When does $hbar rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

• $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

• $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

• $\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

Answer

The theory of deformation quantization provides a framework in which the quantum to classical transition can be carried out and understood.

According to this theory, for (practically any) quantum system, one can find (may be nonuniquely) a Poisson manifold $\mathcal{M}$ (phase space) equipped with an associative product called the "star product" such that the quantum observables are represented by smooth functions on $\mathcal{M}$ and the quantum operator product is given by the star product.

Furthermore, the star product of two functions has a formal power series in $\hbar$

$f\star g = \sum_{k=0}^{\infty} \hbar^k B_k(f,g)$

Such that:

$B_0(f,g) = fg$

$B_1(f,g)-B_1(g, f) = \{f,g\}$, (Poisson bracket)

Thus we obtain:

$f\star g - g\star f = \hbar\{f,g\} + \sum_{k=2}^{\infty} \hbar^k (B_k(f,g)-B_k(g,f))$

Please notice that according to the deformation Philosophy, the quantum observables are just functions on the phase space just as the classical observables and all the quantum noncommutativity is provided by the star product. Thus if we define $\hat{f} = \frac{\hbar}{i} f $, we get the required classical limit.

It should be emphasized that this procedure can be carried out even for quantum systems defined by matrix algebras for example an appropriate phase for spin iis the two-sphere $S^2$, please, see the following article by Moreno and Ortega-Navarro. Morover,

Kontsevich in his seminal work provided a constructive method to construct this star product on every finite dimensional Poisson manifold, Please see the following Wikipedia page.

It is also worthwhile to mention that there are efforts to generalize the deformation construction to field theories and incorporate renormalization into it, please see the following work by Dito.