Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Complex Time Dynamics A Unified Framework for Quantum, Thermal, and Geometric Evolution

Y. Li

Published Mar 8, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18907318

Related papers

A Unified Framework for Quantum, Thermal, and Geometric Evolution: The Complex-Time Equation

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.

•Mar 10, 2026•0 citations

Complex Time Quantum Thermal Geometric Framework

Y. Li

We introduce an extension of the Complex-Time Quantum-Thermal-Geometric (CTQTG) framework to \textit{curved complex-time manifolds}, unifying quantum evolution, thermal effects, and entanglement-driven geometric dynamics. The evolution of the complex-time wavefunction $\Psi(\theta,x)$ is governed by the generalized CTQTG equation:\begin{equation}\boxed{\partial_\theta \Psi(\theta,x) = - H(\theta,x) \Psi(\theta,x) - \alpha \Delta_\theta^{(g)} \Psi(\theta,x)}\end{equation}where $\theta = \tau + i t/\hbar$ combines imaginary time $\tau$ (inverse temperature) and real time $t$, $\Delta_\theta^{(g)}$ is the Laplace-Beltrami operator on the curved complex-time manifold, and $\alpha$ quantifies the quantum-thermal-geometric coupling. Analytical solutions reveal entanglement wave propagation modulated by manifold curvature, and in the strong-coupling limit ($\alpha \gg 1$) the system exhibits emergent spacetime structures. This framework provides a unified approach to explore holographic correspondences, thermally-modulated quantum interference, and the dynamical generation of spacetime geometry from entanglement patterns.

•Mar 8, 2026•0 citations

Complex-Time Quantum Thermodynamic Geometry(CTQTG)

Y. Li

The foundations of modern theoretical physics rest on the interplay between quantum dynamics, thermodynamics, and spacetime geometry. While quantum mechanics describes microscopic dynamics through unitary time evolution, thermodynamic phenomena introduce irreversibility and entropy production. In this work we propose a unified framework in which quantum evolution is formulated on a complex-time manifold that naturally incorporates both dynamical and thermodynamic structures. We introduce a complex-time coordinate combining physical time and thermal flow and derive a generalized evolution equation from a variational principle defined on a complex-time geometric manifold. The resulting equation contains both Hamiltonian dynamics and geometric diffusion terms governed by the Laplace–Beltrami operator. This formulation provides a geometric interpretation of thermodynamic irreversibility within quantum theory and establishes connections with Euclidean quantum field theory, geometric quantum mechanics, and thermal time hypotheses. Possible implications for quantum statistical mechanics and information flow in gravitational systems are discussed.

•Mar 10, 2026•0 citations

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

Y. Li

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.

•Mar 13, 2026•0 citations

Complex Time Quantum Thermal Geometry: A Unified Holomorphic Framework for Dynamics and Thermodynamics

Y. Li

We present a first-principles derivation of a master equation that unifies quantum dynamics, thermal effects, and Riemannian geometry within a holomorphic framework. By extending the temporal parameter into the complex domain $\theta = t + i\frac{\hbar}{2k_B T}$, we demonstrate that the KMS condition in thermal field theory necessitates a specific form of evolution. We correct previous dimensional inconsistencies and derive the unique coupling constant $\alpha = \lambda \sqrt{\hbar G / c}$, which links the Laplace-Beltrami operator to complex time evolution. The resulting Complex Time Quantum Thermal Geometric (CTQTG) equation predicts gravity-induced decoherence and a fundamental ultraviolet regularization of momentum states at finite temperatures.

•Mar 13, 2026•0 citations

A Rigorous Derivation of the Complex-Time Quantum-Thermal-Geometric Equation via Gradient Flow of a Free Energy Functional

Y. Li

We present a rigorous derivation of the Complex-Time Quantum-Thermal-Geometric (CTQTG) equation,\begin{equation}\partial_{\theta}\Psi(\theta) = -H(\theta)\Psi(\theta) - \alpha\,\Delta_{\theta}\Psi(\theta),\label{eq:CTQTG}\end{equation}where \(\theta = \tau + i t/\hbar\) is a complex time coordinate, \(H(\theta)\) a Hamiltonian,\(\Delta_{\theta}\) the Laplace–Beltrami operator on the complex-time manifold, and \(\alpha>0\) a diffusion constant. Starting from a free energy functional that combines a Fisher–Rao type information term with an energy expectation, we show that equation~\eqref{eq:CTQTG} emerges as the \(L^2\)-gradient flow of this functional. The derivation is self-contained, relies only on standard variational calculus and the theory of gradient flows in Hilbert spaces, and provides a clear information–geometric interpretation of the coupling between quantum evolution, thermal diffusion, and geometric feedback.

