Verify that the Lagrangian for the free electromagnetic field, $$ \mathcal{L}=-\frac{1}{4} F^{\mu v} F_{\mu v} $$ is invariant under the gauge transformation \(A_{\mu} \rightarrow A_{\mu}^{\prime}=A_{\mu}-\partial_{\mu} \chi\).

Lagrangian mechanics is a reformulation of classical mechanics. It provides a powerful framework for analyzing systems by focusing on energy rather than force. The Lagrangian, denoted as \(\mathcal{L}\), is the difference between kinetic energy (T) and potential energy (V):
\(\mathcal{L} = T - V\).
In electromagnetic fields, the Lagrangian is constructed to respect the principles of special relativity and contain terms involving the field tensor and the potentials. The primary goal of using Lagrangian mechanics in field theory is to obtain the equations of motion via the Euler-Lagrange equation. This equation is fundamental in deriving Maxwell’s equations for electromagnetism.

The electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in its vicinity. The components of the electromagnetic field include both electric and magnetic fields, which are interrelated through Maxwell’s equations. This relationship can be compactly described using the electromagnetic field tensor \(F^{\mu \u}\). The field tensor is antisymmetric, meaning \(F^{\mu \u} = -F^{\u \mu}\).

Gauge transformations are changes applied to the potentials that describe a field without altering observable quantities. In electromagnetism, the gauge transformation for the four-potential \(A_\mu\) is given by \[{A_\mu \rightarrow A_\mu' = A_\mu - \partial_\mu \chi}\]\. Here, \(\chi\) is any smooth function. Such transformations leave the electric and magnetic fields unchanged. This means the physical state of the system remains invariant under these transformations, emphasizing the redundancy in the description of potentials. For our exercise, checking this invariance for the electromagnetic Lagrangian confirms that the physics described remains the same under gauge transformation.

The field tensor \(F_{\mu \u}\) encodes the information of both the electric and magnetic fields. It is defined using the four-potential \(A_\mu\) as: \[F_{\mu \u} = \partial_\mu A_\u - \partial_\u A_\mu\].
This expression highlights how the field tensor arises from the derivatives of the potentials. Invariance under gauge transformation can be checked by applying transformations to \(A_\mu\) and then simplifying using the property that partial derivatives commute. This invariance signifies the consistency and robustness of the framework describing electromagnetic fields. Understanding and manipulating the field tensor thus allows for deeper insights into the behavior and properties of electromagnetic fields under various transformations.