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. Lagrangian The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is where is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by and running from to ; is the gauge covariant derivative; the γμ are Dirac matrices connecting the spinor representation to the vector representation of the Lorentz group. Herein, the gauge covariant derivative couples the quark field with a coupling strength to the gluon fields via the infinitesimal SU(3) generators in the fundamental representation. An explicit representation of these generators is given by , wherein the are the Gell-Mann matrices. The symbol represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, Fμν, in quantum electrodynamics. It is given by: where are the gluon fields, dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b and c running from to ; and fabc are the structure constants of SU(3). Note that the rules to move-up or pull-down the a, b, or c indices are trivial, (+, ..., +), so that fabc = fabc = fabc whereas for the μ or ν indices one has the non-trivial relativistic rules corresponding to the metric signature (+ − − −). 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 Quarks are massive spin- fermions that 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 − or +) and participate in weak interactions as part of weak isospin doublets. They carry global