Regauging - Classical Gauge Theory - Classical Electromagnetism

Classical Electromagnetism

Historically, the first example of gauge symmetry to be discovered was classical electromagnetism. In electrostatics, one can either discuss the electric field, E, or its corresponding electric potential, V. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, correspond to the same electric field. This is because the electric field relates to changes in the potential from one point in space to another, and the constant C would cancel out when subtracting to find the change in potential. In terms of vector calculus, the electric field is the gradient of the potential, . Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A, with

\begin{align} \mathbf{E} &= -\nabla V - \frac{\partial \mathbf{A}}{\partial t}\\ \mathbf{B} &= \nabla \times \mathbf{A} \end{align}

The general gauge transformations now become not just but

\begin{align} \mathbf{A} &\rightarrow \mathbf{A} + \nabla f\\ V &\rightarrow V - \frac{\partial f}{\partial t} \end{align}

where f is any function that depends on position and time. The fields remain the same under the gauge transformation, and therefore Maxwell's equations are still satisfied. That is, Maxwell's equations have a gauge symmetry.

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