The internal representation (FeynCalcIntenral or FCI) is how FeynCalc internally “sees” the objects. For example, a 4-dimensional 4-vector is represented by
Pair[LorentzIndex[\[Mu]], Momentum[p]]\overline{p}^{\mu }
Pair is one of the most basic FeynCalc objects. Depending on its arguments, it can represent a 4-vector, a metric tensor
Pair[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]]\bar{g}^{\mu \nu }
or a scalar product of two 4-vectors
Pair[Momentum[p], Momentum[q]]\overline{p}\cdot \overline{q}
Another essential object is DiracGamma that is used to represent Dirac matrices. An uncontracted Dirac matrix is
DiracGamma[LorentzIndex[\[Mu]]]\bar{\gamma }^{\mu }
and for a Feynman slash we use
DiracGamma[Momentum[p]]\bar{\gamma }\cdot \overline{p}
The Levi-Civita-Tensor is
Eps[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], LorentzIndex[\[Sigma]]]\bar{\epsilon }^{\mu \nu \rho \sigma }
or, when contracted with 4-momenta
Eps[Momentum[p1], Momentum[p2], Momentum[q1], Momentum[q2]]\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p2}}\;\overline{\text{q1}}\;\overline{\text{q2}}}
This notation (momenta in the index slots) is also used in many other tools (e.g. FORM). The advantage is, that we do not need to canonicalize the indices of the Levi-Civita-Tensor, e.g. to ensure that
diff = Eps[LorentzIndex[\[Mu]], Momentum[p2], Momentum[q1], Momentum[q2]] Pair[LorentzIndex[\[Mu]], Momentum[p1]] -
Eps[LorentzIndex[\[Nu]], Momentum[p2], Momentum[q1], Momentum[q2]] Pair[LorentzIndex[\[Nu]], Momentum[p1]]\overline{\text{p1}}^{\mu } \bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{q1}}\;\overline{\text{q2}}}-\overline{\text{p1}}^{\nu } \bar{\epsilon }^{\nu \overline{\text{p2}}\;\overline{\text{q1}}\;\overline{\text{q2}}}
diff // Contract0
is zero.
The internal representation is useful for the internal programming FeynCalc, but obviously too cumbersome for the user input. This is why FeynCalc also has an external representation (FeynCalcExternal or FCE), that is concise and convenient.
Let us start with the 4-vector. In the FCE-notation it is just FV (“FourVector”)
FV[p, \[Mu]]\overline{p}^{\mu }
It is not hard to guess that the scalar product is SP
SP[p, q]\overline{p}\cdot \overline{q}
while for the metric tensor we write MT
MT[\[Mu], \[Nu]]\bar{g}^{\mu \nu }
To input a Dirac matrix or a Feynman slash, use GA or GS respectively
GA[\[Mu]]\bar{\gamma }^{\mu }
GS[p]\bar{\gamma }\cdot \overline{p}
The Levi-Civita tensor is LC
LC[\[Mu], \[Nu], \[Rho], \[Sigma]]\bar{\epsilon }^{\mu \nu \rho \sigma }
The fully contracted form is entered via
LC[][p1, p2, q1, q2]\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p2}}\;\overline{\text{q1}}\;\overline{\text{q2}}}
It is also possible to enter a mixed form
LC[\[Mu]][p1, p2, q]\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p2}}\overline{q}}
LC[\[Mu], \[Nu]][p1, p2]\bar{\epsilon }^{\mu \nu \overline{\text{p1}}\;\overline{\text{p2}}}
To convert between the two representations we use the functions FCI and FCE, which are shortcuts for FeynCalcInternal and FeynCalcExternal. One cannot distinguish between the notations using the typesetting, i.e. when we see a typeset object in the TraditionalForm, we cannot really tell if it is in the FCI or FCE notation.
ex1 = FV[p, \[Mu]]
ex2 = Pair[Momentum[p], LorentzIndex[\[Mu]]]\overline{p}^{\mu }
\overline{p}^{\mu }
However, we can always use StandardForm to see the difference
ex1 // StandardForm
ex2 // StandardForm
(*FV[p, \[Mu]]*)
(*Pair[LorentzIndex[\[Mu]], Momentum[p]]*)All FeynCalc functions that are meant for users will automatically convert the user input in the FCE notation into the FCI notation. You do not have to do it by yourself.
On the other hand, virtually all FeynCalc functions produce their output in the FCI form. So when you have an expression that was obtained from FeynCalc and want to apply some replacement rules to it, we have to use the FCI form in the rule
ex = Pair[Momentum[p], Momentum[q]]\overline{p}\cdot \overline{q}
No surprise that following does not work
ex /. SP[p, q] -> 1\overline{p}\cdot \overline{q}
But if we wrap the r.h.s of the rule with FCI, then everything is fine
ex /. FCI[SP[p, q]] -> 11