Free Fields

We have arrived at one of the great intellectual achievements of the twentieth century: quantum field theory (QFT). So far, we have quantized individual particles. But particles are created and destroyed in high-energy collisions. An electron and a positron can annihilate into photons; a photon can create an electron-positron pair. No fixed-particle-number quantum mechanics can handle this. We need a framework where particles themselves emerge dynamically from something deeper, and that something is the quantum field.

From Particles to Fields

In classical mechanics, we have particles with positions $x_i(t)$. In classical field theory, we have fields $\phi(x, t)$ defined at every point in space. Think of a vibrating string: instead of tracking individual atoms, we describe the displacement field $\phi(x,t)$ along the string.

The crucial conceptual leap of QFT is this: particles are not fundamental objects. They are quantized excitations of underlying fields that permeate all of space. An electron is a ripple in the electron field. A photon is a ripple in the electromagnetic field. The field is primary; the particle is secondary.

The Klein-Gordon Field

We begin with the simplest case: a real scalar field $\phi(x,t)$, a field with no internal indices, no spin. The classical Lagrangian density is:

$$\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2}m^2\phi^2$$

where we use natural units ($\hbar = c = 1$) and the metric signature $(+,-,-,-)$. The Euler-Lagrange equation gives the Klein-Gordon equation:

$$(\partial_\mu \partial^\mu + m^2)\phi = 0 \quad \Longleftrightarrow \quad (\Box + m^2)\phi = 0$$

This is the relativistic wave equation for a spin-0 particle of mass $m$. In momentum space, the solutions are plane waves with the relativistic dispersion relation:

$$E^2 = \mathbf{p}^2 + m^2$$

Mode Expansion

The general solution is a superposition of plane waves. We write it as:

$$\phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_\mathbf{p}}} \left( a_\mathbf{p} \, e^{-ip \cdot x} + a_\mathbf{p}^* \, e^{ip \cdot x} \right)$$

where $\omega_\mathbf{p} = \sqrt{\mathbf{p}^2 + m^2}$ is the energy of mode $\mathbf{p}$.

Canonical Quantization

To quantize, we promote $\phi$ and its conjugate momentum $\pi = \partial_0 \phi$ to operators, and impose equal-time commutation relations:

$$[\hat{\phi}(\mathbf{x}, t), \hat{\pi}(\mathbf{y}, t)] = i\delta^{(3)}(\mathbf{x} - \mathbf{y})$$

This is the field-theory analogue of $[\hat{x}, \hat{p}] = i\hbar$. The mode coefficients become operators $\hat{a}_\mathbf{p}$ and $\hat{a}_\mathbf{p}^\dagger$ satisfying:

Fock Space

The Hilbert space of a quantum field is called Fock space. It is built by acting with creation operators on the vacuum state $|0\rangle$:

$$|0\rangle \quad \text{(vacuum: no particles)}$$ $$\hat{a}_\mathbf{p}^\dagger |0\rangle = |\mathbf{p}\rangle \quad \text{(one particle with momentum } \mathbf{p}\text{)}$$ $$\hat{a}_\mathbf{p}^\dagger \hat{a}_\mathbf{q}^\dagger |0\rangle = |\mathbf{p}, \mathbf{q}\rangle \quad \text{(two particles)}$$

Fock space is the direct sum of all $n$-particle Hilbert spaces. It accommodates any number of particles. The particle number operator is:

$$\hat{N} = \int \frac{d^3p}{(2\pi)^3} \hat{a}_\mathbf{p}^\dagger \hat{a}_\mathbf{p}$$

The Hamiltonian

The Hamiltonian for the free Klein-Gordon field is:

$$\hat{H} = \int \frac{d^3p}{(2\pi)^3} \omega_\mathbf{p} \left( \hat{a}_\mathbf{p}^\dagger \hat{a}_\mathbf{p} + \frac{1}{2}(2\pi)^3\delta^{(3)}(0) \right)$$

The second term is the vacuum energy, an infinite sum of zero-point energies from each mode. This infinity is our first encounter with the divergences that permeate QFT.

Normal Ordering and Vacuum Energy

The infinite vacuum energy arises because we summed the zero-point energy $\frac{1}{2}\omega_\mathbf{p}$ of every mode. We remove it by normal ordering: placing all creation operators to the left of annihilation operators. The normal-ordered Hamiltonian is:

$$:\hat{H}: = \int \frac{d^3p}{(2\pi)^3} \omega_\mathbf{p} \, \hat{a}_\mathbf{p}^\dagger \hat{a}_\mathbf{p}$$

This simply counts the total energy of all particles present. The vacuum now has zero energy: $:\hat{H}: |0\rangle = 0$. Whether the discarded vacuum energy has physical significance is a deep question. It reappears in the cosmological constant problem.

The Dirac Field

Electrons are not scalars. They have spin $\frac{1}{2}$. The appropriate field is a four-component spinor $\psi_\alpha(x)$, governed by the Dirac equation:

$$(i\gamma^\mu \partial_\mu - m)\psi = 0$$

where $\gamma^\mu$ are the $4 \times 4$ Dirac matrices satisfying the Clifford algebra:

$$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$$

The Dirac equation was Dirac's attempt to write a first-order relativistic wave equation. Its extraordinary consequence: it predicts antiparticles.

