Name: V. Shtabovenko (email_not_shown)
Date: 10/21/17-08:51:12 AM Z

Although FeynCalc 9.3 still cannot go into single components of Lorentz
tensors, it can distinguish between temporal and spatial components, so
that one can do the following:

ex = LC[a, b, c, d] FourVector[k1, a] FourVector[k2, b] FourVector[k3,
c] FourVector[k4, d] // LorentzToCartesian // Contract

which yields

TC[k4] CLC[][k1, k2, k3] - TC[k3] CLC[][k1, k2, k4] +
TC[k2] CLC[][k1, k3, k4] - TC[k1] CLC[][k2, k3, k4]

Here TC[x] stands for x^0, i.e.

?TC

TC[p] is the temporal component of a 4-vector and is transformed into
TemporalPair[TemporalMomentum[p], TemporalIndex[]] by FeynCalcInternal,

while CLC is the Cartesian 3D epsilon tensor

?CLC

CLC[m,n,r] evaluates to Eps[CartesianIndex[m], CartesianIndex[n], \
CartesianIndex[r]] applying FeynCalcInternal. CLC[m,...][p, ...] \
evaluates to Eps[CartesianIndex[m], ..., CartesianMomentum[p], ...] \
applying FeynCalcInternal.

That is, CLC[][a,b,c] is precisely the triple product a. (b x c)

So, given the explicit values of the components of k1, k2, k3 and k4
we can write something like

explicit[k1] = Table[k1[i], {i, 1, 3}];
explicit[k2] = Table[k2[i], {i, 1, 3}];
explicit[k3] = Table[k3[i], {i, 1, 3}];
explicit[k4] = Table[k4[i], {i, 1, 3}];

ex /. FCI[CLC[][a_, b_, c_]] :> Dot[explicit[a], Cross[explicit[b],
explicit[c]]] /. {FCI@TC[x_] :> x[0]}

which gives

(k1[3] (-k2[2] k3[1] + k2[1] k3[2]) +
k1[2] (k2[3] k3[1] - k2[1] k3[3]) +
k1[1] (-k2[3] k3[2] + k2[2] k3[3])) k4[0] -
k3[0] (k1[3] (-k2[2] k4[1] + k2[1] k4[2]) +
k1[2] (k2[3] k4[1] - k2[1] k4[3]) +
k1[1] (-k2[3] k4[2] + k2[2] k4[3])) +
k2[0] (k1[3] (-k3[2] k4[1] + k3[1] k4[2]) +
k1[2] (k3[3] k4[1] - k3[1] k4[3]) +
k1[1] (-k3[3] k4[2] + k3[2] k4[3])) -
k1[0] (k2[3] (-k3[2] k4[1] + k3[1] k4[2]) +
k2[2] (k3[3] k4[1] - k3[1] k4[3]) +
k2[1] (-k3[3] k4[2] + k3[2] k4[3]))

Cheers,

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>
> Hello,
>
> I'm using version 4.1.1 on Mathematica 4.0.
> The problem arises when I have the following expression:
>
> x=LC[a,b,c,d]FourVector[k1,a]FourVector[k2,b]FourVector[k3,c]FourVector[k4,d]
>
> I have previously given the value of all possible scalar products. I want
> Feyncalc to expand x using these scalar products.
> I've tried to use every possible Feyncalc command, without success. On the
> other hand, if I ask it to compute Calc[x*x], it works! I've checked by hand,
> the result is correct, but then I loose the sign of x.
> Is there a way to make Feyncalc expand a single LC contraction? Is it a bug?
> Regards,
>
> - --
> Thibaut Cousin
> email : cousin@in2p3.fr
> web : http://clrwww.in2p3.fr
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