Gauge Transformation/Examples

Examples of Gauge Transformations

Consider the magnetic fields $\mathbf E$ and $\mathbf B$.

They can be written in terms of scalar potential $\phi$ and vector potential $\mathbf A$ as:

$\ds \mathbf E$

$=$

$\ds -\nabla \phi - \dfrac {\partial \mathbf A} {\partial t}$

$\ds \mathbf B$

$=$

$\ds \nabla \times \mathbf A$

These may be changed by means of the gauge transformation:

$\ds \phi$

$\to$

$\ds \phi + \dfrac {\partial \phi} {\partial t}$

$\ds \mathbf A$

$\to$

$\ds \mathbf A - \nabla \phi$

The physical quantities $\mathbf E$ and $\mathbf B$ have not been changed, but the purely mathematical $\phi$ and $\mathbf A$ have.