FeynCalc manual (development version)

 

MTLPD

MTLPD[mu,nu,n,nb] denotes the positive component in the lightcone decomposition of the metric tensor gμνg^{\mu \nu} along the vectors n and nb. It corresponds to 12nˉμnν\frac{1}{2} \bar{n}^{\mu} n^\nu.

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

See also

Overview, Pair, FVLPD, FVLND, FVLRD, SPLPD, SPLND, SPLRD, MTLND, MTLRD.

Examples

MTLPD[\[Mu], \[Nu], n, nb]

nν  nbμ2\frac{n^{\nu } \;\text{nb}^{\mu }}{2}

StandardForm[MTLPD[\[Mu], \[Nu], n, nb] // FCI]

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

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

MTLPD[\[Mu], \[Nu], n, nb] FVD[p, \[Mu]] // Contract

12nν(nbp)\frac{1}{2} n^{\nu } (\text{nb}\cdot p)

MTLPD[\[Mu], \[Nu], n, nb] FVD[p, \[Nu]] // Contract

12  nbμ(np)\frac{1}{2} \;\text{nb}^{\mu } (n\cdot p)

MTLPD[\[Mu], \[Nu], n, nb] FVD[n, \[Nu]] // Contract

n2  nbμ2\frac{n^2 \;\text{nb}^{\mu }}{2}

FCClearScalarProducts[]
SPD[n] = 0;
SPD[nb] = 0;
SPD[n, nb] = 2;
MTLPD[\[Mu], \[Nu], n, nb] FVD[n, \[Nu]] // Contract

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FCClearScalarProducts[]