Name: V. Shtabovenko 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,
Vladyslav
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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_at_HIDDEN-E-MAIL]
> web : http://clrwww.in2p3.fr
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