Complex-Time Quantum–Thermal–Geometric Dynamics and Its Extension to Quantum Field Theory

We propose a unified theoretical framework, Complex-Time Quantum–Thermal–Geometric Dynamics (CTQTG), in which quantum mechanics, thermal physics, and emergent geometry are integrated into a single evolution governed by complex time. The fundamental evolution equation reads{equation}∂_θ φ(θ,x) = -δ H[φ]/δ φ(x) - α Δ_θ⁽ᵍ⁾ φ(θ,x),{equation}where θ = τ + i t / is the complex time, α is a geometric diffusion coupling, and Δ_θ⁽ᵍ⁾ is the Laplace-Beltrami operator on a curved complex-time manifold. We generalize this equation from quantum mechanics to quantum field theory, demonstrating that thermal effects arise naturally from imaginary-time periodicity, and that emergent geometry arises as a low-energy effective consequence of quantum fluctuations. This formalism provides a first-principles pathway to unify quantum mechanics, thermodynamics, and gravity within a single complex-time framework.