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quantum chromodynamics
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{{short descriptionTheory of the strong nuclear interactions}}{{RedirectQCD}}{{more citations needed date=March 2017}}{{Standard model of particle physicscTopic=quantum chromodynamics}}In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a nonabelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.QCD exhibits two main properties:  the content below is remote from Wikipedia
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 Color confinement. This is a consequence of the constant force between two color charges as they are separated: In order to increase the separation between two quarks within a hadron, everincreasing amounts of energy are required. Eventually this energy becomes so great as to spontaneously produce a quarkâ€“antiquark pair, turning the initial hadron into a pair of hadrons instead of producing an isolated color charge. Although analytically unproven, color confinement is well established from lattice QCD calculations and decades of experiments.BOOK
, J. Greensite
, 2011
, An introduction to the confinement problem
, Springer Science+Business Media, Springer
, 9783642143816
, , 2011
, An introduction to the confinement problem
, Springer Science+Business Media, Springer
, 9783642143816
 Asymptotic freedom, a steady reduction in the strength of interactions between quarks and gluons as the energy scale of those interactions increases (and the corresponding length scale decreases). The asymptotic freedom of QCD was discovered in 1973 by David Gross and Frank Wilczek,
, D.J. Gross, F. Wilczek, 1973
, Ultraviolet behavior of nonabelian gauge theories
, Physical Review Letters
, 30, 26, 1343â€“1346
, 1973PhRvL..30.1343G
, 10.1103/PhysRevLett.30.1343
, and independently by David Politzer in the same year.JOURNAL
, Ultraviolet behavior of nonabelian gauge theories
, Physical Review Letters
, 30, 26, 1343â€“1346
, 1973PhRvL..30.1343G
, 10.1103/PhysRevLett.30.1343
, H.D. Politzer
, 1973
, Reliable perturbative results for strong interactions
, Physical Review Letters
, 30, 26, 1346â€“1349
, 1973PhRvL..30.1346P
, 10.1103/PhysRevLett.30.1346
, For this work all three shared the 2004 Nobel Prize in Physics.WEB
, 1973
, Reliable perturbative results for strong interactions
, Physical Review Letters
, 30, 26, 1346â€“1349
, 1973PhRvL..30.1346P
, 10.1103/PhysRevLett.30.1346
,weblink
, The Nobel Prize in Physics 2004
, Nobel Web
, 2004
, 20101024
, no
,weblink" title="web.archive.org/web/20101106025744weblink">weblink
, 20101106
,
, , The Nobel Prize in Physics 2004
, Nobel Web
, 2004
, 20101024
, no
,weblink" title="web.archive.org/web/20101106025744weblink">weblink
, 20101106
,
Terminology
Physicist Murray GellMann coined the word quark in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in Finnegans Wake by James Joyce. On June 27, 1978, GellMann wrote a private letter to the editor of the Oxford English Dictionary, in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.)BOOK , The three kinds of charge in QCD (as opposed to one in quantum electrodynamics or QED) are usually referred to as "color charge" by loose analogy to the three kinds of color (red, green and blue) perceived by humans. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.The force between quarks is known as the colour force (wikt:colour force) (or color force WEB,weblink Archived copy, 20070829, no,weblink" title="web.archive.org/web/20070820075205weblink">weblink 20070820, retrieved 6 May 2017) or strong interaction, and is responsible for the strong nuclear force.Since the theory of electric charge is dubbed "electrodynamics", the Greek word Ï‡Ïá¿¶Î¼Î± chroma "color" is applied to the theory of color charge, "chromodynamics".History
With the invention of bubble chambers and spark chambers in the 1950s, experimental particle physics discovered a large and evergrowing number of particles called hadrons. It seemed that such a large number of particles could not all be fundamental. First, the particles were classified by charge and isospin by Eugene Wigner and Werner Heisenberg; then, in 1953â€“56,JOURNAL, Nakano, T
, Nishijima, N
, 1953
, Charge Independence for Vparticles, Progress of Theoretical Physics
, 10, 5, 581
, 10.1143/PTP.10.581, 1953PThPh..10..581N,
JOURNAL
, Nishijima, N
, 1953
