Modern Physics · Advanced Topics

Introduction to Quantum Field Theory

Quantum field theory (QFT) unifies quantum mechanics with special relativity. Fields replace particles as the fundamental objects; particles are quanta of the field. QFT underlies the Standard Model and is the most precisely tested theory in physics.

Prerequisites

Quantum mechanics (Ch. 20) \cdot Special relativity (Ch. 19) \cdot Lagrangian mechanics (Ch. LA) \cdot

Complex analysis (Ch. CA)

Learning Goals

Quantize the real scalar field using canonical commutation relations and interpret particle states as field quanta.
Derive the Dirac equation from the requirement of a first-order relativistic wave equation and identify its predictions.
$Write the QED Lagrangian, derive the Feynman rules, and compute the tree-level e^{-}\mu^{-} scatte$ ring amplitude.
Explain dimensional regularization and the renormalization procedure for absorbing UV divergences in QED.
Interpret the beta function and describe asymptotic freedom in QCD versus the Landau pole in QED.

QF.1 From Particles to Fields

In non-relativistic quantum mechanics, a single particle is described by a wavefunction ψ(x,t). But in relativistic theory, particle number is not conserved — a high-energy collision can create particle-antiparticle pairs. We need a formalism that allows for variable particle number.

The solution: quantize the field itself. Just as the harmonic oscillator is quantized by replacing x and p with operators satisfying [x̂, p̂] = iℏ, we quantize the field φ(x,t) by imposing canonical commutation relations:

[\phî(x,t), \pî(y,t)] = i\hbar \delta^{3}(x-y) \qquad (canonical commutation relations for fields)

(QF.1)

where π(x,t) = ∂L/∂(∂φ/∂t) is the conjugate momentum density. This promotes the classical field to an operator-valued distribution.

QF.2 The Klein-Gordon Field

The simplest relativistic field: a real scalar field φ(x,t) with Lagrangian density:

L = \frac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2}m^{2}\phi^{2} \qquad (Klein-Gordon Lagrangian density)

(QF.2)

The Euler-Lagrange equations give the Klein-Gordon equation:

(\Box + m^{2})\phi = 0 \qquad where \Box = \partial^{2}/\partialt^{2} - \nabla^{2} (d'Alembertian)

(QF.3)

This is the relativistic wave equation for a spin-0 particle of mass m (in natural units ℏ = c = 1). The dispersion relation: E² = p² + m², i.e., E = √(p² + m²).

Expanding in Fourier modes — each mode is a harmonic oscillator:

\phî(x) = \int d^{3}p/(2\pi)^{3} \times 1/\sqrt(2E_p) \times [â_p e^{ip\cdotx} + â^\dagger_p e^{-ip\cdotx}]

(QF.4)

where â_p and â†_p are annihilation and creation operators satisfying [â_p, â†_q] = (2π)³ δ³(p−q). The vacuum |0⟩ is the state with â_p|0⟩ = 0 for all p. A single particle of momentum p is: |p⟩ = √(2E_p) â†_p |0⟩.

Definition QF.1 — Normal Ordering and Vacuum Energy

The Hamiltonian contains a sum over zero-point energies: H = \int d^{3}p/(2\pi)^{3} \times E_{p} \times (â^\dagger_p â_p

+ ½). The ½ per mode gives a formally infinite vacuum energy — the first encounter with the ultraviolet divergences of QFT.Normal ordering :O: places al

l creation operators left of annihilation operators, dropping the zero-point energy: :H: =

E_{p} \times â^\dagger_p â_p. Physical observables are measured relative to the vacuum, so this subtract

on is physically motivated (though the cosmological constant problem shows it's subtle).

QF.3 The Dirac Field

Spin-½ particles require the Dirac equation. Dirac sought a first-order (in time) relativistic wave equation:

(i\gamma^\mu \partial_\mu - m)\psi = 0 \qquad (Dirac equation)

(QF.5)

where γ^μ are 4×4 gamma matrices satisfying the Clifford algebra{γ^μ, γ^ν} = 2g^(μν)I. The Dirac spinor ψ has 4 components — two spin states each for particle and antiparticle.

