My talk on ideal quantum Brownian motion
I have blogged before about the conceptual importance of ideal, symplectic covariant quantum Brownian motion (QBM). In short: QBM is to open quantum systems as the harmonic oscillator is to closed quantum systems. Like the harmonic oscillator, (a) QBM is universal because it’s the leading-order behavior of a taylor series expansion; (b) QBM evolution has a very intuitive interpretation in terms of wavepackets evolving under classical flow; and (c) QBM is exactly solvable.
If that sounds like a diatribe up your alley, then you are in luck. I recently ranted about it here at PI. It’s just a summary of the literature; there are no new results. As always, I recommend downloading the raw video file so you can run it at arbitrary speed.
Abstract: In the study of closed quantum system, the simple harmonic oscillator is ubiquitous because all smooth potentials look quadratic locally, and exhaustively understanding it is very valuable because it is exactly solvable. Although not widely appreciated, Markovian quantum Brownian motion (QBM) plays almost exactly the same role in the study of open quantum systems. QBM is ubiquitous because it arises from only the Markov assumption and linear Lindblad operators, and it likewise has an elegant and transparent exact solution. QBM is often introduced with specific non-Markovian models like Caldeira-Leggett, but this makes it very difficult to see which phenomena are universal and which are idiosyncratic to the model. Like frictionless classical mechanics or nonrenormalizable field theories, the exact Markov property is aphysical, but handling this subtlety is a small price to pay for the extreme generality. The widest class of QBM dynamics is symplectic invariant and includes Einstein-Smoluchowski diffusion, damped harmonic oscillations, and pure spatial decoherence as special cases, whose close relationship is often obscured.
If you want more reading, see the blog post linked above and the citations in the introduction of my related PRA.
The fact that this isn’t taught in every graduate quantum mechanics class is more evidence that graduate education is awful and stagnant. Professors choose topics based on historical accident and inertia rather than trying to give their students a thorough, deep, and clarifying understanding of quantum mechanics.