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Mathematics for Physics

54 Gauge Field

A field built from the curvature of a connection that is used to measure the physical strength of a gauge interaction, where a connection is a potential that tells how to compare internal frames at nearby points.

The curvature formula. The gauge field is the curvature of the gauge potential. This principle is used to obtain the physical field strength from \(A\).

The curvature is related to the potential by

\[ F = dA + A \wedge A \]

where

  • \(A\) is the gauge potential.
  • \(F\) is the gauge field.
  • \(d\) is the exterior derivative.
  • \(\wedge\) is the wedge product of forms.

The abelian reduction \(F=dA\). In electromagnetism the gauge group is abelian, so \(A \wedge A = 0\) and the field strength reduces to \(dA\). An abelian group is a group whose elements commute. This principle is used to recover \(\mathbf{B}=\nabla\times\mathbf{A}\) and \(F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\).

The abelian field strength is

\[ F = dA \]

where

  • \(A\) is the electromagnetic potential.
  • \(F\) is the electromagnetic field strength.

The tensorial transformation of \(F\). Under a gauge transformation the field strength transforms tensorially. This principle is used to keep \(F\) physically meaningful while \(A\) changes as a connection.

The adjoint transformation of the field strength is

\[ F' = h F h^{-1} \]

where

  • \(h\) is a gauge transformation.
  • \(F\) is the gauge field.

Note: Also called the field strength. Also called the field tensor. Also called the curvature of the gauge potential.