Connection between Poisson bracket and Anti-commutator?

Question

Canonical quantization promotes Poisson brackets in classical mechanics to commutators in quantum mechanics. Is there any classical counterpart similar to the Poisson bracket for the anticommutator?

Answer 1

FWIW, in classical Hamiltonian formalism, the phase space is a supermanifold with a super-Poisson bracket $\{\cdot,\cdot\}_{SPB}$. The super-Poisson bracket corresponds to a super-commutator $\frac{1}{i\hbar}[\cdot,\cdot]_{SC}$ upon quantization. The super-commutator is a commutator (an anticommutator) in the Grassmann-even (Grassmann-odd) sector, respectively.

See also my Phys.SE answer here.