# 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.