Koopman–von Neumann mechanics references

In summary, Koopman's formalism is useful for working with probabilistic outcomes, but it is not the only tool available for studying quantum mechanics.
  • #1
user_12345
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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.
 
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  • #3
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.
 
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  • #4
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.
 
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  • #5
there is no need to study the whole KvN formalism to turn to ergodic theory, just what the Koopman operator is
 
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  • #6
andresB said:
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":smile: 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:
WilcekOnTwitter.jpg

You'll be unsurprised to hear that Wilczek didn't answer me.
 
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1. What is Koopman–von Neumann mechanics?

Koopman–von Neumann mechanics is a mathematical framework for classical mechanics that describes the evolution of a system in terms of a single unitary operator acting on a complex function space, rather than using the traditional approach of equations of motion and phase space. It was developed by B.O. Koopman and J. von Neumann in the 1930s.

2. How does Koopman–von Neumann mechanics differ from traditional classical mechanics?

In traditional classical mechanics, the evolution of a system is described by a set of equations of motion and a phase space. In Koopman–von Neumann mechanics, the evolution is described by a single unitary operator acting on a complex function space. This approach allows for a more elegant and unified description of classical mechanics.

3. What are the advantages of using Koopman–von Neumann mechanics?

One of the main advantages of using Koopman–von Neumann mechanics is that it provides a more intuitive and elegant mathematical framework for describing classical mechanics. It also allows for a more unified treatment of classical and quantum mechanics, making it useful for studying systems that exhibit both classical and quantum behavior.

4. What are some applications of Koopman–von Neumann mechanics?

Koopman–von Neumann mechanics has been applied in various fields such as fluid dynamics, statistical mechanics, and quantum mechanics. It has also been used in the study of complex systems, such as biological networks and financial markets.

5. Are there any limitations to using Koopman–von Neumann mechanics?

One limitation of Koopman–von Neumann mechanics is that it is a purely mathematical framework and does not provide physical interpretations of the operators and functions used. It also assumes that the system is deterministic, which may not always be the case in real-world systems. Additionally, it can be challenging to apply in systems with a large number of degrees of freedom.

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