  We have constructed a theory of dual canonical formalism to study the quantum
competing systems. In such a system, as the relationship between current and
voltage of each, we assumed the duality conditions. We considered competing
system of two types. One type of these system are composed of the sandwich
structure by SC(superconductor)/SI(superinsulator)/SC junction, and its dual
junction is consists of the sandwich structure by SI/SC/SI junction. Another
one type of these system is consists of the sandwich structure by
SC/FM(ferromagnet)/SC junction. and its dual junction is consists of the
sandwich structure by FM/SC/FM junction (spin Josephson junctions). We derived
the relationship between the phase and the number of particles in a dual system
of each other. As an application of the dual competitive systems, we introduced
the quantum spin transistor.
