Representative Geometry of States in the Bilocal Formalism. SU(n) Symmetries from the State-Value Space and Spin Dualism.

We consider the bilocal formalism of quantum mechanics, in which the wave functiondefined on the space Q Q is interpreted as a description of two statesof the same particle. In contrast to earlier approaches, we do not assume the wave function to be scalar-valued, but instead allow it to take values in a finite-dimensional complex vector space. We show that changing the codomain of the wave function does not modify the structureof the configuration space nor its phase-space realization. The entire bilocal geometry, responsible for the emergence of spacetime, mass, and kinematical spin, remains unchanged. New structures appear exclusively at the level of the state-value space. The space of physical states then takes the form of the complex projective spaceCP^n-1, which is a natural symplectic manifold and a coadjoint orbitof the group SU (n). For n=2, the resulting geometry duplicates the structural spin spherealready present in the bilocal formalism, whereas the case n=3 leads tothe first non-degenerate representational symmetry. These results suggest that internal symmetries of a representational naturecan be understood as consequences of the geometry of quantum state spaces, rather than as additional postulated degrees of freedom.