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We present a consistent formalism to describe the dynamics of hybrid systemswith mixed classical and quantum degrees of freedom. The probability functionof the system, which, in general, will be a combination of the classicaldistribution function and the quantum density matrix, is described in terms ofits corresponding moments. We then define a hybrid Poisson bracket, such thatthe dynamics of the moments is ruled by an effective Hamiltonian. Inparticular, a closed formula for the Poisson brackets between any two momentsfor an arbitrary number of degrees of freedom is presented, which correctsprevious expressions derived in the literature for the purely quantum case.This formula is of special relevance for practical applications of theformalism. Finally, we study the dynamics of a particular hybrid system givenby two coupled oscillators, one being quantum and the other classical. Due tothe coupling, specific quantum and classical properties are transferred betweendifferent sectors. In particular, the quantum sector is allowed to violate theuncertainty relation, though we explicitly show that there exists a minimumpositive bound of the total uncertainty of the hybrid system.

Hybrid classical-quantum systems in terms of moments