StandardModel is a Physics's package that implements computational representations for the mathematical objects formulating the Standard Model in particle physics. The package includes field representations for the leptons and quarks, Weinberg's angle, the Higgs boson, the fields and field strengths after breaking symmetries and most of the fields before that, the structure constants ${\mathrm{FSU3}}_{a\,b}$and Gell-Mann matrices ${\mathrm{\lambda }}_a$, the electroweak charges and isospins, and also a command, Lagrangian, to retrieve the Lagrangian of the different sectors of the Standard Model like QED, QCD or Electro-Weak, or of all of it.

Loading the package sets things to proceed computing with the model, as the fields and their computational properties, the commutation rules, covariant derivatives, the values of all the constants and matrices involved, the letters representing different kinds of tensor indices etc. Some of these settings, for instance the kinds of letters used to represent different kinds of indices, can be changed using the Physics Setup command.

$\mathrm{with}\left(\mathrm{Physics}\right)\:\mathrm{with}\left(\mathrm{StandardModel}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{Dirac\; spinor}\mathrm{indices}$

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; adjoint\; representation,\; (1..8)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; fundamental\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; adjoint\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; fundamental\; representation,\; (1..2)}\mathrm{indices}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Defined\; as\; the\; electron,\; muon\; and\; tau\; leptons\; and\; corresponding\; neutrinos:}{\mathrm{e}}_j,{\mathrm{\mu }}_j,{\mathrm{\tau }}_j,{\mathrm{ElectronNeutrino}}_j,{\mathrm{MuonNeutrino}}_j,{\mathrm{TauonNeutrino}}_j$

$\mathrm{Defined\; as\; the\; up,\; charm,\; top,\; down,\; strange\; and\; bottom\; quarks:}{\mathrm{u}}_{j,A},{\mathrm{c}}_{j,A},{\mathrm{t}}_{j,A},{\mathrm{d}}_{j,A},{\mathrm{s}}_{j,A},{\mathrm{b}}_{j,A}$

$\mathrm{Defined\; as\; gauge\; tensors:}{\mathrm{B}}_{\mathrm{\mu }},{\mathrm{𝔹}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{A}}_{\mathrm{\mu }},{\mathrm{𝔽}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{W}}_{\mathrm{\mu },J},{\mathrm{𝕎}}_{\mathrm{\mu },\mathrm{\nu },J},{\mathrm{WPlusField}}_{\mathrm{\μ}},{\mathrm{WPlusFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}},{\mathrm{WMinusField}}_{\mathrm{\μ}},{\mathrm{WMinusFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}},{\mathrm{Z}}_{\mathrm{\mu }},{\mathrm{\mathbb{Z}}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{G}}_{\mathrm{\mu },a},{\mathrm{𝔾}}_{\mathrm{\mu },\mathrm{\nu },a}$

$\mathrm{Defined\; as\; Gell-Mann\; (Glambda),\; Pauli\; (Psigma)\; and\; Dirac\; (Dgamma)\; matrices:}{\mathrm{\lambda }}_a,{\mathrm{\sigma }}_J,{\mathrm{\gamma }}_{\mathrm{\mu }}$

$\mathrm{Defined\; as\; the\; electric,\; weak\; and\; strong\; coupling\; constants:}\mathrm{g\_\_e},\mathrm{g\_\_w},\mathrm{g\_\_s}$

$\mathrm{Defined\; as\; the\; charge\; in\; units\; of\; |g\_\_e|\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; and\; 3)\; the\; down,\; strange\; and\; bottom:}\mathrm{\%q\_\_e}=\mathrm{-1},\mathrm{\%q\_\_u}=\frac{2}{3},\mathrm{\%q\_\_d}=-\frac{1}{3}$

$\mathrm{Defined\; as\; the\; weak\; isospin\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; 3)\; the\; down,\; strange\; and\; bottom,\; and\; 4)\; all\; the\; neutrinos:}\mathrm{\%I\_\_e}=-\frac{1}{2},\mathrm{\%I\_\_u}=\frac{1}{2},\mathrm{\%I\_\_d}=-\frac{1}{2},\mathrm{\%I\_\_n}=\frac{1}{2},$

