  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.
