We present a rigorous geometric framework that unifies spacetime topology with quantum dynamics by relaxing the Riemannian axiom of a torsion-free connection. We define the physical vacuum as an Almost Hermitian Manifold $\mathcal{M} \cong \mathbb{C}^4$ governed by the Chern connection, which preserves the non-integrable complex structure $J$ at the cost of non-vanishing Torsion. We demonstrate that the Schrödinger Equation emerges naturally as the linear hydrodynamic limit of a stochastic diffusion process driven by the imaginary sector of the Torsion tensor (the Lee Form). Furthermore, we prove via Hurwitz's Theorem on Composition Algebras and Adams' Theorem on Parallelizability that the maximal dimensionality of a causal, unitary vacuum is constrained to 8 real dimensions ($\mathcal{V} \cong \mathbb{O}$), with 4-dimensional spacetime emerging as a Lagrangian submanifold. This framework resolves the "measurement problem" by identifying $\hbar$ with the kinematic viscosity of the symplectic manifold, rendering quantum non-locality a consequence of symplectic rigidity (Gromov's Theorem).
We present a rigorous geometric framework that unifies spacetime topology with quantum dynamics by relaxing the Riemannian axiom of a torsion-free connection. We define the physical vacuum as an Almost Hermitian Manifold M≅C4 governed by the Chern connection, which preserves the non-integrable complex structure J at the cost of non-vanishing Torsion. We demonstrate that the Schrödinger Equation emerges naturally as the linear hydrodynamic limit of a stochastic diffusion process driven by the imaginary sector of the Torsion tensor (the Lee Form). Furthermore, we prove via Hurwitz's Theorem on Composition Algebras and Adams' Theorem on Parallelizability that the maximal dimensionality of a causal, unitary vacuum is constrained to 8 real dimensions (V≅O), with 4-dimensional spacetime emerging as a Lagrangian submanifold. This framework resolves the "measurement problem" by identifying ℏ with the kinematic viscosity of the symplectic manifold, rendering quantum non-locality a consequence of symplectic rigidity (Gromov's Theorem).
We present a rigorous geometric framework that unifies spacetime topology with quantum dynamics by relaxing the Riemannian axiom of a torsion-free connection. We define the physical vacuum as an Almost Hermitian Manifold M≅C4 governed by the Chern connection, which preserves the non-integrable complex structure J at the cost of non-vanishing Torsion. We demonstrate that the Schrödinger Equation emerges naturally as the linear hydrodynamic limit of a stochastic diffusion process driven by the imaginary sector of the Torsion tensor (the Lee Form). Furthermore, we prove via the Cayley-Dickson construction and Adams' Theorem on Parallelizability that the maximal dimensionality of a causal, unitary vacuum is constrained to 8 real dimensions (V≅O), with 4-dimensional spacetime emerging as a Lagrangian submanifold. This framework resolves the "measurement problem" by identifying ℏ with the kinematic viscosity of the symplectic manifold, rendering quantum non-locality a consequence of symplectic rigidity (Gromov's Theorem).
We introduce a unifying theoretical framework based on \emph{complex time} defined on a K\"ahler manifold,which coherently integrates quantum unitary evolution, thermodynamic relaxation, and Riemannian geometric dynamicswithin a single dynamical system. By elevating time from a real parameter to a fundamental complex variableθ=τ+iℏt,we demonstrate that quantum mechanics, statistical thermodynamics, and gravitational geometryarise as distinct real-sector projections of an underlying holomorphic structure.A fundamental complex-time evolution equation is derived, which admits consistent Hilbert-space inner products,strict probability conservation, and natural coupling to the spacetime metric gμν.We prove that the theory reduces exactly to the Schrödinger equation, imaginary-time thermal dynamics,and geometric diffusion in their respective physical limits, while yielding testable predictionsconcerning wave-packet broadening, energy-level renormalization by geometry, and complex causal structure.This framework suggests that observed real spacetime is merely the \emph{real slice} of a fundamental complex manifold,providing a mathematically consistent route toward quantum-gravity unification.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.
In this work, we introduce a novel framework for quantum dynamics based on a second-order evolution equation, ∂2/∂t2∣Ψ⟩=K∣Ψ⟩ , with K=i(αD+βV) , where the operators D and V satisfy the Heisenberg algebra [D,V]=dI in d -dimensional space. Unlike conventional quantum mechanics governed by the first-order Schrödinger equation, this formulation emerges from a unified dynamical structure rooted in phase-space symmetry, naturally accommodating non-Hermitian interactions while preserving unitary evolution through the introduction of an η -inner product and a dual-state mapping. We demonstrate that this theory shares an SL(2,R) symmetry with the standard quantum formalism, revealing that both the Schrödinger and our second-order dynamics arise as distinct realizations within the same symplectic framework — the former as a quadratic generator (energy conservation), the latter as a linear generator (driving-dissipation balance). Extending the theory to four dimensions via Dirac matrices yields a covariant, spinor-valued wave equation whose stationary solutions carry instanton topological charge, π3(S3)=Z , thereby unifying the topological structures from one to four spatial dimensions under the homotopy classification πd−1(Sd−1)=Z . This approach not only generalizes non-Hermitian quantum mechanics beyond PT -symmetric potentials but also establishes a direct bridge between quantum dynamics, symplectic geometry, and topological field theory.
The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.
This paper proves that quantum mechanics emerges as a mathematical necessity when a deterministic classical system possesses compact Kähler dimensions inaccessible to observation. The proof rests on three independent results from established mathematics: Spectral discreteness: The spectral theorem for elliptic operators on compact manifolds guarantees a discrete eigenspectrum, producing superposition and complex phase structure. The uncertainty principle: The Gromov non-squeezing theorem establishes an irreducible lower bound on observable uncertainties, with Planck's constant identified as the symplectic capacity of the compact space. The Born rule: The unique Kähler-compatible L² inner product determines the probability rule P = |ψ|² with no alternatives. Together, these results demonstrate that the complete axiomatic structure of quantum mechanics — complex Hilbert space, superposition, interference, the uncertainty principle, the Born rule, and entanglement — is a necessary consequence of dimensional projection rather than a set of independent postulates. The paper further shows that: The complex numbers in quantum mechanics originate from the complex structure (J) of the compact manifold, providing a geometric explanation for recent experiments ruling out real-valued quantum mechanics (Renou et al., Nature 2021; Chen et al., PRL 2022). Unitary time evolution arises because the complex structure converts what would otherwise be dissipative dynamics into probability-preserving oscillation. The number field of quantum mechanics (real, complex, or quaternionic) is determined by the topology of the compact dimensions (Riemannian, Kähler, or hyperkähler respectively), making laboratory quantum experiments probes of extra-dimensional geometry. CPT symmetry corresponds to orientation-invariance of the compact manifold, connecting discrete symmetry tests to the topology of the extra dimensions. The theorem requires no physical assumptions beyond the existence of inaccessible compact Kähler dimensions. It applies to any number of dimensions, any compact Kähler geometry, and any deterministic dynamics. The paper concludes that quantum mechanics is not a law of nature but a law of perspective — the second such law in physics, following relativity. Where relativity revealed that the separation of spacetime into space and time is observer-dependent, this work shows that the separation of dynamics into deterministic and probabilistic is equally observer-dependent. Originally submitted to Foundations of Physics on March 16, 2026