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Mathematics for Physics

89 U(1) Gauge Theory

An abelian gauge theory with symmetry group \(U(1)\) that is used to describe classical and quantum electrodynamics, where \(U(1)\) is the circle group of phase factors, and where an abelian group is a group whose elements commute.

Electrodynamics as a \(U(1)\) gauge theory. Electrodynamics is defined by a rank-one potential, the four-potential, whose field strength is the electromagnetic field. This principle is used to write Maxwell theory as a \(U(1)\) gauge theory.

The electromagnetic field strength is

\[ F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} \]

where

  • \(F_{\mu\nu}\) is the electromagnetic field tensor.
  • \(A_{\mu}\) is the four-potential.
  • \(\partial_{\mu}\) is the spacetime derivative.

Invariance of \(F_{\mu\nu}\). A \(U(1)\) gauge transformation shifts the potential by a gradient and leaves \(F_{\mu\nu}\) invariant. This principle is used to change \(A_{\mu}\) without changing the physical fields.

The \(U(1)\) gauge transformation is

\[ A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\lambda \]

where

  • \(A_{\mu}\) is the four-potential.
  • \(\lambda\) is the gauge function.

The abelian curvature \(F=dA\). In the geometric language the potential is a \(U(1)\) connection and the field strength is its curvature \(F=dA\). This principle is used to identify electromagnetism with the abelian case of a general gauge theory.

The abelian curvature is

\[ F = dA \]

where

  • \(A\) is the \(U(1)\) connection.
  • \(F\) is the curvature.
  • \(d\) is the exterior derivative.

Note: Also called abelian gauge theory.