$\mathrm{You\; can\; use\; the\; active\; form\; without\; the\; \%\; prefix,\; or\; the\; \text{'}value\text{'}\; command\; to\; give\; the\; corresponding\; value\; to\; any\; of\; the\; inert\; representations}\mathrm{\%q\_\_e},\mathrm{\%q\_\_u},\mathrm{\%q\_\_d},\mathrm{\%I\_\_e},\mathrm{\%I\_\_u},\mathrm{\%I\_\_d},\mathrm{\%I\_\_n},$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{Minkowski\; spacetime\; with\; signatre}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{\%I\_\_d}\,\mathrm{\%I\_\_e}\,\mathrm{\%I\_\_n}\,\mathrm{\%I\_\_u}\,\mathrm{\%q\_\_d}\,\mathrm{\%q\_\_e}\,\mathrm{\%q\_\_u}\,\mathrm{BField}\,\mathrm{BFieldStrength}\,\mathrm{Bottom}\,\mathrm{CKM}\,\mathrm{Charm}\,\mathrm{Down}\,\mathrm{ElectromagneticField}\,\mathrm{ElectromagneticFieldStrength}\,\mathrm{Electron}\,\mathrm{ElectronNeutrino}\,\mathrm{FSU3}\,\mathrm{Glambda}\,\mathrm{GluonField}\,\mathrm{GluonFieldStrength}\,\mathrm{HiggsBoson}\,\mathrm{Lagrangian}\,\mathrm{Muon}\,\mathrm{MuonNeutrino}\,\mathrm{Strange}\,\mathrm{Tauon}\,\mathrm{TauonNeutrino}\,\mathrm{Top}\,\mathrm{Up}\,\mathrm{WField}\,\mathrm{WFieldStrength}\,\mathrm{WMinusField}\,\mathrm{WMinusFieldStrength}\,\mathrm{WPlusField}\,\mathrm{WPlusFieldStrength}\,\mathrm{WeinbergAngle}\,\mathrm{ZField}\,\mathrm{ZFieldStrength}\,\mathrm{g\_\_e}\,\mathrm{g\_\_s}\,\mathrm{g\_\_w}\right]$

A key observation in this implementation is that, with the exception of $\mathrm{Lagrangian}\,$$\mathrm{FSU3}$representing the structure constants ${\mathrm{FSU3}}_{a\,b}$ and $\mathrm{Glambda}$ representing the Gell-Mann matrices ${\mathrm{\lambda }}_a$, the package's commands are not actually routines that perform tasks but representations of the physical fields of the model with the mathematical properties set and understood by the Maple system. Once the representations are in place, the actual computations are performed by the Physics package commands, e.g. scattering amplitudes are computed using FeynmanDiagrams, covariant and non-covariant derivatives are computed using D_ and d_, commutators and anticommutators using Commutator and AntiCommutator, and the simplification of tensor indices is performed by the Physics Simplify command. The $\mathrm{Lagrangian}$ command is used to retrieve all or part of the Lagrangian of the Standard Model, as shown in the Examples.

The following is a list of available commands.

The electric part of the electroweak charges and isospins of the model are all represented in inert form following the Maple convention: prefixed with the % symbol displayed in gray as in (1). If any of these constants is entered without that prefix you get its actual value; for the electric part of the electroweak charges of the model, in units of $\mathrm{g\_\_e}$, that is $\mathrm{q\_\_e}=\mathrm{-1}\, \mathrm{q\_\_u}=\frac{2}{3}\, \mathrm{q\_\_d}=\frac{1}{3}$ respectively for the 1) the electron, muon and tauon, 2) the up, charm and top quarks and 3) the down strange and bottom quarks. For the isospins that is $\mathrm{I\_\_e}=-\frac{1}{2}\, \mathrm{I\_\_u}=\frac{1}{2}\, \mathrm{I\_\_d}=-\frac{1}{2}\, \mathrm{I\_\_n}=\frac{1}{2}$, respectively for 1) the electron, muon and tauon, 2) the up, charm and top, 3) the down, strange and bottom, and 4) all the neutrinos.

The electromagnetic coupling constant $\mathrm{g\_\_e}$ is negative, equal to the charge of the electron, and $\mathrm{g\_\_s}$ and $\mathrm{g\_\_w}$ are respectively the the strong and weak coupling constants. These are the constants that enter as a factor in the gauge term of covariant derivatives, and so in the definition of the gauge field strengths ${\mathbf{𝔾}}_{\mathrm{\mu },\mathrm{\nu },a}$ and ${\mathbf{𝕎}}_{\mathrm{\mu },\mathrm{\nu },J}$ and in the interaction terms of the Lagrangian.

