In this chapter, we describe the most general Lagrangian that is both Lorentz invariant and renormalizable. This is the Lagrangian of the Standard Model that describes the interaction of all known particles with all known forces, except for gravity. The group of internal symmetries of the Standard Model is U(1) × SU(2) × SU(3). All particles predicted by the Standard Model have been detected experimentally except for one, the Higgs boson, which plays an essential role, as we will see below. The elementary matter particles are the leptons (electron, electron neutrino, muon, muon neutrino, tau, and tau neutrino and the corresponding antiparticles), the quarks in six different flavors (up, down, charmed, strange, top, bottom) and each in three different colors (red, blue, green), and their antiparticles. The interaction carriers are the photon for the electromagnetic field, three bosons associated with the weak interaction, corresponding to the internal symmetry U(1) × SU(2), and eight gluons associated with the strong interaction, corresponding to the group SU(3) (each also coming in three different colors), plus the hypothetical Higgs boson, which is the only one not yet detected experimentally.

We emphasize that the Standard Model as presented here is a semi-classical model. After the appropriate Lagrangian is written down, it is still necessary to quantize it. No one knows so far how to do this in a mathematically rigorous, constructive way. The next best thing is to use the methods of perturbative QFT (Chapter 7) and renormalization (Chapter 8).