Operator-Algebraic Extension of Quantum Field Theory: Entropic Emergence of Spacetime, Gauge Fields, and Matter

We formulate a unified operator-algebraic framework in which spacetime geometry and gauge dynamics emerge as thermodynamic limits of an extended quantum field theory. The microscopic degrees of freedom live in a Diagram Hilbert Space—the algebraic span of QFT interaction histories—on which a complex operator Z=M+iQ unifies mass–energy and charge. Statistical extremization of the von Neumann entropy under constraints on ⟨M⟩ and ⟨Q⟩ yields Einstein’s equations from the real sector and Maxwell’s equations from the imaginary one. The operator algebra extends conventional QFT without replacing it: Feynman diagrams define the spectral data of M, while coarse-grained entropy projections reproduce gravitational and electromagnetic fields as macroscopic expectation values. The framework parallels AdS/CFT duality by relating microscopic quantum information to emergent geometry, and it shares structural similarities with Loop-Quantum-Gravity spin networks and Verlinde’s entropic gravity. Hidden eigenstates of Z produce halo-like mass profiles consistent with galactic rotation curves, while residual entropic energy acts as a cosmological constant. This operator-entropic extension of QFT thus provides a continuous bridge from gauge field theory to gravitational thermodynamics.