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\section*{geometry of physics -- perturbative quantum field theory}

[[!redirects geometry of physics -- A first idea of quantum field theory]]



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

$\,$

\begin{quote}%
This is one chapter of \emph{[[geometry of physics]]}.

Previous chapters: \emph{[[geometry of physics -- smooth sets|smooth sets]]}, \emph{[[geometry of physics -- supergeometry|supergeometry]]}.


\end{quote}
$\,$

\hypertarget{contents}{}\section*{{Contents}}\label{contents}



$\,$

For broad introduction of the idea of the topic of \emph{[[perturbative quantum field theory]]} see [[perturbative quantum field theory|there]] and see

\begin{itemize}%
\item PhysicsForums-Insights: \emph{\href{https://www.physicsforums.com/insights/paqft-idea-references/}{Introduction to Perturbative Quantum Field Theory}}

\end{itemize}
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 \emph{[[perturbative algebraic quantum field theory]]}.

We aim to give a \emph{fully [[local field theory|local]]} discussion, where all structures arise on the ``[[jet bundle]] over the [[field bundle]]'' (introduced \hyperlink{FieldVariations}{below}) and ``[[transgression|transgress]]'' from there to the [[spaces of field histories]] over [[spacetime]] (discussed \hyperlink{Observables}{further below}). This ``[[schreiber: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. \ref{MinkowskiSpacetime} below), where the [[field bundle]] (def. \ref{FieldsAndFieldBundles} below) is an ordinary [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle} below) and hence the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below) is [[locally variational field theory|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. \ref{GaugeParametersClosed} below). But we aim to organize all concepts such that the \emph{structure} of their generalization to [[AQFT on curved spacetime|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 quantum field theory|perturbative]] [[BV formalism|BRST-BV quantization]] of [[gauge fields]] coupled to [[fermions]] [[AQFT on curved spacetime|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 \emph{[[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]]; eventually combining these to a disussion of [[quantum electrodynamics]].

\textbf{running examples}

\begin{tabular}{l|l|l|l}
[[field (physics)&field]]&[[field bundle]]&[[Lagrangian density]]\\
\hline 
[[real scalar field]]&expl. \ref{RealScalarFieldBundle}&expl. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}&expl. \ref{FreeScalarFieldEOM}\\
[[Dirac field]]&expl. \ref{DiracFieldBundle}&expl. \ref{LagrangianDensityForDiracField}&expl. \ref{EquationOfMotionOfDiracFieldIsDiracEquation}\\
[[electromagnetic field]]&expl. \ref{Electromagnetism}&expl. \ref{ElectromagnetismLagrangianDensity}&expl. \ref{ElectromagnetismEl}\\
[[Yang-Mills field]]&expl. \ref{YangMillsFieldOverMinkowski},  expl. \ref{YangMillsFieldInInstantonSector}&expl. \ref{YangMillsLagrangian}&expl. \ref{YangMillsOnMinkowskiEl}\\
[[B-field]]&expl. \ref{BField}&expl \ref{BFieldLagrangianDensity}&expl. \ref{EulerLagrangeFormBField}\\
\end{tabular}

$\,$

\begin{tabular}{l|l|l|l|l}
[[field (physics)&field]]&[[Poisson bracket]]&[[causal propagator]]&[[Wightman propagator]]\\
\hline 
[[real scalar field]]&expl. \ref{PeierlsBracketEistsForScalarFieldAndDiracField},  expl. \ref{PoissonBracketForRealScalarField}&prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}&def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}&def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}\\
[[Dirac field]]&expl. \ref{PeierlsBracketEistsForScalarFieldAndDiracField},  expl. \ref{PoissonBracketForDiracField}&prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}&def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}&def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}\\
[[electromagnetic field]]&&prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}&&prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}\\
\end{tabular}

