Koopman–von Neumann mechanics references

[user_12345]

Hello everyone, I am new here. I am studying physics as a self-taught student.
I have been studying classical Lagrangian and Hamiltonian mechanics from Goldstein's book and have read that there is an additional formulation of classical mechanics in Hilbert spaces.
Is it worth studying? Do you know of any easy textbooks for learning classical mechanics in Hilbert spaces?
Thank you and sorry for my English.

 

[andresB]

I don't think there is a textbook on the subject, but, luckily, the wikipedia page on KvN mechanics is quite complete and well referenced. The following are quite readable

https://arxiv.org/abs/quant-ph/0301172
http://frankwilczek.com/2015/koopmanVonNeumann02.pdf

And I can't avoid mentioning my own work
https://arxiv.org/abs/2004.08661
https://arxiv.org/abs/2105.13882

 

[wrobel]

I think that that Koopman–von Neumann theory is needed if only you come to it from some another physics topics not from classical mech. I think that a considerable non mechanical background is needed.

 

[andresB]

Well, the basics of the KvN formalism are quite simple for anyone that is familiar with the mathematical structure of QM.
The usefulness of the KvN is perhaps harder to justify outside ergodic theory. And ergodic theory is, well, hard indeed.

 

[wrobel]

there is no need to study the whole KvN formalism to turn to ergodic theory, just what the Koopman operator is

 

[Peter Morgan]

I don't think there is a textbook on the subject, but, luckily, the wikipedia page on KvN mechanics is quite complete and well referenced. The following are quite readable

https://arxiv.org/abs/quant-ph/0301172
http://frankwilczek.com/2015/koopmanVonNeumann02.pdf

And I can't avoid mentioning my own work
https://arxiv.org/abs/2004.08661
https://arxiv.org/abs/2105.13882

Weird that I've hardly been on PF for I think over a year and this morning I drop by to find there's a mention of Koopman. I second AndresB's mention of the Wikipedia page and of the Frank Wilczek notes.

Can I also not avoid mentioning my own work? "An algebraic approach to Koopman classical mechanics" Some of that may be accessible, some of it will not, but in any case I can take this opportunity to thank AndresB for citing it in his 2105.13882. I fear, however, that also neither of his papers can be thought elementary.

In the year since "An algebraic approach to Koopman classical mechanics" was published in Annals of Physics, I've come to think that Koopman's Hilbert space formalism is very useful indeed if you want to work with probabilities, but if you want to work with definite trajectories not so much. If you do want to work with probabilities, Koopman's Hilbert space formalism can model statistics out of experiments as capably as any other Hilbert space formalism.

The physics literature is working through the question of how much or whether Koopman can help us understand quantum mechanics: if it's decided that it can, then a good textbook account of Koopman's formalism will be soon forthcoming, otherwise it won't. One snippet of gossip, from March 29th on Twitter:

You'll be unsurprised to hear that Wilczek didn't answer me.