Space–Time from a Conserved State Vector

Abstract

This paper develops a foundational theory in which the geometry of spacetime and the dynamics of matter emerge from the evolution of a conserved real state vector, Ψ^μ, in an abstract four-dimensional internal space endowed with a Minkowski metric η_{μν}. The theory is constructed from two core axioms: the strict conservation of the state vector's Minkowski norm and the condition of anti-parallelism between its temporal and spatial components. We derive the minimal and covariant action principle consistent with these axioms, which takes the form of a worldline action for a relativistic particle, S = -m c ∫ √(-η_{μν} dΨ^μ dΨ^ν). We demonstrate that the equations of motion describe a Lorentz-rotation of Ψ^μ, with its components Ψ^μ ≡ (c t, x^i) directly identifiable as physical spacetime coordinates. This identification recovers standard relativistic mechanics, with mass m reinterpreted as the frequency of the state vector's internal oscillation. The framework provides a unified geometric interpretation where physical time, space, motion, and mass are seen as derived, phenomenological aspects of a more fundamental, conserved dynamics in state space. The formulation suggests a natural pathway toward a field-theoretic generalization where the spacetime metric emerges as an induced quantity from the gradients of the state vector field.