quantum field, provided that the statistics of the field are incorporated. For instance, path integrals involving fermion fields require the use of Grassmann variables [100]. Once the generating function for a quantum field has been defined, the field has been quantized. Now we will construct the generating functional for the gluon field. By analogy with the generating functional for the field φ (1.27), we might guess that the generating functional for the gluon field is given by where L$_{YM}$ denotes the Yang-Mills Lagrangian (1.25). Unfortunately, the integration over the configurations of the gluon field is ill-defined. This is due to the gauge symmetry of L$_{YM}$ . It can be shown that under an infinitesimal gauge transformation, the gluon field transforms as This reflects a redundancy among the configurations of the field A a $_{µ}$, which spoils the definition of the generating functional (1.30). The generating functional given in Eq. (1.30) cannot be used to quantize the gluon fields in its present form. We will use a method introduced by Faddeev and Popov [49] to modify the generating functional so that quantization is possible. The redundancy in the integration over the field A a µ can be removed by introducing the gauge-fixing function where G $^{a}$( x ) = ∂ $^{µ}$Aa µ − ω $^{a}$( x ) for some arbitrary function ω $^{a}$. Using Eq. (1.31), it can be shown that the gauge-fixing function transforms as Z [ J ] = ∫ DA exp$^{[}$ i$^{∫}$ d $^{4}$x $^{[}$L$_{YM}$ + J µ $_{a}$A a µ ] , (1.30) A a $_{µ}$( x ) → ˜ A a $_{µ}$( x ) = A a $_{µ}$( x ) − D ab µ θ $^{b}$( x ) , D ab µ = δ $^{ab}$∂$_{µ}$ − gf $^{abc}$Ac $_{µ}$. (1.31) Z [ J ] = ∫ DA det$^{(}$ δG δθ ) δ ( G ) exp$^{[}$ i$^{∫}$ d $^{4}$x $^{[}$L$_{YM}$ + J µ $_{a}$A a µ ] , (1.32) G $^{a}$( x ) → ˜ G $^{a}$( x ) = G $^{a}$( x ) − ∂ $^{µ}$Dab µ θ $^{b}$( x ) . (1.33) 20