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

Gianluca Calcagni and Giuseppe Nardelli

Chaos, Solitons & Fractals, 2025, vol. 201, issue P3

Abstract: We construct representations of complex powers of the d’Alembertian operator □ 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.

Keywords: Quantum gravity; Non-local dynamics; Perturbative quantum field theory; Cauchy problem; Initial conditions; Unitarity (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:201:y:2025:i:p3:s0960077925014146

DOI: 10.1016/j.chaos.2025.117401

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