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.

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.

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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.

Related Topics:
Monoid - Ultraviolet attractor

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