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.[1]. 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.[1].
 • 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

 [1] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.
 [2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.
 [3] Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P., Quantum Electrodynamics. Vol.4, Course of Theoretical Physics, 2nd edition, Pergamon Press, 1982.
 [4] 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.