•Mar 22, 2026•0 citations
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 θ=τ+it/ℏ\theta = \tau + i t / \hbarθ=τ+it/ℏ, where τ\tauτ represents imaginary (thermal) time and ttt real (unitary) time, we derive the governing equation for the quantum-thermal-geometric state Ψ(θ)\Psi(\theta)Ψ(θ): \begin{equation}\partial_\theta \Psi(\theta) = - H(\theta) \Psi(\theta) - \alpha , \Delta_\theta \Psi(\theta),\end{equation} where H(θ)H(\theta)H(θ) is the (possibly complex-time dependent) Hamiltonian, Δθ\Delta_\thetaΔθ​ is the Laplace-Beltrami operator on the complex-time manifold, and α\alphaα 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.
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 θ=τ+it/ℏ\theta = \tau + i t/\hbarθ=τ+it/ℏ, where τ\tauτ represents imaginary (thermal) time and ttt represents real (unitary) time, we derive the governing equation for the quantum-thermal-geometric state Ψ(θ)\Psi(\theta)Ψ(θ):\begin{equation}\partial_{\theta} \Psi(\theta) = - H(\theta) \Psi(\theta) - \alpha \Delta_{\theta} \Psi(\theta), \label{eq:CTQTG}\end{equation}where H(θ)H(\theta)H(θ) is the (potentially complex-time dependent) Hamiltonian, Δθ\Delta_{\theta}Δθ​ is the Laplace-Beltrami operator on the complex-time manifold, and α\alphaα 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.
We introduce an extension of the Complex-Time Quantum-Thermal-Geometric (CTQTG) framework to \textit{curved complex-time manifolds}, unifying quantum evolution, thermal effects, and entanglement-driven geometric dynamics. The evolution of the complex-time wavefunction Ψ(θ,x)\Psi(\theta,x)Ψ(θ,x) is governed by the generalized CTQTG equation:\begin{equation}\boxed{\partial_\theta \Psi(\theta,x) = - H(\theta,x) \Psi(\theta,x) - \alpha \Delta_\theta^{(g)} \Psi(\theta,x)}\end{equation}where θ=τ+it/ℏ\theta = \tau + i t/\hbarθ=τ+it/ℏ combines imaginary time τ\tauτ (inverse temperature) and real time ttt, Δθ(g)\Delta_\theta^{(g)}Δθ(g)​ is the Laplace-Beltrami operator on the curved complex-time manifold, and α\alphaα quantifies the quantum-thermal-geometric coupling. Analytical solutions reveal entanglement wave propagation modulated by manifold curvature, and in the strong-coupling limit (α≫1\alpha \gg 1α≫1) the system exhibits emergent spacetime structures. This framework provides a unified approach to explore holographic correspondences, thermally-modulated quantum interference, and the dynamical generation of spacetime geometry from entanglement patterns.
The foundations of modern theoretical physics rest on the interplay between quantum dynamics, thermodynamics, and spacetime geometry. While quantum mechanics describes microscopic dynamics through unitary time evolution, thermodynamic phenomena introduce irreversibility and entropy production. In this work we propose a unified framework in which quantum evolution is formulated on a complex-time manifold that naturally incorporates both dynamical and thermodynamic structures. We introduce a complex-time coordinate combining physical time and thermal flow and derive a generalized evolution equation from a variational principle defined on a complex-time geometric manifold. The resulting equation contains both Hamiltonian dynamics and geometric diffusion terms governed by the Laplace–Beltrami operator. This formulation provides a geometric interpretation of thermodynamic irreversibility within quantum theory and establishes connections with Euclidean quantum field theory, geometric quantum mechanics, and thermal time hypotheses. Possible implications for quantum statistical mechanics and information flow in gravitational systems are discussed.
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 Ψ\PsiΨ is a temperature-dependent effective wave function in the sense of thermal field theory, HHH is the Hamiltonian, Δg\Delta_gΔg​ is the Laplace-Beltrami operator on a Riemannian manifold, and λ\lambdaλ is an order-one dimensionless constant uniquely fixed by the Planck scale. The complex time variable θ\thetaθ 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 λ\lambdaλ 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.
We present a first-principles derivation of a master equation that unifies quantum dynamics, thermal effects, and Riemannian geometry within a holomorphic framework. By extending the temporal parameter into the complex domain θ=t+iℏ2kBT\theta = t + i\frac{\hbar}{2k_B T}θ=t+i2kB​Tℏ​, we demonstrate that the KMS condition in thermal field theory necessitates a specific form of evolution. We correct previous dimensional inconsistencies and derive the unique coupling constant α=λℏG/c\alpha = \lambda \sqrt{\hbar G / c}α=λℏG/c​, which links the Laplace-Beltrami operator to complex time evolution. The resulting Complex Time Quantum Thermal Geometric (CTQTG) equation predicts gravity-induced decoherence and a fundamental ultraviolet regularization of momentum states at finite temperatures.
We present a rigorous derivation of the Complex-Time Quantum-Thermal-Geometric (CTQTG) equation,\begin{equation}\partial_{\theta}\Psi(\theta) = -H(\theta)\Psi(\theta) - \alpha\,\Delta_{\theta}\Psi(\theta),\label{eq:CTQTG}\end{equation}where θ=τ+it/ℏ\theta = \tau + i t/\hbarθ=τ+it/ℏ is a complex time coordinate, H(θ)H(\theta)H(θ) a Hamiltonian,Δθ\Delta_{\theta}Δθ​ the Laplace–Beltrami operator on the complex-time manifold, and α>0\alpha>0α>0 a diffusion constant. Starting from a free energy functional that combines a Fisher–Rao type information term with an energy expectation, we show that equation~\eqref{eq:CTQTG} emerges as the L2L^2L2-gradient flow of this functional. The derivation is self-contained, relies only on standard variational calculus and the theory of gradient flows in Hilbert spaces, and provides a clear information–geometric interpretation of the coupling between quantum evolution, thermal diffusion, and geometric feedback.