Quantum Constraint Formalism for Bilocal Configuration Spaces

We present a quantum formulation of a bilocal constraint framework in whichphysical states are defined by quadratic Casimir constraints on an extendedconfiguration space.The fundamental object of the theory is a bilocal quantum state defined ona product configuration manifold, rather than a wave function evolvingin an external time parameter.Quantization is performed following the Dirac procedure for constrainedsystems, with the constraint acting as a condition selecting admissiblephysical states.Internal Casimir invariants induce a natural decomposition of the Hilbertspace into representation sectors, in which effective parameters such as massemerge as labels of representations rather than fundamental constants.The formalism provides a consistent quantum counterpart of the classicalbilocal constraint geometry and offers a natural arena for relational andtimeless formulations of quantum dynamics.