Entropy, Topology, and Complex Operators: A Diagram Hilbert Space Approach to Unified Gravity and Electromagnetism

We propose a unifying framework in which gravity and electromagnetism emerge simultaneously from a complex operator formalism defined on a diagram Hilbert space. The microscopic operator 𝑍̂ = 𝑀̂ + 𝑖 𝑄̂ , whose real and imaginary parts encode mass and charge respectively, is projected macroscopically by a map : ℋ𝐷 → ℋ𝑚𝑎𝑐𝑟𝑜 . Expectation values of this projection yield the sources of the gravitational and electromagnetic fields. Entropy maximization under energy conservation produces the Einstein equations, while the imaginary component of 𝑍̂ generates the electromagnetic field tensor. The resulting Einstein–Maxwell system is expressed as a single equation derived from entropic principles. Topological invariants of 𝑍̂ stabilize mass–charge quantization and connect directly to the gauge symmetries of the Standard Model and QCD. This framework provides a coherent renormalizable basis for unification, bridges microscopic Hilbert space dynamics with macroscopic field equations, and suggests experimentally testable consequences through coupling unification and charge–mass correlations.