Origin of the Complex Structure of Quantum Mechanics from Stability of Matter

The complex field structure of quantum mechanics is traditionally treated as an irreducible postulate. We derive this structure from the physical requirement of the stability of matter. We show that enforcing a lower-bounded energy spectrum \((E\ge0)\) constrains the temporal dependence of physical states to the positive-frequency sector of tempered distributions. The resulting generalized solution space of the Schrödinger equation is given by the completed tensor product \(\mathcal P_{\mathrm{gen}} \simeq \mathcal S'_+(\mathbb R_t)\,\widehat\otimes\,L^2(\mathbb R^3_{\mathbf x}),\) with the Hardy space \(H^2_+(\mathbb R_t)\) emerging as the Hardy-class subspace of finite analytic norm. This topological factorization separates temporal analyticity-which enforces causality and energy positivity-from spatial probabilistic localization. We prove a unified no-go theorem demonstrating that treating the real and imaginary components of quantum wavefunctions as independent degrees of freedom leads to an energy spectrum unbounded from below and is therefore incompatible with the stability of matter. As a consequence, the imaginary component of the wavefunction is not an independent degree of freedom but is uniquely determined by the real component via the temporal Hilbert transform, \(\psi_I = \mathcal{H}_t[\psi_R]\). The formulation thus admits a description in terms of strictly real-valued fields, with the complex unit arising as an operator representation of time-domain analyticity.