I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.

Basically, classical mechanics and the $\hbar \to 0$ limit of quantum mechanics study the action of the same algebra on very different representations. I'm curious whether there is a good physical explanation for why as you degenerate to the $\hbar \to 0$ limit the algebra of observables degenerates to the same Poisson algebra appearing in classical mechanics, but the relevant representation changes significantly. Specifically, (non-relativistically) classical systems have evolution $\frac{d}{dt} \rho = \{H, \rho\}$ and quantum systems have evolution $\frac{d}{dt}\psi = [H, \psi]$ (up to some constants). So far these look analogous, but in classical mechanics, the density function $\rho$ is itself a function on the phase space (i.e. a vector of the regular representation), whereas in quantum mechanics, $\psi$ is a is just a function (or something) on the $x_i$ themselves - i.e. a vector of a representation of "square-root dimension" (half-dimensional singular support)!

My guess is that this is a many-particle phenomenon, and a fully honest answer to "why do we observe classical mechanics" will probably involve a serious study of deconherence and questions of "what is observation", etc.

But I'm curious if there is a heuristic way to see why the algebra that's acting is the same (and in what way the representation is allowed to change: e.g., is there some embedding of the regular representation in a tensor product of irreducible ones?)

I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.

Basically, classical mechanics and the $\hbar \to 0$ limit of quantum mechanics study the action of the same algebra on very different representations. I'm curious whether there is a good physical explanation for why as you degenerate to the $\hbar \to 0$ limit the algebra of observables degenerates to the same Poisson algebra appearing in classical mechanics, but the relevant representation changes significantly. Specifically, (non-relativistically) classical systems have evolution $\frac{d}{dt} \rho = \{H, \rho\}$ and quantum systems have evolution $\frac{d}{dt}\psi = [H, \psi]$ (up to some constants). So far these look analogous, but in classical mechanics, the density function $\rho$ is itself a function on the phase space (i.e. a vector of the regular representation), whereas in quantum mechanics, $\psi$ is a is just a function (or something) on the $x_i$ themselves - i.e. a vector of a representation of "square-root dimension" (half-dimensional singular support)!

My guess is that this is a many-particle phenomenon, and a fully honest answer to "why do we observe classical mechanics" will probably involve a serious study of deconherence and questions of "what is observation", etc.

But I'm curious if there is a heuristic way to see why the algebra that's acting is the same (and in what way the representation is allowed to change: e.g., is there some embedding of the regular representation in a tensor product of irreducible ones?)

I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.

I know that (non-relativistically) classical systems have evolution $\frac{d}{dt} \rho = \{H, \rho\}$ and quantum systems have evolution $\frac{d}{dt}\phi = [H, \phi]$ (up to some constants). That's great, but in classical mechanics, the density function $\rho$ is itself a function on the phase space (i.e. a vector of the regular representation), whereas in quantum mechanics, $\phi$ is a is just a function (or something) on the $x_i$ themselves - i.e. a vector of a representation of "square-root dimension" (half-dimensional singular support)! From what I understand, an honest answer to "why do we observe classical mechanics" involves a serious study of deconherence and questions of "what is observation", etc. But I was curious whether there is actually a natural family of (possibly "toy") problems in physics that includes both quantum evolution and (in the $\hbar \to 0$ limit) the Hamiltonian point of view on classical mechanics?

I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.

Basically, classical mechanics and the $\hbar \to 0$ limit of quantum mechanics study the action of the same algebra on very different representations. I'm curious whether there is a good physical explanation for why as you degenerate to the $\hbar \to 0$ limit the algebra of observables degenerates to the same Poisson algebra appearing in classical mechanics, but the relevant representation changes significantly. Specifically, (non-relativistically) classical systems have evolution $\frac{d}{dt} \rho = \{H, \rho\}$ and quantum systems have evolution $\frac{d}{dt}\psi = [H, \psi]$ (up to some constants). So far these look analogous, but in classical mechanics, the density function $\rho$ is itself a function on the phase space (i.e. a vector of the regular representation), whereas in quantum mechanics, $\psi$ is a is just a function (or something) on the $x_i$ themselves - i.e. a vector of a representation of "square-root dimension" (half-dimensional singular support)!

My guess is that this is a many-particle phenomenon, and a fully honest answer to "why do we observe classical mechanics" will probably involve a serious study of deconherence and questions of "what is observation", etc. 

But I'm curious if there is a heuristic way to see why the algebra that's acting is the same (and in what way the representation is allowed to change: e.g., is there some embedding of the regular representation in a tensor product of irreducible ones?)