the normal-ordering operator moves all creation operators a s † k ( q ) to the left. For example, This can also be applied to the quark field operators (1.9). A crucial property is that where the dots denote any combination of quantum fields. The quark field operators can be used to determine the amplitude for a quark to propagate between two distinct locations in spacetime. In order to calculate this amplitude we must first define the time-ordering operator, which anticommutes a product of Dirac fields so that the field with the earliest time is the furthest right and the field with the latest time is the furthest left. The time-ordering operator ensures that particles only propagate forward in time. Explicitly calculating Eqn. (1.13) using the expressions for the quark fields (1.9) and their algebra (1.8), it can be shown that where / p = γ $^{µ}$p$_{µ}$ and η → 0 $^{+}$. Equation (1.14) is the Feynman quark propagator, which is also a Green’s function of the Dirac equation (1.1). The iη pole prescription ensures that time-ordering is respected. The Feynman propagator S ( x − y ) is the quantum mechanical amplitude for a quark to travel between the spacetime points y and x , if y 0 < x $^{0}$. The The S -matrix formalism relates physical quantities such as scattering cross sections and decay rates to correlation functions, which are also referred to as Green’s functions or n - 14 1.3.2 Perturbation Theory : a r j ( p ) a s † k ( q ) : = − a s † k ( q ) a r j ( p ) . (1.11) 〈 0 | : . . . : | 0 〉 = 0 , (1.12) T $^{[}$Q ( x ) ¯ Q ( y ) ] =    Q ( x ) ¯ Q ( y ) , x 0 > y 0 − ¯ Q ( y ) Q ( x ) , x 0 < y 0 (1.13) 〈 0 | T [ Q ( x ) ¯ Q ( y ) ] | 0 〉 = S ( x − y ) = i$^{∫}$ d $^{4}$p (2 π ) 4 / p + m p 2 − m 2 + iη e − ip · ( x − y $^{)}$, (1.14)