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.

An Illustrated Handbook