Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part I, Chapter 1 | Page 8 of 8

Interacting Fields and Summary

Beyond free fields: interactions and the road ahead

1.29 Beyond Free Fields: Interactions

So far we've studied free field theories—fields that don't interact with each other. Real physics requires interactions.

Self-Interacting Scalar Field

The simplest interacting theory adds a potential V(φ) to the scalar Lagrangian:

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

Common interaction terms:

  • φ3 theory: $V(\phi) = \frac{g}{3!}\phi^3$
    Used in statistical field theory, not stable in 4D
  • φ4 theory: $V(\phi) = \frac{\lambda}{4!}\phi^4$
    The standard model of scalar interactions, renormalizable in 4D
  • Sine-Gordon: $V(\phi) = \frac{m^2}{\beta^2}(1 - \cos(\beta \phi))$
    Exactly solvable in 2D, has soliton solutions

The equation of motion becomes nonlinear:

$$\Box \phi + m^2 \phi + V'(\phi) = 0$$

For φ4 theory:

$$\Box \phi + m^2 \phi + \frac{\lambda}{6}\phi^3 = 0$$

1.30 Yukawa Interaction: Fermion-Scalar Coupling

The Yukawa interaction couples a Dirac fermion ψ to a scalar field φ:

$$\mathcal{L}_{\text{int}} = -g\bar{\psi}\psi \phi$$

The full Lagrangian is:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m_f)\psi + \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m_b^2 \phi^2 - g\bar{\psi}\psi \phi$$

where mf is the fermion mass and mb is the boson (scalar) mass.

Physical Interpretation

  • The Yukawa term allows fermions to emit/absorb scalar particles
  • This mediates a force between fermions (Yukawa potential in position space)
  • In the Standard Model, the Higgs field couples to fermions via Yukawa interactions, giving them mass
  • The coupling constant g determines the interaction strength

The classical potential mediated by scalar exchange is:

$$V(r) = -\frac{g^2}{4\pi r}e^{-m_b r}$$

This is the Yukawa potential, exponentially screened at distances r \gtrsim 1/mb.

1.31 Quantum Electrodynamics (QED)

The most important interacting field theory is Quantum Electrodynamics—the theory of electrons and photons.

The QED Lagrangian

$$\boxed{\mathcal{L}_{\text{QED}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi - e\bar{\psi}\gamma^\mu \psi A_\mu}$$

This consists of three parts:

  • Maxwell term: -¼FμνFμν (free photon)
  • Dirac term: $\bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi$ (free electron)
  • Interaction: $-e\bar{\psi}\gamma^\mu \psi A_\mu$ (electron-photon coupling)

where e is the electric charge (coupling constant).

Local Gauge Invariance

QED has a profound property: it's invariant under local U(1) gauge transformations:

$$\psi(x) \to e^{ie\Lambda(x)}\psi(x), \quad A_\mu \to A_\mu + \partial_\mu \Lambda(x)$$

Note that Λ now depends on spacetime! This local gauge invariance completely determines the form of the interaction.

QED can be derived by demanding that the free Dirac Lagrangian be invariant under local U(1) transformations. This requires introducing the gauge field Aμ and replacing ∂μ → Dμ = ∂μ + ieAμ (covariant derivative).

1.32 Yang-Mills Theory: Non-Abelian Gauge Fields

QED is an Abelian gauge theory (U(1) group is commutative). The Standard Model requires non-Abelian gauge theories.

For a non-Abelian gauge group G (e.g., SU(2), SU(3)), we have multiple gauge fields Aaμ (a = 1, ..., dim(G)):

$$F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc}A^b_\mu A^c_\nu$$

where fabc are the structure constants of the Lie algebra. The Yang-Mills Lagrangian:

$$\mathcal{L}_{\text{YM}} = -\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}$$

Unlike QED, the field strength is nonlinear in the gauge field. This leads to:

  • Gluon self-interactions: Gauge bosons interact with themselves
  • Asymptotic freedom: Coupling decreases at high energy (crucial for QCD)
  • Confinement: Quarks cannot be isolated (in QCD)

The Standard Model

The Standard Model is based on the gauge group:

$$SU(3)_C \times SU(2)_L \times U(1)_Y$$
  • SU(3)C: Quantum Chromodynamics (QCD), strong force, 8 gluons
  • SU(2)L × U(1)Y: Electroweak theory, W±, Z0, photon
  • Higgs mechanism: Spontaneous symmetry breaking gives mass to W, Z, and fermions

1.33 The Road Ahead

We've established the classical foundations of field theory. The journey to quantum field theory involves:

Part I (Classical Field Theory) - Remaining Topics:

  • Noether's theorem in detail
  • Complex scalar fields and internal symmetries
  • Electromagnetic field theory
  • Dirac field and spinors
  • Discrete symmetries (C, P, T)
  • Energy-momentum tensor and angular momentum

Part II: Canonical Quantization

  • Quantum mechanics of fields
  • Creation and annihilation operators
  • Fock space and particle interpretation
  • Causality and microcausality
  • Quantizing the Dirac field (anticommutators, fermion statistics)
  • Quantizing the electromagnetic field (photons, gauge fixing)

Part III: Interacting Fields

  • Interaction picture and S-matrix
  • Wick's theorem and normal ordering
  • Feynman diagrams and rules
  • Cross sections and decay rates
  • QED processes and calculations

Part IV: Path Integrals

  • Path integral formulation
  • Generating functionals
  • Perturbation theory from path integrals
  • Gauge field path integrals
  • Effective actions and quantum corrections

Part V-VIII: Advanced Topics

  • Renormalization theory
  • Non-Abelian gauge theories (Yang-Mills)
  • Spontaneous symmetry breaking and Higgs mechanism
  • Standard Model of particle physics
  • Anomalies and topology
  • Quantum field theory in curved spacetime
  • Introduction to supersymmetry and string theory

Chapter 1 Summary

In this chapter, we established the Lagrangian formulation of classical field theory, the foundation for all of quantum field theory.

Key Results:

  • • Action principle: S = ∫d4x ℒ(φ, ∂μφ)
  • • Euler-Lagrange equations: Field equations from δS = 0
  • • Klein-Gordon: (□ + m2)φ = 0 for free scalar field
  • • Hamiltonian formalism: (φ, π) phase space, canonical quantization
  • • Noether's theorem: Continuous symmetries → conserved currents
  • • Energy-momentum tensor: Tμν from translation invariance
  • • Maxwell theory: ℒ = -¼FμνFμν, gauge invariance
  • • Dirac field: Spin-1/2 fermions, positive probability
  • • Interactions: φ4, Yukawa, QED, Yang-Mills

These classical field theories form the starting point for quantization. In the next chapters, we'll explore symmetries in detail, then proceed to canonical quantization where fields become operators and particles emerge as quantum excitations.

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