1. It was shown that correlation functions defined at fixed geodesic distance can be renormalised by treating the geodesic embedding coordinates as fields, and performing a wave function renormalisation for them. A scalar field correlation function at fixed geodesic distance was computed in flat space, including one-loop graviton corrections, with the final result being gauge-invariant but dependent on the finite parts of counterterms even for separated points.
2. It was further shown that correlation functions using a field-dependent coordinate system are renormalisable using the usual local counterterms in flat space. A scalar field correlation function was computed including one-loop graviton corrections, and the final result was gauge-invariant and uniquely defined for non-coincident points.
3. The field-dependent coordinate system was generalised to cosmological (Friedmann-Lemaître-Robertson-Walker) spacetimes, and the graviton propagator in a gauge adapted to this coordinate system was determined.
4. A non-local observable corresponding to the local expansion rate was constructed, and its one-loop expectation value was computed for spacetimes of physical interest: matter- and radiation-dominated eras, and slow-roll inflation. For slow-roll inflation, the one-loop quantum corrections lead to a diminution of the slow-roll parameters; in the matter-dominated era a secular decrease of the expansion rate was found, and for radiation domination the result vanishes at one-loop order.
5. Ward identities for general gauge theories (including gauge theories with open gauge algebras) were studied in detail in algebraic quantum field theory, and a gauge-invariant quantum state was constructed to all orders in perturbation theory. These were used to show the renormalisability of the observable mentioned in Point 4 to all orders using the usual local counterterms.
6. Quantum corrections to the Newtonian potential for spinning particles have been computed in flat space and de Sitter space, and it was found that there exists a term (proportional to the Hubble constant squared) that grows logarithmically with the distance to the particle. When resummed, this term leads to a slower fall-off of the gravitational force with distance in de Sitter space (with respect to flat space).
7. It was shown that in cosmological spacetimes local measurements of the graviton are equivalent to local measurements of the linearised Weyl tensor and another local infrared-convergent tensor, or more technically, that that graviton field smeared with a compactly supported test function is equal to a smearing of the linearised Weyl tensor and this other tensor with a compactly supported test function determined in a precise way from the first.
8. It was shown that the two tensors mentioned in Point 7, together with the linearised Einstein tensor, form a complete set of local observables at linear order in cosmological spacetimes.
The above results have been published in 10 research articles in high-impact journals, and presented in 22 conferences, workshops and seminars. They have already attracted the attention of eminent researchers in the field.
