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# Research Interests

## The ERG

### 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
• simply looking around yourself,

• looking through a microscope,

• looking through a telescope,

• etc...
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.

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.

### Technical Description

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 assets:
• 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.)
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