Name: Vladyslav Shtabovenko (email_not_shown)
Date: 12/17/15-11:21:18 AM Z


Hi Mikkel,

both expressions are in fact the same. They look differently only due to
the Schouten identity. See here:

<http://www.feyncalc.org/forum/0859.html>

aTmu1 = c1*
    Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[\[Nu]m].GA[\[Mu]m]] +
   c2*Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[5].GA[\[Nu]m].GA[\[Mu]m]];
aTmu2 = c3*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[\[Nu]m]] +
   c4*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[5].GA[\[Nu]m]];
aTmu3 = c5*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[\[Mu]m]] +
   c6*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[5].GA[\[Mu]m]];

bTmu1 = c1*
    Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[\[Alpha]].GA[\[Mu]m]] +
   c2*Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[
       5].GA[\[Alpha]].GA[\[Mu]m]];
bTmu2 = c3*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[\[Alpha]]] +
   c4*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[5].GA[\[Alpha]]];
bTmu3 = c5*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[\[Mu]m]] +
   c6*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[5].GA[\[Mu]m]];

Simplify[(aTmu1 /. {\[Nu]m -> \[Alpha]}) - bTmu1] // Schouten
Simplify[(aTmu2 /. {\[Nu]m -> \[Alpha]}) - bTmu2] // Schouten
Simplify[(aTmu3 /. {\[Nu]m -> \[Alpha]}) - bTmu3] // Schouten
Simplify[Contract[aTmu1.aTmu2.aTmu3] -
   Contract[bTmu1.bTmu2.bTmu3]] // Schouten

Returns zero in all cases.

As to why this happens: The ordering of the indices of the Levi-Civita
tensor is done using Mathematica's Sort, i.e. it's lexicographic. Having
dummy indices named differently changes their position in the epsilon
tensor so that the final results look differently being in fact
identical by Schouten.

Cheers,
Vladyslav

Am 17.12.2015 um 10:09 schrieb Mikkel Bjoern:
> I have a contraction of three traces, for which the result depends on my naming of indices that are contracted out completely. I suspect the issue is with two-character issues, but I am not sure. And now, of course, I am unsure which result is right.
>
> The two expressions are:
> -- code ---------------------------------
> Tmu1 = c1*Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[\[Nu]m].GA[\[Mu]m]] +
> c2*Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[5].GA[\[Nu]m].GA[\[Mu]m]]
> Tmu2 = c3*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[\[Nu]m]] +
> c4*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[5].GA[\[Nu]m]]
> Tmu3 = c5*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[\[Mu]m]] +
> c6*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[5].GA[\[Mu]m]]
> Simplify[FCE[Contract[Tmu1.Tmu2.Tmu3]]]
> ------------------------------------------
>
> and the same, but where I exchange all "\[Nu]m" with "\[Alpha]" - producing EXTRA TERMS in the final trace (that do not simplify to zero, at least not according to Mathematica). Thus the code is
>
> --- code -------------------------------
> Tmu1 = c1*
> Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[\[Alpha]].GA[\[Mu]m]] +
> c2*Tr[GS[pm].GA[\[Mu]].GA[\[Nu]].GS[p].GA[5].GA[\[Alpha]].GA[\[Mu]m]]
> Tmu2 = c3*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[\[Alpha]]] +
> c4*Tr[GS[l1].GA[\[Nu]].GS[l2].GA[5].GA[\[Alpha]]]
> Tmu3 = c5*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[\[Mu]m]] +
> c6*Tr[GS[k1].GA[\[Mu]].GS[k2].GA[5].GA[\[Mu]m]]
> Simplify[FCE[Contract[Tmu1.Tmu2.Tmu3]]]
> --------------------------------------------------
>
> Since I suspceted the two-character index might be an issue I tried renaming all "\[Nu]m"'s to "\[Alpha]" and all "\[Mu]m"'s to "\[Beta]" - for which THE RESULT AGREES WITH THE ORIGINAL, that is, without the extra term.
>
> Does anyone have an idea what is going on? I guess it is some kind of bug, since names of contracted indices should obviously now matter.
>
> However I am new to Feyncal and could be doing something wrong.
>
> Thank you!
> Mikkel
>



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