The Riemannian Geometry of Order Parameter Spacetime — Field-Theoretic Foundations of Irreversible Dynamics

This paper proves that the irreversible equation of motion from the general framework is an inevitable consequence of the variation of the order parameter spacetime field theory. The unified action contains three fundamental fields—metric, gauge, and order parameter—without any dissipative terms. Variation of the metric field yields the Einstein field equations, variation of the gauge field yields the Yang–Mills equations, and variation of the order parameter field yields conservative dynamical equations. Dissipation emerges from the quantum fluctuations of the order parameter field: under the assumptions of a high-temperature environment and an Ohmic spectral density, the imaginary part of the effective action reduces to a covariant Rayleigh dissipation function, and the generalized Euler–Lagrange equation yields the geodesic equation with dissipation. The one-dimensional reduction exactly recovers the equation of the general framework. The metric field provides a geometric origin for the inertial resistance coefficient, and the fluctuation–dissipation theorem provides a quantum origin for the relaxation time. The dissipation strength is proved to be naturally proportional to the inertial mass. Finite temperature is revealed as the necessary condition for the transition from quantum fluctuations to classical dissipation. This paper recovers Postulates 1 and 3 of the general framework; Postulate 2 is left for future work. This paper builds upon the foundational framework established in [10.5281/zenodo.21325459].