Continuous transition from quantum to classical system

Introducing a coarse-grained position operator based on the SO(3,1) as a representative of macroscopic nature, ive explain within quantum theory the classical appearance of our macroscopic world. The transition from quantum to classical is explicitly presented, in which a dimensionless continuous parameter plays a crucial role. This parameter consists of the ratio of a coarse-grained length to de Broglie wavelength. The macroscopic limit is investigated in detail. As a concrete example of the application of this treatment for the macroscopic system, crystal lattice consisting of N atoms is examined. In this example, energy eigenvalues involving both quantum and classical limits are obtained.