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The Many Scales of Quantum Fields and Gravity https://doi.org/10.17863/CAM.96676 Melo, João Santos, Jorge It is a basic fact that Nature behaves differently at different scales. What happens at the human scale looks rather different to the motion of planets or the dynamics of subatomic particles. Rather remarkably, understanding this seemingly elementary phenomenon led to one of the most profound paradigm shifts in Physics and to the most accurate agreement between theory and experiment in Science. All thanks to the powerful tool of the Renormalisation Group. However, in the presence of gravity this becomes rather challenging as scales become outputs of the dynamics rather than inputs. This thesis covers several studies regarding this intersection, trying to understand the interplay between scales, quantum fields, and gravity. We begin with an overview of renormalisation techniques to set the stage for the rest of the discussion. We then delve into the AdS/CFT correspondence, first studying a local version of the renormalisation group and testing the quantum renormalisation group proposal. After that we analyse the surprising implications conformal and supersymmetry on the field theory side have on string theoretic corrections for its gravitational dual. Finally we explore the world of finite temperature quantum field theory and the subtleties behind correctly setting initial conditions. University of Cambridge 2023-05-01 Thesis Doctoral Doctor of Philosophy (PhD) eng Cambridge Trust - Vice Chancellor's Award Fundação para Ciência e Tecnologia - PhD Studentship https://www.repository.cam.ac.uk/handle/1810/349933 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/dac0b1aa-b25c-418a-bbfb-0d9a8c1e3832/download 1c55a1ee6b37c53c561f905e4bcfb423 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/09a906ad-96f1-441f-8098-e3c5c5cdfca3/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ AdS/CFT Correspondence Quantum Field Theory