ChiralityProjector[+1] denotes 1/2\left(1+\gamma^5\right).
ChiralityProjector[-1] denotes 1/2\left(1+\gamma ^5\right).
The shortcut ChiralityProjector is deprecated, please use GA[6] and GA[7] instead!
{ChiralityProjector[+1], ChiralityProjector[-1]}
DiracSimplify[#, DiracSubstitute67 -> True] & /@ %\left\{\bar{\gamma }^6,\bar{\gamma }^7\right\}
\left\{\frac{\bar{\gamma }^5}{2}+\frac{1}{2},\frac{1}{2}-\frac{\bar{\gamma }^5}{2}\right\}
ChiralityProjector is scheduled for removal in the future versions of FeynCalc. The safe alternative is to use GA[6] and GA[7].
{GA[6], GA[7]}\left\{\bar{\gamma }^6,\bar{\gamma }^7\right\}
FCI[GA[6]] === ChiralityProjector[+1]\text{True}
FCI[GA[7]] === ChiralityProjector[-1]\text{True}