# Can I specify the components of FourVector in FeynCalc?

I'm using FeynCalc to calculate Compton's scattering. I need to use specific values for the components of the photon polarization four-vector. How can I do that?

• Perhaps you can directly ask the authors of FeynCalc on their forum here: feyncalc.github.io/forum. I'm sure you'll get helpful pointers. Commented Dec 11, 2017 at 20:22

I recently came across this problem as well. How I ended up calculating fully contracted amplitudes with explicit four-vectors (let it be momenta, polarization vectors) is as follows:

Define the functions

Mink[t1_, t2_] :=
t1[[1]] t2[[1]] - t1[[2]] t2[[2]] - t1[[3]] t2[[3]] -
t1[[4]] t2[[4]];
LevContracted[a_, b_, c_, d_] :=
Sum[-LeviCivitaTensor[
4][[mu, nu, alpha, beta]]] a[[mu]] b[[nu]] c[[alpha]] d[[beta]], {mu, 1, 4}, {nu, 1, 4}, {alpha, 1,
4}, {beta, 1, 4}];


for a Minkowski scalar product and a fully contracted expression with the Levi-Civita-Tensor. Here, I use the signature $$(+---)$$ and the convention $$\epsilon^{0123}=+1$$ (to improve readability, I replaced the \[Mu] from Mathematica by mu). If you then take your four-vectors to be contravariant, you get the additional minus sign in the definition of the second function. I will provide two examples of how to define the four-vectors. We will use lists, as can also be seen from the definition of the functions above. For the momentum of an on-shell particle with mass $$m$$ in its rest frame, we can define

p = {m, 0, 0, 0};


and for the three corresponding polarization vectors we use

Pol={{0,1,0,0},{0,0,1,0},{0,0,0,1}};


i.e. a list of lists, in order to sum over all its polarizations.

Note that when calculating the spin-/polarization-averaged/-summed (I will refer to this as "pol.") squared amplitude and summing over the different polarization states, you have to add them after squaring, i.e.

$$|\overset{\sim}{M}| = \frac{1}{(2J_1+1)\ldots (2J_N+1)}\sum_{\{\text{pol. }i\}}{|M_i|^2}$$

where $$i$$ refers to one given polarization state, i.e. every particle has a fixed polarization and $$J_i$$ are the spins of the particle(s) in the initial state.

ScalarProduct[p, q] /.
FCI[SP[p, q]] -> {p1, p2, p3, p4}.{q1, -q2, -q3, -q4}


I cannot say more.I found this

Here's how I did it.

MinkSP[vec1_, vec2_] := Times[vec1[[1]], vec2[[1]]] - Times[vec1[[2]], vec2[[2]]] - Times[vec1[[3]], vec2[[3]]] - Times[vec1[[4]], vec2[[4]]]