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Renormalisation and momentum dependence in Quantum Gravity

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Renormalisation in curved spacetimes is an involved subject. In contrast to renormalisation in a flat spacetime, the standard momentum representation is not directly available. Nevertheless, the momentum dependence of correlation functions is crucial to deciding whether a theory is unitary and causal. I will discuss how to define a notion of momentum dependence in gravity on a fundamental level. With this at hand, one can discuss an important quantum field theory observable: scattering cross sections. Taking the example of gravity-mediated scalar scattering, I will discuss conditions that a quantum field theory of gravity has to fulfil to have a well-behaved scattering amplitude. These can be satisfied without the introduction of massive higher spin modes as is done in string theory. Finally, I will review the status of first principle calculations of the non-perturbative momentum dependence of quantum gravity correlation functions.