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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.21485 |
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| _version_ | 1866909851542618112 |
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| author | Calcagni, Gianluca Nardelli, Giuseppe |
| author_facet | Calcagni, Gianluca Nardelli, Giuseppe |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of the fractional d'Alembertian and initial conditions in fractional dynamics Calcagni, Gianluca Nardelli, Giuseppe High Energy Physics - Theory General Relativity and Quantum Cosmology 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 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. |
| title | Representations of the fractional d'Alembertian and initial conditions in fractional dynamics |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2505.21485 |