Abstract
The historical debate surrounding the necessity of complex numbers in quantum mechanics has been revitalized by recent experimental falsifications of standard real-amplitude models. However, Barrios Hita et al. (2026) demonstrated that a physically consistent real-number quantum theory can be constructed by mapping complex Hilbert spaces onto real quotient Hilbert spaces with local flags. In this work, we extensively expand this framework to full quantum field theory and numerous applications. We present explicit, index-level, and operator-level derivations of non-relativistic dynamics (the Schr¨odinger equation, Ehrenfest’s theorem, uncertainty principle), relativistic quantum mechanics (Klein-Gordon and Dirac equations), and quantum field theory (canonical quantization, Fock space, propagators, path integrals). We further apply the real formulation to harmonic oscillators, spin systems, perturbation theory, scattering theory, quantum entanglement, gauge theories and Quantum computing. We show that U(1) gauge transformations are rigorously isomorphic to SO(2) rotations in the real flag space, and that all physical predictions—including Bell violations and scattering amplitudes—are exactly reproduced. All derivations are presented at the explicit index level with full algebraic details, using a symplectic structure J that replaces the complex unit i.