Quantum Field Theory Course

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

Part I, Chapter 1 | Page 2 of 8

Euler-Lagrange Equations for Fields

Deriving the field equations from the action principle

1.4 Euler-Lagrange Equations for Fields

The action for a field theory is:

$$S[\phi] = \int d^4x \, \mathcal{L}(\phi(x), \partial_\mu \phi(x))$$

where the Lagrangian density $\mathcal{L}$ depends on the field φ and its first derivatives $\partial_\mu \phi$.

Variation of the Action

Consider a small variation of the field:

$$\phi(x) \to \phi(x) + \epsilon \eta(x)$$

where ε is infinitesimal and η(x) is an arbitrary function that vanishes at spatial infinity and at the time boundaries: η(x) → 0 as |x| → ∞ and at t = t₁, t₂.

The variation of the action is:

$$\delta S = \epsilon \int d^4x \left[\frac{\partial \mathcal{L}}{\partial \phi}\eta + \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\mu \eta\right]$$

Integration by Parts

For the second term, integrate by parts:

$$\int d^4x \, \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\mu \eta = \int d^4x \, \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \eta\right) - \int d^4x \, \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\right) \eta$$

The first term is a total derivative. By the divergence theorem, it becomes a surface integral at infinity, which vanishes since η(x) → 0 at the boundaries.

Thus:

$$\delta S = \epsilon \int d^4x \, \left[\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\right)\right] \eta(x)$$

Principle of Stationary Action

The physical field extremizes the action: δS = 0 for all variations η(x). Since η is arbitrary, the integrand must vanish:

$$\boxed{\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\right) = 0}$$

These are the Euler-Lagrange equations for fields. This is a partial differential equation (PDE) for φ(x).

1.5 Multiple Fields

For a theory with multiple fields φa(x) (a = 1, 2, ..., N), the Lagrangian density is:

$$\mathcal{L} = \mathcal{L}(\phi_a, \partial_\mu \phi_a)$$

Each field satisfies its own Euler-Lagrange equation:

$$\frac{\partial \mathcal{L}}{\partial \phi_a} - \partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi_a)}\right) = 0, \quad a = 1, \ldots, N$$

This gives a system of N coupled PDEs.

1.6 Example: Real Scalar Field

Consider the simplest relativistic field theory—a real scalar field φ(x) with Lagrangian density:

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

where m is the mass parameter. Expanding:

$$\mathcal{L} = \frac{1}{2}(\partial_t \phi)^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2$$

This has the structure: kinetic term - gradient term - mass term.

Deriving the Equation of Motion

Compute the necessary derivatives:

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

The Euler-Lagrange equation becomes:

$$-m^2 \phi - \partial_\mu \partial^\mu \phi = 0$$

Rearranging:

$$\boxed{(\partial_\mu \partial^\mu + m^2)\phi = (\Box + m^2)\phi = 0}$$

This is the Klein-Gordon equation, the relativistic wave equation for a free massive scalar field. It generalizes the non-relativistic Schrödinger equation. Here □ = ∂μ∂μ is the , which is Lorentz invariant.

Physical Interpretation

Expanding the d'Alembertian:

$$\frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi + m^2 \phi = 0$$

This describes wave propagation with:

  • Dispersion relation: ω² = k² + m²
  • Energy: E = ω (in natural units)
  • Momentum: p = k
  • Relativistic energy-momentum relation: E² = p² + m²

For m = 0 (massless), we recover the standard wave equation $\Box \phi = 0$.

Key Concepts (Page 2)

  • • Euler-Lagrange equations: $\partial \mathcal{L}/\partial \phi - \partial_\mu(\partial \mathcal{L}/\partial(\partial_\mu \phi)) = 0$
  • • Each field component gets its own equation of motion
  • • Klein-Gordon equation: (□ + m²)φ = 0 for free scalar field
  • • Dispersion relation E² = p² + m² emerges naturally
  • • Integration by parts crucial for deriving field equations
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