lecture-8-GaugeInvariance - Matthew Schwartz 2012 II-1 Spin...

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Matthew Schwartz 2012 II-1: Spin 1 and 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 is known 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 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 known that 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 are massless 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 questions follow from an understanding of Lorentz invariance and the requirements of a consistent quantum 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. 2 Unitary representations of the Poincaré group Our universe has a number of apparent symmetries that we would like our quantum field theory to 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 fields ψ ( x ) and replace them by ψ ( x + a ) for any constant 4-vector a ν , the observables should look the same. Another symmetry is 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 Lorentz transformations is called the Poincaré group , ISO(1,3) (the iso metry group of Minkowski space). Our universe also has a bunch of different types of particles in it. Particles have mass and spin and all kinds of other quantum numbers. They also have momentum and the value of spin projected 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 do not. So a particle can be defined as a set of states which mix only among themselves under Poincaré transformations.

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