Gauge invariance 1 Introduction

Up until now, we have dealt with general features of quantum field theories. For example, we have seen how to calculate scattering amplitudes starting from a general Lagrangian. Now we will begin to explore what the Lagrangian of the real world could possibly be. Then, of course, we will discuss what it actually is, or at least what we have figured out about it so far. A good way to start understanding the consistency requirements of the physical universe is with a discussion of spin. There is a deep connection between spin and Lorentz invariance that is totally obscure if you just work in non-relativistic quantum mechanics. For example, well before quantum field theory, it was known from atomic spectroscopy that the electron had two spin states. It was also known that light had two polarizations. The polarizations of light are easy to understand at the classical level, if light is a field, but how can an individual photon be polarized? For the electron, we can at least think of it as a spinning top, so there is a classical analogy, but light is massless, so what exactly is spinning? And can we actually predict from first principles the size of the electrons magnetic dipole moment? The answers to all these questions follow from an understanding of Lorentz invariance and the requirements of a consistent quantum field theory.