$\,$

\begin{tabular}{l|l|l|l}
[[field (physics)&field]]&[[gauge symmetry]]&[[local BRST complex]]\\
\hline 
[[electromagnetic field]]&expl. \ref{InfinitesimalGaugeSymmetryElectromagnetism}&expl. \ref{LocalBRSTComplexForFreeElectromagneticFieldOnMinkowskiSpacetim}&expl. \ref{NLGaugeFixingOfElectromagnetism}\\
[[Yang-Mills field]]&expl. \ref{InfinitesimalGaugeSymmetryOfYangMillsTheory}&expl. \ref{YangMillsLocalBRSTComplex}&\ldots{}\\
[[B-field]]&expl. \ref{InfinitesimalGaugeSymmetryOfTheBField}&expl. \ref{LocalBRSTComplexBFieldMinkowskiSpacetime}&\ldots{}\\
\end{tabular}

$\,$

\begin{tabular}{l|l|l}
[[interacting field theory]]&[[interaction]] [[Lagrangian density]]&[[interaction]] [[Wick algebra]]-element\\
\hline 
[[phi{\tt \symbol{94}}n theory]]&exp. \ref{phintheoryLagrangian}&expl. \ref{InWickAlgebraphinInteraction}\\
[[quantum electrodynamics]]&expl. \ref{LagrangianQED}&expl. \ref{InWickAlgebraElectronPhotonInteraction}\\
\end{tabular}

$\,$

\textbf{References}

Pointers to the literature are given in each chapter, alongside the text. The following is a selection of these references.

The discussion of [[spinors]] in chapter \emph{\hyperlink{Spacetime}{2. Spacetime}} follows \href{spin+representation#BaezHuerta09}{Baez-Huerta 09}.

The [[functorial geometry]] of [[supergeometry|supergeometric]] [[spaces of field histories]] in \emph{\hyperlink{Fields}{3. Fields}} follows \href{higher+prequantum+geometry#Schreiber13}{Schreiber 13}.

For the [[jet bundle]]-formulation of [[variational calculus]] of [[Lagrangian field theory]] in \emph{\hyperlink{FieldVariations}{4. Field variations}}, and \emph{\hyperlink{Lagrangians}{5. Lagrangians}} we follow \href{variational+bicomplex#Anderson89}{Anderson 89} and \hyperlink{Olver86}{Olver 86}; for \emph{\hyperlink{Symmetries}{6. Symmetries}} augmented by \hyperlink{higher+prequantum+geometry#FRS13b}{Fiorenza-Rogers-Schreiber 13b}.

The identification of [[polynomial observables]] with [[distributions]] in \emph{\hyperlink{Observables}{7. Observables}} was observed by \href{distributions+are+the+smooth+linear+functionals#Paugam12}{Paugam 12}.

The discussion of the [[Peierls-Poisson bracket]] in \emph{\hyperlink{PhaseSpace}{8. Phase space}} is based on \href{Peierls+bracket#Khavkine14}{Khavkine 14}.

The derivation of [[wave front sets]] of [[propagators]] in \emph{\hyperlink{Propagators}{9. Propagators}} takes clues from \href{Hadamard+distribution#Radzikowski96}{Radzikowski 96} and uses results from \href{Cauchy+principal+value#GelfandShilov66}{Gelfand-Shilov 66}.

For the general idea of [[BV-BRST formalism]] a good review is \href{BV-BRST+formalism#Henneaux90}{Henneaux 90}.

The [[Lie algebroid]]-perspective on [[BRST complexes]] developed in chapter \emph{\hyperlink{GaugeSymmetries}{10. Gauge symmetries}}, may be compared to \href{BRST+complex#Barnich10}{Barnich 10}.

For the [[local BV-BRST complexes|local BV-BRST theory]] laid out in chapter \emph{\hyperlink{ReducedPhaseSpace}{11. Reduced phase space}} we are following \href{local+BRST+cohomology#BarnichBrandtHenneaux00}{Barnich-Brandt-Henneaux 00}.

For the [[gauge fixing|BV-gauge fixing]] developed in \emph{\hyperlink{GaugeFixing}{12. Gauge fixing}} we take clues from \href{BV-BRST+formalism#FredenhagenRejzner11a}{Fredenhagen-Rejzner 11a}.

