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Complex Time Quantum Thermal Geometry A First-Principles Derivation of Unified Dynamics

Y. Li

Published Mar 13, 2026• 0 citations
Publisher

Abstract

We present a first-principles derivation of a master equation that unifies quantum dynamics, thermal effects, and Riemannian geometry within a holomorphic framework based on complex time. Starting from five fundamental axioms—complex-time holomorphy, geometric consistency, probability conservation, dimensional completeness, and spectral stability—we derive the Complex Time Quantum Thermal Geometric (CTQTG) equation: \begin{equation}\frac{\partial \Psi}{\partial \theta} = -\frac{i}{\hbar}H\Psi - i\lambda \frac{\hbar G}{c^2} \Delta_g \Psi, \qquad \theta = t + i\frac{\hbar}{2k_B T},\end{equation} where $\Psi$ is a temperature-dependent effective wave function in the sense of thermal field theory, $H$ is the Hamiltonian, $\Delta_g$ is the Laplace-Beltrami operator on a Riemannian manifold, and $\lambda$ is an order-one dimensionless constant uniquely fixed by the Planck scale. The complex time variable $\theta$ unifies real-time unitary evolution with imaginary-time thermal relaxation, reflecting the KMS condition of thermal field theory. We prove that the equation preserves a conserved norm, reduces to the Schr\"odinger equation in the flat-space zero-temperature limit, and is globally well-posed on compact manifolds. Exact analytic solutions are constructed, including plane waves and Gaussian wave packets, revealing gravity-induced modifications to quantum dynamics such as temperature-dependent wave-packet broadening and thermal suppression of high-momentum modes. The equation contains no free parameters beyond $\lambda$ and offers a mathematically consistent framework for exploring the interface between quantum mechanics, general relativity, and thermodynamics. We discuss testable predictions in matter-wave interferometry and ultra-cold atom systems.

DOI
10.5281/zenodo.18994147

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