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

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  • Calcagni, Gianluca
  • Nardelli, Giuseppe

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.

Suggested Citation

  • Calcagni, Gianluca & Nardelli, Giuseppe, 2025. "Representations of the fractional d’Alembertian and initial conditions in fractional dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 201(P3).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p3:s0960077925014146
    DOI: 10.1016/j.chaos.2025.117401
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    References listed on IDEAS

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    1. Beghin, L. & Orsingher, E., 2005. "The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 1017-1040, June.
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