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

We construct representations of complex powers of the d’Alembertian operator $\mathrm{\square }$ 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 Källén–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.