Dynamical Emergence of Lorentzian Spacetime from a Non-Spatiotemporal Quantum Many-Body Condensate

Abstract We develop a fully background-free quantum many-body framework in which Lorentzian spacetime arises dynamically from a fundamentally non-spatiotemporal microscopic system. The fundamental degrees of freedom are quantum excitations carrying purely relational and algebraic data, with no presupposed manifold, metric, dimensionality, or causal structure, extending and unifying ideas explored in several approaches to emergent spacetime and quantum geometry. We show that in a large-, low-temperature regime the system undergoes a condensation transition into a collective hydrodynamic phase characterized by a macroscopic order parameter, analogous to condensation phenomena in quantum many-body systems but without geometric interpretation at the microscopic level. Fluctuations around this condensate are governed by an effective kinetic operator whose structure is determined by the spectral properties of microscopic correlation functions. A central result of this work is that, starting from a fully non-spatiotemporal microscopic theory, the sign structure of the condensate Hessian dynamically selects a single time-like direction, yielding a Lorentzian signature in the long-wavelength limit. This mechanism does not rely on fine-tuning, imposed causal conditions, or background spacetime assumptions, and is robust under coarse-graining, reflecting universality properties familiar from renormalization-group analyses. Beyond the emergence of Lorentzian signature, the framework naturally gives rise to additional features associated with continuum spacetime physics, including effective locality, renormalization-group flow of geometric observables, and a controlled effective field theory description at long wavelengths. While these aspects are developed here to the extent necessary to establish internal consistency, the emphasis of the present paper is on the dynamical origin of spacetime structure itself rather than on detailed phenomenological applications. This work establishes a dynamically complete route from non-spatiotemporal quantum many-body dynamics to Lorentzian spacetime and effective continuum physics, providing a unified foundation for further investigations of emergent geometry, matter, and cosmology.