Duality of quantum competing system

Abstract: 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.