Ultraviolet-complete quantum field theories with fractional operators

Abstract

We explore quantum field theories with fractional d'Alembertian ${\square }^{\gamma }$. Both a scalar field theory with a derivative-dependent potential and gauge theory are super-renormalizable for a fractional power $1<\gamma \le 2$, one-loop super-renormalizable for $\gamma >2$ and finite if one introduces killer operators. Unitarity is achieved by splitting the kinetic term into the product of massive fractional operators, eventually sending the masses to zero if so desired. Fractional quantum gravity is also discussed and found to be super-renormalizable for $2<\gamma \le 4$ and one-loop super-renormalizable for $\gamma >4$. To make it unitary, we combine the splitting procedure with a fractional generalization of the Anselmi-Piva procedure for fakeons. Among new technical results with wider applications, we highlight the Leibniz rule for arbitrary powers of the d'Alembertian and the Källén-Lehmann representation for a propagator with an arbitrary number of branch cuts.

86 pages (38 + appendices), 11 figures; v2: some material moved to appendices and streamlined, introduction shortened, motivation and discussions improved and expanded, proof of BPHZ renormalizability included

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• Classicized dynamics and initial conditions in field theories with fakeons
• Noether's currents for conformable fractional scalar field theories
• Discrete time heat kernel and UV modified propagators with Dimensional Deconstruction
• Fractional Einstein field equations in $2+1$ dimensional spacetime
• Fractional Particle and Sigma Model
• Gravitational Foundations and Exact Solutions in $n$-Dimensional Fractional Cosmology
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• Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness
• Quantum gravity with purely virtual particles from asymptotically local quantum field theory
• Inflation and Primordial Perturbations in Fractal Cosmology
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• Diffusion method in field theories with fakeons
• Cosmology of fractional gravity