The Field Lagrangian and Gauge Invariance

The Principle of Least Action for Fields

Classical mechanics is built on the principle of stationary action. For a single particle, we minimise the action \( S = \int L \, dt \). For a field, which is a function of space and time, we generalise this to the action \( S = \int \mathcal{L} \, d^4x \), where \( \mathcal{L} \) is the Lagrangian density. The physical state of the electromagnetic field is the configuration that makes this integral stationary. This is a powerful unification: instead of solving four coupled differential equations (Maxwell’s equations) directly, we define a single density function. The equations of motion then emerge naturally as the condition for the action to be invariant under small variations of the field. This variational principle ensures that the resulting laws are automatically consistent with special relativity, as the action is a Lorentz scalar.

Constructing the Gauge-Invariant Lagrangian

To construct \( \mathcal{L} \), we must respect two constraints: Lorentz invariance and gauge invariance. The field strength tensor \( F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \) is the only object built from the potential \( A_\mu \) that is invariant under the gauge transformation \( A_\mu \to A_\mu + \partial_\mu \chi \). The simplest Lorentz scalar formed from \( F_{\mu\nu} \) is \( F_{\mu\nu}F^{\mu\nu} \). Thus, the free-field part of the Lagrangian density is proportional to \( -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu} \). The interaction with matter is described by the coupling term \( -J^\mu A_\mu \), where \( J^\mu \) is the four-current. The total Lagrangian density is therefore:

Notice that \( \mathcal{L} \) depends on \( A_\mu \) and its first derivatives. This structure is crucial because it allows us to apply the Euler–Lagrange equation for fields, \( \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu A_\nu)} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 \).

Recovering Maxwell’s Equations

Applying the Euler–Lagrange equation to our Lagrangian yields the inhomogeneous Maxwell equations. The term \( \frac{\partial \mathcal{L}}{\partial A_\nu} \) is simply \( -J^\nu \). The term involving the derivative requires careful index manipulation. Since \( F_{\mu\nu} \) is antisymmetric, the variation leads to \( \partial_\mu F^{\mu\nu} \). Setting the variation to zero gives:

This single tensor equation encapsulates both Gauss’s law and the Ampère–Maxwell law. The homogeneous equations, \( \nabla \cdot \mathbf{B} = 0 \) and \( \nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = 0 \), are not derived from the action but are identities satisfied by the definition of \( F_{\mu\nu} \) in terms of \( A_\mu \). Specifically, the Bianchi identity \( \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \) holds automatically because \( F_{\mu\nu} \) is defined as a curl of the potential. This distinction is vital: the inhomogeneous equations are dynamical laws, while the homogeneous equations are kinematic constraints.

Gauge Invariance and Charge Conservation

The most profound consequence of this formulation is the link between symmetry and conservation. Consider the action \( S = \int \mathcal{L} \, d^4x \). Under a local gauge transformation \( A_\mu \to A_\mu + \partial_\mu \chi \), the field strength \( F_{\mu\nu} \) remains unchanged. However, the interaction term changes by \( -\int J^\mu \partial_\mu \chi \, d^4x \). Integrating by parts, this becomes \( +\int (\partial_\mu J^\mu) \chi \, d^4x \) (ignoring boundary terms). For the action to be invariant for arbitrary \( \chi \), we must have \( \partial_\mu J^\mu = 0 \). This is the continuity equation. Thus, local gauge invariance of the action is equivalent to the local conservation of electric charge. This is a specific instance of Noether’s second theorem, which states that local symmetries imply identities among the equations of motion.