The Wetterich equation is an exact functional differential equation for the effective average action 
, a scale-dependent quantum effective action in which
low-momentum fluctuations are suppressed (Wetterich 1993). It describes a flow from
an ultraviolet description at high momentum and short distance to an infrared description
at low momentum and long distance. In standard notation, it is
![(partialGamma_k[Phi])/(partialt)=1/2STr[(Gamma_k^((2))[Phi]+R_k)^(-1)(partialR_k)/(partialt)],](/images/equations/WetterichEquation/NumberedEquation1.svg) |
where 
is the logarithmic scale, 
is the field-dependent Hessian
formed by two functional derivatives, and
is an infrared regulator. The supertrace 
takes the trace over bosonic components and subtracts the
trace over fermionic components.
The infrared regulator is a momentum-dependent quadratic term added to 
. It suppresses modes with momenta small compared
with 
while leaving modes with momenta large compared with 
essentially unchanged. Starting with 
near the microscopic action at a large ultraviolet scale,
lowering 
incorporates fluctuations one momentum scale at a time. As 
, the regulator vanishes and 
becomes the full quantum effective action. Although
the right-hand side has the form of a one-loop trace, the equation is exact because
it contains the full field-dependent Hessian. Repeatedly
taking functional derivatives produces coupled
flow equations for correlation functions, which may be approximated by truncating
the allowed form of 
.
