Gravitationally induced decoherence of a scalar field: investigating the one-particle sector and its interplay with renormalisation

We investigate the one-particle sector for the field-theoretical model of gravitationally induced decoherence for a scalar field in Fahn et al (2023 Class. Quantum Grav.40 094002) with a special focus on the renormalisation of the one-particle master equation. In contrast to existing models in the literature, where the renormalisation is usually performed after the Markov and rotating wave approximation (RWA) and often only for certain limits such as the non- or ultra-relativistic limit, here we apply the renormalisation directly after the one-particle projection. With this strategy, we show that UV-divergent contributions in the one-particle master equation can be identified with the vacuum contributions in the self-energy of the scalar field in the effective quantum field theory and depending on the chosen one-particle projection method, its vacuum bubbles, while the additional thermal contributions in the self-energy are all UV-finite. To obtain the renormalised one-particle master equation, we use an on-shell renormalisation procedure of the underlying effective QFT. We then apply the Markov and RWA, specifying a condition under which the Markov approximation can be applied in the case of the ultra-relativistic limit. We compare our results with those available in the literature. This includes an analysis of two different kinds of one-particle projections, a comparison of the application and effects of renormalisation of quantum mechanical and field theoretical models, the non-relativistic and ultra-relativistic limits of the renormalised one-particle master equations, and a comparison with a quantum mechanical toy model for gravitationally induced decoherence in the context of neutrino oscillations.