, Charge Independence for Vparticles, Progress of Theoretical Physics
, 10, 5, 581
, 10.1143/PTP.10.581, 1953PThPh..10..581N,
, Nishijima, K
, 1955
, Charge Independence Theory of V Particles
, Progress of Theoretical Physics
, 13, 3, 285â€“304
, 10.1143/PTP.13.285, 1955PThPh..13..285N,
JOURNAL
, 1955
, Charge Independence Theory of V Particles
, Progress of Theoretical Physics
, 13, 3, 285â€“304
, 10.1143/PTP.13.285, 1955PThPh..13..285N,
, GellMann, M
, 1956
, The Interpretation of the New Particles as Displaced Charged Multiplets
, Il Nuovo Cimento
, 4
, S2, 848â€“866
, 10.1007/BF02748000
, 1956
, The Interpretation of the New Particles as Displaced Charged Multiplets
, Il Nuovo Cimento
, 4
, S2, 848â€“866
, 10.1007/BF02748000
Strangeness (particle physics)>strangeness by Murray GellMann and Kazuhiko Nishijima (see GellMannâ€“Nishijima formula). To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the ''eightfold way (physics)  '', invented in 1961 by GellMannGellMann, M. (1961). "The Eightfold Way: A Theory of strong interaction symmetry" (No. TID12608; CTSL20). California Inst. of Tech., Pasadena. Synchrotron Lab (online). and Yuval Ne'eman. GellMann and George Zweig, correcting an earlier approach of Shoichi Sakata, went on to propose in 1963 that the structure of the groups could be explained by the existence of three flavour (particle physics)>flavors of smaller particles inside the hadrons: the quarks.Perhaps the first remark that quarks should possess an additional quantum number was madeARXIV
, 0904.0343 , as a short footnote in the preprint of Boris StruminskyB. V. Struminsky, Magnetic moments of barions in the quark model. JINRPreprint P1939, Dubna, Russia. Submitted on January 7, 1965. in connection with Î©âˆ’ hyperon composed of three strange quarks with parallel spins (this situation was peculiar, because since quarks are fermions, such combination is forbidden by the Pauli exclusion principle): {{QuotationThree identical quarks cannot form an antisymmetric Sstate. In order to realize an antisymmetric orbital Sstate, it is necessary for the quark to have an additional quantum number.B. V. StruminskyMagnetic moments of barions in the quark modelJINRPreprint P1939, Dubna, Submitted on January 7, 1965}} Boris Struminsky was a PhD student of Nikolay Bogolyubov. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research. In the beginning of 1965, Nikolay Bogolyubov, Boris Struminsky and Albert Tavkhelidze wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.N. Bogolubov, B. Struminsky, A. Tavkhelidze. On composite models in the theory of elementary particles. JINR Preprint D1968, Dubna 1965. This work was also presented by Albert Tavkhelidze without obtaining consent of his collaborators for doing so at an international conference in Trieste (Italy), in May 1965.A. Tavkhelidze. Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965, Vienna IAEA, 1965, p. 763.V. A. Matveev and A. N. Tavkhelidze (INR, RAS, Moscow) The quantum number color, colored quarks and QCD {{webarchiveurl=https://web.archive.org/web/20070523073026weblink date=20070523 }} (Dedicated to the 40th Anniversary of the Discovery of the Quantum Number Color). Report presented at the 99th Session of the JINR Scientific Council, Dubna, 19â€“20 January 2006.A similar mysterious situation was with the Î”++ baryon; in the quark model, it is composed of three up quarks with parallel spins. In 1964â€“65, GreenbergO. W. Greenberg, "Spin and Unitary Spin Independence in a Paraquark Model of Baryons and Mesons." Phys. Rev. Lett. 13, 598â€“602 (1964). and Hanâ€“NambuM. Y. Han and Y. Nambu, "ThreeTriplet Model with Double SU(3) Symmetry." Phys. Rev. 139, B1006â€“B1010 (1965) independently resolved the problem by proposing that quarks possess an additional SU(3) gauge degree of freedom, later called color charge. Han and Nambu noted that quarks might interact via an octet of vector gauge bosons: the gluons.Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then was defined as a particle which could be separated and isolated, GellMann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context: He meant quarks are confined, but he also was implying that the strong interactions could probably not be fully described by quantum field theory.Richard Feynman argued that high energy experiments showed quarks are real particles: he called them partons (since they were parts of hadrons). By particles, Feynman meant objects which travel along paths, elementary particles in a field theory.The difference between Feynman's and GellMann's approaches reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explained diffractive scattering. Although GellMann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of Smatrix theory.James Bjorken proposed that pointlike partons would imply certain relations in deep inelastic scattering of electrons and protons, which were verified in experiments at SLAC in 1969. This led physicists to abandon the Smatrix approach for the strong interactions.In 1973 the concept of color as the source of a "strong field" was developed into the theory of QCD by physicists Harald Fritzsch and (:de:Heinrich Leutwyler), together with physicist Murray GellMann.JOURNAL, Fritzsch, H., GellMann, M., Leutwyler, H., Advantages of the color octet gluon picture, Physics Letters, 47B, 4, 365â€“368, 1973, 10.1016/03702693(73)906254, 1973PhLB...47..365F, 10.1.1.453.4712, In particular, they employed the general field theory developed in 1954 by Chen Ning Yang and Robert MillsJOURNAL, ChenNing Yang, C. N., Yang, Robert Mills (physicist), R., Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Physical Review, 96, 1, 191â€“195, 1954, 10.1103/PhysRev.96.191, 1954PhRv...96..191Y, (see Yangâ€“Mills theory), in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.)The discovery of asymptotic freedom in the strong interactions by David Gross, David Politzer and Frank Wilczek allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of perturbation theory. Evidence of gluons was discovered in threejet events at PETRA in 1979. These experiments became more and more precise, culminating in the verification of perturbative QCD at the level of a few percent at the LEP in CERN.The other side of asymptotic freedom is confinement. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within lattice QCD computations, but is not mathematically proven. One of the Millennium Prize Problems announced by the Clay Mathematics Institute requires a claimant to produce such a proof. Other aspects of nonperturbative QCD are the exploration of phases of quark matter, including the quarkâ€“gluon plasma.The relation between the shortdistance particle limit and the confining longdistance limit is one of the topics recently explored using string theory, the modern form of Smatrix theory.JOURNAL
, Fyodor Tkachov , A contribution to the history of quarks: Boris Struminsky's 1965 JINR publication , physics.histph , 2009 , J. Polchinski, M. Strassler, 2002
, Hard Scattering and Gauge/String duality , Physical Review Letters , 88, 31601 , 10.1103/PhysRevLett.88.031601, 11801052, 3  bibcode = 2002PhRvL..88c1601P, JOURNAL
, Brower, Richard C.
, Mathur, Samir D. , ChungI Tan , Glueball Spectrum for QCD from AdS Supergravity Duality , 2000 , 10.1016/S05503213(00)004351 , Nuclear Physics B , 587 , 1â€“3 , 249â€“276 , hepth/0003115, 2000NuPhB.587..249B, TheorySome definitions{{unsolvedphysicsQCD in the nonperturbative regime:
Additional remarks: dualityAs mentioned, asymptotic freedom means that at large energy â€“ this corresponds also to short distances â€“ there is practically no interaction between the particles. This is in contrast â€“ more precisely one would say dualâ€“ to what one is used to, since usually one connects the absence of interactions with large distances. However, as already mentioned in the original paper of Franz Wegner,F. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter, J. Math. Phys. 12 (1971) 2259â€“2272.
Reprinted in Claudio Rebbi (ed.), Lattice Gauge Theories and Monte Carlo Simulations, World Scientific, Singapore (1983), p. 60â€“73. Abstract: weblink {{webarchiveurl=https://web.archive.org/web/20110504173247weblink date=20110504 }} a solid state theorist who introduced 1971 simple gauge invariant lattice models, the hightemperature behaviour of the original model, e.g. the strong decay of correlations at large distances, corresponds to the lowtemperature behaviour of the (usually ordered!) dual model, namely the asymptotic decay of nontrivial correlations, e.g. shortrange deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.Perhaps one can guess that in the "original" model mainly the quarks would fluctuate, whereas in the present one, the "dual" model, mainly the gluons do.
