Scaling Analysis and Renormalisation Group for General (Quantum) Many Body Systems in the Critical Regime

With the help of a smooth scaling and coarse-graining approach of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) we perform a rigorous renormalisation group analysis of the critical regime. The approach is quite general, encompassing classical, quantum, discrete and continuous systems, the main thrust going to quantum many body systems. Our central topic is the analysis of the emergent properties of critical systems on the intermediate scales and in the scaling limit. To mention some particularly interesting points, we show that systems typically loose part of their quantum character in the scaling limit (vanishing of commutators) and we rigorously prove, with the help of the KMS-condition, the emergence of the phenomenon of critical slowing down together with the necessity of renormalising the time variable. These general features are then illustrated with the help of an instructive class of models and are related to the singular structure of quasi particle excitation modes for vanishing energy-momentum.