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Continuous versus discrete free Hamiltonian systems

The Euclidean group contains two models of free Hamiltonian evolution: one has a continuous configuration space in which the wavefunctions obey the Helmholtz equation, and require two initial conditions: initial values and initial velocities; the other is based on a discrete position space where the wavefunctions obey a difference equation, and its evolution requires only initial values. Yet the two models are unitarily equivalent. We find that the two initial conditions of the former correspond, according to their parity, with the initial condition of the latter at alternate points, either even or odd.