Classically, the Dirac equation has both positive energy (E = +√(p² + m²)) and negative energy (E = -√(p² + m²)) solutions. Negative energies seem unphysical—you could keep extracting infinite energy!

Dirac's brilliant solution: Interpret negative energy states as antiparticles(positrons) with positive energy moving backward in time, or equivalently, forward in time with opposite charge!

5.3 Naive Quantization Fails!

Let's try to quantize like we did for scalars, using commutators:

$$\hat{\psi}(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_s \left[ \hat{b}_p^s u^s(p) e^{-ip \cdot x} + \hat{d}_p^{s\dagger} v^s(p) e^{ip \cdot x} \right]$$

If we impose commutation relations [b̂, b̂^†] = 1, we get:

❌ Problem: Negative Energy!

The Hamiltonian would be:

$$\hat{H} = \int \frac{d^3p}{(2\pi)^3} E_p \sum_s \left[ \hat{b}_p^{s\dagger}\hat{b}_p^s - \hat{d}_p^s\hat{d}_p^{s\dagger} \right]$$

The second term has the wrong sign! Using [d̂, d̂^†] = 1 gives -E_p contribution → unbounded from below! The vacuum would be unstable.

5.4 The Solution: Anticommutators!

Instead, we impose anticommutation relations:

\begin{align*} \{\hat{b}_p^r, \hat{b}_q^{s\dagger}\} &= (2\pi)^3 \delta^3(\mathbf{p} - \mathbf{q}) \delta^{rs} \\ \{\hat{d}_p^r, \hat{d}_q^{s\dagger}\} &= (2\pi)^3 \delta^3(\mathbf{p} - \mathbf{q}) \delta^{rs} \\ \{\hat{b}_p^r, \hat{b}_q^s\} &= 0, \quad \{\hat{d}_p^r, \hat{d}_q^s\} = 0 \\ \{\hat{b}_p^r, \hat{d}_q^s\} &= 0, \quad \{\hat{b}_p^r, \hat{d}_q^{s\dagger}\} = 0 \end{align*}

where {A, B} = AB + BA is the anticommutator.

Now the Hamiltonian becomes:

$$\hat{H} = \int \frac{d^3p}{(2\pi)^3} E_p \sum_s \left[ \hat{b}_p^{s\dagger}\hat{b}_p^s + \hat{d}_p^{s\dagger}\hat{d}_p^s \right] + \text{const}$$

Perfect! Both terms are positive. The constant is an infinite vacuum energy (like for bosons), removed by normal ordering.

💡Why Anticommutators Work

With anticommutators, {d̂, d̂^†} = 1 means:

d̂ d̂^† + d̂^† d̂ = 1

Rearranging: d̂ d̂^† = 1 - d̂^† d̂

This flips the sign in the Hamiltonian! The "negative energy" term -E_p d̂ d̂^† becomes +E_p d̂^† d̂ after normal ordering. The d̂^† operator now creates antiparticles with positive energy!

5.5 Pauli Exclusion Principle

Anticommutators automatically give us the Pauli exclusion principle!

Consider trying to create two particles in the same state:

\begin{align*} (\hat{b}_p^{s\dagger})^2 |0\rangle &= \hat{b}_p^{s\dagger} \hat{b}_p^{s\dagger} |0\rangle \\ &= -\hat{b}_p^{s\dagger} \hat{b}_p^{s\dagger} |0\rangle \quad \text{(anticommutation)} \\ &= 0 \end{align*}

The state vanishes! You cannot put two fermions in the same state (same p, same spin).

The occupation number for fermions can only be 0 or 1:

$$n_p^s = \hat{b}_p^{s\dagger} \hat{b}_p^s, \quad n_p^s \in \{0, 1\}$$

5.6 Particles and Antiparticles

The quantized Dirac field has two types of excitations:

Particles (electrons)

Created by: b̂†_p,s
Destroyed by: b̂_p,s
Energy: +E_p = +√(p² + m²)
Charge: -e (for electrons)
Spinor: u^s(p)

Antiparticles (positrons)

Created by: d̂†_p,s
Destroyed by: d̂_p,s
Energy: +E_p = +√(p² + m²) (also positive!)
Charge: +e (opposite to particle)
Spinor: v^s(p)

Both have positive energy! The quantum field theory interpretation is:

Dirac Sea → Modern Interpretation:

Old view (Dirac): Vacuum is a "sea" of filled negative energy states. A hole in the sea is a positron.

