B.5 Gauge theories

B.5.1 Why gauge invariance?

Gauge invariance is needed in order to embed massless spin-$1$ particles with only two physical DoFs (i.e., two polarizations), like the photon or gluons, into a spin-$1$ Lorentz tensor with 3 DoFs.¹⁷ It also ensures the renormalizability of spin-$1$ fields (a Nobel-prize-winning result of ‘t Hooft in 1971 [444, 445]). The spin-$1$ tensor itself is simply an abstract mathematical convenience, which is redundant up to gauge transformations; only terms that are gauge invariant can be physical.

Why the charade of inventing fields with extra DoFs and then imposing an abstract symmetry to remove them? The purely pragmatic answer is that it has proven the most expedient and precise way to calculate physical observables. In this sense, it is not so dissimilar to using complex numbers to describe oscillating physical phenomena or renormalizing by imposing a cut-off and taking the limit as it goes to infinity. They are all simply mathematical conveniences without necessarily any deeper physical meaning.

A less abstact alternative proposed in the 1960s, for example, was S-matrix theory which aimed to do away with all this QFT mumbo-jumbo and focus directly on the physical observables; however, to quote Weinberg, “it got nowhere with real calculations” [446]. On the other hand, despite its abstruseness, in the end with QFT we simply draw some pretty pictures and can quickly read off extremely sophisticated results (with some heavy caveats).

A more poetic view is that, on top of their practicality, gauge symmetries offer a beautiful and elegant description of the fundamental forces of nature. It is rather amazing that we need only to require a quantum $U(1)$ gauge theory, with the usual physical properties of Lorentz invariance, causality, renormalizability etc., and QED naturally falls out! To quote O’Raifeartaigh, “gauge symmetry introduces all the physical radiation fields in a natural way and determines the form of their interactions, up to a few coupling constants. It is remarkable that this variety of physical fields, which play such different roles at the phenomenological level, are all manifestations of the same simple principle and even more remarkable that the way in which they interact with matter is prescribed in advance.” [447].

This universality can be extended further with the geometric view of gauge theories, which has a strong connection to general relativity (GR). Namely, invariance under gauge transformations in the SM is analogous to invariance under local diffeomorphisms (an external local symmetry) in GR, and gauge fields are themselves connections on their respective gauge groups’ fiber bundles, similar to the Levi-Civita connection between tangent bundles on a manifold.¹⁸ Indeed, this is why the “covariant derivative” below is named so.

Finally, there is the possibility that gauge invariance is simply one of those mysteries of the SM, like flavor and charge quantization, which point to some deeper underlying physics we are yet to uncover. For example, in string theory, gauge invariance can arise naturally in an EFT of massless spin-$1$ particles [448]. Ultimately, these considerations are not particularly relevant to the experimental physics, but after all this is a dissertation for a doctorate of philosophy...