Home > 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 algebraIn abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K linear map such that the following diagram commutes :. Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε it, The renormalization group in quantum field theory was studied by Gell-MannMurray Gell-Mann (born September 15, 1929) is an American physicist. Murray Gell-Mann received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. He introduced the " eightfold way" as a means to coherently organize the gre and LowLow is: the surname of Francis E. Low, quantum physicist a short term for a meteorological low pressure system the name of a 1977 album by David Bowie the name of a 1994 album by Testament the name of the first track and famous single by Cracker from thei
1 Real-space renormalization
Let's say we have a family of models over a certain space which admits rescalings which are automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an objs but not isometriesIn geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. General definitions The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry''. Both are often ca. For example, in Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers, the isometries preserve the distance between any two points. Even though a rescaling of a Euclidean space is an automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an obj in the sense that a rescaled n-dimensional Euclidean space is simply another n-dimensional Euclidean space, which are isomorphic, it's not an isometry because it changes distances by a constant factor. The same thing goes for Minkowski spaceIn physics and mathematics, Minkowski space (or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a si. 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.
