RenormalizationGroup

Renormalization Group (RG) and RG Flow

Renormalization group (RG) is a conceptual, mathematical tool that helps in understanding and analyzing the behavior of physical systems with many degrees of freedom, specifically at different scales. It's an invaluable tool for studying phenomena in statistical physics and quantum field theory.

Fundamental Concepts

1. Scale Invariance: Some systems appear similar at different scales. For example, if you zoom into a fractal, the smaller piece of the fractal can look very similar to the whole.
2. Fixed Points and Flow: These are fundamental to understanding RG. Fixed points are those points that don't change under a transformation. In RG, these transformations relate to changing the scale of observation. The way systems move under these transformations is known as the RG flow.

Renormalization Group

Let's consider a physical system characterized by a set of parameters (e.g., couplings in a quantum field theory) collected in a vector $g$ in a parameter space $G$.

The idea of the RG is to consider transformations on this parameter space that correspond to changes in the scale on which you are observing the system. Specifically, these transformations will rescale momenta (for field theories) or lengths (for statistical mechanics systems).

Mathematically, this is described by a transformation $T_s$ that depends on a scale factor $s$. That is, given a point $g$ in parameter space, we can find a new point $T_s(g)$ that corresponds to the same physics but observed at a different scale. This transformation is the RG transformation.

These transformations generate a flow in the space of the theories (or more formally in the space of couplings), the so-called RG flow. This flow can be described by a set of differential equations, the RG equations.

RG Flow

The RG Flow can be expressed by a set of equations in the form:

$$\frac{d}{dt} g_i = \beta_i(g),$$

where $t$ is the logarithm of the scale factor $s$, $\beta_i(g)$ are known as the beta functions, and the $g_i$ are the components of the parameter vector $g$. The beta functions essentially dictate how the couplings $g_i$ change under a change of scale. They are typically computed from the theory under consideration (e.g., the Quantum Field Theory).

In many cases of interest, there are points in parameter space $g*$ at which the beta functions vanish, i.e., $\beta_i(g*) = 0$ for all $i$. These points are known as fixed points of the RG flow. Near these points, the theory does not change under a change of scale and is said to be scale-invariant. These points play a crucial role in the understanding of critical phenomena and phase transitions.

The behavior of the system at different scales can be understood by following the flow generated by these equations starting from a point $g$ in parameter space, which typically corresponds to the physical system at the scale we initially observe.

Important Notes

It's important to note that the renormalization group does not apply just to theories with infinities. It's also a powerful tool to study the behavior of perfectly finite theories at different scales, which is particularly relevant for the understanding of critical phenomena in statistical mechanics.

Another point worth mentioning is that the renormalization group provides a link between different theories at different scales. By integrating out high energy (short distance) degrees of freedom, one can derive an effective low energy (large distance) theory, which can be radically different in nature from the original one. A typical example is Quantum Chromodynamics (QCD), which is a free theory at high energies and a confined theory at low energies.

In summary, the renormalization group is a powerful theoretical tool that allows us to understand how physical systems change under different scales. It has profound implications for both high-energy physics and statistical mechanics.