Quantising a Damped Mass on a Spring

Background: this question discusses Lagrangian/Hamiltonian formulation of a dissipative problem. However, I'm not clear if this can be made quantum and would like a more explicit roadmap if possible.

I'm interested if there are quantum systems whose classical limits are not Hamiltonian, and how one would describe such a system if they do exist. I have in mind something like the following:

1) There is a Hilbert space of states.

2) Time evolution is completely positive. This may have to not be unitary/Hamiltonian, but this is acceptable since I have in mind some effective theory of a subsystem.

3) The classical limit has time evolution given by the dissipative EoMs:

$$m\ddot{x} +\gamma \dot{x} +kx = 0$$

How do I define such a system, what makes it quantum, and how do I achieve the quantisation such that I get the correct classical behaviour?

• @Gert this is not the simple harmonic oscillator. I am asking whether classical systems without Hamiltonian formulations can arise from quantum systems and what those quantum systems must look like if so. – jacob1729 Jan 21 '20 at 14:52

$$S_{\text{eff}} = \int_0^\tau dt \left( \frac{1}{2}m\dot{x}^2+V(x)+\frac{\gamma}{4\pi}\int_{-\infty}^\infty dt' \left( \frac{x(t)-x(t')}{t-t'} \right)^2 \right)$$