 Graded Dirac bracket

A Dirac bracket on a graded phase space eliminates second-class constraints while respecting the graded Poisson bracket. The fermionic kinetic term $i{\psi }^m{\dot{\psi }}_m/2$ gives $\{{\psi }^m,{\psi }^n{\}}_D=-i{\eta }^{mn}$. Quantization then gives $\{{\hat{\psi }}^m,{\hat{\psi }}^n{\}}_{+}={\eta }^{mn}$.