# Probing the relation between decoherence theory versus the arguments based on the smallness of $h$ in connection with the quantum classical transition

This question tries to relate the elementary account of quantum effects in the macroscopic domain in terms of the smallness of the quantum of action, which is usually presented in introductory quantum mechanics courses, with the more comprehensive account for the same problem in terms of decoherence theory.

When one first comes across the former account, the classical limit of quantum theory appears to have all the general features that the limiting case of all other types of general theory possess. In particular, we expect the characteristics of the general theory to be of negligible significance in the physical situations well represented by the limiting theory. To illustrate this point, consider the amplitude of oscillation associated with a classical harmonic oscillator supposed to be in the quantum ground state. (This supposition is valid when the classical oscillator is isolated from environmental noise by isolating the system in a vacuum chamber near 0 K)

The amplitude for the oscillator in ground state $$|n=0\rangle$$ is $$A=\sqrt{\hbar/2m\omega}$$. To give a perspective, for a oscillator of mass 1kg, the amplitude is of the order of $$10^{-7}$$

One could also consider other macroscopic physical situations, where quantities like transmission coefficient (in a barrier potential) take negligible values.

However, the point (1) that will be mentioned below appears to suggest that arguments based on the smallness of quantum of action as compared to macroscopic dimensions is not complete in it's account of the classical world interms of quantum theoretic ideas(Unlike other limiting theory-general theory relation, where all the observed features of the limiting situation could be explained by taking appropriate limits of the general theory). Here goes the point(1):

(1) - Although the above considered quantities are restricted by the formalism of quantum theory to take negligible values in the macroscopic realm, there is NO restriction from within quantum theory that constrains the magnitude of the statistical variance of observables associated with macroscopic systems of large masses$$\textbf{(that is considered to be environmentally isolated)}$$. Furthermore, if we consider our environmentally isolated macroscopic system in the position eigenstate $$|r_1\rangle$$, then by the linearity of Schrödinger equation, the coherent superposition state $$a_1|r_1\rangle + a_2|r_2\rangle$$ is also a physically allowed state.(where $$|r_2\rangle$$ is another solution to the Schrödinger equation that is vastly seperated from $$|r_1\rangle$$ ) However, experiments seems to suggest that we never observe such states even when the body is kept in an ultracold vacuum chamber.

Now the question: In connection to the classical-quantum transition, does decoherence theory resolve this problem?