Complex Time and the Unification of Quantum, Thermal, and Geometric Degrees of Freedom A First-Principles Derivation

We propose a novel framework in which time is extended to a complex coordinate on a Kähler manifold, thereby unifying unitary quantum evolution, thermal (imaginary-time) dynamics, and geometric degrees of freedom originating from quantum entanglement. Starting from first principles—namely, the principle of least action, general covariance on the complex time manifold, and the requirement of consistent reduction to known physics—we derive a set of coupled field equations. The matter sector is described by a holomorphic wavefunction $\Psi(z,\bar{z})$ on a complex time surface, while geometry is encoded in a Kähler metric $ds^2 = \Omega(z,\bar{z})\,dz d\bar{z}$. The total action combines the Einstein-Hilbert term in two dimensions with a matter action that naturally yields a modified wave equation. Variation leads to a Liouville equation coupling the conformal factor $\Omega$ to the stress tensor of $\Psi$, and a Laplace-type equation for $\Psi$ itself. In the limit of weak gravity and non-relativistic energies, the theory reproduces the Schrödinger equation; in the thermal (imaginary time) direction, it reduces to the Bloch equation for thermal density matrices. Moreover, the geometric sector encodes the entanglement entropy through the Ryu–Takayanagi formula, providing a dynamical mechanism for the emergence of spacetime from quantum information. We discuss exact solutions, perturbative expansions, and potential experimental signatures, as well as connections to holography and black hole thermodynamics.