The Isomorphism of Complex Numbers and Latent Worlds 1; Symmetry Constraints and the Inevitability of Quantum Structures- From Spacetime Transformations to the Complex Domain

The indispensable nature of complex numbers in quantum mechanics has beenexperimentally confirmed, yet its deep origin — whether complex numbers are anad hoc introduction or a logical consequence of some more fundamental constraint—remains insufficiently clarified. From the perspective of symmetry analysis, thispaper argues that the complex structure is not an accidental choice of quantumtheory, but an inevitable outcome under the joint constraints of spacetime transformation symmetry and algebraic completeness. Through a rigorous analysis ofthe covariance of the Schrödinger equation under Galilean transformations, thispaper reveals: if the transformation rule of the wave function is required to preserve the probability interpretation and satisfy spacetime symmetry, then the wavefunction must take values in the complex number field. This conclusion can beextended to relativistic cases, forming an intrinsic echo with the role of mass asa central charge in Bargmann’s theorem. On this basis, we establish a quantitative criterion for the strength of symmetry constraints, proving that the transitionfrom a real-valued formulation of quantum theory to a complex-valued formulationcorresponds to a central extension of the symmetry algebra, which is the uniqueway for a physical theory to realize symmetry in the sense of projective representations. This framework is compatible with recent experimental advances regardingimaginarity as a quantum resource and multipath interference tests of hypercomplex theories, and proposes a symplectic geometry-based scheme for measuring thestrength of symmetry constraints. This paper advances the foundational issue ofcomplex numbers in quantum theory from an ”ontological debate” to a new levelof ”symmetry constraint analysis.”