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I understand on a surface level that there are these matrices that generate the group SU(3). However, when reading books on gauge theory they appear to make the jump from SU(3) having 8 generators to there are 8 gluons. What is the connection between SU(3) and quantum chromodynamics?

~thanks

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Let $U$ denote a matrix in the usual representation of the $\text{SU}(3)$ Lie algebra. Just as we can motivate electromagnetism to localise a global symmetry of a scalar, in QCD the transformation $\phi\to U\phi$ of a scalar-multiplet $\phi^a$ with $a\in\{1,\,2,\,3\}$ is a global symmetry we seek to localise. The Lagrangian density's kinetic terms use the gauge-covariant derivative $D_\mu:=\partial_\mu-ig \lambda_a A_\mu^a$ with Gell-Mann matrices $\lambda_a$, so that for local $U$ the transformation $\phi\to U\phi$ obtains $D_\mu\phi\to UD_\mu\phi$. In particular the $\lambda_a$ form a basis of the Lie algebra. In theory QCD could be $\text{U}(3)$ theory instead, by just adding one more $\lambda_a$ and one more vector $A_\mu^a$, but empirical evidence says that's wrong. The gluons $A_\mu^a$ therefore number $8$ instead of $9$. (The hypothetical ninth "gluon" of a $\text{U}(3)$ alternative would be a photon-like colour singlet, which we don't observe.)

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