# 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.