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\begin{document}

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\section*{Feynman diagram}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_qunantum_field_theory}{}\paragraph*{{Algebraic Qunantum Field Theory}}\label{algebraic_qunantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_perturbative_quantum_field_theory}{In perturbative quantum field theory}\dotfill \pageref*{in_perturbative_quantum_field_theory} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{ForFinitelyManyDegreesOfFreedom}{For finitely many degrees of freedoms}\dotfill \pageref*{ForFinitelyManyDegreesOfFreedom} \linebreak
\noindent\hyperlink{DetailsForPerturbativeQuantumFieldTheory}{In perturbative quantum field theory}\dotfill \pageref*{DetailsForPerturbativeQuantumFieldTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesInCausalPerturbationTheory}{In causal perturbation theory}\dotfill \pageref*{ReferencesInCausalPerturbationTheory} \linebreak
\noindent\hyperlink{ReferencesInTermsOfMotivicStructures}{In terms of motivic structures}\dotfill \pageref*{ReferencesInTermsOfMotivicStructures} \linebreak
\noindent\hyperlink{ReferencesInHomologicalBVQuantization}{In homological BV-quantization}\dotfill \pageref*{ReferencesInHomologicalBVQuantization} \linebreak
\noindent\hyperlink{AsStringDiagrams}{As string diagrams (morphisms in monoidal categories)}\dotfill \pageref*{AsStringDiagrams} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For $\mu_S$ a [[Gaussian probability measure]] and $\exp(-I)\mu_S$ a perturbation with $I$ [[polynomial ring|polynomial]] at least of degree 3, there is a combinatorial expression for the [[moments]]/[[expectation values]] of $\exp(-I)\mu_S$ as a sum over certain [[graphs]] whose $k$-ary vertices are labeled by the monomials of degree $k$ in $I$. (This is such that for $I = 0$ it reduces to [[Wick's lemma]].)

These graphs are called \emph{Feynman diagrams}.

\hypertarget{in_perturbative_quantum_field_theory}{}\subsubsection*{{In perturbative quantum field theory}}\label{in_perturbative_quantum_field_theory}

Feynman graphs play a central role in [[perturbative quantum field theory]], where $\exp(I)\mu_S$ plays the role of an [[action functional]] on a space of [[field (physics)|fields]], $\mu_S$ is the exponentiated [[kinetic action]] and hence the [[measure]] for [[free fields]], while $\exp(I)$ is the [[interaction]] part of the [[action functional]]: the order-$k$ monomials in $I$ encode an interaction of $k$ [[field (physics)|fields]]. In this context the corresponding Feynman diagrams are traditionally thought of as depicting interaction processes of quanta of these fields, with propagation along the edges and interaction at the vertices. But this interpretation has its limits, which is partly reflected in speaking of ``[[virtual particles]]''. A precise interpretation is given by \emph{[[worldline formalism]]}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{ForFinitelyManyDegreesOfFreedom}{}\subsubsection*{{For finitely many degrees of freedoms}}\label{ForFinitelyManyDegreesOfFreedom}

We discuss Feynman diagrams for a single real [[scalar field]] on a [[discrete space]] of $k\in \mathbb{N}$ points. This contains in it already all the aspects of real Feynman diagrams in [[perturbative quantum field theory]] and in this context everything is easily well-defined. The generalization to more field components is immediate and simply obtained by thinking of all ``$\phi$'' in the following as taking values in some appropriate [[representation]] space and all products of $\phi$s as given by suitable [[intertwiners]] and [[inner products]], otherwise the form of the formulas remains the same. Similarly, in generalization to continuous space (non-finitely many degrees of freedom) all the diagrammatics remains the same, the only issue now is to make sense (namely via [[renormalization]]) of the numerical value that is assigned to any one Feynman diagram.

So a [[field (physics)|field]] $\phi$ here is a map $[k]\to \mathbb{R}$ from the $k$-element [[finite set]] to the [[real numbers]], and hence the space of all field configurations is $\mathbb{R}^k$.

Fix then a $k \times k$ real-valued [[matrix]] $A \coloneqq (A_{x y}) \in Mat_{k\times k}(\mathbb{R})$ of non-vanishing [[determinant]] $det A \neq 0$.

