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This paper formulates a new particle-in-cell method for the Vlasov-Maxwell system. Under the Lorenz gauge condition, Maxwell's equations for the electromagnetic fields can be written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a Hamiltonian formulation for particles that employs the generalized momentum. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. An analytical method for constructing these spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. Moreover, these derivatives are shown to converge at the same rate as the fields in the both time and space. The field solver we consider in this work is first-order accurate in time and fifth-order accurate in space and belongs to a larger class of methods which are unconditionally stable, can address geometry, and leverage fast summation methods for efficiency. We demonstrate the method on several well-established benchmark problems, and the efficacy of the proposed formulation is demonstrated through a comparison with standard methods presented in the literature. The new method shows mesh-independent numerical heating properties even in cases where the plasma Debye length is close to the grid spacing. The use of high-order spatial approximations in the new method means that fewer grid points are required in order to achieve a fixed accuracy. Our results also suggest that the new method can be used with fewer simulation particles per cell compared to standard explicit methods, which permits further computational savings.