The Dirac equation automatically predicts:

1. Spin-½: the spinor structure gives s = ½ with g-factor g = 2 (QED corrects this to g = 2.00231930436..., the most precise prediction in physics).

2. Antiparticles: the negative-energy solutions, reinterpreted via Dirac sea or QFT, give the positron (Anderson, 1932 — first antiparticle detected).

3. Spin-statistics theorem: Dirac fields must be quantized with anticommutators {ψ, ψ†} = δ³(x−y), giving the Pauli exclusion principle. Scalar fields use commutators (Bose-Einstein statistics).

QF.4 Interaction and Feynman Diagrams

Interacting field theories add terms to the Lagrangian. For QED (quantum electrodynamics):

L_{QED} = \psī(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_\mu\nu F^\mu\nu \qquad where D_\mu = \partial_\mu + ieA_\mu

(QF.6)

The covariant derivative D_μ introduces the interaction: the electron (ψ) couples to the photon field (A_μ) with coupling constant e. This is the minimal coupling principle — replacing ∂_μ with D_μ is gauge invariance in action.

Perturbation theory in powers of the fine structure constant α = e²/(4π) ≈ 1/137 generates Feynman diagrams. Each diagram is a term in the perturbation series for the scattering amplitude M:

— Electron propagator: iS_F(p) = i(p̸ + m)/(p² − m² + iε) (Feynman propagator)

— Photon propagator: iD_F^μν = −ig^μν/(k² + iε) (in Feynman gauge)

— Vertex: ieγ^μ (from L_int = −eψ̄γ^μ ψ A_μ)

Example QF.1 — Electron-Muon Scattering (Tree Level)

$Compute the leading-order amplitude for e^{-}\mu^{-} \to e^{-}\mu^{-} scattering via photon exchange$ .

Diagram:One photon exchanged (t-channel):

one vertex on electron line, one on muon line, connected by photon propagator. This is O(\alpha

4\pi\alpha

Amplitude:

M = (ū(p_{3})(-ie \gamma^\mu) u(p_{1})) \times (-ig_\mu\nu/(q^{2})) \times (ū(p_{4})(-ie \gamma^\nu) u(p_{2})) where q = p_{1} - p_{3} is t

he momentum transfer.

Simplify:

M = ie^{2}(ū(p_{3})\gamma^\mu u(p_{1}))(ū(p_{4})\gamma_\mu u(p_{2}))/q^{2}

Cross section:|

M|^{2} averaged/summed over spins using trace techniques: Tr[(p̸_{1}+m)\gamma^\mu(p̸_{3}+m)\gamma^\nu] \times \cdots The

result gives the Mott scattering cross section, reducing to Rutherford at low energy.

QED precision:

Higher-order corrections (loops) give \alpha/\pi \approx 0.23% corrections. QED predictions tested to 1

0 significant figures — no other theory matches this precision.

QF.5 Renormalization

Loop diagrams in QFT contain integrals over all momenta that diverge in the ultraviolet (high k → ∞). This is not a disaster — it means the theory requires a cutoffΛ (the scale where new physics enters), and physical quantities must be expressed in terms of measured (renormalized) parameters, not bare (divergent) ones.

Renormalization procedure:(1) Regularize: dim-reg replaces d⁴k with d^(4−ε)k, turning ∞ into 1/ε poles. (2) Absorb divergences into counterterms: δm, δZ, δe. (3) Fix counterterms by renormalization conditions (measured mass, charge at some scale μ). (4) Predict everything else — finite, unambiguous.

A theory is renormalizable if only finitely many counterterms are needed. QED, QCD, and the electroweak theory are all renormalizable (proved by 't Hooft, 1971 — Nobel Prize 1999). Gravity is non-renormalizable — this is why quantum gravity is hard.

