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.
Table of Contents
- 1. Geometry
- 2. Spacetime
- 3. Fields
- 4. Field variations
- 5. Lagrangians
- 6. Symmetries
- 7. Observables
- 8. Phase space
- 9. Propagators
- 10. Gauge symmetries
- 11. Reduced phase space
- 12. Gauge fixing
- 13. Quantization
- 14. Free quantum fields
- 15. Interacting quantum fields
- 16. Renormalization
- PhysicsForums-Insights: Introduction to Perturbative Quantum Field Theory
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 3.4 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. 10.6 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).
|field||field bundle||Lagrangian density||equation of motion|
|real scalar field||expl. 3.5||expl. 5.4||expl. 5.17|
|Dirac field||expl. 3.50||expl. 5.9||expl. 5.30|
|electromagnetic field||expl. 3.6||expl. 5.6||expl. 5.18|
|Yang-Mills field||expl. 3.7,
|expl. 5.7||expl. 5.19|
|B-field||expl. 3.9||expl 5.8||expl. 5.20|
|field||Poisson bracket||causal propagator||Wightman propagator||Feynman propagator|
|real scalar field||expl. 8.8,
|prop. 9.54||def. 9.57||def. 9.61|
|Dirac field||expl. 8.8,
|prop. 9.70||def. 9.71||def. 9.72|
|electromagnetic field||prop. 12.9||prop. 12.9|
|field||gauge symmetry||local BRST complex||gauge fixing|
|electromagnetic field||expl. 10.14||expl. 10.30||expl. 12.8|
|Yang-Mills field||expl. 10.15||expl. 10.31||…|
|B-field||expl. 10.16||expl. 10.32||…|
|interacting field theory||interaction Lagrangian density||interaction Wick algebra-element|
|phi^n theory||exp. 5.5||expl. 14.13|
|quantum electrodynamics||expl. 5.11||expl. 14.14|
Pointers to the literature are given in each chapter, alongside the text. The following is a selection of these references.
For the jet bundle-formulation of variational calculus of Lagrangian field theory in 4. Field variations, and 5. Lagrangians we follow Anderson 89 and Olver 86; for 6. Symmetries augmented by Fiorenza-Rogers-Schreiber 13b.
For the free quantum BV-operators in 13. Free quantum fields and the interacting quantum master equation in 15. Interacting quantum fields we are following Fredenhagen-Rejzner 11b, Rejzner 11, which in turn is taking clues from Hollands 07.
The discussion of quantization in 13. Quantization takes clues from Hawkins 04, Collini 16 and spells out the derivation of the Moyal star product from geometric quantization of symplectic groupoids due to Gracia-Bondia & Varilly 94.
The perspective on the Wick algebra in 14. Free quantum fields goes back to Dito 90 and was revived for pAQFT in Dütsch-Fredenhagen 00. The proof of the folklore result that the perturbative Hadamard vacuum state on the Wick algebra is indeed a state is cited from Dütsch 18.
The discussion of causal perturbation theory in 15. Interacting quantum fields follows the original Epstein-Glaser 73. The relevance here of the star product induced by the Feynman propagator was highlighted in Fredenhagen-Rejzner 12. The proof that the interacting field algebra of observables defined by Bogoliubov’s formula is a causally local net in the sense of the Haag-Kastler axioms is that of Brunetti-Fredenhagen 00.
Our derivation of Feynman diagrammatics follows Keller 10, chapter IV, our derivation of the quantum master equation follows Rejzner 11, section 5.1.3, and our discussion of Ward identities is informed by Dütsch 18, chapter 4.
In chapter 16. Renormalization we take from Brunetti-Fredenhagen 00 the perspective of Epstein-Glaser renormalization via extension of distributions and from Brunetti-Dütsch-Fredenhagen 09 and Dütsch 10 the rigorous formulation of Gell-Mann & Low renormalization group flow, UV-regularization, effective quantum field theory and Polchinski’s flow equation.
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.