we pass to a quantum field theory. Let us now consider canonical quantization of Dirac fields. Quantum theory uses the Hamiltonian to determine the time evolution of a system. In the Heisenberg picture of quan- tum theory, time-dependence is carried by operators governed by the Heisenberg equation of motion, In order to quantize the Dirac fields, we must first know what Hamiltonian operator to use in (1.4). Since the Hamiltonian and Lagrangian are related, we may determine a suitable Lagrangian for the Dirac fields and use this to find the corresponding Hamiltonian. The simplest form of the Lagrangian can be written as where ¯ Q = Q $^{†}$γ $^{0}$. When this Lagrangian is substituted into the Euler-Lagrange equation (1.3) with Q and ¯ Q treated as dynamical fields, the correct equations of motion for Q and ¯ Q result, so this is a suitable Lagrangian for the quark fields. Using the relationship between the Hamiltonian and the Lagrangian along with (1.5), the Hamiltonian for Dirac fields can be shown to be We may now use the Heisenberg equation of motion (1.4) to quantize Dirac fields. Using the Hamiltonian (1.6) and the identity [ A , BC ] = { A , B } C − B { A , C } , 12 [ Q ( x ) , H ] = i$^{∫}$ d $^{3}$y [ Q ( x ) , ¯ Q ( y ) γ 0 ∂ ∂y $_{0}$Q ( y )$^{]}$ = i$^{∫}$ d $^{3}$y ( { Q ( x ) , ¯ Q ( y ) } γ 0 ∂ ∂y $_{0}$Q ( y ) − ¯ Q ( y ) { Q ( x ) , γ 0 ∂ ∂y $_{0}$Q ( y ) }$^{)}$ = i$^{∫}$ d $^{3}$y ( { Q ( x ) , ¯ Q ( y ) } γ 0 ∂ ∂y $_{0}$Q ( y ) − ¯ Q ( y ) γ 0 ∂ ∂y $_{0}${ Q ( x ) , Q ( y ) }$^{)}$ = i ∂ ∂x $_{0}$Q ( x ) . (1.7) H = ∫ d $^{3}$x H = ∫ d $^{3}$x$^{[}$ ∂ L ∂ [ ∂$_{0}$ Q ( x )] ∂$_{0}$Q ( x ) − L$^{]}$ = i$^{∫}$ d $^{3}$x ¯ Q ( x ) γ $^{0}$∂$_{0}$ Q ( x ) . (1.6) L = ¯ Q $^{[}$i / ∂ − m $^{]}$Q , (1.5) i∂$_{0}$Q ( x ) = [ Q ( x ) , H ] . (1.4)