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We derive renormalised finite functional flow equations for quantum fieldtheories in real and imaginary time that incorporate scale transformations ofthe renormalisation conditions, hence implementing a flowing renormalisation.The flows are manifestly finite in general non-perturbative truncation schemesalso for regularisation schemes that do not implement an infrared suppressionof the loops in the flow. Specifically, this formulation includes finitefunctional flows for the effective action with a spectral Callan-Symanzikcutoff, and therefore gives access to Lorentz invariant spectral flows. Thefunctional setup is fully non-perturbative and allows for the spectraltreatment of general theories. In particular, this includes theories that donot admit a perturbative renormalisation such as asymptotically safe theories.Finally, the application of the Lorentz invariant spectral functionalrenormalisation group is briefly discussed for theories ranging from realscalar and Yukawa theories to gauge theories and quantum gravity.

Renormalised spectral flows