The $\mathrm{CKM}$ command is a representation for the Cabibbo Kobayashi Maskawa matrix.

Further information relevant to the use of these commands is found under Conventions used in the Physics package (spacetime tensors, not commutative variables and functions, quantum states and Dirac notation).

The typesetting of the package's representations for fields and constant follows as much as possible the one in textbooks and works under copy and paste - so that expressions from the output can be reused as input. To transform Maple worksheets using this package into LaTeX files see LaTeX.

$\mathrm{Setup}\left(\mathrm{massless}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{massless}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{masslessfields}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{masslessfields}=\left\{\mathbf{G}\,\mathrm{MuonNeutrino}\,\mathrm{TauonNeutrino}\,\mathbf{A}\,\mathrm{ElectronNeutrino}\right\}\right]$

$\mathrm{StandardModel}:-\mathrm{Leptons}$

$\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\,\mathrm{ElectronNeutrino}\,\mathrm{MuonNeutrino}\,\mathrm{TauonNeutrino}\right]$

$\mathrm{StandardModel}:-\mathrm{Quarks}$

$\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\,\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]$

$\mathrm{StandardModel}:-\mathrm{GaugeFields}$

$\left[\mathbf{A}\,\mathbf{𝔽}\,\mathbf{B}\,\mathbf{𝔹}\,\mathbf{W}\,\mathbf{𝕎}\,\mathbf{G}\,\mathbf{𝔾}\,\mathrm{WMinusField}\,\mathrm{WMinusFieldStrength}\,\mathrm{WPlusField}\,\mathrm{WPlusFieldStrength}\,\mathbf{Z}\,\mathbf{\mathbb{Z}}\right]$

$\mathrm{CompactDisplay}\left(\left(\mathrm{StandardModel}:-\mathrm{Leptons}\,\mathrm{StandardModel}:-\mathrm{Quarks}\,\mathrm{StandardModel}:-\mathrm{GaugeFields}\, \mathrm{HiggsBoson}\right)\left(X\right)\,\mathrm{quiet}\right)$

$\mathrm{interface}\left(\mathrm{imaginaryunit} \= i\right)\:$

$\mathrm{ElectromagneticField}\left[\mathrm{definition}\right]$

${\mathbf{A}}_{\mathrm{\mu }}=\sin \left(\mathrm{WeinbergAngle}\right){\mathbf{W}}_{\mathrm{\mu }\phantom{3}}^{\phantom{\mathrm{\mu }}3}+\cos \left(\mathrm{WeinbergAngle}\right){\mathbf{B}}_{\mathrm{\mu }}$

$\mathrm{map}\left(u \to u\left[\mathrm{definition}\right]\,\mathrm{StandardModel}:-\mathrm{GaugeFields}\right)$

