Representations of the fractional d’Alembertian and initial conditions in fractional dynamics

We construct representations of complex powers of the d'Alembertian operator square in Lorentzian signature and pinpoint one which is self-adjoint and suitable for classical and quantum fractional field theory. This self-adjoint fractional d'Alembertian is associated with complex-conjugate poles, which are removed from the physical spectrum via the Anselmi-Piva prescription. As an example of empty spectrum, we consider a purely fractional propagator and its Kallen-Lehmann representation. Using a cleaned-up version of the diffusion method, we formulate and solve the problem of initial conditions of the classical dynamics with a standard plus a fractional d'Alembertian, showing that the number of initial conditions is two. We generalize this result to a much wider class of nonlocal theories and discuss its applications to quantum gravity.