Abstract
Gauge theories play a central role in our current understanding of the fundamental interactions. The weak, electromagnetic and strong interactions are well described by gauge theories. We introduce them in this chapter for the first time. Although we often talk about gauge invariance, or gauge symmetry, these terms are a bit misleading. The gauge symmetry is more a redundancy in the description of the physical degrees of freedom than a symmetry, as will be shown later on. The redundancy is of course very useful because it makes Lorentz invariance and locality explicit, but it is not a symmetry in the same sense as rotations or translations. These theories have an incredible richness and complexity. Many aspects of their dynamics are still poorly understood. In our presentation we just scratch the surface of a deep subject.
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Notes
- 1.
The quantization of the electric charge has another consequence, which is that the gauge transformation of the wave function (4.10) is periodic. Using technical jargon one says that the U(1) gauge group gets compactified (see Appendix B). Although this might seem just a technical point, it has important physical consequences for the production of monopoles in gauge theories.
- 2.
Some basics facts about Lie groups have been summarized in Appendix B.
- 3.
In principle it is also possible that the procedure finishes because some kind of inconsistent identity is found. In this case the system itself is inconsistent as it happens with the Lagrangian \(L(q,\dot{q})=q.\)
- 4.
This constraint can also be obtained from the requirement that \(\pi^{0}=0\) be preserved by the time evolution, \(\{\pi^{0},H\}_{\rm PB}=0.\) A detailed analysis of Maxwell electrodynamics using the general formalism for constrained systems can be found in [9].
- 5.
Although we present for simplicity only the case of SU(2), similar arguments apply to any simple group.
- 6.
The existence of topological sectors in (1 + 1)-dimensional electrodynamics is a consequence of the nontrivial character of the first homotopy group of \(S^{1},\) namely \(\pi_{1}(S^{1})={\mathbb{Z}}.\)
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Álvarez-Gaumé, L., Vázquez-Mozo, M.Á. (2012). Theories and Lagrangians II: Introducing Gauge Fields. In: An Invitation to Quantum Field Theory. Lecture Notes in Physics, vol 839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23728-7_4
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