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.

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.

Related Topics:
Rescalings - Automorphism - Isometries - Euclidean space - Minkowski space - Manifold

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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.

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