For standard applications this $A$ is a discretized version of the [[Laplacian]] and then the expression

\begin{displaymath}
S_{kin} = E_{kin} \coloneqq 
  \tfrac{1}{2}\sum_{x,y = 1}^k \phi_x A_{x y} \phi_y
\end{displaymath}
is the [[kinetic energy]] and [[kinetic action]] of the field configuration $\phi$. More concretely, think of the set of $k$ space points as being a discretization of a circle and write $(\phi_p)$ for the corresponding ([[discrete Fourier transform|discrete]]) [[Fourier transform]] of $(\phi_x)$. Then the kinetic term of the [[free field theory|free]] [[scalar field]] on this space is given by $A$ which is the [[diagonal matrix]] in the $p$-basis with

\begin{displaymath}
A_{p,p} = p^2 - m^2
  \,,
\end{displaymath}
where $m$ is a constant called the [[mass]] of the [[field (physics)|field]] $\phi$.

A [[sum]] over all values of $\phi$ is the finite (and hence well-defined) analog of a [[path integral]]. The [[Gaussian integral]] of $A$ is called the \emph{[[partition function]]}:

\begin{displaymath}
\begin{aligned}
  Z_0
  & \coloneqq 
  \int  \exp(- S_{kin}(\phi)) \,D\phi
  \\
  & \coloneqq
  \int_{\mathbb{R}^k}
  \,
  \exp(- \tfrac{1}{2}\sum_{x,y = 1}^k \phi_{x} A_{x,y} \phi_y  )
  \;
  d\phi_{1} \cdots d\phi_{k} 
  \\
  & =
  (2\pi)^{k/2} (det A)^{-1/2}
  \end{aligned}
\end{displaymath}
An \emph{[[n-point function]]} $\langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle$ is an $n$th [[moment]] of this Gaussian distribution:

\begin{displaymath}
\begin{aligned}
  \langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle
  & \coloneqq
  \frac{1}{Z_0}
  \int  
    \left( 
    \exp(-S_{kin}(\phi))
    \phi_{x_1}\phi_{x_2}\cdots \phi_{x_n}
    \right)
    D \phi
  \\
  & \coloneqq
  \frac{1}{Z_0} 
  \int_{\mathbb{R}^k} 
  \left(
  \exp(-\tfrac{1}{2} \phi_x A_{x,y}\phi_y)
  \,
  \phi_{x_1}\phi_{x_2} \cdots\phi_{x_n}
  \right)
  d\phi_1 \cdots d\phi_k
  \end{aligned}
\end{displaymath}
In order to compute these conveniently, pass to the [[generating function]] obtained by adding a [[source]] [[variable]] $J = (J_x)$:

\begin{displaymath}
\begin{aligned}
  Z(J)
  & \coloneqq
  \int( \exp(- S_{kin}(\phi) + J \cdot\phi) ) D\phi
  \\
  & \coloneqq
  \int_{\mathbb{R}^k}  
    \exp(- \tfrac{1}{2}\sum_{x = 1}^k \phi_{x} A_{x,y} \phi_y 
    + \sum_{x = 1}^k J_x \phi_x
   )
  \;
  d\phi_{1} \cdots d\phi_{k}
  \\
    & =
    Z_0 \exp(\tfrac{1}{2} \sum_{x = 1}^k J_x A^{-1}_{x,y} J_y)
  \end{aligned}
  \,,
\end{displaymath}
where $A^{-1} = (A^{-1}_{x,y})$ is the [[inverse matrix]] of $A$, which appears by computing the new integral here again as a [[Gaussian integral]] after [[completing the square]] in the exponent.

In applications to field theory this $A^{-1}$ is called the \emph{[[Feynman propagator]]}. In the standard example where $A$ is the kinetic term of the [[free field theory|free field]] of [[mass]] $m$ then in Fourier-transformed components $A^{-1}$ is diagonal with components

\begin{displaymath}
A^{-1}_{p,p} = \frac{1}{p^2 - m^2}
  \,.
\end{displaymath}
This means, incidentally. that in the non-finite case of interest in physics, $A$ is not actually naively invertible after all, as $p^2 = m^2$ on the mass [[shell]]. In this case one has to replace the naive $A^{-1}$ with its [[zeta function regularization|zeta-function regularized version]].

