Quantum thermal equation

Jiřı́ J. Mareš, Jaroslav S̈esták

Y. Li

We introduce the \textbf{Complex-Time Quantum-Thermal-Geometric (CTQTG) Equation}, a novel framework unifying quantum dynamics, thermal diffusion, and information-geometric feedback within a single complex-time formalism. By defining the complex-time coordinate $\theta = \tau + i t / \hbar$, where $\tau$ represents imaginary (thermal) time and $t$ real (unitary) time, we derive the governing equation for the quantum-thermal-geometric state $\Psi(\theta)$: \begin{equation}\partial_\theta \Psi(\theta) = - H(\theta) \Psi(\theta) - \alpha \, \Delta_\theta \Psi(\theta),\end{equation} where $H(\theta)$ is the (possibly complex-time dependent) Hamiltonian, $\Delta_\theta$ is the Laplace-Beltrami operator on the complex-time manifold, and $\alpha$ is a coupling constant controlling the strength of quantum-thermal-geometric interaction. The CTQTG Equation generalizes standard Schr\"odinger and heat equations, naturally incorporates unitary evolution and thermalization, and predicts novel phenomena such as oscillatory entanglement-wave modes driven by geometric feedback. We present explicit analytic solutions for constant and perturbed Hamiltonians, discuss limiting behaviors, and highlight potential experimental and theoretical implications for emergent spacetime, entanglement dynamics, and black hole microstate fluctuations. This work establishes a new class of dynamical equations in which quantum, thermal, and geometric effects are inseparable, providing a foundational framework for future research in complex-time physics.

Y. Li

We present the Complex-Time Quantum-Thermal-Geometric (CTQTG) framework, a novel unification of quantum dynamics, thermal diffusion, and information-geometric feedback within a single complex-time formalism. By defining the complex-time coordinate $\theta = \tau + i t/\hbar$, where $\tau$ represents imaginary (thermal) time and $t$ represents real (unitary) time, we derive the governing equation for the quantum-thermal-geometric state $\Psi(\theta)$:\begin{equation}\partial_{\theta} \Psi(\theta) = - H(\theta) \Psi(\theta) - \alpha \Delta_{\theta} \Psi(\theta), \label{eq:CTQTG}\end{equation}where $H(\theta)$ is the (potentially complex-time dependent) Hamiltonian, $\Delta_{\theta}$ is the Laplace-Beltrami operator on the complex-time manifold, and $\alpha$ is a fundamental coupling constant controlling the strength of the quantum-thermal-geometric interaction. The CTQTG Equation generalizes the Schr\"odinger and heat equations, naturally incorporating unitary evolution and thermalization. It predicts novel phenomena such as oscillatory entanglement-wave modes driven by geometric feedback. We present explicit analytic solutions for constant and perturbed Hamiltonians, discuss limiting behaviors, and explore profound implications for emergent spacetime, entanglement dynamics, black hole microstate fluctuations, and the nature of fundamental laws. This work establishes a new class of dynamical equations in which quantum, thermal, and geometric effects are inseparable, providing a foundational framework for future research in complex-time physics.

Jaroslav S̈esták

Hong-Yi Fan, Yue Fan

Based on the thermal field dynamics, by introducing the thermal entangled state we find a new approach for converting density operators' quantum master equations into c-number equations. This approach, in many cases, is concise and convenient for deriving the solutions of these equations, because the thermal entangled states possess favourable properties.

F. C. Khanna, Adolfo P. C. Malbouisson, J. M. C. Malbouisson +1 more