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

In summary, classical physics can be reformulated in an operator-observable form, similar to quantum mechanics, thanks to the work of Koopman and von Neumann. This is known as Koopman-von Neumann mechanics and there has been research on smooth transitions between this theory and orthodox quantum mechanics. It is not an easy task, but it has been achieved.
  • #1
jshrager
Gold Member
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1
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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  • #2
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.
 
  • #3
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!
 
  • #4
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.
 

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

1. What is the significance of reinterpreting classical physics in terms of operators, amplitudes, and eigenvalues?

Reinterpreting classical physics in terms of operators, amplitudes, and eigenvalues is important because it allows us to bridge the gap between classical and quantum mechanics. By using these mathematical tools, we can better understand the underlying principles and behaviors of physical systems, and make predictions about their behavior.

2. How does the use of operators differ from traditional classical physics equations?

In traditional classical physics, equations are typically written in terms of variables such as position, velocity, and time. However, in the reinterpretation of classical physics, these variables are replaced with operators, which represent physical observables. This allows us to express the behavior of a system in terms of its measurable properties, rather than just its classical parameters.

3. What is the role of amplitudes in the reinterpretation of classical physics?

Amplitudes are a fundamental concept in quantum mechanics, representing the probability of a particle or system being in a certain state. In the reinterpretation of classical physics, amplitudes are used to describe the quantum behavior of a classical system, allowing us to understand how classical systems can exhibit wave-like behavior under certain conditions.

4. How are eigenvalues used in the reinterpretation of classical physics?

Eigenvalues are used to describe the possible outcomes of a measurement on a physical system. In the reinterpretation of classical physics, eigenvalues are used to describe the different states that a classical system can exist in, and how these states can change over time. This allows us to make predictions about the behavior of classical systems and how they may evolve over time.

5. What are some applications of reinterpreting classical physics in terms of operators, amplitudes, and eigenvalues?

There are many applications of this reinterpretation, including in fields such as quantum computing, quantum information processing, and quantum simulations. This approach also allows us to better understand the behavior of complex systems, such as molecules and materials, and make more accurate predictions about their properties and behaviors.

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