content_image/312750.jpg content_image/312749.jpg content_image/312748.jpg content_image/312747.jpg
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, 

<img src='content_image/312750.jpg'>

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 

<img src='content_image/312749.jpg'>

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 

<img src='content_image/312748.jpg'>

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 } , 

<img src='content_image/312747.jpg'>