For the free quantum [[BV-operators]] in \emph{\hyperlink{FreeQuantumFields}{13. Free quantum fields}} and the interacting [[quantum master equation]] in \emph{\hyperlink{InteractingQuantumFields}{15. Interacting quantum fields}} we are following \href{BV-BRST+formalism#FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b}, \href{BV-BRST+formalism#Rejzner11}{Rejzner 11}, which in turn is taking clues from \href{BV-BRST+formalism#Hollands07}{Hollands 07}.

The discussion of [[quantization]] in \emph{\hyperlink{Quantization}{13. Quantization}} takes clues from \href{geometric+quantization+of+symplectic+groupoids#EH}{Hawkins 04}, \href{star+product#Collini16}{Collini 16} and spells out the derivation of the [[Moyal star product]] from [[geometric quantization of symplectic groupoids]] due to \href{Moyal+deformation+quantization#GBV}{Gracia-Bondia \& Varilly 94}.

The perspective on the [[Wick algebra]] in \emph{\hyperlink{FreeQuantumFields}{14. Free quantum fields}} goes back to \href{Wick+algebra#Dito90}{Dito 90} and was revived for [[pAQFT]] in \href{Wick+algebra#DuetschFredenhagen00}{Dütsch-Fredenhagen 00}. The proof of the folklore result that the perturbative [[Hadamard vacuum state]] on the [[Wick algebra]] is indeed a [[state on a star-algebra|state]] is cited from \href{Wick#algebra#Duetsch18}{Dütsch 18}.

The discussion of [[causal perturbation theory]] in \emph{\hyperlink{InteractingQuantumFields}{15. Interacting quantum fields}} follows the original \href{causal+perturbation+theory#EpsteinGlaser73}{Epstein-Glaser 73}. The relevance here of the [[star product]] induced by the [[Feynman propagator]] was highlighted in \href{perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12}{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 \href{pAQFT#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}.

Our derivation of [[Feynman diagrams|Feynman diagrammatics]] follows \href{Feynman+diagram#Keller10}{Keller 10, chapter IV}, our derivation of the [[quantum master equation]] follows \href{BV-BRST+formalism#Rejzner11}{Rejzner 11, section 5.1.3}, and our discussion of [[Ward identities]] is informed by \href{Ward+identity#Duetsch18}{Dütsch 18, chapter 4}.

In chapter \emph{\hyperlink{Renormalization}{16. Renormalization}} we take from \href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00} the perspective of [[Epstein-Glaser renormalization]] via [[extension of distributions]] and from \href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09} and \href{renormalization#Duetsch10}{Dütsch 10} the rigorous formulation of [[Gell-Mann-Low renormalization cocycle|Gell-Mann \& Low renormalization group flow]], [[UV-regularization]], [[effective quantum field theory]] and [[Polchinski's flow equation]].

$\,$

\textbf{Acknowledgement}

These notes profited greatly from discussions with [[Igor Khavkine]] and [[Michael Dütsch]].

Thanks also to [[Marco Benini]], [[Klaus Fredenhagen]], [[Arnold Neumaier]] and [[Kasia Rejzner]] for helpful discussion.

$\,$

$\,$

[[!include A first idea of quantum field theory -- Geometry]]

[[!include A first idea of quantum field theory -- Spacetime]]

[[!include A first idea of quantum field theory -- Fields]]

[[!include A first idea of quantum field theory -- Field variations]]

[[!include A first idea of quantum field theory -- Lagrangians]]

[[!include A first idea of quantum field theory -- Symmetries]]

[[!include A first idea of quantum field theory -- Observables]]

[[!include A first idea of quantum field theory -- Phase space]]

[[!include A first idea of quantum field theory -- Propagators]]

[[!include A first idea of quantum field theory -- Gauge symmetries]]

[[!include A first idea of quantum field theory -- Reduced phase space]]

[[!include A first idea of quantum field theory -- Gauge fixing]]

[[!include A first idea of quantum field theory -- Quantization]]

[[!include A first idea of quantum field theory -- Free quantum fields]]

[[!include A first idea of quantum field theory -- Interacting quantum fields]]

[[!include A first idea of quantum field theory -- Renormalization]]

[[!redirects A first idea of quantum field theory]]



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