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Mathematics for Physics

35 Gauge Field

A field built from the curvature of a connection that is used to measure the physical strength of a gauge interaction.

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

definition [d] (Gauge Field = Field Tensor = Curvature) From Nash and Sen: the connection \(A_{i}(x)\) is what is called the gauge potential in physics, and the curvature \(F_{ij}\) is called the gauge field or field tensor. The curvature is related to the potential by

  • \(F = dA + A \wedge A\) .

where

  • \(A\) is the gauge potential, also called the connection.
  • \(F\) is the gauge field, also called the curvature.
  • \(d\) is the exterior derivative.
  • \(\wedge\) is the wedge product of forms.

35.1 Elementary Example

35.1.1 Simple

In electromagnetism the gauge group is abelian, so \(A \wedge A = 0\) and the field strength reduces to

\[ F = dA \]

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

where

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

35.1.2 General

In a nonabelian gauge theory the quadratic term is kept.

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

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

where

  • \(h\) is a gauge transformation.
  • \(F\) transforms tensorially, while \(A\) transforms as a connection.