Perturbative unitarity of fractional field theories and gravity

TL;DR

This paper investigates the unitarity of fractional field theories and gravity with nonlocal kinetic terms, demonstrating conditions under which these theories remain consistent and unitary at all perturbative orders.

Contribution

It introduces a detailed analysis of the spectral properties and unitarity conditions of fractional kinetic operators in quantum field theories and gravity, including the effects of different self-adjoint extensions.

Findings

Standard mass singularity appears for >
in the spectrum.

Different self-adjoint fractional kinetic terms lead to the same physical spectrum for 0<<3.

Unitarity is preserved with specific prescriptions depending on  range.

Abstract

Motivated by quantum gravity on spacetimes with multi-scale geometry, we analyze quantum field theories with a self-adjoint fractional power $(\Box^2)^{\gamma/2}$ of the d'Alem\-bert\-ian in the kinetic term, for any real $\gamma>0$. Selecting a particularly simple version of the kinetic term which we call hermitian polynomial, we study the spectral decomposition of the propagator and, when $\gamma>1$, obtain the standard mass singularity $-k^2=m^2$. This is the only mode in the perturbative spectrum of asymptotic states, since the only other content of the theory is a cloud of purely virtual particles with complex masses. We also show that other versions of the self-adjoint fractional kinetic term lead to a different distribution of the virtual complex modes but to the same physical spectrum for $0<\gamma<3$, thus addressing the issue of uniqueness in this class of nonlocal theories. The non-hermitian version of the theory has the $-k^2=m^2$ particle plus a continuum of standard massive modes. Finally, we prove that unitarity of scalar, gauge and gravity models is respected at all perturbative orders if, in the hermitian cases, one adopts the fakeon prescription on scattering amplitudes or, in the non-hermitian case, $0<\gamma<1$ or $2<\gamma<3$ with the standard Feynman prescription. These results drastically simplify previous characterizations of fractional quantum gravity, which is super-renormalizable for $\gamma>2$.