Complex Time Quantum Thermal Geometric Dynamics

We present a comprehensive formulation of Complex-Time Quantum--Thermal--Geometric Dynamics (CTQTG), a first-principles framework unifying quantum mechanics, thermal physics, and emergent spacetime geometry through analytic continuation in the time parameter. The central dynamical equation is given by{equation}∂θφ(θ, x) = -δ H[φ]/δ φ(x) - α Δθ⁽ᵍ⁾φ(θ, x),{equation}where \(θ = τ + i t/\) defines a complex temporal coordinate, \(α\) is a fundamental coupling constant governing geometric diffusion, and \(Δθ⁽ᵍ⁾\) denotes the Laplace--Beltrami operator on a curved complex-time manifold. We extend this construction from quantum mechanics to quantum field theory, demonstrating that thermal field theory emerges naturally from compactification of the imaginary-time dimension, while gravitational dynamics arise from the backreaction of quantum fields on the complex-time geometry. The path integral formulation reveals the deep connection between complex-time evolution and the statistical mechanics of quantum fields. The formalism yields testable predictions for low-energy quantum systems and provides a pathway toward a unified description of quantum phenomena, thermodynamics, and spacetime geometry.