$\left[{\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\=\sin \left(\mathrm{WeinbergAngle}\right){\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~3}}\+\cos \left(\mathrm{WeinbergAngle}\right){\mathrm{BField}}_{\mathrm{\μ}}\,{\mathrm{ElectromagneticFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{ElectromagneticField}}_{\mathrm{\ν}}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\left(X\right)\,\left[X\right]\right)\,{\mathrm{BField}}_{\mathrm{\μ}}\=\left(\left[\begin{array}{cccc}{\mathrm{BField}}_1& {\mathrm{BField}}_2& {\mathrm{BField}}_3& {\mathrm{BField}}_4\end{array}\right]\right)\,{\mathrm{BFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{BField}}_{\mathrm{\ν}}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{BField}}_{\mathrm{\μ}}\left(X\right)\,\left[X\right]\right)\,{\mathrm{WField}}_{\mathrm{\μ}\,J}\=\left(\left[\begin{array}{ccc}{\mathrm{WField}}_{1\,1}& {\mathrm{WField}}_{1\,2}& {\mathrm{WField}}_{1\,3}\\ {\mathrm{WField}}_{2\,1}& {\mathrm{WField}}_{2\,2}& {\mathrm{WField}}_{2\,3}\\ {\mathrm{WField}}_{3\,1}& {\mathrm{WField}}_{3\,2}& {\mathrm{WField}}_{3\,3}\\ {\mathrm{WField}}_{4\,1}& {\mathrm{WField}}_{4\,2}& {\mathrm{WField}}_{4\,3}\end{array}\right]\right)\,{\mathrm{WFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}\,J}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{WField}}_{\mathrm{\ν}\,J}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{WField}}_{\mathrm{\μ}\,J}\left(X\right)\,\left[X\right]\right)\+\mathrm{g\_\_w}{\mathrm{LeviCivita}}_{J\,K\,L}\mathrm{`*`}\left({\mathrm{WField}}_{\mathrm{\μ}\,K}\left(X\right)\,{\mathrm{WField}}_{\mathrm{\ν}\,L}\left(X\right)\right)\,{\mathrm{GluonField}}_{\mathrm{\μ}\,a}\=\left(\left[\begin{array}{cccccccc}{\mathrm{GluonField}}_{1\,1}& {\mathrm{GluonField}}_{1\,2}& {\mathrm{GluonField}}_{1\,3}& {\mathrm{GluonField}}_{1\,4}& {\mathrm{GluonField}}_{1\,5}& {\mathrm{GluonField}}_{1\,6}& {\mathrm{GluonField}}_{1\,7}& {\mathrm{GluonField}}_{1\,8}\\ {\mathrm{GluonField}}_{2\,1}& {\mathrm{GluonField}}_{2\,2}& {\mathrm{GluonField}}_{2\,3}& {\mathrm{GluonField}}_{2\,4}& {\mathrm{GluonField}}_{2\,5}& {\mathrm{GluonField}}_{2\,6}& {\mathrm{GluonField}}_{2\,7}& {\mathrm{GluonField}}_{2\,8}\\ {\mathrm{GluonField}}_{3\,1}& {\mathrm{GluonField}}_{3\,2}& {\mathrm{GluonField}}_{3\,3}& {\mathrm{GluonField}}_{3\,4}& {\mathrm{GluonField}}_{3\,5}& {\mathrm{GluonField}}_{3\,6}& {\mathrm{GluonField}}_{3\,7}& {\mathrm{GluonField}}_{3\,8}\\ {\mathrm{GluonField}}_{4\,1}& {\mathrm{GluonField}}_{4\,2}& {\mathrm{GluonField}}_{4\,3}& {\mathrm{GluonField}}_{4\,4}& {\mathrm{GluonField}}_{4\,5}& {\mathrm{GluonField}}_{4\,6}& {\mathrm{GluonField}}_{4\,7}& {\mathrm{GluonField}}_{4\,8}\end{array}\right]\right)\,{\mathrm{GluonFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}\,a}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{GluonField}}_{\mathrm{\ν}\,a}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{GluonField}}_{\mathrm{\μ}\,a}\left(X\right)\,\left[X\right]\right)\+\mathrm{g\_\_s}{\mathrm{FSU3}}_{a\,b\,c}\mathrm{`*`}\left({\mathrm{GluonField}}_{\mathrm{\μ}\,b}\left(X\right)\,{\mathrm{GluonField}}_{\mathrm{\ν}\,c}\left(X\right)\right)\,{\mathrm{WMinusField}}_{\mathrm{\μ}}\=\frac{1}{2}\left({\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~1}}\+\mathrm{i}{\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~2}}\right)\sqrt{2}\,{\mathrm{WMinusFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{WMinusField}}_{\mathrm{\ν}}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{WMinusField}}_{\mathrm{\μ}}\left(X\right)\,\left[X\right]\right)\,{\mathrm{WPlusField}}_{\mathrm{\μ}}\=\frac{1}{2}\left({\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~1}}-\mathrm{i}{\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~2}}\right)\sqrt{2}\,{\mathrm{WPlusFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{WPlusField}}_{\mathrm{\ν}}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{WPlusField}}_{\mathrm{\μ}}\left(X\right)\,\left[X\right]\right)\,{\mathrm{ZField}}_{\mathrm{\μ}}\=\cos \left(\mathrm{WeinbergAngle}\right){\mathrm{WField}}_{\mathrm{\μ}\,\mathrm{~3}}-\sin \left(\mathrm{WeinbergAngle}\right){\mathrm{BField}}_{\mathrm{\μ}}\,{\mathrm{ZFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{ZField}}_{\mathrm{\ν}}\left(X\right)\,\left[X\right]\right)-{\mathrm{d\_}}_{\mathrm{\ν}}\left({\mathrm{ZField}}_{\mathrm{\μ}}\left(X\right)\,\left[X\right]\right)\right]$

$$\begin{gathered} \mathrm{D\_}\left[\mathrm{mu}\right]\left(\mathrm{Electron}\left[j\right]\left(X\right)\right)\: \\ \% \= \mathrm{expand}\left(\%\right) \end{gathered}$$

