  We construct exact functional renormalization group (RG) flow equations for
non-relativistic fermions in arbitrary dimensions, taking into account not only
mode elimination but also the rescaling of the momenta, frequencies and the
fermionic fields. The complete RG flow of all relevant, marginal and irrelevant
couplings can be described by a system of coupled flow equations for the
irreducible n-point vertices. Introducing suitable dimensionless variables, we
obtain flow equations for generalized scaling functions which are continuous
functions of the flow parameter, even if we consider quantities which are
dominated by momenta close to the Fermi surface, such as the density-density
correlation function at long wavelengths. We also show how the problem of
constructing the renormalized Fermi surface can be reduced to the problem of
finding the RG fixed point of the irreducible two-point vertex at vanishing
momentum and frequency. We argue that only if the degrees of freedom are
properly rescaled it is possible to reach scale-invariant non-Fermi liquid
fixed points within a truncation of the exact RG flow equations.
