**Abstract**：

Statistical mechanics is related to quantum mechanics at the root. Classical statistical mechanics

has several problems such as ergodic hypothesis, Gibbs factor, necessity of Planck constant,

principle of a priori probability and correspondence between classical and quantum related to

temperature.

Our purpose is to find one of the argument which lead from quantum mechanics to statistical

mechanics. In this research, microcanonical and canonical ensemble are derived from stationary

Schrödinger equation for boson and fermion. The important aims are to realize perfect

decoherence and a priori probability for isolated system. However, these require special condition

for entanglement of stationary. For the sake of generation such entanglement, it is necessary to

introduce interactions between the particles. The concept of classical ideal gas also includes weak

interactions which disappear after carrying out thermodynamic limit. Thus, this discussion starts

from determination of interaction Hamiltonian. The hidden gauge structure plays central role,

here. We adopt the phase operator of special type as introduction of gauge transformation. This

phase operator come into play abelian gauge field. We set coupling constant of this interaction to

vanishing at thermodynamic limit. By doing this, the situation which is similar to random phase

appear, moreover, perfect decoherence for isolated system and a priori probability is realized.

Once the above is accomplished, it is possible that a microcanonical ensemble for isolated system

is constructed. Furthermore, we constitute canonical ensemble theory from thus obtained

microcanonical ensemble theory.