You might have wondered why 4-vectors, scalar products and Dirac matrices all have a bar, like or . The bar is there to specify that they are 4-dimensional objects. Objects that live in dimensions do not have a bar, cf.
[p, \[Mu]]
FVD% // FCI // StandardForm
(*Pair[LorentzIndex[\[Mu], D], Momentum[p, D]]*)
[\[Mu], \[Nu]]
MTD% // FCI // StandardForm
(*Pair[LorentzIndex[\[Mu], D], LorentzIndex[\[Nu], D]]*)
This origin of this notation is a publication by Breitenlohner and Maison on the treatment of in dimensions in the t’Hooft-Veltman scheme. The main idea was that we can decompose indexed objects into - and -dimensional pieces, e.g. . Consequently, in FeynCalc we can also enter -dimensional objects
[p, \[Mu]]
FVE% // FCI // StandardForm
(*Pair[LorentzIndex[\[Mu], -4 + D], Momentum[p, -4 + D]]*)
[p, q]
MTE% // FCI // StandardForm
(*Pair[LorentzIndex[p, -4 + D], LorentzIndex[q, -4 + D]]*)
When we contract Lorentz tensors from different dimensions, the contractions are resolved according to the rules from the paper of Breitenlohner and Maison, e. g.
[p, \[Mu]] FV[q, \[Mu]]
FVD[%] Contract
[p, \[Mu]] FVE[q, \[Mu]]
FV[%] Contract
Sometimes we need to switch from one dimension to another, e.g. to convert a 4-dimensional object to a -dimensional one or vice versa. This is done via
[p, \[Mu]]
FVD[%, 4] ChangeDimension
The second argument of ChangeDimension
is the new dimension . The most common choices are , or
[p, \[Mu]]
FVD[%, D - 4] ChangeDimension