${\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_j\left(X\right)\,\left[X\right]\right)\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_j\left(X\right)\,\left[X\right]\right)\+\mathrm{i}\mathrm{g\_\_e}\mathrm{`*`}\left({\mathrm{Electron}}_j\left(X\right)\,{\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\left(X\right)\right)$

$$\begin{gathered} \mathrm{D\_}\left[\mathrm{mu}\right]\left(\mathrm{Up}\left[j\,A\right]\left(X\right)\right)\: \\ \% \= \mathrm{expand}\left(\%\right) \end{gathered}$$

${\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{Up}}_{j\,A}\left(X\right)\,\left[X\right]\right)\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{Up}}_{j\,A}\left(X\right)\,\left[X\right]\right)-\frac{1}{2}\mathrm{i}\mathrm{g\_\_s}\mathrm{`*`}\left({{\mathrm{Glambda}}_a}_{A\,B}\,{\mathrm{Up}}_{j\,B}\left(X\right)\,{\mathrm{GluonField}}_{\mathrm{\μ}\,a}\left(X\right)\right)$

$\mathrm{Glambda}\left[\right]$

${\mathrm{Glambda}}_a\=\left(\left[\begin{array}{cccccccc}{\mathrm{Glambda}}_1& {\mathrm{Glambda}}_2& {\mathrm{Glambda}}_3& {\mathrm{Glambda}}_4& {\mathrm{Glambda}}_5& {\mathrm{Glambda}}_6& {\mathrm{Glambda}}_7& {\mathrm{Glambda}}_8\end{array}\right]\right)$

$\mathrm{seq}\left(\mathrm{Glambda}\left[a\,\mathrm{matrix}\right]\, a\=1..8\right)$

${\mathrm{\lambda }}_1=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],{\mathrm{\lambda }}_2=\left[\begin{array}{ccc}0& \mathrm{-i}& 0\\ \mathrm{i}& 0& 0\\ 0& 0& 0\end{array}\right],{\mathrm{\lambda }}_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{-1}& 0\\ 0& 0& 0\end{array}\right],{\mathrm{\lambda }}_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],{\mathrm{\lambda }}_5=\left[\begin{array}{ccc}0& 0& \mathrm{-i}\\ 0& 0& 0\\ \mathrm{i}& 0& 0\end{array}\right],{\mathrm{\lambda }}_6=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],{\mathrm{\lambda }}_7=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& \mathrm{-i}\\ 0& \mathrm{i}& 0\end{array}\right],{\mathrm{\lambda }}_8=\left[\begin{array}{ccc}\frac{\sqrt{3}}{3}& 0& 0\\ 0& \frac{\sqrt{3}}{3}& 0\\ 0& 0& -\frac{2\sqrt{3}}{3}\end{array}\right]$

$\mathrm{Library}:-\mathrm{DefaultAlgebraRules}\left(\mathrm{Glambda}\right)$

$\mathrm{\%Commutator}\left({\mathrm{Glambda}}_b\,{\mathrm{Glambda}}_c\right)\=2\mathrm{i}{\mathrm{FSU3}}_{a\,b\,c}{\mathrm{Glambda}}_a$

$\mathrm{FSU3}\left[1\,b\,c\,\mathrm{matrix}\right]$

${\mathrm{FSU3}}_{1\,b\,c}\=\left(\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{2}& 0\\ 0& 0& 0& 0& 0& -\frac{1}{2}& 0& 0\\ 0& 0& 0& 0& \frac{1}{2}& 0& 0& 0\\ 0& 0& 0& -\frac{1}{2}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\right)$

$\mathrm{TensorArray}\left(\mathrm{FSU3}\left[a\,b\,c\right]\,\mathrm{explore}\right)$

${\mathrm{FSU3}}_{a\,b\,c}\left(\mathrm{ordering\; of\; free\; indices}=\left[a\,b\,c\right]\right)$

$\mathrm{\%Commutator}\left({\mathrm{Glambda}}_b\,{\mathrm{Glambda}}_c\right)\=2\mathrm{i}{\mathrm{FSU3}}_{a\,b\,c}{\mathrm{Glambda}}_a$

