Bee,
Thanks for the comment! I never discussed the representation as being an operator in the comment. If you chose to make that connection yourself that is fine, but you're projecting your own thoughts into the meaning of these things, which is fine too, but it does lead to a lot of miscommunication.
All you have to do is go look at how h is defined and used in the quantum mechanics versus quantum mechanics.
If classical mechanics is arrived at when we reduce h, what does that mean? First of all this is already well known (page 19 of A. Zee QFT), so we shouldn't confuse that continuous spectrum emerge in classical limits against the effect of dividing through by a constant.
If we seriously look at h as it is to describe a single entity it does in fact describe wave like properties in terms of expected position and momentum. However, the only way we can reduce h in the classical realm is through process related to mutual information, defined as:
[math](\rho^{ab}) = S(\rho^{a}) S(\rho^{b}) - S(\rho^{ab})[/math]
Which is understood in terms of relative entropy as:
[math](\rho^{ab}) = S(\rho^{ab}|| \rho^{a} \otimes \rho^{b})[/math]
As the wikipedia article on quantum mutual information states:
"if we assume the two variables x and y to be uncorrelated, mutual information is the discrepancy in uncertainty resulting from this (possibly erroneous) assumption."
It is easy to assume that when we are talking about classical variables, such as position and momentum, uncertainty does not scale with the number of systems, so as more and more systems are added, mutual information increases, so the uncertainty in larger systems decreases and the system becomes more classical...e.g. the classical world emerges as we scale up with more systems.
I might even be tempted to declare it a law, but that would be an easy way out.
In any case, this is sufficient to begin discussions about how the objective world of Einstein is a world dependent all the component density matrices, and the world as we know it is an emergent property in the limit of vanishing uncertainty.
This is also best explained by understanding the relationship of Wigner's function and the Moyal equation to Liouville's equation (http://en.wikipedia.org/wiki/Density_matrix#.22Quantum_Liouville.22.2C_Moyal.27s_equation)
As mutual information increases with the number of systems, the uncertainty decreases, this would appear as a decrease of the uncertainty (represented with h) in the equivalent classical phase space. So the classical world eventually starts to emerge in the limit of large systems.
This is probably closer to the concept you are trying to articulate in the article.
