 represents a Feynman integral appearing in the expansion of the Scattering matrix (in coordinates representation) or in its matrix elements (in momentum representation) - Maple Programming Help

Physics[FeynmanIntegral] - represents a Feynman integral appearing in the expansion of the Scattering matrix (in coordinates representation) or in its matrix elements (in momentum representation)

 Calling Sequence FeynmanIntegral(integrand, [[X1], [X2], ... [Xn]]) FeynmanIntegral(integrand, [[p1], [p2], ... [pn]])

Parameters

 integrand - the integrand that appears in the expansion of the Scattering matrix $S=1+{S}_{1}+{S}_{2}+\mathrm{...}$ in coordinates representation, or its matrix elements $⟨f|S|i⟩$ in momentum representation. [[X1], ... [Xn]] - a list of lists, here each Xj is a label representing a coordinates system; it can also be one set using Setup or Coordinates. In this case the integral is in coordinates representation and displayed with a preceding factor $\frac{{i}^{n}}{n!}$, where n is the order of ${S}_{n}$ in the expansion of S. [[p1], ... [pn]] - a list of lists, here each pj is the name of a tensor, automatically defined as such when the integral appears in the output of FeynmanDiagrams; it can also be defined using Define. In this case, the integral is in momentum representation and has no preceding factor.

Description

 • A scattering matrix $S$ relates the initial and final states, $\left|i\right⟩$ and $\left|f\right⟩$, of an interacting system. In an N-dimensional spacetime with coordinates $X$, $S$ can be written as:

$S=T\left({ⅇ}^{i\int L\left(X\right)ⅆ{X}^{4}}\right)$

 where $i$ is the imaginary unit and $L$ is the interaction Lagrangian, written in terms of quantum fields depending on the spacetime coordinates $X$. The T symbol means time-ordered. For the terminology used in this page, see for instance chapters IV and V, "The Scattering Matrix" and "The Feynman Rules and Diagrams", in ref.. This exponential can be expanded as

$S=1+{S}_{1}+{S}_{2}+{S}_{3}+\mathrm{...}$

 where

${S}_{n}=\frac{{i}^{n}}{n!}\int \dots \int T\left(L\left(\mathrm{X__1}\right),\dots ,L\left(\mathrm{X__n}\right)\right)ⅆ{\mathrm{X__1}}^{4}\dots ⅆ{\mathrm{X__n}}^{4}$

 and $T\left(L\left({X}_{1}\right),\mathrm{...},L\left({X}_{n}\right)\right)$ is the time-ordered product of n interaction Lagrangians evaluated at different points. Note the factor $\frac{{i}^{n}}{n!}$ in this definition of ${S}_{n}$ used here.
 • The FeynmanIntegral(integrand, [[X1], [X2], ... [Xn]]) command is thus a computational representation for ${S}_{n}$ in coordinates representation. Note the factor $\frac{{i}^{n}}{n!}$ included in the definition used here for ${S}_{n}$. That factor is automatically displayed when the second argument, [[X1], [X2], ... [Xn]], is a list of lists of coordinate systems. For details on the algebraic structure of integrand see FeynmanDiagrams.
 • The FeynmanIntegral(integrand, [[p1], [p2], ... [pn]]) command, where [[p1], [p2], ... [pn]] is a list of lists of spacetime tensors, is a computational representation for the integrals in momentum representation that enter the S-matrix elements $⟨f|S|i⟩$ with initial and final states $\left|i\right⟩$ and $\left|f\right⟩$, respectively with s initial particles with defined momentum ${p}_{i}$ and r final particles with defined momentum ${p}_{f}$.
 • To obtain the S-matrix elements $⟨f|S|i⟩$ in momentum representation for each ${S}_{n}$ entering the expansion of S in coordinates representation see sec. 20.1 of ref..
 • In both coordinates and momentum representation, to each element in the list of lists of the second argument in FeynmanIntegral(integrand, [[..], [..], ... [..]]) corresponds a 4-dimensional integral, and the number of elements in that list of lists indicates the number of vertices of the Feynman diagrams corresponding to the Feynman integral.
 • The inert form of this command, %FeynmanIntegral, is used by the FeynmanDiagrams command to represent each ${S}_{j}$ entering $S=1+{S}_{1}+\mathrm{...}+{S}_{n}$ in coordinates representation, as well as the integrals in momentum representation entering $⟨f|S|i⟩$. For details on the algebraic structure of integrand in both representations see FeynmanDiagrams.