Symmetry groupsThe color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1) which is gauged to give QED: this is an abelian group. If one considers a version of QCD with Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group SUL(Nf) × SUR(Nf) × UB(1) × UA(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SUV(Nf) with the formation of a chiral condensate. The vector symmetry, UB(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry UA(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly.There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry which rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of up, down and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD.LagrangianThe dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is{{Equation box 1indent =:  borderborder colour = #50C878background colour = #ECFCF4}}where psi_i(x) , is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by i,,j,,ldots; D_mu is the gauge covariant derivative; the Î³Î¼ are Dirac matrices connecting the spinor representation to the vector representation of the Lorentz group.The symbol G^a_{mu nu} , represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, FÎ¼Î½, in quantum electrodynamics. It is given by:NEWS, The field strength correlator from QCD sum rules  author2=H.G. Dosch, M. Jamin, Heidelberg, Germany  volume=86  year=2000  bibcode=2000NuPhS..86..421E, 10.1016/S09205632(00)005983,
G^a_{mu nu} = partial_mu mathcal{A}^a_nu  partial_nu mathcal{A}^a_mu + g f^{abc} mathcal{A}^b_mu mathcal{A}^c_nu ,,
where mathcal{A}^a_mu(x) , are the gluon fields, dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b,...; and fabc are the structure constants of SU(3). Note that the rules to moveup or pulldown the a, b, or c indices are trivial, (+, ..., +), so that fabc = fabc = f'a'bc whereas for the Î¼ or Î½ indices one has the nontrivial relativistic rules corresponding to the metric signature (+ âˆ’ âˆ’ âˆ’).The variables m and g correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.An important theoretical concept is the Wilson loop (named after Kenneth G. Wilson). In lattice QCD, the final term of the above Lagrangian is discretized via Wilson loops, and more generally the behavior of Wilson loops can distinguish confined and deconfined phases.Fields(File:QCD.svg300pxrightthumbThe pattern of strong charges for the three colors of quark, three antiquarks, and eight gluons (with two of zero charge overlapping).)Quarks are massive spin{{frac12}} fermions which carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU(3). They also carry electric charge (either −{{frac13}} or +{{frac23}}) and participate in weak interactions as part of weak isospin doublets. They carry global quantum numbers including the baryon number, which is {{frac13}} for each quark, hypercharge and one of the flavor quantum numbers.Gluons are spin1 bosons which also carry color charges, since they lie in the adjoint representation 8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups.Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark.DynamicsAccording to the rules of quantum field theory, and the associated Feynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with QED, in which only the first kind of interaction occurs, since photons have no charge. Diagrams involving Faddeevâ€“Popov ghosts must be considered too (except in the unitarity gauge).Area law and confinementDetailed computations with the abovementioned LagrangianSee all standard textbooks on the QCD, e.g., those noted above show that the effective potential between a quark and its antiquark in a meson contains a term that increases in proportion to the distance between the quark and antiquark (propto r), which represents some kind of "stiffness" of the interaction between the particle and its antiparticle at large distances, similar to the entropic elasticity of a rubber band (see below). This leads to confinement Confinement gives way to a quarkâ€“gluon plasma only at extremely large pressures and/or temperatures, e.g. for T approx 5cdot 10^{12} K or larger. of the quarks to the interior of hadrons, i.e. mesons and nucleons, with typical radii Rc, corresponding to former "Bag models" of the hadronsKenneth A. Johnson. (July 1979). The bag model of quark confinement. Scientific American. The order of magnitude of the "bag radius" is 1 fm (= 10−15 m). Moreover, the abovementioned stiffness is quantitatively related to the socalled "area law" behaviour of the expectation value of the Wilson loop product PW of the ordered coupling constants around a closed loop W; i.e. ,langle P_Wrangle is proportional to the area enclosed by the loop. For this behaviour the nonabelian behaviour of the gauge group is essential.MethodsFurther analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.Perturbative QCDThis approach is based on asymptotic freedom, which allows perturbation theory to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.Lattice QCDmissing image!
Among nonperturbative approaches to QCD, the most well established one is lattice QCD. This approach uses a discrete set of spacetime points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation which is then carried out on supercomputers like the QCDOC which was constructed for precisely this purpose. While it is a slow and resourceintensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, the numerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars). Fluxtube meson.png  A quark and an antiquark (red color) are glued together (green color) to form a meson (result of a lattice QCD simulation by M. Cardoso et al.M. Cardoso et al., "Lattice QCD computation of the colour fields for the static hybrid quarkâ€“gluonâ€“antiquark system, and microscopic study of the Casimir scaling", Phys. Rev. D 81, 034504 (2010) ).) {{frac1N}} expansionA wellknown approximation scheme, the {{frac1N}} expansion, starts from the idea that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now, it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include