The way this works is insightful even when the naive $A^{-1}$ does exist. Notice that using ``[[Schwinger parameterization]]'' the [[propagator]] is equivalently rewritten as a [[Mellin transform]] [[integral]]:

\begin{displaymath}
A^{-1}_{p,p} = \int_0^\infty \exp(- \tau A_{p,p}) d\tau
  \,.
\end{displaymath}
Again in the example of the standard [[scalar field]] kinetic term expressed in Fourier diagonalization $A_{p,p}= p^2 - m^2$ then with $X \colon [0,\tau] \to \mathbb{R}$ a parameterization of the straight line with slope $p$, the exponent is equivalently

\begin{displaymath}
\begin{aligned}
    \tau A_{p,p}
    & = 
    \tau (p^2 - m^2)
    \\
    & = 
    \int_0^\tau (\dot X^2 - m^2)
  \end{aligned}
  \,.
\end{displaymath}
This now happens to be the standard [[action functional]] ([[Polyakov action]]) for a [[sigma model]] describing the propagation of a particle along its [[worldline]]. This means that the propagator of the [[scalar field]] may be thought of as coming from the [[path integral]] of a scalar particle along its [[worldline]]. This perspective is called the ``[[worldline formalism]]'', it is a formalization of [[second quantization]], expressing the dynamics of [[field (physics)|fields]] in terms of that of their particle ``quanta'' running along [[worldlines]] of the form of the corresponding Feynman diagrams (to which we finally come in a moment).

Back to the computation of the $n$-point function. By construction, it is now equally expressed by [[partial derivatives]] of the [[generating function]] with respect to the [[source]] [[variable]] $J$ and evaluated at $J = 0$, and this in turn is a combinatorial expression just in products of the [[propagator]]:

\begin{displaymath}
\begin{aligned}
    \langle \phi_{x_1} \cdots \phi_{x_n}  \rangle
    &=
    \frac{1}{Z_0}
    \left(
    \frac{\partial}{\partial J_{x_1}}
    \cdots
    \frac{\partial}{\partial J_{x_n}}
    Z(J)
    \right)_{\vert J = 0}
    \\
    & =
   \left(
   \frac{\partial}{\partial J_{x_1}}
   \cdots
   \frac{\partial}{\partial J_{x_n}}
   \exp(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y)
   \right)_{\vert J = 0}
   \\
   & = 
   \underset{pairings}{\sum}  A^{-1}_{x_{j_1} x_{j_2}} \cdot A^{-1}_{x_{j_3} x_{j_4}} \cdots A^{-1}_{x_{j_{n-1}}x_{j_n}}
  \end{aligned}
  \,.
\end{displaymath}
Here the last [[equality]] -- known as [[Wick's theorem]] -- comes from simple inspection: take the derivatives inside the [[exponential series]] and observe that then the only summands non-vanishing at $J = 0$ appears for even $n$ and are those where all derivatives hit the monomial $\left(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y\right)^{n/2}$.

Thinking of $A^{-1}_{x y}$ here as labeling an [[edge]] (a ``[[worldline]]'') from [[vertex]] $x$ to vertex $y$ This is the source of all Feynman digrammatics.

Now consider a [[polynomial]] $V(\phi)$ of degree $\geq 3$. In applications to field theory this represents the [[potential energy]] or (self)[[interaction]] of the field configuration. The difference of the [[kinetic energy]] and the [[potential energy]] is called the (here: ``Wick rotated''/``Euclidean'') [[action]]

\begin{displaymath}
\begin{aligned}
    S & =  S_{kin} + S_{int} 
    \\
    & = \tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y - g V(\phi)
  \end{aligned}
  \,.
\end{displaymath}
The prefactor $g$ is called the \emph{[[coupling constant]]}.

Putting everything together, the integral over the full [[action]] may be expressed as a [[power series]] in the [[coupling constant]] $g$ of the [[moments]] with respect to the kinetic action of the powers of the interaction term:

\begin{displaymath}
\begin{aligned}
  Z(g)
  & = 
  \frac{1}{Z_0}\int \exp(- S(\phi)) \, D\phi
  \\
  & \coloneqq
  \frac{1}{Z_0}
  \int \left(\exp(-\tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y + g V(\phi)) \right)
  \, d\phi_1 \cdots d\phi_k
  \\
  & =  
  \langle \exp(g V(\phi))  \rangle
  \\
  & = 
  1 + g \langle V(\phi)\rangle + \frac{g^2}{2} \langle V(\phi) ^2\rangle + \cdots
  \end{aligned}
\end{displaymath}
By [[Wick's theorem]] stated above, each $\langle V(\phi)^\ell\rangle$ is equivalently expressed as a sum over products of components of the [[propagator]] $A^{-1}_{x y}$. Thinking of each such propagator term as an [[edge]] produces a diagram, this is the corresponding Feynman diagram.