$\mathrm{SumOverRepeatedIndices}\left(\right)$

$\mathrm{\%Commutator}\left({\mathrm{Glambda}}_b\,{\mathrm{Glambda}}_c\right)\=2\mathrm{i}\left({\mathrm{FSU3}}_{1\,b\,c}{\mathrm{Glambda}}_1\+{\mathrm{FSU3}}_{2\,b\,c}{\mathrm{Glambda}}_2\+{\mathrm{FSU3}}_{3\,b\,c}{\mathrm{Glambda}}_3\+{\mathrm{FSU3}}_{4\,b\,c}{\mathrm{Glambda}}_4\+{\mathrm{FSU3}}_{5\,b\,c}{\mathrm{Glambda}}_5\+{\mathrm{FSU3}}_{6\,b\,c}{\mathrm{Glambda}}_6\+{\mathrm{FSU3}}_{7\,b\,c}{\mathrm{Glambda}}_7\+{\mathrm{FSU3}}_{8\,b\,c}{\mathrm{Glambda}}_8\right)$

$\mathrm{eval}\left(\,\left[b\=4\,c\=5\right]\right)$

$\mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_5\right)\=2\mathrm{i}\left(\frac{1}{2}{\mathrm{Glambda}}_3\+\frac{1}{2}\sqrt{3}{\mathrm{Glambda}}_8\right)$

$\mathrm{value}\left(\right)$

$2\mathrm{i}{\mathrm{FSU3}}_{4\,5\,a}{\mathrm{\lambda }}_a=2\mathrm{i}\left(\frac{{\mathrm{\lambda }}_3}{2}+\frac{\sqrt{3}{\mathrm{\lambda }}_8}{2}\right)$

$\mathrm{expand}\left(\mathrm{SumOverRepeatedIndices}\left(\right)\right)$

$\mathrm{i}{\mathrm{\lambda }}_3+\mathrm{i}\sqrt{3}{\mathrm{\lambda }}_8=\mathrm{i}{\mathrm{\lambda }}_3+\mathrm{i}\sqrt{3}{\mathrm{\lambda }}_8$

$\mathrm{TensorArray}\left(\right)$

$\left[\begin{array}{cccccccc}\mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_2\right)\=2\mathrm{I}{\mathrm{Glambda}}_3& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_3\right)\=-2\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_4\right)\=\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_5\right)\=-\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_6\right)\=\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_7\right)\=-\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_1\,{\mathrm{Glambda}}_8\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_1\right)\=-2\mathrm{I}{\mathrm{Glambda}}_3& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_3\right)\=2\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_4\right)\=\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_5\right)\=\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_6\right)\=-\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_7\right)\=-\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_2\,{\mathrm{Glambda}}_8\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_1\right)\=2\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_2\right)\=-2\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_3\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_4\right)\=\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_5\right)\=-\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_6\right)\=-\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_7\right)\=\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_3\,{\mathrm{Glambda}}_8\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_1\right)\=-\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_2\right)\=-\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_3\right)\=-\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_4\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_5\right)\=\mathrm{I}\left(\sqrt{3}{\mathrm{Glambda}}_8\+{\mathrm{Glambda}}_3\right)& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_6\right)\=\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_7\right)\=\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_4\,{\mathrm{Glambda}}_8\right)\=-\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_5\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_1\right)\=\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_2\right)\=-\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_3\right)\=\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_4\right)\=-\mathrm{I}\left(\sqrt{3}{\mathrm{Glambda}}_8\+{\mathrm{Glambda}}_3\right)& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_5\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_6\right)\=-\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_7\right)\=\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_5\,{\mathrm{Glambda}}_8\right)\=\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_4\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_1\right)\=-\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_2\right)\=\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_3\right)\=\mathrm{I}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_4\right)\=-\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_5\right)\=\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_6\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_7\right)\=\mathrm{I}\left(-{\mathrm{Glambda}}_3\+\sqrt{3}{\mathrm{Glambda}}_8\right)& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_6\,{\mathrm{Glambda}}_8\right)\=-\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_7\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_1\right)\=\mathrm{I}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_2\right)\=\mathrm{I}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_3\right)\=-\mathrm{I}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_4\right)\=-\mathrm{I}{\mathrm{Glambda}}_1& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_5\right)\=-\mathrm{I}{\mathrm{Glambda}}_2& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_6\right)\=-\mathrm{I}\left(-{\mathrm{Glambda}}_3\+\sqrt{3}{\mathrm{Glambda}}_8\right)& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_7\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_7\,{\mathrm{Glambda}}_8\right)\=\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_6\\ \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_3\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_4\right)\=\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_5& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_5\right)\=-\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_4& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_6\right)\=\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_7& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_7\right)\=-\mathrm{I}\sqrt{3}{\mathrm{Glambda}}_6& \mathrm{\%Commutator}\left({\mathrm{Glambda}}_8\,{\mathrm{Glambda}}_8\right)\=0\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativeprefix} \= \left\{\mathrm{f\_\_L}\,\mathrm{f\_\_Q}\right\}\right)$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{f\_\_L}\,\mathrm{f\_\_Q}\right\}\right]$