Modern view (QFT): Vacuum is the state |0⟩ with no particles or antiparticles. Both electrons (b̂^†) and positrons (d̂^†) are excitations above the vacuum, both with positive energy!

5.7 Equal-Time Anticommutation Relations

For the field operators themselves:

\begin{align*} \{\hat{\psi}_\alpha(\mathbf{x}, t), \hat{\psi}_\beta^\dagger(\mathbf{y}, t)\} &= \delta^3(\mathbf{x} - \mathbf{y}) \delta_{\alpha\beta} \\ \{\hat{\psi}_\alpha(\mathbf{x}, t), \hat{\psi}_\beta(\mathbf{y}, t)\} &= 0 \\ \{\hat{\psi}_\alpha^\dagger(\mathbf{x}, t), \hat{\psi}_\beta^\dagger(\mathbf{y}, t)\} &= 0 \end{align*}

where α, β = 1, 2, 3, 4 are spinor indices.

These are the canonical anticommutation relations (CAR) for fermions, analogous to CCR for bosons.

5.8 Dirac Propagator

The Feynman propagator for the Dirac field is:

$$S_F(x - y) = \langle 0 | T\{\hat{\psi}(x) \bar{\hat{\psi}}(y)\} | 0 \rangle$$

In momentum space:

$$\tilde{S}_F(p) = \frac{i(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i\epsilon} = \frac{i(\slashed{p} + m)}{p^2 - m^2 + i\epsilon}$$

where /p = γ^μp_μ is the Feynman slash notation.

Note: This is a 4×4 matrix (in spinor space), not a scalar!

5.9 Bosons vs. Fermions

Scalar Fields vs. Dirac Fields

Key differences between bosonic and fermionic quantization

Aspect	Scalar Field (Bosons)	Dirac Field (Fermions)
Spin	Spin 0	Spin 1/2
Field Components	1 (scalar φ)	4 (spinor ψᵅ)
Statistics	Bose-Einstein	Fermi-Dirac
Algebra	Commutators [â, â†]	Anticommutators {b̂, b̂†}
Exclusion	Multiple occupation allowed	Pauli exclusion: 0 or 1 particle per state
Equation of Motion	Klein-Gordon: (□ + m²)φ = 0	Dirac: (iγᵘ∂ᵤ - m)ψ = 0

Code Example: Dirac Spinor Modes

🐍

Dirac Field Mode Expansion

Visualize electron and positron spinor modes

python

import numpy as np
import matplotlib.pyplot as plt

# Dirac spinor mode decomposition
def dirac_field_mode(x, t, p, m, s, particle_type='particle'):
    """
    Calculate Dirac field contribution from a single mode

    Parameters:
    - x: spatial position
    - t: time
    - p: momentum (3-vector)
    - m: mass
    - s: spin projection (±1/2)
    - particle_type: 'particle' or 'antiparticle'
    """
    E_p = np.sqrt(np.dot(p, p) + m**2)

    # Plane wave
    phase = np.dot(p, x) - E_p * t

    if particle_type == 'particle':
        # Particle mode: positive energy
        spinor = positive_energy_spinor(p, m, s)
        psi = spinor * np.exp(-1j * phase)
    else:
        # Antiparticle mode: uses v spinor
        spinor = negative_energy_spinor(p, m, s)
        psi = spinor * np.exp(1j * phase)  # Note: +i phase for antiparticle

    return psi

def positive_energy_spinor(p, m, s):
    """u spinor for particles (simplified)"""
    E_p = np.sqrt(np.dot(p, p) + m**2)
    # Normalization: u†u = 2m
    if s == 0.5:  # spin up
        return np.array([1, 0, p[2]/(E_p + m), (p[0] + 1j*p[1])/(E_p + m)]) * np.sqrt(E_p + m)
    else:  # spin down
        return np.array([0, 1, (p[0] - 1j*p[1])/(E_p + m), -p[2]/(E_p + m)]) * np.sqrt(E_p + m)

def negative_energy_spinor(p, m, s):
    """v spinor for antiparticles (simplified)"""
    E_p = np.sqrt(np.dot(p, p) + m**2)
    # Normalization: v†v = 2m
    if s == 0.5:  # spin up
        return np.array([p[2]/(E_p + m), (p[0] + 1j*p[1])/(E_p + m), 1, 0]) * np.sqrt(E_p + m)
    else:  # spin down
        return np.array([(p[0] - 1j*p[1])/(E_p + m), -p[2]/(E_p + m), 0, 1]) * np.sqrt(E_p + m)

