Work

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

Year: 2025

Type: article

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.

Source: Chaos Solitons & Fractals

DOI: 10.1016/j.chaos.2025.117401

Authors: Gianluca Calcagni, Giuseppe Nardelli

Institutions: Instituto de Estructura de la Materia, Università Cattolica del Sacro Cuore, Istituto Nazionale di Fisica Nucleare, University of Trento, Istituto Nazionale di Fisica Nucleare, Trento Institute for Fundamental Physics And Applications

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,970

Funders: Ministerio de Ciencia, Innovación y Universidades, Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación