nLab Lagrangian quantum field theory

Idea

In physics, a Lagrangian quantum field theory is a quantum field theory (QFT) which arises via some version of quantization from a Lagrangian density on (the jet bundle of) some field bundle, hence from a prequantum field theory.

Most QFTs that are being considered are Lagrangian quantum field theories. Indeed all of traditional perturbative quantum field theory, subsuming in particular QED, EW, QCD, pQG and hence the standard model of particle physics, is Lagrangian.

But not all QFTs are Lagrangian; or at least they need not have been explicitly constructed via quantization from a Lagrangian density, even if they could be constructed this way:

For example self-dual higher gauge theory (such as the chiral WZW model, which is “one half” of the Lagrangian WZW model) is thought not to be Lagrangian. Still, these self-dual higher gauge theories are thought be be defined via the holographic principle as the holographic boundary of a QFT that is Lagrangian: higher dimensional Chern-Simons theory.

Many topological quantum field theories may be constructed by abstract algebraic means. Famous examples are the A-model/B-model topological string worldvolume theories, which may abstractly be constructed from Calabi-Yau A-∞ categories. Still, these were originally conceived of as topological twists of a Lagrangian field theory, namely of the 2-dimensional supersymmetric sigma-model with target space a 3-dimensional Calabi-Yau manifold.

Similarly, some classes of 2d CFTs may be constructed by purely algebraic means, in particular the construction of 2d rational conformal field theory is completely reduced to an algebraic construction in suitable modular tensor categories by the FRS-theorem on rational 2d CFT. This includes the (chiral or non-chiral) WZW model mentioned above, but there are also 2d CFTs that are thought to be genuinely “non-geometric”, such as the Gepner model.

Related concepts

• non-Lagrangian field theory

• local quantum field theory

• locally variational field theory

• algebraic quantum field theory

• perturbative algebraic quantum field theory

References

Almost all traditional texts on field theory are actually tacitly focused on Lagrangian field theory, such as:

• Marc Henneaux, Claudio Teitelboim: Quantization of Gauge Systems, Princeton University Press (1992) [ISBN:9780691037691, doi:10.2307/j.ctv10crg0r, jstor:j.ctv10crg0r]

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, Advanced classical field theory, World Scientific (2009) [doi:10.1142/7189]

Some are more explicit about it:

• C. W. Kilmister: Lagrangian Dynamics: An Introduction for Students, Logos Press (1967?)

• Jean-Louis Basdevant: Lagrangian Field Theory, chapter 5 in Variational Principles in Physics, Springer (2007) 97-106 [doi:10.1007/978-0-387-37748-3_5]

On smooth sets as a convenient category for variational calculus of Lagrangian classical field theory:

• Grigorios Giotopoulos, Hisham Sati, Field Theory via Higher Geometry I: Smooth Sets of Fields, Journal of Geometry and Physics (2025) 105462 [arXiv:2312.16301, doi:10.1016/j.geomphys.2025.105462]

• Grigorios Giotopoulos: Sheaf Topos Theory: A powerful setting for Lagrangian Field Theory [arXiv:2504.08095]