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I have a large expression of a sum of gamma matrices which resulted from using Casimir's Trick. It is quite tedious to evaluate so I thought about using Mathematica. However, I have no idea how, especially, when the contraction of Lorentz indices is involved.

I'm looking to evaluate an expression of the following form:

$(\gamma^\mu \moveright 6pt /p_a \gamma^\nu + \gamma^0 \moveright 6pt / p_b \gamma^\nu \gamma^\mu)(\gamma_\nu +p_\nu)(\moveright 6pt / p_c+m)$

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You could use FeynCalc ( I leave out the \gamma^0, but you could add GA[0] ):

Import["https://raw.githubusercontent.com/FeynCalc/feyncalc/master/\
install.m"]
InstallFeynCalc[]

TR[(GA[\[Mu]].GS[Subscript[p, a]].GA[\[Nu]] + 
    GS[Subscript[p, b]].GA[\[Nu]].GA[\[Mu]]).(GA[\[Nu]] + 
    FV[p, \[Nu]]).(GS[Subscript[p, c]] + m)]

resulting in

enter image description here

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  • $\begingroup$ Ah, wow. That is exactly what I needed. Thank you! $\endgroup$ – infinitezero Dec 29 '17 at 18:47
  • $\begingroup$ You are welcome. There is a slight potential problem: After installation of the current stable FeynCalc version you will see all output cells having TraditionalForm. You can reset this by CurrentValue[$FrontEnd, {CommonDefaultFormatTypes, "Output"}] = StandardForm. I was too lazy to fix it myself, but at least added an issue. $\endgroup$ – Rolf Mertig Dec 29 '17 at 20:59
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May I also advertise Package-X, which I wrote? (Is this allowed by rules of this forum?) You can use the function Spur which takes traces of Dirac matrices in $d=4-2\epsilon$ dimensions.

enter image description here enter image description here


Copy and paste into your notebook, and select "Yes" to interpret the text below:

Cell[BoxData[
 RowBox[{
  RowBox[{"Contract", "[", 
   RowBox[{"Spur", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"DiracMatrix", "[", 
       RowBox[{
        SubscriptBox["\[Gamma]", "\[Mu]"], ",", 
        RowBox[{"pa", ".", "\[Gamma]"}], ",", 
        SubscriptBox["\[Gamma]", "\[Nu]"]}], "]"}], "+", 
      RowBox[{"DiracMatrix", "[", 
       RowBox[{
        SubscriptBox["\[Gamma]", "\[Alpha]"], ",", 
        RowBox[{"pb", ".", "\[Gamma]"}], ",", 
        SubscriptBox["\[Gamma]", "\[Nu]"], ",", 
        SubscriptBox["\[Gamma]", "\[Mu]"]}], "]"}]}], ",", 
     RowBox[{
      SubscriptBox["\[Gamma]", "\[Nu]"], "+", 
      RowBox[{
       SubscriptBox["p", "\[Nu]"], "\[DoubleStruckOne]"}]}], ",", 
     RowBox[{
      RowBox[{"pc", ".", "\[Gamma]"}], "+", 
      RowBox[{"m", " ", "\[DoubleStruckOne]"}]}]}], "]"}], "]"}], "/.", 
  RowBox[{"\[Alpha]", "->", "0"}]}]], "Input"]
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