The term gauge transformation refers to two related notions in this context. Let $P$ be a principal $G$-bundle over a manifold $M$, and let $\cup_i U_i$ be a cover of $M$. A connection on $P$ is specified by a collection of $\mathfrak{g}=\mathrm{Lie}(G)$ valued 1-forms $\{A_i\}$ defined in each patch $\{U_i\}$, together with $G$-valued functions $g_{ij} : U_i \cap U_j \to G$ on each double overlap, such that overlapping gauge fields are related by

$$A_j = g_{ij} A_i g_{ij}^{-1} + g_{ij} \mathrm{d} g_{ij}^{-1}.\tag{1}$$

The transition functions must also satisfy the cocycle condition on triple overlaps, $g_{ij}g_{jk}g_{ki} =1$. This is the first notion of a gauge transformation, relating local gauge fields on overlapping charts.

Second, there is a notion of gauge equivalence on the space of connections. Two connections $\{ A_i, g_{ij} \}$ and $\{A_i',g_{ij}'\}$ are called gauge-equivalent if there exist $G$-valued functions $h_i : U_i \to G$ defined on each patch such that
$$A_i' = h_i A_i h_i^{-1} + h_i \mathrm{d}h_i^{-1} ~~\text{and}~~ g_{ij}' = h_j g_{ij} h_i^{-1}\tag{2}$$

In terms of the globally defined connection 1-form $\omega$ on $P$, the local gauge fields $\{A_i\}$ are defined by choosing a collection of sections $\{\sigma_i\}$ on each patch of $M$. The local gauge fields are obtained by pulling back the global 1-form, $A_i = \sigma_i^* \omega$. On overlapping patches, such pullbacks are related by (1). On the other hand, the choice of sections was arbitrary; a different collection of sections $\{\sigma'_i\}$ related to the first by $\sigma'_i = \sigma_i h_i$ leads to the gauge-equivalence (2).

Given a map $f: M \to M'$ between two manifolds and a bundle $P'$ over $M'$, we obtain a bundle over $M$ by pullback, $f^* P'$. Moreover, the pullback bundle depends only on the homotopy class of $f$. Suppose we have a contractible manifold $X$. By definition, there exists a homotopy between the identity map $\mathbf{1}:X \to X$ and the trivial map $p: X \to X$ which takes the entire manifold to a single point $p\in X$. Let $P$ be a bundle over $X$. The identity pullback of course defines the same bundle, $\mathbf{1}^* P = P$. On the other hand, the pullback $p^* P$ is a trivial bundle; it maps the same fiber above $p$ to every point on $X$. But the bundles $\mathbf{1}^*P$ and $p^*P$ are equivalent since $\mathbf{1}$ and $p$ are homotopic maps. Thus, a bundle over a contractible space is necessarily trivial (i.e. a direct product).

In particular, a $G$-bundle over $\mathbb{R}^4$ is trivial, whether $G$ is abelian or non-abelian. The cover $\cup_i U_i$ has a single chart, $\mathbb{R}^4$ itself. There is a single gauge field $A$, which is a globally defined $\mathfrak{g}$-valued 1-form. It is obtained from the 1-form $\omega$ on $P$ by pullback, $A = \sigma^* \omega$, where $\sigma$ is a globally defined section. Picking another section $\sigma' = \sigma g(x)$ produces a gauge-equivalent connection, related to $A$ by the usual gauge transformation law given above.

For more details, see e.g. Nakahara "Topology, Geometry, and Physics," chapter 10.

This post imported from StackExchange Physics at 2016-07-07 18:16 (UTC), posted by SE-user user81003