MTLND[mu,nu,n,nb] denotes the positive component in the
lightcone decomposition of the metric tensor g^{\mu \nu} along the vectors n
and nbin D dimensions. It
corresponds to \frac{1}{2} n^{\mu}
\bar{n}^\nu.
If one omits n and nb, the program will use
default vectors specified via $FCDefaultLightconeVectorN
and $FCDefaultLightconeVectorNB.
Overview, Pair, FVLPD, FVLND, FVLRD, SPLPD, SPLND, SPLRD, MTLPD, MTLRD.
MTLND[\[Mu], \[Nu], n, nb]\frac{n^{\mu } \;\text{nb}^{\nu }}{2}
StandardForm[MTLND[\[Mu], \[Nu], n, nb] // FCI]\frac{1}{2} \;\text{Pair}[\text{LorentzIndex}[\mu ,D],\text{Momentum}[n,D]] \;\text{Pair}[\text{LorentzIndex}[\nu ,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
MTLND[\[Mu], \[Nu], n, nb] FVD[p, \[Mu]] // Contract\frac{1}{2} \;\text{nb}^{\nu } (n\cdot p)
MTLND[\[Mu], \[Nu], n, nb] FVD[p, \[Nu]] // Contract\frac{1}{2} n^{\mu } (\text{nb}\cdot p)
MTLND[\[Mu], \[Nu], n, nb] FVD[n, \[Nu]] // Contract\frac{1}{2} n^{\mu } (n\cdot \;\text{nb})
FCClearScalarProducts[]
SPD[n] = 0;
SPD[nb] = 0;
SPD[n, nb] = 2;MTLND[\[Mu], \[Nu], n, nb] FVD[n, \[Nu]] // Contractn^{\mu }
FCClearScalarProducts[]