It might be worth to add, that Planck found "his" constant $h$ by considering a many-body-problem indeed.

At this time, it was quite a standard method in statistical mechanics, to divide the physical phase space in discrete small phase volumes ("cells") and to count them. So, statistical mechanics at this time already contained a double limit procedure, by letting the number of particles $N \to \infty$ (and the total volume $V \to \infty$ as well, to obtain a thermodynamic limit), and by letting the discretizing phase space volume unit $\nu \to 0+$.

From a today's point of view, to obtain classical many-body statistical mechanics, one first takes $N \to \infty$ in the thermodynamic limit of quantum statistical mechanics, and then $\hbar \to 0+$ afterwards to obtain the (semi-)classical limit.
This is in a sense, and mathematically spoken from a today's point of view, what Planck found out (without saying so).
Classical statistical mechanics would have been obtained by first taking the "(semi-)classical limit" $\hbar \to 0+$, and the thermodynamics limit $N \to \infty$ afterwards.

A nice reference is Longair: Theoretical Concepts in Physics, 2ndEd, 2003, chap 13 (see also chap 12).

One also has to understand, that physics is not a formal science, like mathematics, but its ultimate goal is to understand real phenomena. There's no one-particle-system anywhere in the world. There exist only many-body-systems. Considering a system with one particle only or with one degree of freedom etc. stems from big reduction procedures and perturbation theory.

John von Neumann's Quantum Mechanics' book has a chapter about the coupling of the observed quantity to the environment.

In elemetary form, the procedure to obtain "physically treatable" models is as follows. One considers a certain part of reality an tries to specify it by declaring a boundary, separating the "observed system" and the "environment". One takes a thermodynamic limit for the environment. Then we can speak about intensive quantities such as temperature. These quantities determine a (one or maybe several) equilibrium state of the observed system. We make a perturbation ansatz for the observed system by considering states near the equilibrium. We try to get rid of all detailed interactions of the observed system with the detailed "particles" of the environment, by considering only averaged interactions with the environment. In general, the obtained equations for a subsystem would be non-local (both in space and time). We localize near the equilibrium to obtain equations for our observed subsystem. The data of the environment enter only through some (not too many) parameters.
Here is where most models of physics start.

These considerations make also clear, why one has to take into account mixed states (both in quantum statistical mechanics and in classical statistical mechanics) and not only pure states. The many averaging procedures mentioned above will in general provide a mixed state for the observed subsystem, not a pure state.

Pure states are mainly for helping in modelling, and for analyzing some phenomena - because mixed states can in general be written as a combination of pure states.

I would not consider quantum mechanics to "quantize" classical mechanics, because there is no unique procedure classical -> quantum. It's rather the other way round, that quantum mechanics models have also a (semi-)classical limit.

It would be physically misleading to think, that geometric quantization or deformation quantization are the answer to the question of how to model a classical system quantum mechanically. That's basically not true. The quantization procedures help only in identifying which kind of quantum models could be appropriate, and which are clearly to be discarded.
The real procedure in physics is, to guess a quantum mechanical model (most often by experience, and as a combination of simpler models that are already understood) and to look by experiment how good it describes reality. Classical mechanics only helps in guiding modelling eg. via geometry (for instance, a geometry of indistinguishable fermions is not so easy to visualise...).

Physicists' thinking is in some sense opposite to the thinking of mathematicians. A mathematician usually has a system of axioms for the "general case" and the specializes by making further assumptions.
In physical modelling for new situations, there are basically no axioms (except very general ones), but there are very specific models, that have been tested in specific situations. These models are then combined, like in a modular construction system. That's also, why physics textbooks almost never have axioms, but a plethora of examples and exercises to be calculated. Physical modelling in this sense is "bottom up", where the method of how a mathematician models is rather "top down". Mathematicians model only in situations, where the physics has been settled, eg. take the already know general equations of continuum mechanics and specifiy by choosing appropriate material constants (or functionals). Physical modelling usually occurrs in situations, where the "general equations" are unknown, so they can't be specialized to the specific situation.

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