57 Gauge Transformations

A change of the electromagnetic potentials that leaves the physical fields unchanged that is used to exploit freedom in choosing \(V\) and \(\mathbf{A}\).

The gauge transformation of \(V\) and \(\mathbf{A}\). Adding the gradient of a scalar to \(\mathbf{A}\) and subtracting the time derivative of that scalar from \(V\) leaves \(\mathbf{E}\) and \(\mathbf{B}\) unchanged. This principle is used to pass from one allowed pair of potentials to another.

A gauge transformation is

\[ \mathbf{A}' = \mathbf{A} + \nabla\lambda \]

\[ V' = V - \dfrac{\partial\lambda}{\partial t} \]

where

\(\lambda\) is an arbitrary scalar function of position and time.
\(V\) and \(\mathbf{A}\) are the original potentials.
\(V'\) and \(\mathbf{A}'\) are the new potentials.
\(t\) is time.

The invariance of \(\mathbf{B}\). In magnetostatics the curl of a gradient vanishes, so \(\mathbf{B}\) is automatically invariant. This principle is used to check the transformation on \(\mathbf{B}\) alone.

The invariance of \(\mathbf{B}\) is

\[ \mathbf{B}' = \nabla\times\mathbf{A}' = \nabla\times\mathbf{A} = \mathbf{B} \]

where

\(\mathbf{B}\) is the magnetic field.
\(\mathbf{A}\) is the vector potential.

The geometric reading as a change of fiber frame. The same freedom is a local change of frame in the fibers of a bundle. A bundle is a space that assigns an internal space to each point of spacetime. This principle is used to identify the electromagnetic gauge transformation with the geometric one.

Note: Also called a gauge transformation of the potentials.

57.1 References

Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — gauge transformations of \(V\) and \(\mathbf{A}\).
Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory. Basic Books, 2017. — gauge invariance of the vector potential.
Frankel, T. The Geometry of Physics. Cambridge University Press, 2012. — gauge transformation as change of fiber frame.

56 Gauge Theory

58 Hall Effect