✎ Koopman-von-Neumann Mechanics [The Physics Travel Guide]

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====== Koopman-von-Neumann Mechanics ======

Koopman-von-Neumann Mechanics is a reformulation of [[theories:classical_mechanics:newtonian|classical mechanics]] using the same language that we usually only use in [[theories:quantum_mechanics:canonical|quantum mechanics]].

Usually in classical mechanics, our main focus is the trajectories of objects.

In contrast, in quantum mechanics, our main focus are observables, states and measurements. This means, instead of describing the trajectories of objects, we now ask instead for example: If we measure the momentum of this ball, what's the probability that the result will be $10$ kg m/s?

<tabbox Concrete> 
The equation of motion in Koopmann-von-Neumann mechanics is a complexified reformulation of the classical Liouville equation (which handles probability distributions over the phase space).

* See the notes by Frank Wilczek on [[https://web.archive.org/web/20170206103038/http://frankwilczek.com/2015/koopmanVonNeumann02.pdf|"Koopman von Neumann Mechanics, and a Step Beyond"]]
  * See also this [[https://physics.stackexchange.com/a/80822|nice answer at StackExchange]]
  * [[https://arxiv.org/pdf/quant-ph/0301172.pdf|Topics in Koopman-von Neumann Theory]] -a  Ph.D. Thesis by Danilo Mauro
 
<tabbox Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox Why is it interesting?> 
The Koopman-von-Neumann reformulation of classical mechanics is extremely helpful to understand the differences between classical and quantum mechanics. Usually these theories are describes in completely different languages which makes it hard to compare them. Using the Koopman-von-Neumann reformulation we describe classical mechanics in exactly the same language that we use in quantum mechanics and thus can perfectly analyze what is different.

----

<blockquote>Ordinary mechanics must also be statistically formulated: the determinism
of classical physics turns out to be an illusion, it is an idol, not an ideal in
scientific research.
<cite>Max Born, 1954 Nobel Prize Lecture.</cite></blockquote>

</tabbox>

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