Pre-Quantum Electrodynamics

• Gr 10.1.2

When rewriting our fields into potentials, we passed from six unknowns (the three components of the electric and magnetic fields) to four (the scalar potential, and the three components of the vector potential).

There is however a certain ambiguity left when representing fields in terms of potentials. Namely, it is easy to verify that new fields \[ \phi^\prime = \phi - \frac{\partial \lambda}{\partial t}, \hspace{10mm} {\boldsymbol A}^\prime = {\boldsymbol A} + {\boldsymbol \nabla} \lambda \] give the same \({\boldsymbol E}\) and \({\boldsymbol B}\) fields as \(\phi, {\boldsymbol A}\) for any scalar function \(\lambda ({\boldsymbol r}, t)\).

This very interesting fact that electrodynamics is invariant under the gauge choice is called gauge freedom. Going from one choice to another is done by implementing a gauge transformation.