Reinterpretation of classical physics in operator/amplitude/e.v.?

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Discussion Overview

The discussion revolves around the possibility of reformulating classical physics using operator formalism, similar to quantum mechanics. Participants explore concepts such as the Koopman-von Neumann mechanics and the potential for using amplitude notation in classical contexts.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant suggests that classical physics can be reformulated in an operator-observable form, potentially using bra/kets and amplitude notation, but questions the feasibility given the nature of macroscopic objects.
  • Another participant references the Koopman-von Neumann mechanics, indicating that it has been established that classical mechanics can be expressed as an operatorial theory akin to quantum mechanics.
  • A later reply acknowledges the relevance of the Koopman-von Neumann mechanics and mentions a concept called Operational Dynamic Modeling, which involves smooth transitions between this theory and orthodox quantum mechanics.
  • One participant expresses excitement about the connection to von Neumann's work, reflecting on the tendency to discover ideas that have already been explored by prominent figures in physics.

Areas of Agreement / Disagreement

Participants express differing views on the ease of reformulating classical physics in an operator framework, with some suggesting it has been done while others remain skeptical about its simplicity.

Contextual Notes

There are unresolved questions regarding the specific methods and implications of using operator formalism in classical physics, as well as the limitations of applying quantum-like concepts to macroscopic phenomena.

jshrager
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Presumably it's easy to re-formulate non-quantum (i.e., classical) physics entirely in operator -> observable form (perhaps even to the point of using bra/key and amplitude^2 notation, etc -- although since macroscopic objects are supposed to be in a single location, everything would end up being delta functions, or something like that). Can someone point me to someplace that summarizes how this is done? (Maybe this should be a question for classical physics topic, but they might not know what I'm talking about.)
 
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I don't think it's easy, but if I understand your question right, it has been done:

Koopman-von Neumann mechanics

As Koopman and von Neumann demonstrated, a Hilbert space of complex, square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.

This might also be relevant: I saw a talk from some people who worked on smooth transitions between KvN theory and orthodox quantum mechanics. They called it Operational Dynamic Modeling.
 
Awesome! That *exactly* what I was thinking of. (I'd say something about great minds thinking alike, but it seems way too pretentious to compare my mind to von Neumann's! :-)

Anyway, thanks!
 
jshrager said:
I'd say something about great minds thinking alike, but it seems way too pretentious to compare my mind to von Neumann's!

I do that all the time. Whoa, cool, I've just made a big discovery! followed by Oh ****, [famous physicist or mathematician] already figured that out several decades ago and I just didn't know about it.

Von Neumann's name comes up a lot, though I think the all-time champion of I-already-thought-of-that has to be Gauss.
 

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