FVLPD[p,mu,n,nb] denotes the positive component in the
lightcone decomposition of the Lorentz vector p^{\mu } along the vectors n and
nb in D dimensions. It
corresponds to \frac{1}{2} \bar{n}^{\mu} (p
\cdot n).
If one omits n and nb, the program will use
default vectors specified via $FCDefaultLightconeVectorN
and $FCDefaultLightconeVectorNB.
Overview, Pair, FVLND, FVLRD, SPLPD, SPLND, SPLRD, MTLPD, MTLND, MTLRD.
FVLPD[p, \[Mu], n, nb]\frac{1}{2} \;\text{nb}^{\mu } (n\cdot p)
StandardForm[FVLPD[p, \[Mu], n, nb] // FCI]\frac{1}{2} \;\text{Pair}[\text{LorentzIndex}[\mu ,D],\text{Momentum}[\text{nb},D]] \;\text{Pair}[\text{Momentum}[n,D],\text{Momentum}[p,D]]
Notice that the properties of n and nb
vectors have to be set by hand before doing the actual computation
FVLPD[p, \[Mu], n, nb] FVLND[q, \[Mu], n, nb] // Contract\frac{1}{4} (n\cdot \;\text{nb}) (n\cdot p) (\text{nb}\cdot q)
FVLPD[p, \[Mu], n, nb] FVLPD[q, \[Mu], n, nb] // Contract\frac{1}{4} \;\text{nb}^2 (n\cdot p) (n\cdot q)
FCClearScalarProducts[]
SPD[n] = 0;
SPD[nb] = 0;
SPD[n, nb] = 2;FVLPD[p, \[Mu], n, nb] FVLND[q, \[Mu], n, nb] // Contract\frac{1}{2} (n\cdot p) (\text{nb}\cdot q)
FVLPD[p, \[Mu], n, nb] FVLPD[q, \[Mu], n, nb] // Contract0
FCClearScalarProducts[]