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Renormalization Group - Wikipedia, the Free Encyclopedia 3/17/12 5:27 PM Renormalization Group from Wikipedia, the Free Encyclopedia Renormalization group - Wikipedia, the free encyclopedia 3/17/12 5:27 PM Renormalization group From Wikipedia, the free encyclopedia In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle (cf. Compton wavelength). A change in scale is called a "scale transformation". The renormalization group is intimately related to "scale invariance" and "conformal invariance", symmetries in which a system appears the same at all scales (so-called self-similarity). (However, note that scale transformations are included in conformal transformations, in general: the latter including additional symmetry generators associated with special conformal transformations.) As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so- called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable "couplings" which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances. For example, in quantum electrodynamics (QED), an electron appears to be composed of electrons, positrons (anti- electrons) and photons, as one views it at higher resolution, at very short distances. The electron at such short distances has a slightly different electric charge than does the "dressed electron" seen at large distances, and this change, or "running," in the value of the electric charge is determined by the renormalization group equation. Contents 1 History of the renormalization group 2 Block spin renormalization group 3 Elements of RG theory 4 Relevant and irrelevant operators, universality classes 5 Momentum space RG 6 Appendix: Exact Renormalization Group Equations 7 Threshold effect 8 See also 9 References 9.1 Pedagogical and Historical reviews 9.2 Books 10 External links History of the renormalization group The idea of scale transformations and scale invariance is old in physics. Scaling arguments were commonplace for the Pythagorean school, Euclid and up to Galileo.[citation needed] They became popular again at the end of the 19th century, http://en.wikipedia.org/wiki/Renormalization_group Page 1 of 14 Renormalization group - Wikipedia, the free encyclopedia 3/17/12 5:27 PM perhaps the first example being the idea of enhanced viscosity of Osborne Reynolds, as a way to explain turbulence. The renormalization group was initially devised in particle physics, but nowadays its applications extend to solid-state physics, fluid mechanics, cosmology and even nanotechnology. An early article[1] by Ernst Stueckelberg and Andre Petermann in 1953 anticipates the idea in quantum field theory. Stueckelberg and Petermann opened the field conceptually. They noted that renormalization exhibits a group of transformations which transfer quantities from the bare terms to the counterterms. They introduced a function h(e) in QED, which is now called the beta function (see below). Murray Gell-Mann and Francis E. Low in 1954 restricted the idea to scale transformations in QED,[2] which are the most physically significant, and focused on asymptotic forms of the photon propagator at high energies. They determined the variation of the electromagnetic coupling in QED, by appreciating the simplicity of the scaling structure of that theory. They thus discovered that the coupling parameter g(!) at the energy scale ! is effectively given by the group equation g(!) = G"1( (!/M)d G(g(M)) ) , for some function G and a constant d, in terms of the coupling at a reference scale M. Gell-Mann and Low realized in these results that the effective scale can be arbitrarily taken as !, and can vary to define the theory at any other scale: g(#) = G"1( (#/!)d G(g(!)) ) = G"1( (#/M)d G(g(M)) ) . The gist of the RG is this group property: as the scale ! varies, the theory presents a self-similar replica of itself, and any scale can be accessed similarly from any other scale, by group action, a formal conjugacy of couplings[3] in the mathematical sense (Schröder's equation). On the basis of this (finite) group equation, Gell-Mann and Low then focussed on infinitesimal transformations, and invented a computational method based on a mathematical flow function $(g) = G d/(%G/%g) of the coupling parameter g, which they introduced. Like the function h(e) of Stueckelberg and Petermann, their function determines the differential change of the coupling g(!) with respect to a small change in energy scale ! through a differential equation, the renormalization group equation: %g / %ln(!) = $(g) = &(g) . The modern name is also indicated, the beta function, introduced by C. Callan and K. Symanzik in the early 1970s. Since it is a mere function of g, integration in g of a perturbative estimate of it permits specification of the renormalization trajectory of the coupling, that is, its variation with energy, effectively the function G in this perturbative approximation. The renormalization group prediction (cf Stueckelberg-Petermann and Gell-Mann-Low works) was confirmed 40 years later at the LEP accelerator experiments: the fine structure "constant" of QED was measured to be about 1/127 at energies close to 200 GeV, as opposed to the standard low-energy physics value of 1/137. (Early applications to quantum electrodynamics are discussed in the influential book of Nikolay Bogolyubov and Dmitry Shirkov in 1959.[4]) The renormalization group emerges from the renormalization of the quantum field variables, which normally has to address the problem of infinities in a quantum field theory (although the RG exists independently of the infinities). This problem of systematically handling the infinities of quantum field theory to obtain finite physical quantities was solved for QED by Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga, who received the 1965 Nobel prize for these contributions. They effectively devised the theory of mass and charge renormalization, in which the infinity in the momentum scale is cut-off by an ultra-large regulator, ' (which could ultimately be taken to be infinite — infinities reflect the pileup of contributions from an infinity of degrees of freedom at infinitely high energy scales.). The dependence of physical quantities, such as the electric charge or electron mass, on the scale ' is hidden, effectively swapped for the longer-distance scales at which the physical quantities are measured, and, as a result, all observable quantities end up being finite, instead, even for an infinite '. Gell-Mann and Low thus realized in these results that, while, infinitesimally, a tiny http://en.wikipedia.org/wiki/Renormalization_group Page 2 of 14 Renormalization group - Wikipedia, the free encyclopedia 3/17/12 5:27 PM change in g is provided by the above RG equation given $(g), the self-similarity is expressed by the fact that $(g) depends explicitly only upon the parameter(s) of the theory, and not upon the scale !. Consequently, the above renormalization group equation may be solved for (G and thus) g(!). A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[5] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances. This approach covered the conceptual point and was given full computational substance[6] in the extensive important contributions of Kenneth Wilson. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982. Meanwhile, the RG in particle physics had been reformulated in more practical terms by C. G. Callan and K. Symanzik in 1970. [7] The above beta function, which describes the "running of the coupling" parameter with scale, was also found to amount to the "canonical trace anomaly", which represents the quantum-mechanical breaking of scale (dilation) symmetry in a field theory. (Remarkably, quantum mechanics itself can induce mass through the trace anomaly and the running coupling.) Applications of the RG to particle physics exploded in number in the 1970s with the establishment of the Standard Model. In 1973, it was discovered that a theory of interacting colored quarks, called quantum chromodynamics had a negative beta function. This means that an initial high-energy value of the coupling will eventuate
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