1.3.1 Canonical Quantization of Quark Fields QCD begins with quantizing spin-1 / 2 fermion fields that represent quarks. These satisfy the Dirac equation, where the quark field Q ( x ) is a complex four-component spinor field and we are using natural units (see Appendix A). The set of four matrices γ µ satisfy the algebra { γ µ , γ $^{ν}$} = 2 g $^{µν}$. A peculiarity of the Dirac equation is that it permits both positive and negative energy solu- tions for free particles. Negative energy solutions represent antiparticles, which are identical in every way to their particle counterparts, except that they have opposite electric charge. Particles and antiparticles can interact to annihilate one another and particle-antiparticle pairs can be created spontaneously. When interactions are included, the statistical interpre- tation of non-relativistic quantum theory cannot be applied to the Dirac equation. The solution to this problem is to reinterpret the Dirac equation as a field equation, rather than a single particle wave equation, and then quantize the field. In this way, a consistent quantum field theory that incorporates interactions can be constructed. The dynamics of fields are governed by the principle of least action, where the action is defined as where L ( Q, ∂$_{µ}$Q ) is the Lagrangian density, which is commonly referred to as the Lagrangian. The principle of least action states that as the field Q ( x ) evolves in spacetime it does so in a way that minimizes the action (1.2). It can be shown that in order to satisfy the principle of least action, the field must satisfy the Euler-Lagrange equation, Note that this must be satisfied by each distinct field in a given Lagrangian. Given the Lagrangian for a field, the equations of motion for the field can be determined using (1.3). It is important to emphasize that at this stage the fields are still classical quantities. Only when the fields have been reinterpreted as operators that satisfy an appropriate algebra will 11 ∂ L ∂Q − ∂$_{µ}$ ( ∂ L ∂ ( ∂$_{µ}$Q )$^{)}$ = 0 . (1.3) S = ∫ d $^{4}$x L ( Q, ∂$_{µ}$Q ) , (1.2) $^{[}$i / ∂ − m $^{]}$Q ( x ) = 0 , / ∂ = γ µ ∂ ∂x µ , (1.1)