the AdS/CFT approach.Effective theoriesFor specific problems effective theories may be written down which give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameter of the QCD Lagrangian. One such effective field theory is chiral perturbation theory or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u, d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks which are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT . Other effective theories are heavy quark effective theory (which expands around heavy quark mass near infinity), and softcollinear effective theory (which expands around large ratios of energy scales). In addition to effective theories, models like the Nambuâ€“JonaLasinio model and the chiral model are often used when discussing general features.QCD sum rulesBased on an Operator product expansion one can derive sets of relations that connect different observables with each other.Nambuâ€“JonaLasinio modelIn one of his recent works, KeiIchi Kondo derived as a lowenergy limit of QCD, a theory linked to the Nambuâ€“JonaLasinio model since it is basically a particular nonlocal version of the Polyakovâ€“Nambuâ€“JonaLasinio model.JOURNAL, KeiIchi Kondo
The Nambuâ€“JonaLasinio model in itself is, among many other things, used because it is a 'relatively simple' model of chiral symmetry breaking, phenomenon present up to certain conditions (Chiral limit i.e. massless fermions) in QCD itself.In this model, however, there is no confinement. In particular, the energy of an isolated quark in the physical vacuum turns out well defined and finite., 2010 , Toward a firstprinciple derivation of confinement and chiralsymmetrybreaking crossover transitions in QCD , Physical Review D , 82, 065024 , 10.1103/PhysRevD.82.065024 , 1005.0314, 2010PhRvD..82f5024K , 6, The later being in its local version, nothing but the Nambuâ€“JonaLasinio model in which one has included the Polyakov loop effect, in order to describe a 'certain confinement'. Experimental testsThe notion of quark flavors was prompted by the necessity of explaining the properties of hadrons during the development of the quark model. The notion of color was necessitated by the puzzle of the {{SubatomicParticleDelta++}}. This has been dealt with in the section on the history of QCD.The first evidence for quarks as real constituent elements of hadrons was obtained in deep inelastic scattering experiments at SLAC. The first evidence for gluons came in three jet events at PETRA.Several good quantitative tests of perturbative QCD exist:
Crossrelations to solid state physicsThere are unexpected crossrelations to solid state physics. For example, the notion of gauge invariance forms the basis of the wellknown Mattis spin glasses,D.C. Mattis, Phys. Lett. 56a (1976) 421 which are systems with the usual spin degrees of freedom s_i=pm 1, for i =1,...,N, with the special fixed "random" couplings J_{i,k}=epsilon_i ,J_0,epsilon_k,. Here the Îµi and Îµk quantities can independently and "randomly" take the values Â±1, which corresponds to a mostsimple gauge transformation (,s_ito s_icdotepsilon_iquad,J_{i,k}to epsilon_i J_{i,k}epsilon_k,quad s_kto s_kcdotepsilon_k ,),. This means that thermodynamic expectation values of measurable quantities, e.g. of the energy {mathcal H}:=sum s_i,J_{i,k},s_k,, are invariant.However, here the coupling degrees of freedom J_{i,k}, 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 J0 the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "frustration" at all. This term is a basic measure in spin glass theory.J. Vanninemus and G. Toulouse, J. Phys. C 10 (1977) 537 Quantitatively it is identical with the loop product P_W:,=,J_{i,k}J_{k,l}...J_{n,m}J_{m,i} along a closed loop W. However, for a Mattis spin glass â€“ in contrast to "genuine" spin glasses â€“ the quantity PW never becomes negative.The basic notion "frustration" of the spinglass 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 nonfavorable 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,JOURNAL, Fradkin, Eduardo, 1978, Gauge symmetries in random magnetic systems, Physical Review B, 18, 9, 4789â€“4814, 10.1103/physrevb.18.4789, 1978PhRvB..18.4789F, which also stresses the notion of duality.A further analogy consists in the already mentioned similarity to polymer physics, where, analogously to Wilson Loops, socalled "entangled nets" appear, which are important for the formation of the entropyelasticity (force proportional to the length) of a rubber band. The nonabelian character of the SU(3) corresponds thereby to the nontrivial "chemical links", which glue different loop segments together, and "asymptotic freedom" means in the polymer analogy simply the fact that in the shortwave limit, i.e. for 0leftarrowlambda_wll R_c (where Rc is a characteristic correlation length for the glued loops, corresponding to the abovementioned "bag radius", while Î»w is the wavelength of an excitation) any nontrivial correlation vanishes totally, as if the system had crystallized.A. Bergmann, A. Owen, "Dielectric relaxation spectroscopy of poly[(R)3Hydroxybutyrate] (PHD) during crystallization", Polymer International 53 (7) (2004) 863â€“868, weblink{{dead linkdate=February 2019bot=medic}}{{cbignorebot=medic}}There is also a correspondence between confinement in QCD â€“ the fact that the color field is only different from zero in the interior of hadrons â€“ and the behaviour of the usual magnetic field in the theory of typeII superconductors: there the magnetism is confined to the interior of the Abrikosov fluxline lattice,Mathematically, the fluxline lattices are described by Emil Artin's braid group, which is nonabelian, since one braid can wind around another one. i.e., the London penetration depth λ of that theory is analogous to the confinement radius Rc of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, propto g G^a_mu bar{psi}_i gamma^mu T^a_{ij} psi_j,, on the r.h.s. of the Lagrangian.See also
References{{reflist35em}}Further reading
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