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The covariant derivative (1.24) leads to the quark-gluon interaction term (1.21) discussed earlier. Also, notice that the definition of the gluon field strength tensor (1.25) leads to self- interactions among gluon fields. Interactions between gauge fields with a universal coupling are a distinguishing feature of non-Abelian gauge theories. 

## 1.3.4 Path Integral Quantization of Gluon Fields 

Although the Lagrangian (1.25) contains interactions, the gluon field still has to be quantized. The methods of canonical quantization that was used in Section 1.3.1 to quantize quark fields are ill-suited for this purpose. Instead, we will utilize the path integral to quantize the gluon field. The discussion in this section closely follows that of Ref. [111]. 

The generating functional for a quantum field φ is defined as 

<img src='content_image/317575.jpg'>

where L = L ( φ , ∂$_{µ}$φ ) is the Lagrangian for the field φ . The integration in (1.27) is over the space of configurations of the field φ . An integral of this form is called a path integral. The path integral is a functional, that is, a function that acts upon functions and returns numbers. The term J ( x ) in the exponential is known as a source term. It is useful to define the functional derivative 

<img src='content_image/317574.jpg'>

It can be shown that correlation functions involving the field φ can be calculated as functional derivatives of the generating functional [100]. For example, 

<img src='content_image/317573.jpg'>

where Z$_{0}$ = Z [ J = 0]. Correlation functions involving more φ fields can be calculated simply by calculating more functional derivatives of the generating functional. Note that the field φ has been quantized: the path integral can be used to calculate correlation functions of the field φ , which are quantum mechanical amplitudes. This procedure generalizes to any 

