# Matter fields and gauges

Gauge theories associate each point $${x}$$ on the spacetime manifold $${M}$$ with a (usually complex) vector space $${V_{x}\cong\mathbb{C}^{n}}$$, called the internal space. A $${V}$$-valued 0-form $${\vec{\Phi}}$$ on $${M}$$ is called a matter field (AKA particle field). A matter field lets us define analogs of quantities related to a change of frame, as follows.

A basis for each $${V_{x}}$$ is called a gauge, and is the analog of a frame; choosing a gauge is sometimes called gauge fixing. Like the frame, a gauge is generally considered on a region $${U\subseteq M}$$. The analog of a change of frame is then a (local) gauge transformation (AKA gauge transformation of the second kind), a change of basis for each $${V_{x}}$$ at each point $${x\in U}$$. This is viewed as a representation of a gauge group (AKA symmetry group, structure group) $${G}$$ acting on $${V}$$ at each point $${x\in U}$$, so that we have

\displaystyle \begin{aligned}\gamma^{-1}\colon & U\rightarrow G\\ \rho\colon & G\rightarrow GL(V)\\ \Rightarrow\check{\gamma}^{-1}\equiv\rho\gamma^{-1}\colon & U\rightarrow GL(V), \end{aligned}

and if we choose a gauge it can thus be associated with a matrix-valued 0-form or tensor field

$$\displaystyle (\gamma^{-1})^{\beta}{}_{\alpha}\colon U\rightarrow GL(n,\mathbb{C}),$$

so that the components of the matter field $${\Phi^{\alpha}}$$ transform according to

$$\displaystyle \left(\Phi^{\beta}\right)^{\prime}=\gamma^{\beta}{}_{\alpha}\Phi^{\alpha}.$$

Recalling from the section on compact Lie groups that all reps of a compact $${G}$$ are similar to a unitary rep, for compact $${G}$$ we can then choose a unitary gauge, which is defined to make gauge transformations unitary, so $${\check{\gamma}^{-1}\colon U\rightarrow U(n)}$$; this is the analog of choosing an orthonormal frame, where a change of orthonormal frame then consists of a rotation at each point. A global gauge transformation (AKA gauge transformation of the first kind) is a gauge transformation that is the same at every point. If the gauge group is non-abelian (i.e. most groups considered beyond $${U(1)}$$), the matter field is called a Yang-Mills field (AKA YM field).

 Δ The term “gauge group” can refer to the abstract group $${G}$$, the matrix rep of this group within $${GL(V)}$$, the matrix rep within $${U(n)}$$ under a unitary gauge, or the infinite-dimensional group of maps $${\gamma^{-1}}$$ under composition. This last is sometimes called the global gauge group, with $${G}$$ or its reps then called the local gauge group.
 Δ As with vector fields, the matter field $${\vec{\Phi}}$$ is considered to be an intrinsic object, with only the components $${\Phi^{\alpha}}$$ changing under gauge transformations.
 Δ Unlike with the frame, whose global existence is determined by the topology of $${M}$$, there can be a choice as to whether a global gauge exists or not. This is the essence of fiber bundles, as we will see.