FeynCalc manual (development version)

DiracGamma

DiracGamma[x, dim] is the head of all Dirac matrices and slashes (in the internal representation). Use GA, GAD, GS or GSD for manual (short) input.

DiracGamma[x, 4] simplifies to DiracGamma[x].

DiracGamma[5] is γ5\gamma ^5.

DiracGamma[6] is (1+γ5)/2(1+\gamma ^5)/2.

DiracGamma[7] is (1γ5)/2(1-\gamma ^5)/2.

See also

Overview, DiracGammaExpand, GA, DiracSimplify, GS, DiracTrick.

Examples

DiracGamma[5]

γˉ5\bar{\gamma }^5

DiracGamma[LorentzIndex[\[Alpha]]]

γˉα\bar{\gamma }^{\alpha }

A Dirac-slash, i.e., γμqμ\gamma ^{\mu }q_{\mu}, is displayed as γq\gamma \cdot q.

DiracGamma[Momentum[q]] 

γˉq\bar{\gamma }\cdot \overline{q}

DiracGamma[Momentum[q]] . DiracGamma[Momentum[p - q]]

(γˉq).(γˉ(pq))\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \left(\overline{p}-\overline{q}\right)\right)

DiracGamma[Momentum[q, D], D] 

γq\gamma \cdot q

GS[p - q] . GS[p] 
 
DiracGammaExpand[%]

(γˉ(pq)).(γˉp)\left(\bar{\gamma }\cdot \left(\overline{p}-\overline{q}\right)\right).\left(\bar{\gamma }\cdot \overline{p}\right)

(γˉpγˉq).(γˉp)\left(\bar{\gamma }\cdot \overline{p}-\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{p}\right)

ex = GAD[\[Mu]] . GSD[p - q] . GSD[q] . GAD[\[Mu]]

γμ.(γ(pq)).(γq).γμ\gamma ^{\mu }.(\gamma \cdot (p-q)).(\gamma \cdot q).\gamma ^{\mu }

DiracTrick[ex]

4((pq)q)+(D4)(γ(pq)).(γq)4 ((p-q)\cdot q)+(D-4) (\gamma \cdot (p-q)).(\gamma \cdot q)

DiracSimplify[ex]

D(γp).(γq)Dq24(γp).(γq)+4(pq)D (\gamma \cdot p).(\gamma \cdot q)-D q^2-4 (\gamma \cdot p).(\gamma \cdot q)+4 (p\cdot q)

DiracGamma may also carry Cartesian indices or appear contracted with Cartesian momenta.

DiracGamma[CartesianIndex[i]]

γi\overline{\gamma }^i

DiracGamma[CartesianIndex[i, D - 1], D]

γi\gamma ^i

DiracGamma[CartesianMomentum[p]]

γp\overline{\gamma }\cdot \overline{p}

DiracGamma[CartesianMomentum[p, D - 1], D]

γp\gamma \cdot p

Temporal indices are represented using ExplicitLorentzIndex[0]

DiracGamma[ExplicitLorentzIndex[0]]

γˉ0\bar{\gamma }^0