Editing Quantum chromodynamics (section)

==Cross-relations to condensed matter physics==
There are unexpected cross-relations to [[condensed matter physics]]. For example, the notion of [[gauge invariance]] forms the basis of the well-known Mattis [[spin glass]]es,<ref>{{cite journal |first=D. C. |last=Mattis |title=Solvable Spin Systems with Random Interactions |journal=Phys. Lett. A |volume=56 |issue=5 |year=1976 |pages=421–422 |doi=10.1016/0375-9601(76)90396-0 |bibcode=1976PhLA...56..421M }}</ref> which are systems with the usual spin degrees of freedom <math>s_i=\pm 1\,</math> for ''i''&nbsp;=1,...,N,  with the special fixed  "random" couplings <math>J_{i,k}=\epsilon_i \,J_0\,\epsilon_k\,.</math>  Here the ε<sub>i</sub>  and ε<sub>k</sub> quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation  <math>(\,s_i\to s_i\cdot\epsilon_i\quad\,J_{i,k}\to \epsilon_i J_{i,k}\epsilon_k\,\quad s_k\to s_k\cdot\epsilon_k \,)\,.</math> This means that thermodynamic expectation values of measurable quantities, e.g. of the energy <math display="inline">{\mathcal H}:=-\sum s_i\,J_{i,k}\,s_k\,,</math> are invariant.

However, here the ''coupling degrees of freedom'' <math>J_{i,k}</math>, which in the QCD correspond to the ''gluons'', are "frozen" to fixed values  (quenching).  In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom the [[entropy]] plays an important role (see below).

For positive ''J''<sub>0</sub> the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "[[Geometrical frustration|frustration]]" at all. This term is a basic  measure in spin glass theory.<ref>{{cite journal |first1=J. |last1=Vannimenus |first2=G. |last2=Toulouse |title=Theory of the frustration effect. II. Ising spins on a square lattice |journal=Journal of Physics C: Solid State Physics |year=1977 |volume=10 |issue=18 |pages=537 |doi=10.1088/0022-3719/10/18/008 |bibcode=1977JPhC...10L.537V }}</ref> Quantitatively it is identical with the loop product <math>P_W:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}</math>  along a closed loop ''W''. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantity ''P<sub>W</sub>'' never becomes negative.

The basic notion "frustration" of the spin-glass is actually similar to the Wilson loop quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop product to the Hamiltonian, by some kind of term representing a "punishment". In the QCD the Wilson loop is essential for the Lagrangian rightaway.

The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman and Shenker,<ref>{{Cite journal|last=Fradkin|first=Eduardo|date=1978|title=Gauge symmetries in random magnetic systems|journal=Physical Review B|volume=18|issue=9|pages=4789–4814|doi=10.1103/physrevb.18.4789| bibcode=1978PhRvB..18.4789F| osti=1446867|url=https://www.slac.stanford.edu/cgi-bin/getdoc/slac-pub-2112.pdf}}</ref> which also stresses the notion of [[Kramers–Wannier duality|duality]].

A further analogy consists in the already mentioned similarity to  [[polymer physics]], where, analogously to  Wilson loops, so-called "entangled nets" appear, which are important for the formation of the [[entropic force|entropy-elasticity]]  (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links", which  glue different loop segments together, and "[[asymptotic freedom]]" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for <math>0\leftarrow\lambda_w\ll R_c</math>  (where ''R<sub>c</sub>'' is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λ<sub>w</sub> is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.<ref>{{cite journal |first1=A. |last1=Bergmann |first2=A. |last2=Owen |title=Dielectric relaxation spectroscopy of poly[(R)-3-Hydroxybutyrate] (PHD) during crystallization |journal=Polymer International |volume=53 |issue=7 |year=2004 |pages=863–868 |doi=10.1002/pi.1445 }}</ref>

There is also a correspondence between confinement in QCD – the fact that the color field is only different from zero in the interior of hadrons – and the behaviour of the usual magnetic field in the theory of [[type-II superconductor]]s: there the magnetism is confined to the interior of the [[Abrikosov vortex|Abrikosov flux-line lattice]],<ref>Mathematically, the flux-line lattices are described by [[Emil Artin]]'s braid group, which is nonabelian, since one braid can wind around another one.</ref> i.e., the London penetration depth ''λ'' of that  theory is analogous to the confinement radius ''R<sub>c</sub>'' of  quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, <math>\propto  g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j\,,</math> on the r.h.s. of the Lagrangian.