FeynCalc manual (development version)

 

GALPD

GALPD[mu,n,nb] denotes the positive component in the lightcone decomposition of the Dirac matrix γμ\gamma^{\mu } along the vectors n and nb in DD-dimensions. It corresponds to 12nˉμ(γn)\frac{1}{2} \bar{n}^{\mu} (\gamma \cdot n).

If one omits n and nb, the program will use default vectors specified via $FCDefaultLightconeVectorN and $FCDefaultLightconeVectorNB.

See also

Overview, DiracGamma, GALND, GALRD, GSLPD, GSLND, GSLRD.

Examples

GALPD[\[Mu], n, nb]

12  nbμγn\frac{1}{2} \;\text{nb}^{\mu } \gamma \cdot n

StandardForm[GALPD[\[Mu], n, nb] // FCI]

12  DiracGamma[Momentum[n,D],D]  Pair[LorentzIndex[μ,D],Momentum[nb,D]]\frac{1}{2} \;\text{DiracGamma}[\text{Momentum}[n,D],D] \;\text{Pair}[\text{LorentzIndex}[\mu ,D],\text{Momentum}[\text{nb},D]]

Notice that the properties of n and nb vectors have to be set by hand before doing the actual computation

GALPD[\[Mu], n, nb] . GALPD[\[Nu], n, nb] // DiracSimplify

14n2  nbμ  nbν\frac{1}{4} n^2 \;\text{nb}^{\mu } \;\text{nb}^{\nu }

FCClearScalarProducts[]
SPD[n] = 0;
SPD[nb] = 0;
SPD[n, nb] = 2;
GALPD[\[Mu], n, nb] . GALPD[\[Nu], n, nb] // DiracSimplify

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FCClearScalarProducts[]