Nonunitarity problem in quantum gravity corrections to quantum field theory with Born-Oppenheimer approximation

The problem of time is one of the most relevant open issues in canonical quantum gravity. Although there is a huge literature about this topic, a commonly accepted solution has not been found yet. Here, we focus on the semiclassical approach to the problem of time, that has the main goal of reproducing quantum field theory on a fixed Wentzel-Kramers-Brillouin (WKB) background accounting also for quantum gravity corrections. We analyze the different choices of the expansion parameter and discuss the problems arising in previous proposals, where a nonunitary evolution emerges as an effect of quantum gravity corrections. In this work, we develop a new approach to solve this problem by performing the WKB expansion with the introduction of the so-called kinematical action as a clock for quantum matter, that allows to recover a unitary dynamics