Canonical quantization is a fundamental procedure in quantum field theory (QFT) used to transition from classical field descriptions to their quantum counterparts. For fermion fields, this process involves promoting classical fields to operators and imposing specific commutation or anti-commutation relations that reflect their fermionic nature.

The starting point for quantizing fermion fields is their classical description, typically governed by the Dirac Lagrangian. This Lagrangian describes spin-1/2 particles, such as electrons and quarks, and is invariant under Lorentz transformations. The field itself is a multi-component spinor, denoted by $\psi(x)$.

The Dirac Lagrangian density is given by:

$\mathcal{L} = i \bar{\psi}(x) \gamma^\mu \partial_\mu \psi(x) - m \bar{\psi}(x) \psi(x)$

where $\psi$ is the Dirac spinor, $\bar{\psi} = \psi^\dagger \gamma^0$, $\gamma^\mu$ are the Dirac matrices, and $m$ is the mass of the fermion.

To apply canonical quantization, we first need to identify the canonical conjugate momenta associated with the field. The momentum conjugate to $\psi(x)$ is defined as:

$\pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \psi(x))}$

Calculating this derivative from the Dirac Lagrangian yields:

$\pi(x) = i \psi^\dagger(x)$

A crucial distinction for fermions is that they obey Fermi-Dirac statistics, which implies that their field operators must satisfy equal-time anti-commutation relations. This is a direct consequence of the spin-statistics theorem. The canonical anti-commutation relations are:

$\left\{ \psi_a(\mathbf{x}, t), \pi_b(\mathbf{y}, t) \right\} = i \delta_{ab} \delta^{(3)}(\mathbf{x} - \mathbf{y})$

and

$\left\{ \psi_a(\mathbf{x}, t), \psi_b(\mathbf{y}, t) \right\} = 0$

$\left\{ \pi_a(\mathbf{x}, t), \pi_b(\mathbf{y}, t) \right\} = 0$

where $a$ and $b$ are spinor indices, and $\{\cdot, \cdot\}$ denotes the anti-commutator: $\{A, B\} = AB + BA$. These relations ensure that the theory correctly describes particles that obey the Pauli exclusion principle.

The Dirac field operator can be expanded in terms of creation and annihilation operators. For a free Dirac field, this expansion is:

$\psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_{s} \left( b_s(p) u_s(p) e^{-ip \cdot x} + d_s^\dagger(p) v_s(p) e^{ip \cdot x} \right)$

where $b_s^\dagger(p)$ and $d_s^\dagger(p)$ are creation operators for particles and antiparticles, respectively, $b_s(p)$ and $d_s(p)$ are annihilation operators, $u_s(p)$ and $v_s(p)$ are Dirac spinors for positive and negative energy solutions, and $E_p = \sqrt{|\mathbf{p}|^2 + m^2}$.

The anti-commutation relations for these operators are:

$\left\{ b_s(p), b_r^\dagger(q) \right\} = (2\pi)^3 \delta_{sr} \delta^{(3)}(\mathbf{p} - \mathbf{q})$

$\left\{ d_s(p), d_r^\dagger(q) \right\} = (2\pi)^3 \delta_{sr} \delta^{(3)}(\mathbf{p} - \mathbf{q})$

All other anti-commutators involving $b, b^\dagger, d, d^\dagger$ are zero. This structure naturally leads to the Pauli exclusion principle, as applying a creation operator twice results in zero: $b_s^\dagger(p) b_s^\dagger(p) = 0$.

The Hamiltonian density for the Dirac field can be derived from the Lagrangian. After performing the canonical quantization and expressing it in terms of creation and annihilation operators, the Hamiltonian for a free fermion field is:

$H = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} \sum_{s} E_p \left( b_s^\dagger(p) b_s(p) + d_s^\dagger(p) d_s(p) \right) + \text{Zero-point energy}$

The terms $b_s^\dagger(p) b_s(p)$ and $d_s^\dagger(p) d_s(p)$ represent the number operators for particles and antiparticles, respectively. The presence of both positive and negative energy solutions in the classical theory, and their interpretation as particles and antiparticles through canonical quantization, is a hallmark of relativistic quantum mechanics and QFT.