The Electromagnetic Field

The electromagnetic field is a gauge field with gauge group U(1). For the purpose of our approach it is useful to describe the EM charge as a combination of two different charges:

The EM charge Q is, now, a simple linear combination of these two charges: \[ Q = 2I_B + (I_3-\frac12)\]

The point of this decomposition is that the two parts appear in a different way in our model:

Correspondence between strong plus EM fields and U(3)

Nonetheless, the EM field seems in its most important properties more close to the strong interaction. In particular, the photon is, like the gluons of strong interaction, massless, while the W- and Z-bosons of weak interaction are massive.

But there is also a more subtle connection between the combination of \(SU(3)_c\) and \(U(1)_{EM}\), on the one hand, and the group of Wilson gauge fields which has been found to be U(3) on the other hand. The point is that the combination of the SM gauge groups \(SU(3)_c\) of the strong interaction and \(U(1)_{EM}\) of the EM interaction is not exactly \(SU(3) \times U(1)\). There is a subgroup \(\mathbb{Z}_3\) of three elements of \(SU(3) \times U(1)\) which acts trivially on all fermions. If we factor out this trivially acting subgroup, we obtain \[ (SU(3)_c\times U(1)_{EM})/\mathbb{Z}_3 \cong U(3) \]

as the group which really describes the combination of strong and EM fields.

But this group is U(3), thus, the same group found to be the maximal possible Wilson gauge group of the model. The only difference between these two gauge actions is that the EM part acts differently, has the additional \(I_3-\frac12\) part in its charge.