Fixed point

Fixed point (Mathematics)

The naming convention in Scholarpedia is that fixed points are for mappings \[x_{n+1} = f(x_n)\] (\(x\) is a fixed point if \(x=f(x)\)), whereas equilibria are for flows (ODEs) \[x'=f(x)\] (\(x\) is an equilibrium point if \(f(x)=0\)).

Fixed point (Quantum and Statistical Field Theory)

In the context of QSFT, the renormalization group equation for a field theory on a discretized momentum or position space (e.g. block-spin transformations), is --mathematically speaking-- a non-linear functional mapping and the use of the term fixed point is consistent with the mathematical terminology (as defined above).

Nevertheless, it is widely diffused in QFT community to call fixed point the equilibrium points of the renormalization group flow equation for a field theory on a continuous momentum or position space, that is a non-linear functional flow equation in terms of a regulator (infra red cut-off) of the theory. The term fixed point is also used for the equilibrium points of the flow of the renormalized parameters arising from the renormalization group (linear) PDEs for the correlators of the renormalized theory.