How to Contract Indices Using Feyncalc

Question

I am slowly learning how to use Feyncalc, and I have a quick question.
I have a third rank tensor B[mu,nu,tao] and I also have a four-vector FV[W,mu].  I would like to contract the first two indices of "B" to get a four-vector, and then contract the result with "W" to obtain a scalar result.  So I wrote this:
res=Contract[B[mu,mu,tao]*FV[W,tao]]
But this does not seem to be working, I get error messages  and/or results that don't make sense.  Is there a better way to write the formula for "res"?  Thanks for any assistance.

vsht · Accepted Answer

If B is a generic tensor, you have to explicitly wrap the indices
into LorentzIndex
DeclareFCTensor[B]; 
Contract[B[LorentzIndex[mu], LorentzIndex[mu], LorentzIndex[tao]]* FV[W, tao]]

Notice that DeclareFCTensor is only needed if you also want to be able to
uncontract the indices as in
Uncontract[B[LorentzIndex[mu], LorentzIndex[mu], Momentum[W]], W, 
 Pair -> All]

Otherwise you could also omit DeclareFCTensor.
Now, if B is just a shortcut for a tensor made out of products of 4-vectors and metric
tensors, things become much simpler: you can define it as one would define
a generic function in Mathematica
B[mu_,nu_,tao_]:=MT[mu,nu]*FV[v,tao]
Contract[B[mu, mu, tao]*FV[W, tao]]

Notice that
Contract[B[LorentzIndex[mu], LorentzIndex[mu], LorentzIndex[tao]]*
  FV[W, tao]]

also works, because LorentzIndex[LorentzIndex[mu]] is evaluated to LorentzIndex[mu].
Extended FeynCalc-related questions are probably best asked in the new forum on GitHub (which is much easier to use than the old, now retired, mailing list)
https://github.com/FeynCalc/feyncalc/discussions

How to Contract Indices Using Feyncalc

One Answer

Add your own answers!

Ask a Question