# Example: electron and positron
m_e = 1  # electron mass (natural units)
p = np.array([0.5, 0, 0])  # momentum in x-direction

x_vals = np.linspace(-10, 10, 200)
t = 0

electron = [dirac_field_mode(np.array([x, 0, 0]), t, p, m_e, 0.5, 'particle')[0] for x in x_vals]
positron = [dirac_field_mode(np.array([x, 0, 0]), t, p, m_e, 0.5, 'antiparticle')[0] for x in x_vals]

plt.figure(figsize=(12, 5))
plt.subplot(1, 2, 1)
plt.plot(x_vals, np.real(electron), label='Re(ψ)')
plt.plot(x_vals, np.imag(electron), label='Im(ψ)', linestyle='--')
plt.title('Electron (particle) mode')
plt.xlabel('x')
plt.legend()
plt.grid(True, alpha=0.3)

plt.subplot(1, 2, 2)
plt.plot(x_vals, np.real(positron), label='Re(ψ)')
plt.plot(x_vals, np.imag(positron), label='Im(ψ)', linestyle='--')
plt.title('Positron (antiparticle) mode')
plt.xlabel('x')
plt.legend()
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("Electron (e⁻) created by b̂†ₚ operator")
print("Positron (e⁺) created by d̂†ₚ operator")
print("Both have positive energy E_p > 0!")

💡 Run with: python script.py

⚠️Common Mistakes to Avoid

❌

Mistake:

Using commutators instead of anticommutators for fermions

🤔

Why it's wrong:

Commutators would give wrong sign in Hamiltonian and violate Pauli exclusion.

✅

Correct approach:

Fermions satisfy {ψ̂, ψ̂†} = 1, NOT [ψ̂, ψ̂†] = 1. Anticommutators {A, B} = AB + BA are essential for the Pauli exclusion principle!

❌

Mistake:

Forgetting that ψ is a 4-component spinor, not a scalar

🤔

Why it's wrong:

Spinor structure is essential for describing spin-1/2 particles and Lorentz transformations.

✅

Correct approach:

The Dirac field ψ = (ψ₁, ψ₂, ψ₃, ψ₄)ᵀ has 4 components. Each component is an independent field operator!

❌

Mistake:

Confusing particle and antiparticle modes

🤔

Why it's wrong:

Both particles and antiparticles have positive energy after proper quantization.

✅

Correct approach:

b̂†ₚ creates particles (positive energy), d̂†ₚ creates antiparticles (positive energy, opposite charge). Both have E > 0 in the final theory!

🎯 Key Takeaways

Dirac field ψ: 4-component spinor describing spin-1/2 particles
Anticommutators {b̂, b̂^†} = 1 (NOT commutators!)
Pauli exclusion: (b̂^†)² = 0 → max 1 fermion per state
Particles: b̂^† creates electrons with u spinors
Antiparticles: d̂^† creates positrons with v spinors
Both have positive energy E_p > 0!
Dirac propagator: S_F(p) = i(/p + m)/(p² - m² + iε)
CAR: {ψ̂, ψ̂^†} = δ³(x - y) at equal times
Next: Why do bosons commute and fermions anticommute?

Quantum Field Theory Course

Quantizing the Dirac Field

🔗Course Connections

📚Prerequisites

Video Lecture

5.1 Review: Classical Dirac Field

5.2 Mode Expansion

💡The Negative Energy Problem

5.3 Naive Quantization Fails!

❌ Problem: Negative Energy!

5.4 The Solution: Anticommutators!

💡Why Anticommutators Work

5.5 Pauli Exclusion Principle

5.6 Particles and Antiparticles

Particles (electrons)

Antiparticles (positrons)

5.7 Equal-Time Anticommutation Relations

5.8 Dirac Propagator

5.9 Bosons vs. Fermions

Scalar Fields vs. Dirac Fields

Code Example: Dirac Spinor Modes

Dirac Field Mode Expansion

⚠️Common Mistakes to Avoid

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

🎯 Key Takeaways