Editing Renormalization (section)

== In statistical physics ==

===History===
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.<ref>[[Leo Kadanoff|L.P. Kadanoff]] (1966): "Scaling laws for Ising models near <math>T_c</math>", ''Physics (Long Island City, N.Y.)'' '''2''', 263.</ref> 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<ref name=Wilson1975 /> 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 effect|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 in Physics]] for these decisive contributions in 1982.

===Principles===
In more technical terms, let us assume that we have a theory described
by a certain function <math>Z</math> of the state variables
<math>\{s_i\}</math> and a certain set of coupling constants
<math>\{J_k\}</math>. This function may be a [[partition function (quantum field theory)|partition function]],
an [[Action (physics)|action]], a [[Hamiltonian (quantum mechanics)|Hamiltonian]], etc. It must contain the
whole description of the physics of the system.

Now we consider a certain blocking transformation of the state
variables <math>\{s_i\}\to \{\tilde s_i\}</math>,
the number of <math>\tilde s_i</math> must be lower than the number of
<math>s_i</math>. Now let us try to rewrite the <math>Z</math>
function ''only'' in terms of the <math>\tilde s_i</math>. If this is achievable by a
certain change in the parameters, <math>\{J_k\}\to
\{\tilde J_k\}</math>, then the theory is said to be
'''renormalizable'''.

The possible
macroscopic states of the system, at a large scale, are given by this
set of fixed points.

===Renormalization group fixed points===
The most important information in the RG flow is its '''fixed points'''. A fixed point is defined by the vanishing of the [[beta function (physics)|beta function]] associated to the flow.  Then, fixed points of the renormalization group are by definition scale invariant. In many cases of physical interest scale invariance enlarges to conformal invariance.  One then has a [[conformal field theory]] at the fixed point.

The ability of several theories to flow to the same fixed point leads to [[Universality (dynamical systems)|universality]].

If these fixed points correspond to free field theory, the theory is said to exhibit [[quantum triviality]].  Numerous fixed points appear in the study of [[Lattice gauge theory#Quantum triviality|lattice Higgs theories]], but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs">{{cite journal| author=D. J. E. Callaway | year=1988
| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?| journal=[[Physics Reports]]
|volume=167| issue=5 | pages=241–320| doi=10.1016/0370-1573(88)90008-7
|bibcode = 1988PhR...167..241C | author-link=David J E Callaway
}}</ref>