Everything about Renormalization Group totally explained
In
theoretical physics,
renormalization group (RG) refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different
distance scales. In particle physics it reflects the changes in the underlying force laws as one varies the
energy scale at which physical processes occur. A change in scale is called a "scale transformation" or "conformal transformation." The renormalization group is intimately related to "conformal invariance" or "scale invariance," a symmetry by which the system appears the same at all scales (so-called
self-similarity).
As one varies the scale, it's as if changing the magnifying power of a microscope viewing the system. The system will generally make a self-similar copy of itself, with slightly different parameters describing the components of the system. The components, or fundamental variables, may be atoms, or fundamental particles, or atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be "coupling constants" that 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, an electron appears to be composed of electrons, anti-electrons and photons as one views it at very short distances. The electron at very 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.
History of the renormalization group
The idea of scale transformations and scale invariance is old and venerable in physics. Scaling arguments were commonplace for the
Pythagorean school,
Euclid and up to
Galileo. They became popular again at the end of the
19th century, 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 within particle
physics, but nowadays its applications are extended to
solid-state physics,
fluid mechanics,
cosmology and even
nanotechnology. An early article by
Ernst Stueckelberg and Andre Peterman in 1953 anticipates the idea in
quantum field theory.
Stueckelberg and Peterman opened the field. They noted that
renormalization comes with a group of transformations which transfer
quantities from the bare terms to the counterterms.
Murray Gell-Mann and F.E. Low in
1954 restricted it to scaling
transformations, which are the most interesting. They proposed the existence of a mathematical function of the coupling
parameter
of a theory,
.
This function determines the differential change of the coupling constant
with a small change in energy scale
by
the "renormalization group equation:"
is the ERGE.
As there are infinitely many choices of
Rk, there are also infinitely many different interpolating ERGEs.
Generalization to other fields like spinorial fields is straightforward.
Although the Polchinski ERGE and the effective average action ERGE look similar, they're based upon very different philosophies. In the effective average action ERGE, the bare action is left unchanged (and the UV cutoff scale -- if there's one -- is also left unchanged) but we neglect the IR contributions to the effective action whereas in the Polchinski ERGE, we fix the QFT once and for all but vary the "bare action" at different energy scales to reproduce the prespecified model. Polchinski's version is certainly much closer to Wilson's idea in spirit. Note that one uses "bare actions" whereas the other uses effective (average) actions.
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