Renormalization group

In physics, the term renormalization refers to a variety of theoretical concepts and computational techniques revolving either around the idea of rescaling transformations, or around the process of removing infinities from the calculated quantities (see also regularization). Renormalization in more or less its modern form originated in quantum field theory, where it is usually credited to Julian Schwinger, Shin'ichiro Tomonaga, Richard Feynman, and Freeman Dyson. An alternative formulation suitable for statistical field theory was later given by Wilson and Kadanoff.

The idea of renormalization is that, while some continuous physical systems are by necessity described by models with a characteristic smallest length scale (or largest energy scale), the large-scale physical predictions of the theory should not depend on that characteristic length scale. In some cases, the characteristic smallest length scale is manifestly unphysical. Physical consequences of this scale-independence are explored by considering the effect of changing the characteristic scale on various physical calculations. Depending on the formulation, the collection of all scale transformations, called by physicists "the renormalization group", has the mathematical structure of a group, semigroup or quantum group/Hopf algebra. The renormalization group in quantum field theory was studied by Gell-Mann and Low.

Real-space renormalization

Let's say we have a family of models over a certain space which admits rescalings which are automorphisms but not isometries. For example, in Euclidean space, the isometries preserve the distance between any two points. Even though a rescaling of a Euclidean space is an automorphism in the sense that a rescaled n-dimensional Euclidean space is simply another n-dimensional Euclidean space (the two being isomorphic), it's not an isometry because it changes distances by a constant factor. The same thing goes for Minkowski space. However, this isn't true for conformal geometries because rescalings are isometries there. The set of all models of the family is called the parameter space, which is sometimes a manifold. At any rate, it usually admits a differentiable structure. Because of the rescaling automorphisms of the underlying space, given any particular model in the family, by rescaling the space, we get another model which may or may not be the same as the original model. Here, we make the further assumption that by rescaling the underlying space, any rescaled model of the family also belongs to the family. The group of rescalings is isomorphic to R⁺, the group of positive real numbers under multiplication.

This amounts to saying that there is a group action of the rescaling group on the parameter space. In addition, we will assume this group action is differentiable (or maybe continuous/smooth, depending on the needs the renormalization group is put to). The rescaling group is called the renormalization group and the group action is called the renormalization group flow.

Relevant, marginal and irrelevant

Under the action of enlarging rescalings, a parameter could have a positive, zero or negative Lyapunov exponent. That parameter is then called relevant, marginal or irrelevant respectively. In the limit as the rescaling parameter approaches infinity, the RG flows converge to infrared attractors. The points on this attractor are called universality classes because many different models in parameter space start to look like this model at large enough scales, which basically means small scale effects only affect large scale effects insignificantly (a scale independence of sorts). Oftentimes, the parameter space is infinite-dimensional (very huge), but the infrared attractors are only finite dimensional, so that the space of universality classes is much much smaller than the original parameter space.

This means, provided we work at large enough scales and don't mind using approximations, we can reduce the entire parameter space to the space of universality classes. The group action of the RG restricted to this attractor is still a group action. So, for models within a sufficiently small neighborhood of the attractor in parameter space, we can project this neighborhood to the attractor, so that running the renormalization group action forward leads to even better approximations but running it backwards eventually leads to divergence out of the neighborhood for almost every point in the neighborhood. This means the RG should really be treated as a monoid in this restriction. Similarly, RG flows can have ultraviolet attractors.

See also Critical exponent, Lyapunov exponent, Density matrix renormalization group

In statistical mechanics, a second order phase transition corresponds to an infrared repellor (i.e. an "unstable" infrared fixed point).

References

• Shirkov, Dimitrij V. (1999): Evolution of the Bogoliubov Renormalization Group. arXiv.org:hep-th/9909024 (http://arxiv.org/abs/hep-th/9909024). A mathematical introduction and historical overview with a stress on group theory and the application in high-energy physics.
• Delamotte, Bertrand (2004): A hint of renormalization. American Journal of Physics, Vol. 72, No. 2, pp. 170–184, February 2004 (http://scitation.aip.org/journals/doc/AJPIAS-ft/vol_72/iss_2/170_1.html). A pedestrian introduction to renormalization and the renormalization group.
• Maris, Humphrey J. and Leo P. Kadanoff (1978): Teaching the renormalization group. American Journal of Physics, June 1978, Volume 46, Issue 6, pp. 652-657 (http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=AJPIAS&CURRENT=NO&ONLINE=YES&smode=strresults&sort=rel&maxdisp=25&threshold=0&pjournals=AJPIAS&pyears=2001%2C2000%2C1999&possible1=652&possible1zone=fpage&fromvolume=46&SMODE=strsearch&OUTLOG=NO&viewabs=AJPIAS&key=DISPLAY&docID=1&page=1&chapter=0). A pedestrian introduction to the renormalization group as applied in condensed matter physics.

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