FeynCalc manual (development version)

Dimensions

See also

Overview.

Notation

You might have wondered why 4-vectors, scalar products and Dirac matrices all have a bar, like \bar{p}^\mu or \bar{p} \cdot \bar{q}. The bar is there to specify that they are 4-dimensional objects. Objects that live in D dimensions do not have a bar, cf.

FVD[p, \[Mu]]
% // FCI // StandardForm

p^{\mu }

(*Pair[LorentzIndex[\[Mu], D], Momentum[p, D]]*)

MTD[\[Mu], \[Nu]]
% // FCI // StandardForm

g^{\mu \nu }

(*Pair[LorentzIndex[\[Mu], D], LorentzIndex[\[Nu], D]]*)

This origin of this notation is a publication by Breitenlohner and Maison on the treatment of \gamma^5 in D dimensions in the t’Hooft-Veltman scheme. The main idea was that we can decompose indexed objects into 4- and D-4-dimensional pieces, e.g. p^\mu = \bar{p}^\mu + \hat{p}^\mu. Consequently, in FeynCalc we can also enter D-4-dimensional objects

FVE[p, \[Mu]]
% // FCI // StandardForm

\hat{p}^{\mu }

(*Pair[LorentzIndex[\[Mu], -4 + D], Momentum[p, -4 + D]]*)

MTE[p, q]
% // FCI // StandardForm

\hat{g}^{pq}

(*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.

FVD[p, \[Mu]] FV[q, \[Mu]]
Contract[%]

p^{\mu } \overline{q}^{\mu }

\overline{p}\cdot \overline{q}

FV[p, \[Mu]] FVE[q, \[Mu]]
Contract[%]

\hat{q}^{\mu } \overline{p}^{\mu }

0

Sometimes we need to switch from one dimension to another, e.g. to convert a 4-dimensional object to a D-dimensional one or vice versa. This is done via

FVD[p, \[Mu]]
ChangeDimension[%, 4]

p^{\mu }

\overline{p}^{\mu }

The second argument of ChangeDimension is the new dimension . The most common choices are 4, D or D-4

FVD[p, \[Mu]]
ChangeDimension[%, D - 4]

p^{\mu }

\hat{p}^{\mu }