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 been
experimentally confirmed, yet its deep origin — whether complex numbers are an
ad hoc introduction or a logical consequence of some more fundamental constraint
—remains insufficiently clarified. From the perspective of symmetry analysis, this
paper argues that the complex structure is not an accidental choice of quantum
theory, but an inevitable outcome under the joint constraints of spacetime trans
formation symmetry and algebraic completeness. Through a rigorous analysis of
the covariance of the Schrödinger equation under Galilean transformations, this
paper reveals: if the transformation rule of the wave function is required to pre
serve the probability interpretation and satisfy spacetime symmetry, then the wave
function must take values in the complex number field. This conclusion can be
extended to relativistic cases, forming an intrinsic echo with the role of mass as
a central charge in Bargmann’s theorem. On this basis, we establish a quantita
tive criterion for the strength of symmetry constraints, proving that the transition
from a real-valued formulation of quantum theory to a complex-valued formulation
corresponds to a central extension of the symmetry algebra, which is the unique
way for a physical theory to realize symmetry in the sense of projective representa
tions. This framework is compatible with recent experimental advances regarding
imaginarity as a quantum resource and multipath interference tests of hypercom
plex theories, and proposes a symplectic geometry-based scheme for measuring the
strength of symmetry constraints. This paper advances the foundational issue of
complex numbers in quantum theory from an ”ontological debate” to a new level
of ”symmetry constraint analysis.”