FeynCalc manual (development version)

Dimensions

See also

Overview.

Notation for tensors living in different dimensions

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

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

pμp^{\mu }

(*Pair[LorentzIndex[\[Mu], D], Momentum[p, D]]*)
MTD[\[Mu], \[Nu]]
% // FCI // StandardForm

gμνg^{\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 γ5\gamma^5 in DD dimensions in the t’Hooft-Veltman scheme. The main idea was that we can decompose indexed objects into 44- and D−4D-4-dimensional pieces, e.g. pμ=pˉμ+p^μp^\mu = \bar{p}^\mu + \hat{p}^\mu. Consequently, in FeynCalc we can also enter D−4D-4-dimensional objects

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

p^μ\hat{p}^{\mu }

(*Pair[LorentzIndex[\[Mu], -4 + D], Momentum[p, -4 + D]]*)
MTE[p, q]
% // FCI // StandardForm

g^pq\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[%]

pμq‾μp^{\mu } \overline{q}^{\mu }

p‾⋅q‾\overline{p}\cdot \overline{q}

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

q^μp‾μ\hat{q}^{\mu } \overline{p}^{\mu }

00

(FVD[p, \[Mu]] + FVE[p, \[Mu]]) (FVD[q, \[Mu]] + FVE[q, \[Mu]])
Contract[%]

(pμ+p^μ)(qμ+q^μ)\left(p^{\mu }+\hat{p}^{\mu }\right) \left(q^{\mu }+\hat{q}^{\mu }\right)

3(p^⋅q^)+p⋅q3 \left(\hat{p}\cdot \hat{q}\right)+p\cdot q

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

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

pμp^{\mu }

p‾μ\overline{p}^{\mu }

The second argument of ChangeDimension is the new dimension . The most common choices are 44, DD or D−4D-4

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

pμp^{\mu }

p^μ\hat{p}^{\mu }

SP[p, q]
ChangeDimension[%, D]

p‾⋅q‾\overline{p}\cdot \overline{q}

p⋅qp\cdot q

To check the dimension of the given expression one can use FCGetDimensions

FVD[p, \[Mu]] FV[q, \[Mu]]
FCGetDimensions[%, {}]

pμq‾μp^{\mu } \overline{q}^{\mu }

{4,D}\{4,D\}

If one needs to replace the dimensional symbols D in the prefactors of the Lorentz tensors, it is better to use FCReplaceD instead of a replacement rule. Otherwise, the dimensions of the tensors will get messed up

FCI[(D + 2) MTD[\[Mu], \[Nu]]] 
% /. D -> 4 - 2 Epsilon

(D+2)gμν(D+2) g^{\mu \nu }

(6−2ε)g{4−2ε,4−2ε}μν(6-2 \varepsilon ) g_{\{4-2 \varepsilon ,4-2 \varepsilon \}}{}^{\mu \nu }

```mathematica (D + 2) MTD[[Mu], [Nu]] FCReplaceD[%, D -> 4 - 2 Epsilon]

```mathematica

(D+2)gμν(D+2) g^{\mu \nu }

(6−2ε)gμν(6-2 \varepsilon ) g^{\mu \nu }