Multivector Time Generator of Evolution in Phase Plane

The nature of time and its role in physical evolution remain central open questions in theoretical physics, particularly in the presence of irreversibility. In this study, a geometric framework for time evolution is introduced based on a multivector time generator acting on the phase plane. Rather than extending time as a parameter, this approach focuses on the structure of the time derivative and its associated symmetries. Using geometric algebra, the generator decomposes naturally into scalar, bivector, and vector components. The bivector part generates Hamiltonian, symplectic evolution and corresponds to reversible dynamics, while the scalar part produces uniform contraction or expansion and provides a geometric interpretation of irreversibility and entropy production. In addition, vector components generate reversible but anti-symplectic transformations, such as reflections, revealing symmetries that are not captured by the standard Hamiltonian or complex-time approaches. The general solutions of linear systems follow an exponential form, and the reversible generator admits a natural classification into elliptic, hyperbolic, and nilpotent cases, yielding a clear geometric interpretation of oscillatory, overdamped, and critically damped behavior. The framework further clarifies the status of complex time, showing that it arises only as a restricted case when vector components are absent, and the time derivative admits a Wirtinger representation. Outside this regime, the time evolution is a multivector and cannot be described by a single complex parameter. Overall, the proposed framework provides a unified geometric language for analyzing reversible and irreversible dynamics, and highlights the central role of the time generator in shaping temporal evolution.