The pseudodifferential operator square root of the Klein–Gordon equation

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Abstract
A nonlocal square root of the Klein–Gordon equation is proposed. This nonlocal equation is a special relativistic equation for a scalar field of first order in the time derivative. Its space derivative part is described by a pseudodifferential operator. The usual quantum mechanical formalism can be set up. The nonrelativistic limit and the classical limit in the form of plane wave solutions and the Ehrenfest theorem are correctly included. The nonlocality of the wave equation does not disturb the light cone structure, and the relativity principle of special relativity is fulfilled. Uniqueness and existence of solutions of the Cauchy problem for this equation can be proved. The second quantized version of this theory turns out to be macrocausal.

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