Quantum mechanics and Classical limit(s)

Question

I have tried to make sense of this and i am not sure i get it. What i gather from this page about the classical limit is:

You need coherent states something like $\hbar \to 0$ is not really enaugh. Which makes sense to me because i always though it to be a strange thing to do. Like assuming $c=\infty$ istead of $ {v \over c} \to 0$.

But statements on this seem to vary greatly here are a few statements of upvoted answers and one from my statistical mechanics professor:

1) "The short answer: No, classical mechanics is not recovered in the ℏ→0 limit of quantum mechanics." - juanrga

There seem to be contradictory statements to this, and people trying to get people to read their paper on this, these are cases of this:

2)"It is natural, and intuitive, as explained above, to assume that the classical limit is a property of a certain class of states.

As it happens, that view is incorrect. You can actually obtain an exact recovery of Hamiltonian Classical Point Mechanics for any value of ℏ using a different wave-equation:" - Kingsley Jones

3) "What is the limit ℏ→0 of quantum theory?" is that the classical limit of quantum theory is not classical mechanics but a classical statistical theory. - U. Klein

4) This is what shows up in my stat. mech. lecture as:

"Häufig spielen jedoch quantenmechanische Effekte keine Rolle; dies sollte der Fall sein, wenn $\hbar $ kleiner als alle relevanten Wirkungen im System ist und wir den Grenz ̈ubergang $\hbar $ → 0 machen konnen. Dann sollten die quantenmechanischen Formeln in die klassischen Formeln übergehen."

my translation :

"Many times quantum effects can be neglected, if this is the case, if $\hbar$ is smaller than any relevant actions in the system and we can take the limit $ \hbar \to 0 $. Then all the Q.M formulas should transform into the classical ones" (Talking about statistical avarages $\langle O \rangle = Tr(O\rho)$ and von-Neumann enthropy here.)

I would like to know what is going on. Is this true for statistical mechanics? I am looking forward to your takes on this stuff. Just type "classical limit" in the search and look at some threads, it is quite strange (to me) how many "please don't just link your own work" type comments show up.

Answer 1

The classical limit of quantum theories is understood quite well from a mathematical standpoint nowadays. The so-called semiclassical analysis covers the QM (finite dimensional phase-space) cases, the Hepp method and infinite-dimensional semiclassical analysis cover the systems with classically infinitely many degrees of freedom.

The ideas can be summed up in the following quantum-classical dictionary, that can be made rigorous in the limit $\hbar\to 0$ with some technical assumptions:

• Space. Quantum: Infinite dimensional Hilbert space $\mathscr{H}$, in the easier example $L^2(\mathbb{R}^d)$. Classical: finite or infinite dimensional phase space $Z$. In the example above, $Z=\mathbb{R}^{2d}$.

• States. Quantum: $(\rho_{h})_{h\in (0,\bar{h})}\subset \mathcal{L}^1(\mathscr{H})$ is a family of quantum normal states (positive trace class self-adjoint operators with trace one). The dependence on the semiclassical parameter $h$, that has the role of the Planck's constant, is made explicit because the corresponding classical quantity is obtained as a suitable limit $h\to 0$ in the above family. Classical: Probability measures $\mu\in \mathcal{P}(Z)$ of the classical phase space. A probability measure is a positive Borel measure such that $\mu(Z)=1$.

• Observables. Quantum: Families of densely defined operators $A_{h}:D_q\subset\mathscr{H}\to \mathscr{H}$ that depend on the semiclassical parameter $h$ in a suitable (controllable) way. Classical: Densely defined function(al)s of the classical phase space $a:D_c\subset Z\to \mathbb{C}$.

• Evolution. Quantum: Family of strongly continuous unitary groups $U_h(\cdot)=e^{-\frac{i}{h}(\cdot) H_{h}}:\mathbb{R}\times \mathscr{H}\to \mathscr{H}$ that depend on the semiclassical parameter ($H_h$ Hamiltonian of the system). Classical: Nonlinear one-parameter evolution group $\Phi(\cdot):\mathbb{R}\times Z\to Z$; that solves the classical evolution equations (in the case of $\mathbb{R}^{2d}$ the usual Hamilton-Jacobi equations of classical mechanics).

In the limit $h\to 0$, the quantum objects converge to the classical ones, in the following sense (as I said under suitable technical assumptions, and "natural" scaling conditions that ensure everything in the limit is finite): at time zero $$\lim_{h\to 0}\mathrm{Tr}[\rho_h A_h]=\int_Z a(z)d\mu(z)\; ,$$ where $\mu$ is the classical measure corresponding to the family of states $\rho_h$, and the classical observable is averaged over the phase space w.r.t. the classical probability measure; at time $t$ $$\lim_{h\to 0}\mathrm{Tr}[e^{-\frac{i}{h}t H_h}\rho_h e^{\frac{i}{h}t H_h} A_h]=\int_Z a(z)d\Phi(t)_{\#}\mu(z)\; ,$$ where $\Phi(t)_{\#}\mu$ is the push-forward of the measure $\mu$ by the classical non-linear flow $\Phi(t)$ that solves the classical equations.

In $L^2(\mathbb{R}^d)$, to families of coherent states of the type $C\bigl((q/\sqrt{h},p/\sqrt{h})\bigr)$, it corresponds a delta classical measure on the phase space $\mathbb{R}^{2d}\ni (x,\xi)$ centered in the point $(q,p)$, i.e. $d\mu(x,\xi)=\delta(x-q)\delta(\xi-p)dxd\xi$. This means that in the classical limit to coherent evolution it corresponds punctual evolution of the point $(q,p)$ on the classical phase space.

Comment: the answers given in the question you linked are at least inaccurate or incomplete. I would like to stress that the results I sketched above are obtained in a rigorous fashion in a very vast literature of mathematical papers, for a huge class of interesting physical systems of QM and also (bosonic) QFT (in the few cases where it can be defined on a mathematically rigorous standpoint). The picture that emerges is also, in my opinion, quite natural: to a "probabilistic" knowledge that is intrinsic to quantum mechanics, classically it corresponds a similar probabilistic knowledge on the phase space; however the indeterminacy constraint of QM does not hold classically, and for suitable initial quantum states (coherent), a deterministic evolution of observables is recovered in the classical limit.