Perturbative unitarity of fractional field theories and gravity
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 $\gamma>2$.