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