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Matthew Schwartz2012II-1: Spin 1 and Gauge Invariance1IntroductionUp until now, we have dealt with general features of quantum field theories. For example, wehave seen how to calculate scattering amplitudes starting from a general Lagrangian. Now wewill 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 is known about it so far.A good way to start understanding the consistency requirements of the physical universe iswith a discussion of spin. There is a deep connection between spin and Lorentz invariance thatis obscure 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 knownthat light had two polarizations. The polarizations of light are easy to understand at the clas-sical level since 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 photons aremassless and structureless, so what exactly is spinning? Can we actually predict from first prin-ciples the size of the electron’s magnetic dipole moment? The answers to all these questionsfollowfromanunderstandingofLorentzinvarianceandtherequirementsofaconsistentquantum field theory.Our discussion of spin and the Lorentz group is divided into a discussion of integer spin par-ticles (tensor representations) in this lecture and half-integer spin particles (spinor representa-tions) in Lecture II-3. The associated mathematics is also divided, with the gentler, more phys-ical introduction given here.2Unitary representations of the Poincaré groupOur universe has a number of apparent symmetries that we would like our quantum field theoryto respect. One is that no place in space-time seems any different from any other place. Thus,our theory should be translation invariant: if we take all our fieldsψ(x)and replace them byψ(x+a)for any constant 4-vectoraν, the observables should look the same. Another symmetryis Lorentz invariance: physics should look the same whether we point our measurement appa-ratus to the left or to the right, or put it on a train. The group of translations and Lorentztransformations is called thePoincaré group, ISO(1,3) (theisometry group of Minkowskispace).Our universe also has a bunch of different types of particles in it. Particles have mass andspin and all kinds of other quantum numbers. They also have momentum and the value of spinprojected on some axis. If we rotate or boost to change frame, only the momenta and the spin-projection change, as determined by the Poincaré group, but the other quantum numbers donot. So aparticlecan be defined as a set of states which mix only among themselves underPoincaré transformations.