DiracEquation[exp] applies the Dirac equation without expanding exp. If expansions are necessary, use DiracSimplify.
GS[p] . SpinorU[p, m]
DiracSimplify[%]\left(\bar{\gamma }\cdot \overline{p}\right).u(p,m)
m \left(\varphi (\overline{p},m)\right)
GS[p] . SpinorU[p, m]
DiracEquation[%]\left(\bar{\gamma }\cdot \overline{p}\right).u(p,m)
m \left(\varphi (\overline{p},m)\right)
GS[p] . SpinorV[p, m]
DiracEquation[%]\left(\bar{\gamma }\cdot \overline{p}\right).v(p,m)
-m \left(\varphi (-\overline{p},m)\right)
SpinorUBar[p, 0] . GS[p]
DiracEquation[%]\bar{u}(p).\left(\bar{\gamma }\cdot \overline{p}\right)
0
DiracEquation also works in D-dimensions
SpinorVBarD[p, m] . GSD[p]
DiracEquation[%]\bar{v}(p,m).(\gamma \cdot p)
-m (\varphi (-p,m))