FeynCalc manual (development version)

 

MTLND

MTLND[mu,nu,n,nb] denotes the positive component in the lightcone decomposition of the metric tensor gμνg^{\mu \nu} along the vectors n and nbin DD dimensions. It corresponds to 12nμnˉν\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.

See also

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

Examples

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

nμ  nbν2\frac{n^{\mu } \;\text{nb}^{\nu }}{2}

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

12  Pair[LorentzIndex[μ,D],Momentum[n,D]]  Pair[LorentzIndex[ν,D],Momentum[nb,D]]\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

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

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

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

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

12nμ(n  nb)\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]] // Contract

nμn^{\mu }

FCClearScalarProducts[]