effort is made to do justice to LFT as a whole or to alternative approaches and frameworks. Having made this disclaimer, I will be most grateful for feedback of any kind, which will improve the notes for all readers.

current version

  • Lagrangian_Field_Theory_v24.0.pdf Version 24.0 (February 2, 2024). Chapter 8 on the multisymmetric structure was added. Due to popular demand, the incomplete Chapter 9 on examples was added as is.

The link https://people.mpim-bonn.mpg.de/blohmann/Lagrangian_Field_Theory.pdf always points to the current version.

older versions

  • Lagrangian_Field_Theory_v23.0.pdf Version 23.0 (July 10, 2023). Chapter 7 on symmetries and Jacobi fields was added.

  • Lagrangian_Field_Theory_v22.0.pdf Version 22.0 (June 28, 2023). This is a major revision, reorganization, and rewrite carried out while teaching a course on the subject in the summer semester 2023. Contains the material up to Chapter 6 on the cohomological action principle.

  • Lagrangian_Field_Theory_v15.pdf Version 15 (April 5, 2022). Some minor changes with respect to version 14.

  • Lagrangian_Field_Theory_v14.pdf Version 14 (February 17, 2021). This old version has been on my website for a while and has been picked up inadvertently by google search.