A First Idea of Quantum Field Theory – 20 Part Series

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These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically those that arise as the perturbative quantization of a Lagrangian field theory — such as quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity appearing in the standard model of particle physics.

For broad introduction of the idea of the topic of perturbative quantum field theory see there and see

Here, first we consider classical field theory (or rather pre-quantum field theory), complete with BV-BRST formalism; then its deformation quantization via causal perturbation theory to perturbative quantum field theory. This mathematically rigorous (i.e. clear and precise) formulation of the traditional informal lore has come to be known as perturbative algebraic quantum field theory.

We aim to give a fully local discussion, where all structures arise on the “jet bundle over the field bundle” (introduced below) and “transgress” from there to the spaces of field histories over spacetime (discussed further below). This “Higher Prequantum Geometry” streamlines traditional constructions and serves the conceptualization in the theory. This is joint work with Igor Khavkine.

In full beauty these concepts are extremely general and powerful; but the aim here is to give a first precise idea of the subject, not a fully general account. Therefore we concentrate on the special case where spacetime is Minkowski spacetime (def. 2.17 below), where the field bundle (def. 3.1 below) is an ordinary trivial vector bundle (example 3.3 below) and hence the Lagrangian density (def. 5.1 below) is globally defined. Similarly, when considering gauge theory we consider just the special case that the gauge parameter-bundle is a trivial vector bundle and we concentrate on the case that the gauge symmetries are “closed irreducible” (def. 10.5 below). But we aim to organize all concepts such that the structure of their generalization to curved spacetime and non-trivial field bundles is immediate.

This comparatively simple setup already subsumes what is considered in traditional texts on the subject; it captures the established perturbative BRST-BV quantization of gauge fields coupled to fermions on curved spacetimes — which is the state of the art. Further generalization, necessary for the discussion of global topological effects, such as instanton configurations of gauge fields, will be discussed elsewhere (see at homotopical algebraic quantum field theory).

Alongside the theory we develop the concrete examples of the real scalar field, the electromagnetic field and the Dirac field:

running examples

field field bundle Lagrangian density equation of motion gauge fixing Poisson bracket
real scalar field expl. 3.4 expl. 5.4 expl. 5.13 none expl. 8.15
electromagnetic field expl. 3.5 expl. 5.5 expl. 5.14 expl. 12.1
Dirac field expl. 3.48 expl. 5.6 expl. 5.24 none expl. 8.16

The electromagnetic field and the Dirac field combined are the fields of quantum electrodynamics which we turn to at the end below.


These notes profited greatly from discussions with Igor Khavkine. Further helpful comments on the notes were made by Arnold Neumaier, as well as by Severin Bunk, David Corfield and David Roberts. Thanks also to Klaus Fredenhagen and Kasia Rejzner for helpful replies.


Table of Contents

  • 1. Geometry
  • 2. Spacetime
  • 3. Fields 
  • 4. Field variations 
  • 5. Lagrangians 
  • 6. Symmetries 
  • 7. Observables 
  • 8. Phase space (coming soon)
  • 9. Propagators (coming soon)
  • 10. Gauge symmetries (coming soon)
  • 11. Reduced phase space (coming soon)
  • 12. Gauge fixing (coming soon)
  • 13. Quantization (coming soon)
  • 14. Free quantum fields (coming soon)
  • 15. Scattering (coming soon)
  • 16. Quantum observables (coming soon)
  • 17. Feynman diagrams (coming soon)
  • 18. Renormalization (coming soon)
  • 19. Quantum Electrodynamics (coming soon)
  • I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague. Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.
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