Examples

Load the package, set three coordinate systems and set $\mathrm{\phi }$ to represent a quantum operator

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true},\mathrm{coordinates}=\left[X,Y,Z\right],\mathrm{quantumoperators}=\mathrm{\phi }\right)$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(\mathrm{x1}{,}\mathrm{x2}{,}\mathrm{x3}{,}\mathrm{x4}\right){,}Y=\left(\mathrm{y1}{,}\mathrm{y2}{,}\mathrm{y3}{,}\mathrm{y4}\right){,}Z=\left(\mathrm{z1}{,}\mathrm{z2}{,}\mathrm{z3}{,}\mathrm{z4}\right)\right\}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}{,}{Y}{,}{Z}\right\}{,}{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}{,}{\mathrm{quantumoperators}}{=}\left\{{\mathrm{\phi }}\right\}\right]$ (1)

Let $L$ be the interaction Lagrangian

 > $L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^{4}$
 ${L}{≔}{\mathrm{\lambda }}{}{{\mathrm{\phi }}{}\left({X}\right)}^{{4}}$ (2)

The ${2}^{\mathrm{nd}}$ term of the expansion $S=1+{S}_{1}+{S}_{2}+\mathrm{...}$ in coordinates representation is represented using the inert form of the FeynmanIntegral command. All the corresponding Feynman diagrams contain two vertices

 > $S\left[2\right]=\mathrm{FeynmanDiagrams}\left(L,\mathrm{numberofvertices}=2,\mathrm{diagrams}\right)$   ${{S}}_{{2}}{=}{\mathrm{%FeynmanIntegral}}{}\left({16}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{_GF}}{}\left({\mathrm{_NP}}{}\left({\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right){,}{\mathrm{φ}}{}\left({Y}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right){,}\left[\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]\right]\right){+}{96}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{_GF}}{}\left({\mathrm{_NP}}{}\left({\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right){,}\left[\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]{,}\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]{,}\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]\right]\right){+}{72}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{_GF}}{}\left({\mathrm{_NP}}{}\left({\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right){,}\left[\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]{,}\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]\right]\right){,}\left[\left[{X}\right]{,}\left[{Y}\right]\right]\right)$ (3)

From the diagrams above, the possible configurations involve 2, 4 or 6 external legs, with respectively 2, 1 and 0 loops. To get the Feynman integral with the configuration that has only n external legs use the numberofexternallegs option of FeynmanDiagrams

 > $\mathrm{%eval}\left(S\left[2\right],\mathrm{legs}=2\right)=\mathrm{FeynmanDiagrams}\left(L,\mathrm{numberofvertices}=2,\mathrm{numberofexternallegs}=2\right)$
 ${\mathrm{%eval}}{}\left({{S}}_{{2}}{,}{\mathrm{legs}}{=}{2}\right){=}{\mathrm{%FeynmanIntegral}}{}\left({96}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{_GF}}{}\left({\mathrm{_NP}}{}\left({\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right){,}\left[\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]{,}\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]{,}\left[{\mathrm{φ}}{}\left({X}\right){,}{\mathrm{φ}}{}\left({Y}\right)\right]\right]\right){,}\left[\left[{X}\right]{,}\left[{Y}\right]\right]\right)$ (4)

Corresponding to this result, one possible specific process involving 2 external legs is that where there is an initial state $\left|i\right⟩$ with 1 incoming particle, and a final state $\left|f\right⟩$ with 1 outgoing particle. The S-matrix element $⟨f|S|i⟩$ for that process is expressed in terms of a FeynmanIntegral in momentum representation via

