 I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at [the answer to this question](https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics), 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?)



Does quantum mechanics ever really quantize classical mechanics?
It is perhaps helpful to distinguish between four types of mechanics here:


1. *Pure-state classical mechanics*. Here, the mechanics are classical, and the system is described by a single point $(q,p)$ in phase space. This point evolves via Hamilton's equations of motion $\partial\_t q = \frac{\partial H}{\partial p}; \partial\_t p = - \frac{\partial H}{\partial q}$.
2. *Mixed-state classical mechanics*. Here, the mechanics are classical, and the system is described by a probability density function $\rho(q,p)$ on phase space (this density may be a generalised function, e.g. a Dirac delta, rather than a classical function). This density function evolves via the advection equation $\partial\_t \rho = \partial\_p ( \rho \partial\_q H ) - \partial\_q (\rho \partial\_p H ) = \{H,\rho\}$.
3. *Pure-state quantum mechanics*. Here, the mechanics are quantum, and the system is described by a wave function $|\psi\rangle$ in a Hilbert space. This wave function evolves via Schrödinger's equation of motion $\partial\_t |\psi \rangle = \frac{1}{i\hbar} H |\psi\rangle$.
4. *Mixed-state quantum mechanics*. Here, the mechanics are quantum, and the system is described by a [density matrix](https://en.wikipedia.org/wiki/Density_matrix) $\rho$ (a positive semi-definite trace one operator on a Hilbert space). This density matrix evolves by the von Neumann evolution equation $\partial\_t \rho = \frac{1}{i\hbar} [H,\rho]$.


In both the classical and quantum regimes, a mixed state can be viewed as a convex (or classical) superposition of pure states (with a pure classical state $(q,p)$ identified with the Dirac probability density function $\delta\_{(q,p)}$, and a pure quantum state $|\psi \rangle$ identified with a pure density matrix $|\psi \rangle \langle \psi|$). So in principle the pure-state mechanics describes the mixed-state mechanics completely (albeit with the caveat that in the quantum case, in contrast to the classical case, the decomposition of a mixed state as a superposition of pure states is non-unique). However, the correspondence principle is clearest to see at the mixed state level, i.e. to compare 2. with 4. in the semiclassical limit $\hbar \to 0$, rather than comparing 1. with 3.. Indeed, any density matrix $\rho$ has a [Wigner transform](https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution) $\tilde \rho$, which is a function on phase space defined via duality as $\int \tilde \rho(q,p) A(q,p)\ dq dp = \hbox{tr}( \rho \hbox{Op}(A) )$ for any classical observable $A$, where $\hbox{Op}(A)$ is the (Weyl) quantisation of $A$ (i.e. the Wigner transform is the adjoint of the quantisation operator). This Wigner transform $\tilde \rho$ will usually not be non-negative, and hence will not be a classical probability density function, but in semiclassical regimes it is often the case that $\tilde \rho$ will tend (in a suitable weak sense) to a classical probability density when $\hbar \to 0$, which will then evolve by the classical advection equation. This is the dual to the assertion that the quantum Heisenberg equation $\partial\_t A = \frac{i}{\hbar} [H,A]$ for the evolution of quantum observables converges to the classical Poisson equation $\partial\_t A = -\{ H,A\}$ for the evolution of classical observables in the semiclassical limit $\hbar \to 0$.


There is still a correspondence at the level of 1. and 3., but it is a bit trickier to see; one has to restrict to things like "gaussian beam" type solutions $|\psi \rangle$ to the Schrödinger equation that are well localised in both position and momentum space, in order to get a classical limit that is a pure state rather than a mixed state. (An arbitrary wave function would instead get a "phase space portrait" which in the semiclassical limit becomes [assuming some equicontinuity and tightness, and possibly after passing to a subsequence, as noted in comments] a mixed state from 2., rather than a pure state from 1.).


 


