Kyriaki-Evangelia Aslani
A Multivector Generator of Time Evolution in the Phase Plane
A geometric formulation of time evolution is proposed in which the time derivative is represented by a multivector generator acting on the phase plane within two-dimensional geometric (Clifford) algebra. Rather than extending time as an additional scalar or complex parameter, the approach focuses on the algebraic structure of the generator responsible for dynamical evolution. Using the isomorphism between linear phase-space generators and multivectors in Cl(2,0), the time evolution operator is shown to admit a natural decomposition into scalar, vector, and bivector components, each associated with a distinct geometric action. The bivector part generates symplectic, Hamiltonian evolution and preserves the oriented phase-space area, while the scalar part produces uniform contraction or expansion and provides a direct geometric characterization of irreversibility. Vector components generate reversible but anti-symplectic transformations, corresponding to reflections that reverse the orientation of the symplectic form while preserving its magnitude. The general solution of linear systems follows from the exponential of the multivector generator and admits a transparent geometric interpretation. The reversible sector is naturally classified into elliptic, hyperbolic, and nilpotent cases, yielding oscillatory, overdamped, and critically damped behavior without invoking ad hoc assumptions. Within this framework, complex time emerges only as a restricted case corresponding to the even subalgebra; in the general setting, time evolution is intrinsically multivectorial. The proposed formulation provides a unified geometric language for reversible and irreversible dynamics and highlights the role of generator structure, rather than time itself, in shaping temporal evolution.
