Fun And Games ]
The basics idea
There follows a simple description of the ERG. For a more technical discussion, click here.
The basic idea of the ERG is very familiar from everyday experiences: the way in which we describe the
world around us depends on the scale at which we observe it. For example, suppose that you were asked
to describe the world around you. Although it is always the same thing that it being
described, the description would be radically different depending on whether you were
For example, if looking at the Earth from space, it would be described in terms of oceans and
continents; but looking through a microscope the description would be in terms of very different things.
In more technical language, the degrees of freedom relevant to different scales change with the scale.
- simply looking around yourself,
- looking through a microscope,
- looking through a telescope,
Separating out what is relevant at one scale from what is relevant at a different scale
is a very useful way of decomposing hard problems into easier problems. The mathemetical
formalism that allows us to do this is the ERG.
The Exact Renormalization Group (ERG) is essentially the continuous
version of Wilson's RG. Thus, having defined a bare Lagrangian at
the bare scale, we integrate out degrees of freedom between this
scale and a lower, effective scale. This is done in such a way that
the partition function (and hence the physics) is unchanged. The Exact
Renormalization Group Equation (ERGE) states how the effective action
changes with the effective scale. This equation is defined nonperturbatively
and its solution would amount to the solution of the field theory in question.
Needless to say, it is only known how to do this in very special circumstances!
The ERG provides a very powerful framework for
dealing with renormalization and possesses a number of particular
- Renormalizable theories follow directly from fixed points of the ERGE (so long as all
variables have been appropriately rescaled using the effective cutoff).
- Scale dependent renormalizable theories follow by constructing trajectories which,
in the neighbourhood of a fixed point, are spanned by the relevant and marginally
relevant directions. These trajectories are called Renormalized Trajectories (RTs).
At all points along a RT, the effective action can be written in self-similar
form meaning that it depends on the effective scale only through the aforementioned
couplings and the anomolous dimension of the field.
- The existence of RTs is preserved by any sensible approximation
scheme, meaning that approximations of the ERG equations preserve
the renormalizability of the theory.
- There are many ways to formulate the ERG; this flexibility has
been exploited to construct manifestly gauge invariant formulations
of gauge theories.
(For a FORM program which computes the two-loop beta function
without fixing the gauge, click here.)
Last modified: Thu Jan 10 2008