$\mathrm{CompactDisplay}\left(\left(\mathrm{f\_\_L}\,\mathrm{f\_\_Q}\right)\left(X\right)\right)$

$\mathrm{f\_\_L}\left(X\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{f\_\_L}$

$\mathrm{f\_\_Q}\left(X\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{f\_\_Q}$

$\mathrm{L\_\_QED} ≔ -\mathrm{g\_\_e} \mathrm{conjugate}\left(\mathrm{Electron}\left[i\right]\left(X\right)\right)\cdot \mathrm{Dgamma}\left[~\mathrm{mu}\right]\left[i\, j\right]\cdot \mathrm{Electron}\left[j\right]\left(X\right)\cdot \mathrm{ElectromagneticField}\left[\mathrm{mu}\right]\left(X\right)$

$-\mathrm{g\_\_e}{{\mathrm{Dgamma}}_{\mathrm{~mu}}}_{\mathrm{i}\,j}\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_{\mathrm{i}}\left(X\right)\right)\,{\mathrm{Electron}}_j\left(X\right)\,{\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\left(X\right)\right)$

$\mathrm{FeynmanDiagrams}\left(\mathrm{L\_\_QED}\,\mathrm{incomingparticles}=\left[\mathrm{Electron}\right]\,\mathrm{outgoing}=\left[\mathrm{Electron}\right]\,\mathrm{numberofloops}=1\,\mathrm{diagrams}\right)$

$-\mathrm{\%FeynmanIntegral}\left(-\frac{1}{8}\frac{{{\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{Uspinor}}_{\mathrm{Electron}}}_k\left(\mathrm{P\_\_1\_}\right)\mathrm{conjugate}\left({{\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{Uspinor}}_{\mathrm{Electron}}}_l\left(\mathrm{P\_\_2\_}\right)\right){\mathrm{g\_\_e}}^2{{\mathrm{Dgamma}}_{\mathrm{~alpha}}}_{l\,m}{{\mathrm{Dgamma}}_{\mathrm{~nu}}}_{n\,k}\left(\left({\mathrm{P\_\_1}}_{\mathrm{\β}}\+{\mathrm{p\_\_2}}_{\mathrm{\β}}\right){{\mathrm{Dgamma}}_{\mathrm{~beta}}}_{m\,n}\+m_{\mathrm{Electron}}{\mathrm{KroneckerDelta}}_{m\,n}\right){\mathrm{g\_}}_{\mathrm{\α}\,\mathrm{\ν}}\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^3\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\right)}^2-m_{\mathrm{Electron}}^2\+\mathrm{i}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_2}}^2\+\mathrm{i}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)$

$\mathrm{FeynmanDiagrams}\left(\mathrm{L\_\_QED}\,\mathrm{incomingparticles}=\left[\mathrm{ElectromagneticField}\right]\,\mathrm{outgoing}=\left[\mathrm{ElectromagneticField}\right]\,\mathrm{numberofloops}=1\,\mathrm{diagrams}\right)$

$-\mathrm{\%FeynmanIntegral}\left(\frac{1}{16}\frac{{{\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{PolarizationVector}}_{\mathrm{ElectromagneticField}}}_{\mathrm{\ν}}\left(\mathrm{P\_\_1\_}\right)\mathrm{conjugate}\left({{\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{PolarizationVector}}_{\mathrm{ElectromagneticField}}}_{\mathrm{\α}}\left(\mathrm{P\_\_2\_}\right)\right){\mathrm{g\_\_e}}^2{{\mathrm{Dgamma}}_{\mathrm{~alpha}}}_{n\,k}{{\mathrm{Dgamma}}_{\mathrm{~nu}}}_{m\,l}\left(\left({\mathrm{P\_\_1}}_{\mathrm{\β}}\+{\mathrm{p\_\_2}}_{\mathrm{\β}}\right){{\mathrm{Dgamma}}_{\mathrm{~beta}}}_{k\,m}\+m_{\mathrm{Electron}}{\mathrm{KroneckerDelta}}_{k\,m}\right)\left(m_{\mathrm{Electron}}{\mathrm{KroneckerDelta}}_{l\,n}\+{\mathrm{p\_\_2}}_{\mathrm{\κ}}{{\mathrm{Dgamma}}_{\mathrm{~kappa}}}_{l\,n}\right)\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\right)}^2-m_{\mathrm{Electron}}^2\+\mathrm{i}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_2}}^2-m_{\mathrm{Electron}}^2\+\mathrm{i}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)$

