Editing Renormalization group (section)

==Elementary theory<!--'RG flow' redirects here-->==
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'''.

Most fundamental theories of physics such as [[quantum electrodynamics]], [[quantum chromodynamics]] and [[Electroweak force|electro-weak]] interaction, but not gravity, are exactly renormalizable. Also, most theories in condensed matter physics are approximately renormalizable, from [[superconductivity]] to fluid turbulence.

The change in the parameters is implemented by a certain beta function: <math>\{\tilde J_k\}=\beta(\{ J_k \})</math>, which is said to induce a '''renormalization group flow''' (or '''RG flow''') on the <math>J</math>-space. The values of <math>J</math> under the flow are called '''running couplings'''.

As was stated in the previous section, the most important information in the RG flow are its '''fixed points'''. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to exhibit [[quantum triviality]], possessing what is called a [[Landau pole]], as in quantum electrodynamics.  For a {{mvar|φ}}<sup>4</sup> interaction, [[Michael Aizenman]] proved that this theory is indeed trivial, for space-time dimension {{mvar|D}} ≥ 5.<ref name="Aiz81">{{cite journal |last=Aizenman |first=M. |author-link=Michael Aizenman |year=1981 |title=Proof of the triviality of ''Φ{{su|b=d|p=4}}'' field theory and some mean-field features of Ising models for ''d'' > 4  |journal=[[Physical Review Letters]] |volume=47 |issue=1 |pages=1–4 |doi=10.1103/PhysRevLett.47.1 |bibcode=1981PhRvL..47....1A }}</ref> For {{mvar|D}} = 4, the triviality has yet to be proven rigorously, but [[Quantum triviality|lattice computations]] have provided strong evidence for this.  This fact is important as [[quantum triviality]] can be used to bound or even ''predict'' parameters such as the [[Higgs boson]] mass in [[Physics applications of asymptotically safe gravity#The mass of the Higgs boson|asymptotic safety]] scenarios. 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 |first=David J.E. |last=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>

Since the RG transformations in such systems are '''lossy''' (i.e.: the number of variables decreases - see as an example in a different context, [[Lossy data compression]]), there need not be an inverse for a given RG transformation. Thus, in such lossy systems, the renormalization group is, in fact, a [[semigroup]], as lossiness implies that there is no unique inverse for each element.