Antiparticles

The mode expansion of the Dirac field contains two types of creation operators:

$$\hat{\psi}(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_\mathbf{p}}} \sum_{s=1,2} \left( \hat{b}_\mathbf{p}^s \, u^s(p) \, e^{-ip \cdot x} + \hat{d}_\mathbf{p}^{s\dagger} \, v^s(p) \, e^{ip \cdot x} \right)$$

Here $\hat{b}_\mathbf{p}^{s\dagger}$ creates a particle (electron) and $\hat{d}_\mathbf{p}^{s\dagger}$ creates an antiparticle (positron) with spin $s$. The spinors $u^s(p)$ and $v^s(p)$ encode the spin degrees of freedom.

The spin-statistics theorem is one of the deepest results in physics: integer-spin particles are bosons (commutation relations, any number can share a state), half-integer-spin particles are fermions (anticommutation relations, exclusion principle). This is not an empirical observation to be accepted. It is a theorem following from relativity and quantum mechanics.

The Electromagnetic Field

The electromagnetic field $A^\mu = (\phi, \mathbf{A})$ is a vector field (spin-1). Its Lagrangian density is:

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the field strength tensor. The equations of motion are the source-free Maxwell equations.

Gauge Invariance and Quantization

The electromagnetic field has a crucial feature: gauge invariance. The transformation $A_\mu \to A_\mu + \partial_\mu \chi$ leaves the physics unchanged. This means $A_\mu$ has redundant degrees of freedom. Not all four components are physical.

A massless vector field in four dimensions has only two physical polarization states, corresponding to the two transverse polarizations of a photon. After fixing a gauge (say, the Coulomb gauge $\nabla \cdot \mathbf{A} = 0$), we quantize:

$$\hat{\mathbf{A}}(\mathbf{x}, t) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_\mathbf{k}}} \sum_{\lambda=1,2} \boldsymbol{\epsilon}_\lambda(\mathbf{k}) \left( \hat{a}_\mathbf{k}^\lambda \, e^{-ik \cdot x} + \hat{a}_\mathbf{k}^{\lambda\dagger} \, e^{ik \cdot x} \right)$$

where $\boldsymbol{\epsilon}_\lambda(\mathbf{k})$ are the two transverse polarization vectors, $\omega_\mathbf{k} = |\mathbf{k}|$ (photons are massless), and:

$$[\hat{a}_\mathbf{k}^\lambda, \hat{a}_\mathbf{q}^{\lambda'\dagger}] = (2\pi)^3 \delta^{\lambda\lambda'} \delta^{(3)}(\mathbf{k} - \mathbf{q})$$

Photons are bosons. Any number can share the same quantum state. This is why lasers work: coherent states of many photons in the same mode.

Summary: The Free Field Zoo

Field	Spin	Statistics	Equation	Particle
Klein-Gordon $\phi$	0	Boson	$(\Box + m^2)\phi = 0$	Scalar (e.g., Higgs)
Dirac $\psi$	1/2	Fermion	$(i\gamma^\mu\partial_\mu - m)\psi = 0$	Electron, quarks
Maxwell $A^\mu$	1	Boson	$\partial_\mu F^{\mu\nu} = 0$	Photon

The Canonical Quantization Procedure

The recipe for quantizing any free field follows a systematic pattern:

1. Write the classical Lagrangian $\mathcal{L}[\phi, \partial_\mu\phi]$
2. Find the conjugate momentum $\pi = \partial\mathcal{L}/\partial(\partial_0\phi)$
3. Solve the classical equations of motion: expand in plane-wave modes
4. Promote to operators: impose canonical (anti)commutation relations
5. Identify creation and annihilation operators from the mode expansion
6. Build Fock space: the vacuum $|0\rangle$ plus all multi-particle states
7. Normal order the Hamiltonian: subtract the vacuum energy

This procedure works for any free field. The real challenge, and the real physics, comes when we add interactions.

Key Insights

• Combining quantum mechanics with special relativity requires quantum field theory, a framework where particle number is not fixed
• Particles are quantized excitations of fields; the field is the fundamental entity
• Each momentum mode of a free field is an independent harmonic oscillator; Fock space is built from their excitations
• The Klein-Gordon equation describes spin-0 particles, the Dirac equation describes spin-1/2 fermions, and Maxwell's equations describe spin-1 photons
• The spin-statistics theorem (bosons commute, fermions anticommute) follows from the consistency of relativistic quantum theory
• Normal ordering removes infinite vacuum energy, but the conceptual question of vacuum energy remains deep and unresolved

Looking Ahead

Free fields are exactly solvable, and exactly boring. Real physics involves interactions: electrons scattering off each other, photons being emitted and absorbed, particles being created and destroyed. In the next lesson, we will introduce interaction terms into the Lagrangian and develop the most powerful tool in theoretical physics: Feynman diagrams.