Complex-Time Quantum-Thermal-Geometric (CTQTQ) Equation

We propose a novel fundamental dynamical equation named the Complex-Time Quantum-Thermal-Geometric (CTQTQ) Equation, which unifies unitary quantum dynamics, thermal relaxation and information geometry within a single complex-time formalism. By introducing a complex-time coordinate\[θ = τ + it/,\]where t is the real unitary time and τ is the imaginary thermal time, we derive the master equation for a quantum-thermal-geometric state Ψ(θ):\[ Ψ(θ) = -H(θ)Ψ(θ) - α\, Ψ(θ).\] Here H(θ) is the Hamiltonian, the Laplace–Beltrami operator on the complex-time manifold, and α a coupling constant controlling the strength of quantum-thermal-geometric interaction. We obtain exact analytic solutions for free and two-level systems, and find a new phenomenon: geometrically driven entanglement oscillations. In the limit α→ 0, the CTQTQ equation reduces to the Schrödinger equation; in the limit of vanishing quantum coherence, it reduces to the heat equation. We further apply the CTQTQ framework to the black hole information paradox: information is not lost, but evolves unitarily in the thermal-time branch of the complex-time plane. This resolves the apparent tension between unitarity and thermalization. The CTQTQ equation represents a new foundational structure at the interface of quantum mechanics, thermodynamics, geometry and gravity.