We present a fermion model characterized by an anticommuting-parameter shift symmetry. The Hamiltonian formulation exhibits a combination of first-class and second-class constraints. We derive the well-known Dirac equation by fixing the gauge in a covariant manner, enabling the fields to propagate accordingly. Notably, the model inherently possesses invariance under reparametrizations of time. Consequently, the Hamiltonian vanishes, setting it apart from the conventional framework of Dirac's theory. Furthermore, we establish a correspondence between these particles and the zero energy modes of the massless Rarita-Schwinger system, bringing forth the intriguing implication that they may describe a supergravity ground state.