Quantization of dissipative systems?

Question

In undergraduate courses the introduction to Hamiltonian mechanics usually starts from a Newtonian view point. One has equations of motions of the form (not sure if it is ok to use covariant notation for forces, but I will do it anyway):

$F_\mu = -\frac{\partial V}{\partial x^\mu}$

Then assuming certain properties of the potential (e.g. that it is independent of the velocity coordinates) one can show that it can be represented by Hamiltonian mechanics.

Now my question is if the systems that do not fulfill these conditions (e.g. dissipative systems, two energy conserving examples are given @JohnSidles answer to this question). Can such systems be quantized in any meaningful sense?

What I am really trying to achieve with this question is to understand better what quantization actually is. We usually have a Hamiltonian system and replace the Poisson bracket by commutation or anti-commutation relations and promote the functions to operators. But is this necessary for quantization or is there an underlying principle that can also be applied to other things?

Answer 1

Dissipative quantum mechanics does not preserve the pureness of a state, hence must be formulated in terms of density operators.

Conservative dynamics is classically described by a conservative dynamics obtained through an action principle. The quantum version is given on the level of density operators by the von Neumann equation expressing the derivative of the density operator as a commutator with the Hamiltonian.

Dissipative autonomous systems are classically described by modifying the conservative dynamics through adding dissipative terms. Similarly, the quantum version adds to the von Neumann equation dissipative terms of double commutator type. Most prominent is the Lindblad equation, extensively used in dissipative quantum optics.