In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in quantum electrodynamics – the gluon field constructs the gluon field strength tensor. Throughout this article, Latin indices take values 1, 2, ..., 8 for the eight gluon color charges, while Greek indices take values 0 for timelike components and 1, 2, 3 for spacelike components of four-dimensional vectors and tensors in spacetime. Throughout all equations, the summation convention is used on all color and tensor indices, unless explicitly stated otherwise.
Introduction Gluons can have eight colour charges so there are eight fields, in contrast to photons which are neutral and so there is only one photon field. The gluon fields for each color charge each have a "timelike" component analogous to the electric potential, and three "spacelike" components analogous to the magnetic vector potential. Using similar symbols:
A n ( r , t ) = [ A 0 n ( r , t ) ⏟ timelike , A 1 n ( r , t ) , A 2 n ( r , t ) , A 3 n ( r , t ) ⏟ spacelike ] = [ ϕ n ( r , t ) , A n ( r , t ) ] {\displaystyle {\boldsymbol {\mathcal {A}}}^{n}(\mathbf {r} ,t)=[\underbrace {{\mathcal {A}}_{0}^{n}(\mathbf {r} ,t)} _{\text{timelike}},\underbrace {{\mathcal {A}}_{1}^{n}(\mathbf {r} ,t),{\mathcal {A}}_{2}^{n}(\mathbf {r} ,t),{\mathcal {A}}_{3}^{n}(\mathbf {r} ,t)} _{\text{spacelike}}]=[\phi ^{n}(\mathbf {r} ,t),\mathbf {A} ^{n}(\mathbf {r} ,t)]}
where n = 1, 2, ... 8 are not exponents but enumerate the eight gluon color charges, and all components depend on the position vector r of the gluon and time t. Each A α a {\displaystyle {\mathcal {A}}_{\alpha }^{a}} is a scalar field, for some component of spacetime and gluon color charge. The Gell-Mann matrices λa are eight 3 × 3 matrices which form matrix representations of the SU(3) group. They are also generators of the SU(3) group, in the context of quantum mechanics and field theory; a generator can be viewed as an operator corresponding to a symmetry transformation (see symmetry in quantum mechanics). These matrices play an important role in QCD as QCD is a gauge theory of the SU(3) gauge group obtained by taking the color charge to define a local symmetry: each Gell-Mann matrix corresponds to a particular gluon color charge, which in turn can be used to define color charge operators. Generators of a group can also form a basis for a vector space, so the overall gluon field is a "superposition" of all the color fields. In terms of the Gell-Mann matrices (divided by 2 for convenience),
t a = λ a 2 , {\displaystyle t_{a}={\frac {\lambda _{a}}{2}}\,,}
the components of the gluon field are represented by 3 × 3 matrices, given by:
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