$$\phi(x,t) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} \left[a_k e^{-i(k \cdot x - \omega_k t)} + a_k^* e^{i(k \cdot x - \omega_k t)}\right]$$

where the dispersion relation is:

$$\omega_k = \sqrt{\mathbf{k}^2 + m^2}$$

This is Einstein's energy-momentum relation E² = p² + m²! Each mode k is like a relativistic particle.

2.2 Mode Expansion of Field Operator

To quantize, we promote the classical field to an operator and the coefficients a_k become operators:

$$\hat{\phi}(\mathbf{x},t) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} \left[\hat{a}_k e^{i\mathbf{k} \cdot \mathbf{x}} e^{-i\omega_k t} + \hat{a}_k^\dagger e^{-i\mathbf{k} \cdot \mathbf{x}} e^{i\omega_k t}\right]$$

Here:

â_k is the annihilation operator - destroys a particle with momentum k
â_k^† is the creation operator - creates a particle with momentum k
The 1/√(2ω_k) factor ensures proper commutation relations

💡Why Creation and Annihilation Operators?

The quantum field is like an infinite collection of harmonic oscillators, one for each momentum mode k. Just like the QM harmonic oscillator has â† (creation) and â (annihilation), each field mode has its own pair of operators!

Key insight: Particles are excitations. When we apply â_k^† to the vacuum, we create a particle with momentum k. Apply it n times, we get n particles!

2.3 Commutation Relations

The creation and annihilation operators satisfy:

\begin{align*} [\hat{a}_k, \hat{a}_{k'}^\dagger] &= (2\pi)^3 \delta^3(\mathbf{k} - \mathbf{k}') \\ [\hat{a}_k, \hat{a}_{k'}] &= 0 \\ [\hat{a}_k^\dagger, \hat{a}_{k'}^\dagger] &= 0 \end{align*}

These are exactly like the harmonic oscillator algebra, but with a Dirac delta because we have a continuous spectrum of momentum modes!

Verification:

These commutation relations are derived by requiring that the field operators satisfy the canonical commutation relations [φ̂(x), π̂(y)] = iδ³(x-y). It's a beautiful consistency check!

2.4 Hamiltonian in Terms of Modes

The Hamiltonian can be rewritten in terms of creation/annihilation operators:

$$\hat{H} = \int \frac{d^3k}{(2\pi)^3} \omega_k \left(\hat{a}_k^\dagger \hat{a}_k + \frac{1}{2}(2\pi)^3\delta^3(0)\right)$$

The first term ω_k â_k^†â_k is the number operator times the energy ω_k. The second term is the infamous vacuum energy - infinite!

2.5 Normal Ordering and Vacuum Energy

The vacuum energy diverges: E_vac = ∫ d³k ½ω_k = ∞. We fix this by normal ordering - always putting creation operators to the left:

$$:\hat{H}: = \int \frac{d^3k}{(2\pi)^3} \omega_k \hat{a}_k^\dagger \hat{a}_k$$

This removes the infinite constant and gives E_vac = 0. The colons : : denote normal ordering.

💡Is the Vacuum Energy Real?

The vacuum energy is infinite, but in physics we only measure energy differences, not absolute energies. Normal ordering effectively sets the vacuum energy to zero by convention.

However, the vacuum energy may have real effects! Some believe it's related to dark energy and the cosmological constant Λ in general relativity.

