FeynCalc manual (development version)

FermionSpinSum

FermionSpinSum[exp] converts products of closed spinor chains in exp into Dirac traces. Both Dirac and Majorana particles are supported. It is understood, that exp represents a squared amplitude.

See also

Overview, Spinor, ComplexConjugate, DiracTrace.

Examples

FeynCalc uses the customary relativistic normalization of the spinors.

SpinorUBar[Momentum[p], m] . SpinorU[Momentum[p], m] 
 
FermionSpinSum[%] 
 
DiracSimplify[%]

uˉ(p,m).u(p,m)\bar{u}\left(\overline{p},m\right).u\left(\overline{p},m\right)

tr(γˉp+m)\text{tr}\left(\bar{\gamma }\cdot \overline{p}+m\right)

4m4 m

SpinorVBar[Momentum[p], m] . SpinorV[Momentum[p], m] 
 
FermionSpinSum[%] 
 
DiracSimplify[%]

vˉ(p,m).v(p,m)\bar{v}\left(\overline{p},m\right).v\left(\overline{p},m\right)

tr(γˉpm)\text{tr}\left(\bar{\gamma }\cdot \overline{p}-m\right)

4m-4 m

amp = SpinorUBar[k1, m] . GS[p] . GA[5] . SpinorU[p1, m] 
 
ampSq = amp ComplexConjugate[amp]

uˉ(k1,m).(γˉp).γˉ5.u(p1,m)\bar{u}(\text{k1},m).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.u(\text{p1},m)

uˉ(k1,m).(γˉp).γˉ5.u(p1,m)((φ(p1,m)).γˉ5.(γˉp).(φ(k1,m)))\bar{u}(\text{k1},m).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.u(\text{p1},m) \left(-\left(\varphi (\overline{\text{p1}},m)\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{p}\right).\left(\varphi (\overline{\text{k1}},m)\right)\right)

FermionSpinSum[ampSq] 
 
DiracSimplify[%]

tr((γˉk1+m).(γˉp).γˉ5.(γˉp1+m).γˉ5.(γˉp))-\text{tr}\left(\left(\bar{\gamma }\cdot \overline{\text{k1}}+m\right).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{\text{p1}}+m\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{p}\right)\right)

4p2(k1p1)+8(k1p)(pp1)4m2p2-4 \overline{p}^2 \left(\overline{\text{k1}}\cdot \overline{\text{p1}}\right)+8 \left(\overline{\text{k1}}\cdot \overline{p}\right) \left(\overline{p}\cdot \overline{\text{p1}}\right)-4 m^2 \overline{p}^2