Emergent Spacetime from Geometric Condensate with Complex Time

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.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, 20260 citations

Complexified Spacetime

B. G. Sidharth

Foundations of Physics LettersFeb 1, 200314 citations

Chapter 4 Complex Spacetime

North-Holland mathematics studiesJan 1, 19900 citations

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

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.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.

Mar 8, 20260 citations

\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.

\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.