$\mathrm{Lagrangian}\left(\mathrm{QED}\right)$

$\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,\mathrm{i}{{\mathrm{Dgamma}}_{\mathrm{\μ}}}_{j\,k}{\mathrm{D\_}}_{\mathrm{~mu}}-m_{\mathrm{Electron}}{\mathrm{KroneckerDelta}}_{j\,k}\,{\mathrm{Electron}}_k\left(X\right)\right)-\frac{1}{4}\mathrm{`*`}\left({\mathrm{ElectromagneticFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\,{\mathrm{ElectromagneticFieldStrength}}_{\mathrm{~mu}\,\mathrm{~nu}}\right)$

$\mathrm{Library}:-\mathrm{ApplyProductsOfDifferentialOperators}\left(\right)$

$\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,\mathrm{i}{{\mathrm{Dgamma}}_{\mathrm{\μ}}}_{j\,k}{\mathrm{D\_}}_{\mathrm{~mu}}\left({\mathrm{Electron}}_k\left(X\right)\,\left[X\right]\right)-\mathrm{`*`}\left(m_{\mathrm{Electron}}\,{\mathrm{KroneckerDelta}}_{j\,k}\,{\mathrm{Electron}}_k\left(X\right)\right)\right)-\frac{1}{4}\mathrm{`*`}\left({\mathrm{ElectromagneticFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\,{\mathrm{ElectromagneticFieldStrength}}_{\mathrm{~mu}\,\mathrm{~nu}}\right)$

$\mathrm{Lagrangian}\left(\mathrm{QED}\, \mathrm{applied}\right)$

$\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,\mathrm{i}{\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_k\left(X\right)\,\left[X\right]\right){{\mathrm{Dgamma}}_{\mathrm{~mu}}}_{j\,k}-\mathrm{`*`}\left(m_{\mathrm{Electron}}\,{\mathrm{KroneckerDelta}}_{j\,k}\,{\mathrm{Electron}}_k\left(X\right)\right)\right)-\frac{1}{4}\mathrm{`*`}\left({\mathrm{ElectromagneticFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\,{\mathrm{ElectromagneticFieldStrength}}_{\mathrm{~mu}\,\mathrm{~nu}}\right)$

$$\begin{gathered} \mathrm{D\_}\left[\mathrm{mu}\right]\left(\mathrm{Electron}\left[k\right]\left(X\right)\right)\: \\ \% \=\mathrm{expand}\left(\%\right) \end{gathered}$$

${\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_k\left(X\right)\,\left[X\right]\right)\={\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_k\left(X\right)\,\left[X\right]\right)\+\mathrm{i}\mathrm{g\_\_e}\mathrm{`*`}\left({\mathrm{Electron}}_k\left(X\right)\,{\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\left(X\right)\right)$

$\mathrm{expand}\left(\right)$

$\mathrm{i}{{\mathrm{Dgamma}}_{\mathrm{~mu}}}_{j\,k}\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,{\mathrm{d\_}}_{\mathrm{\μ}}\left({\mathrm{Electron}}_k\left(X\right)\,\left[X\right]\right)\right)-{{\mathrm{Dgamma}}_{\mathrm{~mu}}}_{j\,k}\mathrm{g\_\_e}\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,{\mathrm{Electron}}_k\left(X\right)\,{\mathrm{ElectromagneticField}}_{\mathrm{\μ}}\left(X\right)\right)-m_{\mathrm{Electron}}{\mathrm{KroneckerDelta}}_{j\,k}\mathrm{`*`}\left(\mathrm{conjugate}\left({\mathrm{Electron}}_j\left(X\right)\right)\,{\mathrm{Electron}}_k\left(X\right)\right)-\frac{1}{4}\mathrm{`*`}\left({\mathrm{ElectromagneticFieldStrength}}_{\mathrm{\μ}\,\mathrm{\ν}}\,{\mathrm{ElectromagneticFieldStrength}}_{\mathrm{~mu}\,\mathrm{~nu}}\right)$