Multivector Time Generator: A Novel Algebraic Framework for Temporal Symmetries

In this presentation, a multivector time generator is introduced in which temporal evolution is represented as the action of a multivector on the phase plane, formulated within two-dimensional geometric (Clifford) algebra. Rather than extending time as an additional scalar or complex parameter, the proposed approach reinterprets the time derivative itself as a structured generator whose scalar, vector, and bivector components encode distinct temporal symmetries. Within this framework, bivector components generate symplectic, Hamiltonian evolution and correspond to reversible dynamics, while scalar components produce uniform contraction or expansion in the phase plane, providing a direct geometric characterization of irreversibility. Vector components generate reversible but anti-symplectic transformations, such as reflections, revealing temporal symmetries that lie outside the standard Hamiltonian and complex-time formalisms. This decomposition yields a natural classification of linear dynamical systems into elliptic, hyperbolic, and nilpotent regimes, corresponding to oscillatory, overdamped, and critically damped behavior. General solutions are expressed through multivector exponentials, allowing reversible and irreversible contributions to factorize transparently. The framework further clarifies the status of complex time, showing that it emerges only as a restricted case when vector components vanish and the generator reduces to an even subalgebra admitting a Wirtinger representation. Outside this regime, temporal evolution is intrinsically a multivector and cannot be described by a single complex parameter. By focusing on the algebraic structure of the time generator rather than on time as a primitive parameter, the proposed framework provides a unified geometric language for temporal symmetries in dynamical systems and offers new tools for analyzing reversible and irreversible evolution within a purely algebraic setting.