  Co-existence of different states is a profound concept, which possibly
underlies the phase transition and the symmetry breaking. Because of a property
inherent to quantum mechanics (cf. uncertainty), the co-existence is expected
to appear more naturally in quantum-microscopic systems than in macroscopic
systems. In this paper a mathematical theory describing co-existence of states
in quantum systems is presented, and the co-existence is classified into 9
types.
