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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?

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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.

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  • $\begingroup$ As a person who just started learning QFT, are there some gentle introductions to supersymmetry such that I can understand your answer? $\endgroup$ – Leo L. Jan 16 '20 at 17:46

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