Renormalization Group Techniques

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Renormalization Group Techniques#

Here we present a practical introduction to the key concepts renormalization group (RG) techniques in physics by way of a few examples, followed by a bit of philosophy.

The Essence#

The essence of RG techniques is to understand how the structure of a physical theory changes as we consider a system at different scales. I.e. at a microscopic scale, water concerns the dynamics of a bunch of particles, but if we scale out, we can describe many behaviors using the equations of fluid dynamics, for example, as embodied in the Navier-Stokes equations.

One possibility is that the theory looks very different at different scales. Even the appropriate degrees of freedom might be different – water molecules in one case, fluid density, velocity, and temperature in another. How a theory changes as we change the scale is called RG flow.

If the structure of the theory remains the same, but the parameters values change, then these are said to run. In field-theories, the parameters “couple” different fields together, and so are called “couplings”: in this context the term running coupling is often used.

Sometimes, one finds regions of a theory where the couplings run very slowly, and the same theory applies over a wide range of scales. In the extreme case, we may be able to change the scale while rescaling the parameters in such a way that the theory does not change. If we can find a point where this behavior is seen, then we have found a fixed point of the RG flow. At or near a fixed point, the theory will exhibit self-similarity. This is common near second-order phase transitions, an in these cases RG techniques really shine, providing accurate quantitative techniques for calculating properties of the system such as critical exponents.

Taken to an extreme, one might demand that a fundamental theory be well defined at all scales. This was a common strategy in the reductionist approach towards a theory of everything (ToE), and an ideal realized in QCD – the theory of quarks and gluons that describes the strong interactions. QCD exhibits the a property called asymptotic freedom whereby the coupling constants get weaker as one goes to smaller and smaller length scales (equivalently, to higher and higher energies). Asymptotic freedom causes the theory to become non-interacting at small distances (high energies), allowing one to consider the theory at arbitrarily small scales. Asymptotically free theories like QCD can in principle be complete, providing a complete description of everything.

This reductionist ideal, however, is thwarted by QED which is not asymptotically free. Instead, the QED coupling constant (the fine-structure constant \(\alpha\)) becomes larger as one moves to smaller scales. The coupling constant diverges at the Landau pole, rendering the theory invalid. Thus, QED – one of the most accurate physical theory to date – is known to be incomplete, and the quest for physics beyond the Standard Model continues, with billions of dollars being directed to smash particles together with higher and higher energies to try to figure out what the heck is wrong.[1]

The view we shall take here, as clearly described in [Huang, 2013], is that simple physical theories generically arise from RG flow through “theory space” which gets “stuck” near (but not at) fixed-points. Near these fixed-points, most of the terms in the theory becomes small (irrelevant) and the theory takes on a simple form, with only a handful of relevant or marginal terms remaining. Simplicity in physics, thus emerges naturally and somewhat generically from RG flow: it does not necessarily imply that there is a simple underlying theory.

Conversely, this picture also suggests that strange behavior may exist if we finely tune parameters or the scales. For example, I conjecture that the advantages offered by quantum computing and technologies will arise because of such unusual behaviours, however, realizing these advantages will be difficult precisely because of the required fine-tuning.

Renormalization group techniques are not limited to particle physics. They apply to many different systems where the behaviour changes slowly from one scale to another. In these lectures, we will consider a few representative example, but keep in mind that many applications exists to, percolation theory (i.e. how forest fires spread, see [Creswick et al., 1992] for examples), traffic flow (see Kshitij Jeraths work, or his iSciMath seminar – request an account if you would like access and email Michael), fluid dynamics (see Jorge Noronha’s WSU Colloquium or Pavel Kovtun’s Aspen seminar), statistical mechanics, condensed matter, and many other.

The key RG process involves two steps:

  1. Coarse Graining: The first aspect of an RG analysis is to consider a system at different scales. Often this involves some type of averaging, or “decimation” that implies loss of information (in which cased the RG should be considered as a semigroup with no exact inverse).

  2. Rescaling: The second step is to rescale the theory so that it “looks like” the original theory. Most of the quantitative power of RG comes when the theory flows to a fixed point where the structure of the theory is almost invariant under an appropriate coarse-graining + rescaling.

Random Walks#

The first example we will consider (Renormalizing Random Walks) is that of random walks. Much of the content comes from an excellent, but generally hard-to-obtain book [Creswick et al., 1992], but the salient points can be found in the following notes by John McGreevy [McGreevy, 2018]:

We start with a gaussian random walk, which provides an example of a fixed point. Coarse graining amounts to taking \(N\) steps, for which the probability distribution is again a gaussian (products of gaussians are gaussians), but with a different variance. The rescaling step brings the variance back to the starting value, demonstrating exact self-similarity in this case.

Moving away from gaussian distribution, we discuss RG flow by looking at the cumulants as parameters, and show that a wide range of different distributions flow towards a gaussian distribution – a manifestation of universality reminiscent of the central limit theorem.

How to Renormalize the Schrödinger Equation#

The second example we will consider in How to Renormalize The Schrödinger Equation works through parts of [Lepage, 1997]. Here the central techniques of modern effective field theory (especially in the context of nuclear physics) are demonstrated by considering the radial Schrödinger equation. The analogies made in [Lepage, 1997] are accurate and deep: although understanding everything will likely be challenging, it is highly worth working through the examples and trying to understand as much as possible.

This example explicitly demonstrates the nature of many of the divergences seen in field theory, and provides a constructive approach to “regularizing” these divergences to obtain accurate physical results by developing and fitting an appropriate low-energy effective theory.

Discussion points:

  • RG Limit cycles and the \(1/r^2\) potential. (Start with \(1/r\) and show that a dimensionful scale emerges. \(1/r^2\) has no scale. What does this mean? How does a scale for \(E_0\) get set?

Ising Model#

Time-permitting, we will work though some examples related to the Ising model. Exactly solvable examples are available in 1D and 2D, allowing one to test the RG approach for studying critical phenomena and phase transitions. This discussion follows McGreevy’s Lecture Notes (PDF) [McGreevy, 2018] and the discussion in [Creswick et al., 1992].

Discussion points:

  • The coarse-graining procedure for the Ising model is not what one might naturally expect. This is so that the structure of the theory remains fixed. Can one do an RG analysis with the natural coarse graining?

  • Good example of mean field theory.

Philosophy#

Finally, we will end with a discussion of renormalization group techniques based on [Huang, 2013], putting forth the idea that our relatively simple physical theories result from RF flow through the “space of theories” which linger near fixed-points where physics is well approximated by a simple theory over a large range of scales.

Discussion points: