The gauge potential and field strength

We can then define the parallel transporter for matter fields to be a linear map \({\parallel_{C}\colon V_{p}\rightarrow V_{q}}\), where \({C}\) is a curve in \({M}\) from \({p}\) to \({q}\). Choosing a gauge, the parallel transporter can be viewed as a (gauge-dependent) map \({\parallel^{\beta}{}_{\alpha}\colon\left\{ C\right\} \rightarrow GL(n,\mathbb{C})}\). This determines the (gauge-dependent) matter field connection 1-form \({\left(\Gamma_{A}\right)^{\beta}{}_{\alpha}\left(v\right)\colon T_{x}M\to gl(n,\mathbb{C})}\), which can also be written when acting on a \({\mathbb{C}^{n}}\)-valued 0-form as \({\check{\Gamma}_{A}\left(v\right)\vec{\Phi}}\). The values of the parallel transporter are again viewed as a rep of the gauge group \({G}\), so that the values of the connection are a rep of the Lie algebra \({\mathfrak{g}}\), and if \({G}\) is compact we can choose a unitary gauge so that \({\mathfrak{g}}\) is represented by anti-hermitian matrices. We then define the gauge potential (AKA gauge field, four-potential, four-vector potential, vector potential) \({\check{A}}\) by

\(\displaystyle \check{\Gamma}_{A}\equiv-iq\check{A}, \)

where \({q}\) is called the coupling constant (AKA charge, interaction constant, gauge coupling parameter). As is common, we use units in which the physical constant \({\hbar=1}\); in other units, throughout our treatment \({q}\) should be replaced by \({q/\hbar}\). Note that \({A^{\beta}{}_{\alpha}}\) are hermitian matrices in a unitary gauge.

The covariant derivative is then

\(\displaystyle \mathrm{D}_{v}\vec{\Phi}=\mathrm{d}\vec{\Phi}\left(v\right)-iq\check{A}\left(v\right)\vec{\Phi}, \)

which is sometimes called the gauge covariant derivative, and can be generalized to \({\mathbb{C}^{n}}\)-valued \({k}\)-forms as an exterior covariant derivative

\(\displaystyle \mathrm{D}\vec{\Phi}=\mathrm{d}\vec{\Phi}-iq\check{A}\wedge\vec{\Phi}. \)

For a matter field (0-form), this is often written after being applied to \({e_{\mu}}\) as

\(\displaystyle \mathrm{D}_{\mu}\vec{\Phi}=\partial_{\mu}\vec{\Phi}-iq\check{A}{}_{\mu}\vec{\Phi}, \)

where \({\mu}\) is then a spacetime index and \({\check{A}_{\mu}\equiv\check{A}(e_{\mu})}\) are \({gl(n,\mathbb{C})}\)-valued components. This expression is not coordinate-dependent, and we may therefore also write it using an abstract index.

This connection defines a curvature \({\check{R}_{A}\equiv\mathrm{d}\check{\Gamma}_{A}+\check{\Gamma}_{A}\wedge\check{\Gamma}_{A}}\), which lets us define the field strength (AKA gauge field) \({\check{F}}\) by

\(\displaystyle \begin{aligned}\check{R}_{A} & \equiv-iq\check{F}\\ \Rightarrow\check{F} & =\mathrm{d}\check{A}-iq\check{A}\wedge\check{A}. \end{aligned} \)

Δ The definition \({\check{\Gamma}_{A}\equiv-iq\check{A}}\) is the convention with a mostly pluses metric signature; with a mostly minuses signature the sign is reversed. However, one also finds this definition in terms of an elementary charge \({e\equiv\pm q}\), which may be positive or negative depending on convention, again reversing the sign.

An Illustrated Handbook