Theorem QF.1 — Running Coupling Constants

Renormalization introduces a scale \mu; physical predictions must be \mu-independent. This giv

es the renormalization group equation:

\mu d\alpha/d\mu = \beta(\alpha) \qquad (beta function

In QED: \beta(\alpha) = +2\alpha^{2}/(3\pi) + \cdots > 0 — coupling grows at high energy (Landau pole at unreach

ably high scale). In QCD: \beta(g) = -(11-2n_f/3)g^{3}/(16\pi^{2}) < 0 for n_{f} < 16 — coupling shrinks

at high energy (asymptotic freedom,

Nobel 2004). Unification: \alpha_{1}, \alpha_{2}, \alpha_{3} all meet at ~10^{16} GeV in supersymmetric extensions —

int of grand unification.

QF.6 Path Integrals

Feynman's path integral formulation provides an elegant route to QFT. The vacuum-to-vacuum amplitude (generating functional):

Z[J] = \int D\phi exp(i \intd^{4}x (L + J\phi)) \qquad (path integral / generating functional)

(QF.7)

All correlation functions follow by functional differentiation: ⟨φ(x₁)...φ(xₙ)⟩ = (−i)^n δ^n Z/δJ(x₁)...δJ(xₙ)|_(J=0). For a free field, Z is a Gaussian integral — exactly computable. Interactions are treated perturbatively by expanding e^(iL_int) and applying Wick's theorem to evaluate the Gaussian integrals — reproducing Feynman diagrams.

In Euclidean space (t → −iτ), the path integral Z = ∫Dφ e^(−S_E) resembles a partition function in statistical mechanics with S_E playing the role of βH. This QFT ↔ stat mech correspondence is deep: phase transitions and critical phenomena are described by the same renormalization group as QFT.

Definition QF.2 — Common Traps

Fields are primary in QFT: particles are excitations of fields.
Gauge choice is not physics: observables must be gauge invariant.
Renormalization tracks scale dependence: infinities are handled by measured parameters and running couplings.
Vacuum is not empty: it is the lowest-energy field state with fluctuations.

Exercises — QF.1–QF.6 Quantum Field Theory

Show that the plane wave \phi = e^{-ip\cdotx} satisfies the Klein-Gordon equation if and only if

E^{2} = |p|^{2} + m^{2}. What is the non-relativistic limit

Straightforward

Derive the Casimir force between two parallel plates separated by distance a = 1 \mum. What

is the force per unit area? How is this related to zero-point energy?

Intermediate

Analyze the spontaneous symmetry breaking for V(\phi) = -\mu^{2}\phi^{2}/2 + \lambda\phi^{4}/4. Find the vacuum expe

ctation value, the mass of the Higgs-like mode, and identify the Goldstone boson.

Intermediate

Explain why the electron's g-facto

r differs from 2 in QED. What diagram contributes at lowest order? Why is the result g/2 =

reatest triumph of theoretical physics?

Challenging

Key Takeaways

QFT quantizes fields, not particles. Particles are quanta of field modes. Variable particle number arises naturally.
$Klein-Gordon: (\Box+m^{2})\phi = 0 for spin-0. Dirac: (i\gamma^\mu\partial_\mu-m)\psi = 0 for spin-\frac{1}{2}. Both follow from$ relativistic Lagrangians.
$Spin-statistics theorem: integer spin \to bosons (commutators), half-integer \to fermions (ant$ icommutators).
$Feynman rules: propagators, vertices, and loop integrals from the Lagrangian. Tree level =$ classical limit.
Renormalization: absorb UV divergences into measured parameters. QED, QCD, electroweak all renormalizable.
$Running couplings: QED \alpha grows (Landau pole); QCD \alpha_s shrinks (asymptotic freedom). \beta func$ tion governs the flow.
$Path integrals: Z[J] = \intD\phi e^{iS+iJ\phi}. QFT ↔ stat mech: Euclidean path integral = partitio$ n function.