Combined algebra for quantum and classical mechanics

For a canonical Hamiltonian system, an algebra is constructed in which all the observables are realized by ordinary functions $A(p,q)$ of the momenta and coordinates and are simultaneously classical and quantum observables. The classical and quantum states are realized by density matrices $\rho(p,q)$ that are either coincident for the quantum and the classical theory or exist only in one of the theories. The entire difference between the quantum and classical descriptions reduces to the difference between the quantum and classical operations of multiplication of observables, their Poisson brackets, and thus between the evolutions of the observables (or states) in time. A transition from the quantum to the classical theory is proposed and investigated in which the observables and states do not change and the operations of quantum multiplication and taking of the quantum Poisson brackets go over as $\hbar\to0$ into the corresponding classical operations in a perfectly definite sense. It is shown that the quantum operations are infinitely differentiable with respect to $\hbar$ at zero. The transition to classical mechanics is possible for all observables but not for all states. Pure quantum states become mixed in the classical case. The quantum corrections destroy the Hamiltonicity of the classical equations of motion. For the space of observables a topology which admits unbounded operators is used.