# 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. Jan 21, 2020 at 14:52

## 1 Answer

In order to have dissipation you need somewhere for the energy to go, and this somewhere has to be included in the quantization process. In addition to absorbing the energy, the extra system causes quantum decoherence and so makes the problem quite tricky.

The damped harmonic oscillator is one of the problems that were attacked and solved by Caldeira and Leggett (A.Caldiera, A.J.Leggett, Phys. Rev. Lett. vol 46 (1981) p 211). Calderia and Leggett showed that you can almost always model the "somewhere" as a bath of infinitely many harmonic oscillators. In this they derive that the physics is given by the effective (Euclidean) action:

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

• Can you clarify what the approach in that paper is? Looking at it, it appears they assume the linear friction can be represented by a bath of linear oscillators (sounds reasonable) and then express the quantity they care about in terms of path integrals and integrate out the unnecessary DoFs? Jan 21, 2020 at 14:51
• You understanding is broadly correct. Their "assuption" about the oscillator bath being adaquate is actually almost universally valid. There are a few cases when the oscillators have to be replavced a bath of spins. A google search turns up arXiv:0808.1377 as recent paper on spin baths. Jan 21, 2020 at 15:09
• Okay, I need to go away and work through how they obtain their final action, but this looks like a helpful reference, thank you! Jan 21, 2020 at 15:10