Gravitational Lensing as a Manifestation of Nonlinear Critical Relativistic Quantum Wave Fields

We develop a general Hamiltonian formulation of gravitational lensing within the framework of Critical Field Theory (CFT), a self-adjoint non-linear relativistic field theory in which quantum and gravitational phenomena arise as critical transitions. In this approach gravity modifies a dispersion relation that plays the role of an effective refractive index for wave propagation in an underlying flat background, and light rays are realized as characteristics of the corresponding three-dimensional Hamiltonian flow. For photons we derive a weak-field 3D Hamiltonian with three parameters (α, β1, J) and show that, in the small-angle thin-lens regime, it reduces to an effective two-dimensional thin-lens Hamiltonian that is structurally equivalent to the standard GR-based lensing formalism. In the α-only limit we recover the familiar Singular Isothermal Sphere (SIS) and Singular Isothermal Ellipsoid (SIE) deflection laws, critical curves, and caustics, while the β1 and J terms give, respectively, a predominantly radial rescaling of the deflection and an anisotropic, centrifugal-like contribution tied to the ellipsoidal radius of the lens. Full 3D ray tracing through finite-thickness SIS and SIE lenses confirms that the CFT Hamiltonian reproduces classic strong-lensing configurations such as Einstein rings and crosses and yields realistic image morphologies. The Hamiltonian phase-space viewpoint naturally unifies spatial deflections and time delays, clarifies the geometric origin of critical curves and caustics as features of a 3D flow, and provides a systematic route to quantify corrections beyond the standard thin-lens approximation, enabling direct confrontation of CFT with current and forthcoming lensing data.