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

TL;DR

This paper develops representations of the fractional d'Alembertian operator suitable for classical and quantum fractional field theories, clarifies initial condition requirements, and discusses implications for nonlocal theories and quantum gravity.

Contribution

It constructs self-adjoint fractional d'Alembertian operators with complex poles and analyzes initial conditions in fractional dynamics, extending to nonlocal theories.

Findings

Self-adjoint fractional d'Alembertian with complex-conjugate poles

Two initial conditions needed for classical fractional dynamics

Application to quantum gravity models

Abstract

We construct representations of complex powers of the d'Alembertian operator $\Box$ 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 K\"all\'en--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.