For instance, for a cubic point interaction

\begin{displaymath}
V(\phi) = \sum_x \phi_x^3
\end{displaymath}
then

\begin{displaymath}
\begin{aligned}
    \langle V(\phi)^2 \rangle
    &\coloneqq
    \langle (\sum_{x= 1}^k \phi_x^3)^2 \rangle
    \\
    & = \sum_{x_1, x_2 = 1}^k
    \langle \phi_{x_1}\phi_{x_1} \phi_{x_1}\phi_{x_2}\phi_{x_2}\phi_{x_2}\rangle
    \\ 
    & =
    prefactor
    \;
    \sum_{x_1,x_2 = 1}^k 
    \underset{dumbbell\, diagram}{\underbrace{A^{-1}_{x_1 x_1} A^{-1}_{x_1 x_2} A^{-1}_{x_2 x_2}}}
    +
    prefactor
    \;
    \sum_{x_1, x_2 = 1}^k 
    \underset{theta\, diagram}{\underbrace{A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2}}}
  \end{aligned}
\end{displaymath}
Here the first summand corresponds to the ``dumbbell'' Feynman diagram of the form



and the second summand corresponds to the ``theta'' Feynman diagram of the form

.

\hypertarget{DetailsForPerturbativeQuantumFieldTheory}{}\subsubsection*{{In perturbative quantum field theory}}\label{DetailsForPerturbativeQuantumFieldTheory}

For details see at

\begin{itemize}%
\item [[S-matrix]] the section \emph{\href{S-matrix#FeynmanDiagrams}{Feynman perturbation series}}

\end{itemize}
$\,$

[[!include Wick algebra -- table]]

$\,$

[[!include Feynman diagrams in causal perturbation theory -- summary]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[vacuum diagram]]


\item [[perturbative quantum field theory]]


\item [[S-matrix]], [[perturbation series]]


\item [[causal perturbation theory]]


\item [[path integral quantization]]


\item [[scattering amplitude]], [[string scattering amplitude]]


\item [[Feynman propagator]]


\item [[worldline formalism]]


\item [[Schwinger-Dyson equation]]


\item [[tadpole]]


\item [[loop order]]


\item [[radiative correction]]


\item [[graph complex]], [[formality of the little n-disk operad]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Traditional review includes

\begin{itemize}%
\item [[Radovan Dermisek]], \emph{Path integral for interacting field} (\href{http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-5-4p.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Alain Connes]], [[Matilde Marcolli]], section 3 of \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Feynman_diagram}{Feynman diagram}}



\end{itemize}
\hypertarget{ReferencesInCausalPerturbationTheory}{}\subsubsection*{{In causal perturbation theory}}\label{ReferencesInCausalPerturbationTheory}

Discussion of Feynman diagrams in the rigorous formulation of [[causal perturbation theory]] and [[perturbative AQFT]] is due to

\begin{itemize}%
\item [[Kai Keller]], chapter IV of \emph{Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization}, PhD thesis (\href{https://arxiv.org/abs/1006.2148}{arXxiv:1006.2148})

\end{itemize}
parts of which also appears as

\begin{itemize}%
\item [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], \emph{Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization}, J. Math. Phy. 55(12), 122303 (2014) (\href{https://arxiv.org/abs/1311.5424}{arXiv:1311.5424})

\end{itemize}
An exposition of this is in

\begin{itemize}%
\item [[Christian Brouder]], \emph{Multiplication of distributions}, 2010 ([[BrouderProductOfDistributions.pdf:file]])

\end{itemize}
Relation to the [[Hopf algebra]] structure on Feynman diagrams due to \href{renormalization#Kreimer97}{Kreimer 97} is discussed in

\begin{itemize}%
\item [[Jose Gracia-Bondia]], S. Lazzarini, \emph{Connes-Kreimer-Epstein-Glaser Renormalization} (\href{https://arxiv.org/abs/hep-th/0006106}{arXiv:hep-th/0006106})