2.6 Particle States and Fock Space

The vacuum state |0⟩ satisfies:

$$\hat{a}_k|0\rangle = 0 \quad \text{for all } \mathbf{k}$$

Single-particle states are created by acting with â_k^†:

$$|\mathbf{k}\rangle = \hat{a}_k^\dagger|0\rangle$$

Multi-particle states:

$$|\mathbf{k}_1, \mathbf{k}_2, ..., \mathbf{k}_n\rangle = \hat{a}_{k_1}^\dagger \hat{a}_{k_2}^\dagger \cdots \hat{a}_{k_n}^\dagger |0\rangle$$

The energy of an n-particle state is:

$$E = \sum_{i=1}^n \omega_{k_i} = \sum_{i=1}^n \sqrt{\mathbf{k}_i^2 + m^2}$$

2.7 Field Mode Visualization

See how different momentum modes contribute to the quantum field:

🌊Scalar Field Visualization

Mode Number: 1

Field Type: real

• Real scalar field: φ(x,t) ∈ ℝ
• Describes neutral spin-0 particles (like Higgs)
• Blue = positive field values, Red = negative
• Higher modes = higher energy states
• Animation shows time evolution of field configuration

2.8 Computational Example

🐍

Free Scalar Field Mode Expansion

Simulate quantum field as superposition of plane wave modes

python

import numpy as np
import matplotlib.pyplot as plt

# Parameters
L = 10.0  # Box size
N = 100   # Number of grid points
m = 1.0   # Mass
hbar = 1.0

# Momentum modes (periodic boundary conditions)
k_max = 5
k_values = 2*np.pi*np.arange(-k_max, k_max+1)/L
x = np.linspace(0, L, N)

# Energy-momentum dispersion relation
omega_k = np.sqrt(k_values**2 + m**2)

# Field operator expansion (classical limit for visualization)
def field_mode(x, t, k, omega):
    """Single mode contribution to field"""
    # φ_k = (1/√(2ω)) [a_k e^(-iωt+ikx) + a_k† e^(iωt-ikx)]
    # Taking real part and setting a_k = 1
    return (1/np.sqrt(2*omega)) * np.cos(omega*t - k*x)

# Plot field at different times
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
times = [0, 1, 2, 3]

for idx, t in enumerate(times):
    ax = axes[idx//2, idx%2]

    # Sum over modes (quantum superposition)
    phi = np.zeros_like(x)
    for k in k_values:
        omega = np.sqrt(k**2 + m**2)
        phi += field_mode(x, t, k, omega)

    ax.plot(x, phi, 'b-', linewidth=2)
    ax.set_xlabel('Position x')
    ax.set_ylabel('φ(x,t)')
    ax.set_title(f'Field at t = {t}')
    ax.grid(True, alpha=0.3)
    ax.axhline(y=0, color='k', linestyle='-', linewidth=0.5)

plt.tight_layout()
plt.savefig('free_scalar_field.png', dpi=150)
plt.show()

# Energy spectrum
plt.figure(figsize=(10, 6))
plt.plot(k_values, omega_k, 'o-', linewidth=2, markersize=8)
plt.xlabel('Momentum k')
plt.ylabel('Energy ω_k')
plt.title('Dispersion Relation: E² = k² + m²')
plt.grid(True, alpha=0.3)
plt.axhline(y=m, color='r', linestyle='--', label=f'Rest mass m={m}')
plt.legend()
plt.show()

print("Energy levels:")
for k, omega in zip(k_values[:5], omega_k[:5]):
    print(f"k = {k:.2f}: ω = {omega:.3f}")

💡 Run with: python script.py

🎯 Key Takeaways

Field operator: φ̂(x) = ∫ d³k [â_ke^ikx + â_k^†e^-ikx]/√(2ω_k)
Dispersion relation: ω_k = √(k² + m²) (relativistic energy-momentum)
Commutators: [â_k, â_k'^†] = (2π)³δ³(k-k')
Hamiltonian: Ĥ = ∫ d³k ω_k â_k^†â_k (normal ordered)
Vacuum: â_k|0⟩ = 0, lowest energy state
Particles: States |k⟩ = â_k^†|0⟩ have definite momentum
Energy: E = nω_k for n particles in mode k
Particles are quanta - discrete excitations of the quantum field!

Quantum Field Theory Course

Free Scalar Field Quantization

🔗Course Connections

📚Prerequisites

Video Lecture

2.1 Classical Solution: Plane Waves