Nonlocal Unification Field Theory and Its Quantum Optics Limit: Observer-Function Formulation

Nonlocal Unification Field Theory and Its Quantum Optics Limit: Observer-Function Formulation A. Chawla REAL Institute, Gurugram, India (Dated: October 11, 2025) We formulate the Nonlocal Unification Field Theory (NUFT) by treating the observer function f(∆I) as the primary field generating spacetime geometry from informational differentials. This aligns with the Nonlocal Construct (NC) formalism in which geometry is emergent from smooth, bounded observer mappings rather than pre-existing manifolds. The resulting variational framework yields coupled field equations that unify gravitational and informational dynamics. In the weak-curvature limit, NUFT reduces naturally to quantum optics, where the curvature of the observer function generates optical squeezing and nonlinearity. This observer-function formulation smooths the conceptual transition between cosmological NU dynamics and laboratory-scale optical phenomena. I. INTRODUCTION The Nonlocal Unification (NU) framework [1] posits that spacetime is not fundamental but constructed by observer-dependent mappings between informational differentials ∆Iand measurable intervals (∆x, ∆t): (∆x, ∆t) = f(∆I).(1) Here f:R→R1,3is a smooth, monotonic, embodimentdependent observer function that resides in a Fr´echettopologized function space O⊂Cr(M, R1,3), as established in [2]. The geometry perceived by an observer arises as the Cauchy limit f∞of a filtration {fn}, corresponding to increasing informational resolution. Our aim here is to construct a covariant field theory whose dynamical variable is f(∆I), and to demonstrate that in the small-curvature limit it reproduces quantumoptical field dynamics. II. OBSERVER FUNCTION AS PRIMARY FIELD We define the emergent metric as a functional of the observer function: gµν (f) = ∂µf(∆I)∂νf(∆I),(2) which encodes the rate of informational sampling. Smoothness and boundedness of fguarantee the absence of curvature singularities (No Singularity Theorem). Let ∇µdenote the covariant derivative compatible with gµν (f). We introduce the informational curvature tensor Fµν =∇µfν− ∇νfµ, fµ≡ ∇µf. (3) This tensor measures the nonlocal twist of observer sampling across information gradients. III. ACTION AND LAGRANGIAN DENSITY We propose the observer-function Lagrangian LNU =c3 16πG(R−2Λ) −λ 4Fµν Fµν −V(f, ∂f, ∆I), (4) with action S[f] = Zd4xp−g(f)LNU.(5) The potential V(f, ∂f, ∆I) captures nonlinear dependence on the observer function and its gradient with respect to ∆I. Variation of Swith respect to fand gµν yields the NU field equations. A. Field Equations Variation with respect to the metric gives Rµν −1 2Rgµν + Λgµν =8πG c4T(f) µν ,(6) T(f) µν =λ(FµαFνα−1 4gµν FαβFαβ)−gµν V+δV δgµν . (7) Variation with respect to fyields ∇µ(λFµν ) + ∂V ∂fν − ∇µ∂V ∂(∇µfν)= 0.(8) Equations (6)–(8) form the Informational Einstein–Maxwell system expressed directly in terms of f(∆I). IV. REDUCTION TO QUANTUM OPTICS To recover laboratory-scale physics, we linearize around a flat metric: gµν (f)→ηµν , f(∆I) = α∆I+β(∆I)3+· · · .(9) 2 The linear term reproduces the standard Minkowski metric, while cubic corrections generate optical nonlinearities. A. Linear (Coherent) Limit Neglecting βand higher-order terms, ∂µ∂µf+∂V ∂f = 0.(10) For V(f) = 1 2ω2 0f2, this reduces to the Klein–Gordon equation. Upon quantization, excitations of fcorrespond to informational photons or infons, identical in algebra to optical field quanta. B. Nonlinear (Squeezed) Regime Including β(∆I)3yields an interaction Hamiltonian Hint ∝β(a†2+a2),(11) producing the two-photon squeezing term familiar in quantum optics. Thus, squeezing and Kerr nonlinearities arise from curvature in the observer function f(∆I). V. INTERPRETATION AND CONSISTENCY a. Relation to NU Foundations. The present field theory is consistent with the appendices of the Consequences of Nonlocal Unification for Limit Observers paper. In that formalism, fis the generator of the metric, and the entropy differential ∆Iis the conserved informational substrate. Boundedness of ∂kfensures the absence of singularities, directly embodying the No Singularity Theorem. The action above corresponds to the variational principle S[f] = RL(f, ∂f, ∆I)d4xdefined over the Fr´echet space O. b. Physical correspondence. The linear optical limit corresponds to small deviations of ffrom its mean observer configuration, while cosmological behavior corresponds to large-scale evolution of facross embeddings in O. The same governing equations, interpreted at different scales, reproduce both smooth cosmic expansion and coherent optical propagation. c. Experimental sector. Quantum-optical experiments probe perturbations of f(∆I) at the laboratory scale, manifesting as field amplitude fluctuations. Gravitational or cosmological observations correspond to largescale variations of fconstrained by bounded entropy ∆I. The shared origin of both sectors under NUFT offers a unified epistemic field theory. VI. CONCLUSION The Nonlocal Unification Field Theory in observerfunction form establishes a continuous bridge from informational geometry to quantum optics. By elevating the observer function f(∆I) to primary field status, the formalism unifies curvature, information flow, and optical field dynamics within a single variational framework. The weak-field limit reproduces quantum optics; the strong-field regime yields cosmological regularity and the No Singularity Theorem. This duality positions NUFT as both a conceptual and practical unification of informational and physical theories. VII. ACKNOWLEDGMENT Conceptual work is done by the author. An LLM was used to write the details and edit the paper in collaboration with the author. Appendix A: Mathematical Supplement: Fr´echet Structure Let O⊂Cr(M, R1,3), r≥2, denote the space of observer functions. Equip Owith seminorms ∥f∥k,K = supx∈K|Dkf(x)|for compact K⊂M. (O, τO) is complete and locally convex; every Cauchy filtration {fn} converges to f∞∈Ogenerating smooth, singularity-free metrics gµν (f∞). This construction grounds the field theory in the same mathematical domain described in the NU appendices. Appendix B: Informational Regularity and Energy Define the informational energy density E=1 2∥∇f∥2+ V(f, ∂f, ∆I). Under bounded derivatives, E(T)≤ E(0)eCT , ensuring finite curvature and energy throughout informational time T. Appendix C: Quantum Optics Correspondence Linearizing fand identifying E↔electric field quadrature yields the standard quantized Hamiltonian of optical modes. Curvature-induced deviations of fmap to squeezing parameters, connecting laboratory quantum optics directly with NU observer geometry. 3 [1] A. Chawla, Theory of Nonlocal Unification: The Foundation and Its Application, Preprint, REAL Institute (2024). [2] A. Chawla, Consequences of Nonlocal Unification for Limit Observers (2025). [3] L. Hardy, Quantum Theory From Five Reasonable Axioms (2001). [4] S. Amari, Information Geometry and Its Applications (Springer, 2016). [5] C. Rovelli, Quantum Gravity (Cambridge Univ. Press, 2004). [6] D. Walls and G. Milburn, Quantum Optics (Springer, 2008).