\end{itemize}
Relation to [[periods]] (in the sense \href{period#ReferencesInPerturbativeQuantumFieldTheory}{here}) is discussed in

\begin{itemize}%
\item [[Kasia Rejzner]], \emph{Renormalization and periods in perturbative Algebraic Quantum Field Theory} (\href{https://arxiv.org/abs/1603.02748}{arXiv:1603.02748})

\end{itemize}
See also

\begin{itemize}%
\item [[Eli Hawkins]], [[Kasia Rejzner]], section 5.2 of \emph{The Star Product in Interacting Quantum Field Theory} (\href{https://arxiv.org/abs/1612.09157}{arXiv:1612.09157})

\end{itemize}
Review is in

\begin{itemize}%
\item [[Katarzyna Rejzner]], section 6.5.2 of \emph{Perturbative Algebraic Quantum Field Theory}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{pdf})

\end{itemize}
\hypertarget{ReferencesInTermsOfMotivicStructures}{}\subsubsection*{{In terms of motivic structures}}\label{ReferencesInTermsOfMotivicStructures}

In terms of [[motives]] (see also \emph{[[motives in physics]]})

\begin{itemize}%
\item [[Spencer Bloch]], [[Helene Esnault]], [[Dirk Kreimer]], \emph{On motives associated to graph polynomials, (\href{http://preprints.ihes.fr/2013/P/P-13-24.pdf}{pdf})}


\item [[Spencer Bloch]], Pierre Vanhove, The elliptic dilogarithm for the sunset graph, IHES preprint P-13-24, pdf



\end{itemize}
\hypertarget{ReferencesInHomologicalBVQuantization}{}\subsubsection*{{In homological BV-quantization}}\label{ReferencesInHomologicalBVQuantization}

A clean derivation of the [[Feynman rules]] for finite-dimensional spaces of fields in terms of [[quantization]] by passing to [[cochain cohomology]] of [[BV-complexes]] (see at \emph{\href{geometric%20quantization#AsIndexOfSpinCDiracOperator}{BV-formalism -- Homological quantization})} is in

\begin{itemize}%
\item [[Owen Gwilliam]], [[Theo Johnson-Freyd]], \emph{How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism} (2011) (\href{http://arxiv.org/abs/1202.1554}{arXiv:1202.1554})

\end{itemize}
with a review in the broader context of [[factorization algebras of observables]] in section 2.3 of

\begin{itemize}%
\item [[Owen Gwilliam]], \emph{Factorization algebras and free field theories} ([[GwilliamThesis.pdf:file]])

\end{itemize}
\hypertarget{AsStringDiagrams}{}\subsubsection*{{As string diagrams (morphisms in monoidal categories)}}\label{AsStringDiagrams}

It has been observed that Feynman diagrams, notably in [[gauge theory]], are in particular [[string diagrams]] (in the sense of [[category theory]], not here in any sense of [[string theory]]!) in the given [[category of representations]]: the edges are labelled by [[particle]] [[species]], hence by [[Wigner classification]] by [[irreps]], and the vertices are labeled by representation [[homomorphisms]] (``[[intertwiners]]'') which, indeed, label the [[interaction]] of particles in the Feynman diagram.

A review of this formulation is in

\begin{itemize}%
\item [[John Baez]], [[Aaron Lauda]], \emph{A Prehistory of n-Categorical Physics}, Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 (\href{http://arxiv.org/abs/0908.2469}{arXiv:0908.2469})

\end{itemize}
Moreover, one may think of [[string diagrams]] in [[monoidal categories]] as providing [[categorical semantics]] for [[proof nets]] (see there for more) in multiplicative [[linear logic]]. Under this identification then Feynman diagrams have a relation to [[proof nets]]. Something like this is discussed in

\begin{itemize}%
\item [[Richard Blute]], [[Prakash Panangaden]], \emph{Proof nets as formal Feynman diagrams} (\href{http://www.indiana.edu/~iulg/qliqc/phi.pdf}{pdf})

\end{itemize}
[[!redirects Feynman diagrams]]

[[!redirects Feynman rule]] [[!redirects Feynman rules]]

[[!redirects Feynman graph]] [[!redirects Feynman graphs]]



\end{document}