 > $\mathrm{%eval}\left(\mathrm{%Bracket}\left(f,S\left[2\right],i\right),\mathrm{loops}=2\right)=\mathrm{FeynmanDiagrams}\left(L,\mathrm{incomingparticles}=\left[\mathrm{\phi }\right],\mathrm{outgoingparticles}=\left[\mathrm{\phi }\right],\mathrm{numberofloops}=2\right)$
 ${\mathrm{%eval}}{}\left({\mathrm{%Bracket}}{}\left({f}{,}{{S}}_{{2}}{,}{i}\right){,}{\mathrm{loops}}{=}{2}\right){=}{\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left(\frac{\frac{{3}}{{8}}{}{I}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{Dirac}}{}\left({-}\mathit{P__2}{+}\mathit{P__1}\right)}{{{\mathrm{π}}}^{{7}}{}\sqrt{\mathit{E__1}{}\mathit{E__2}}{}\left({\mathit{p__2}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__3}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{P__1}{-}\mathit{p__2}{-}\mathit{p__3}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathit{p__2}\right]\right]\right){,}\left[\left[\mathit{p__3}\right]\right]\right)$ (5)

The Feynman integrals, whose corresponding diagrams have 3 loops, for the same process $\mathrm{\phi }↦\mathrm{\phi }$

 > $\mathrm{%eval}\left(\mathrm{%Bracket}\left(f,S\left[2\right],i\right),\mathrm{loops}=3\right)=\mathrm{FeynmanDiagrams}\left(L,\mathrm{incomingparticles}=\left[\mathrm{\phi }\right],\mathrm{outgoingparticles}=\left[\mathrm{\phi }\right],\mathrm{numberofloops}=3\right)$
 ${\mathrm{%eval}}{}\left({\mathrm{%Bracket}}{}\left({f}{,}{{S}}_{{2}}{,}{i}\right){,}{\mathrm{loops}}{=}{3}\right){=}{\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left(\frac{{9}}{{32}}{}\frac{{{\mathrm{λ}}}^{{3}}{}{\mathrm{Dirac}}{}\left({-}\mathit{P__2}{+}\mathit{P__1}\right)}{{{\mathrm{π}}}^{{11}}{}\sqrt{\mathit{E__1}{}\mathit{E__2}}{}\left({\mathit{p__3}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__4}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__5}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{p__3}{-}\mathit{p__4}{-}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{P__1}{+}\mathit{P__2}{+}\mathit{p__3}{+}\mathit{p__4}{+}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathit{p__3}\right]\right]\right){,}\left[\left[\mathit{p__4}\right]\right]\right){,}\left[\left[\mathit{p__5}\right]\right]\right){+}{2}{}{\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left(\frac{{9}}{{32}}{}\frac{{{\mathrm{λ}}}^{{3}}{}{\mathrm{Dirac}}{}\left({-}\mathit{P__2}{+}\mathit{P__1}\right)}{{{\mathrm{π}}}^{{11}}{}\sqrt{\mathit{E__1}{}\mathit{E__2}}{}\left({\mathit{p__3}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__4}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__5}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{p__3}{-}\mathit{p__4}{-}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{P__1}{+}\mathit{p__4}{+}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathit{p__3}\right]\right]\right){,}\left[\left[\mathit{p__4}\right]\right]\right){,}\left[\left[\mathit{p__5}\right]\right]\right){+}{\mathrm{%FeynmanIntegral}}{}\left({\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{2048}}{}\frac{{\mathrm{λ}}{}{\mathrm{Dirac}}{}\left({-}\mathit{P__2}{+}\mathit{P__1}\right){}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{576}{}{{\mathrm{λ}}}^{{2}}}{\left({\mathit{p__2}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{p__2}{-}\mathit{p__4}{-}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathit{p__2}\right]\right]\right)}{{{\mathrm{π}}}^{{11}}{}\sqrt{\mathit{E__1}{}\mathit{E__2}}{}\left({\mathit{p__4}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathit{p__5}}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathit{P__1}{+}\mathit{p__4}{+}\mathit{p__5}\right)}^{{2}}{-}{\mathit{m__φ}}^{{2}}{+}{I}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathit{p__4}\right]\right]\right){,}\left[\left[\mathit{p__5}\right]\right]\right)$ (6)
 > 

For details about the integrands entering these Feynman integrals see FeynmanDiagrams.

References

  Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.
  Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.
  Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P., Quantum Electrodynamics. Vol.4, Course of Theoretical Physics, 2nd edition, Pergamon Press, 1982.
  Xiao, B., Wang, H., Zhu, S., A simple algorithm for automatic Feynman diagrams generation. Computer Physics Communications, Volume 184, Issue 8, (2013).

Compatibility

 • The Physics[FeynmanIntegral] command was introduced in Maple 2020.