note-math

May migrate to principal-bundle-connection in the future?

cf. #link(<motivation-of-gauge-field>)[]

There are many possible connections

There are connections that cannot be local flat. For any local region, the connection field cannot be eliminated by changing the gauge.

Similar to #link(<flat-metric-iff-curvature-0>)[]

There exist local bundle coordinates or phases, where the connection is zero $𝐴 = 0$ <==> the curvature is zero $𝐹_{𝑖 𝑖^{'}} = [𝐷_{𝑖}, 𝐷_{𝑖^{'}}] = 0$ where $𝐷 = 𝑑 + 𝐴$ , in coordinates $𝐹_{𝑖 𝑖^{'}} = \partial_{𝑖} 𝐴_{𝑖^{'}} - \partial_{𝑖^{'}} 𝐴_{𝑖} + [𝐴_{𝑖}, 𝐴_{𝑖^{'}}]$

When not exist flat connection coordinate, choose $𝐿^{2}$ minimal curvature, $𝐿^{2}$ based on metric-volume-form

Example

$U (1)$ case. $𝐴$ is $u (1)$ valued, and it is commutative $[𝐴, 𝐴^{'}] = 0$ . In this case $𝐹 = 𝑑 𝐴$ , in coordinates $𝐹_{𝑖 𝑗} = \partial_{𝑖} 𝐴_{𝑗} - \partial_{𝑗} 𝐴_{𝑖}$

Transforming from local flat-connection coordinates $𝐴^{'} = 0$ to general coordinates gives a PDE

𝐴 = 𝑑 𝜃

PDE solvable condition $0 = 𝐹 = 𝑑 𝐴$

From the solution of the PDE $𝐴 = 𝑑 𝜃$ , integration gives $𝜃$ and phase transformation $𝑒^{𝜃}$

electromagnetic-field_(tag)

In $ℝ^{1, 3}$

There are many possible connections; select by minimizing curvature.

Note that $Vol$ means the definition of this action requires spacetime metric

\int 𝑑 Vol | 𝐹 |^{2}

𝑑^{†} 𝑑 𝐴 = 0

In coordinates

\begin{matrix} \sum_{𝑖} \partial_{𝑖} (\partial_{𝑖} 𝐴_{𝑗} - \partial_{𝑗} 𝐴_{𝑖}) & = & 0 \\ or & \sum_{𝑖} \partial_{𝑖} 𝐹_{𝑖 𝑗} & = & 0 \end{matrix}

In spacetime decomposition coordinates

𝐹 = 𝐸_{𝑖} (𝑑 𝑡 \land 𝑑 𝑥^{𝑖}) - 𝐵_{𝑖 𝑗} (𝑑 𝑥^{𝑖} \land 𝑑 𝑥^{𝑗}) with 1 \leq 𝑖 < 𝑗 \leq 3

Of course, this decomposition method is not $SO (1, 3)$ invariant

Question How to make $\sum_{𝑖} \partial_{𝑖} (\partial_{𝑖} 𝐴_{𝑗} - \partial_{𝑗} 𝐴_{𝑖}) = 0$ obviously imply the $div, curl$ form of the electromagnetic field $𝐸, 𝐵$ equations? Maxwell-equation_(tag)

\begin{matrix} div 𝐵 & = & 0 & \partial_{𝑡} 𝐵 + curl 𝐸 & = & 0 \\ div 𝐸 & = & 𝜌 & - \partial_{𝑡} 𝐸 + curl 𝐵 & = & 𝑗 \end{matrix}

Also

\begin{matrix} 𝐸 & = & - (\partial_{𝑡} A + grad ϕ) \\ 𝐵 & = & curl A \end{matrix}

Where $ϕ = 𝐴 . time, A = 𝐴 . space$

Note that $curl, div$ and $ℝ^{3}$ exterior derivative are related to $ℝ^{3}$ Hodge star. $𝑑^{†} \sim ⋆ 𝑑 ⋆$ can also be related to Hodge star. It may be related to the behavior of Hodge star in the spatial $ℝ^{3} \subset ℝ^{1, 3}$ coordinates of spacetime decomposition. $⋆ (𝑑 𝑥^{𝑖} \land 𝑑 𝑥^{𝑗}) = \pm 𝑑 𝑡 \land 𝑑 𝑥^{𝑘}$

Note that Hodge star requires a metric

Using the special specification Lorentz-gauge_(tag) $𝑑^{†} 𝐴 = 0$ , the equation $𝑑^{†} 𝑑 𝐴 = 0$ becomes

0 = (𝑑^{†} 𝑑 + 𝑑 𝑑^{†}) 𝐴 = ∆ 𝐴 = 𝜂^{𝑖 𝑖^{'}} \partial_{𝑖} \partial_{𝑖^{'}} 𝐴