08-gaugeinvariance - 253a QFT1 Fall 2008 Matthew Schwartz Lecture 8 Gauge invariance 1 Introduction Up until now we have dealt with general features of

# 08-gaugeinvariance - 253a QFT1 Fall 2008 Matthew Schwartz...

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253a: QFT1 Fall 2008 Matthew Schwartz Lecture 8: 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. Spin should be familiar from quantum mechanics. Spin is a quantum number, labelled by J , which refers to representations of the rotation group SO (3) . Fermions have half integer spin J = 1 2 , 3 2 , and bosons have integer spin J = 0 , 1 , 2 , 3 , . Spin 1 particles are critical to quantum electrodynamics, because the photon is spin 1. But they are also essential to understanding quantum field theory in general. I cannot emphasize enough how important spin 1 fields are – it has hard to do any calculation or study any theoret- ical question in quantum field theory without having to deal with spin 1. In order to understand spin 1, we first have to understand some of the requirements of a consistent quantum field theory. 2 Unitary representations of the Poincare group In quantum field theory, we are interested particles, which transform covariantly under transla- tions and Lorentz transformations. That is, they should form representations of the Poincare group. This means, there is some way to write the Poincare transformation so that | ψ ) → P| ψ ) (1) More explicitly, there is some kind of basis we can choose for our states | ψ ) , call it | ψ i ) so that | ψ i ) → P ij | ψ j ) (2) So when we say representation we mean a way to write the transformation so that when it acts the basis closes in to itself. In addition, we want unitary representations. The reason for this is that the things we com- pute in field theory are matrix elements M = ( ψ 1 | ψ 2 ) (3) which should be Poincare invariant. If M is Poincare invariant, and | ψ 1 ) and | ψ 2 ) transform covariantly under a Poincare transformation P , we find M = (big ψ 1 |P P| ψ 2 )big (4) So we need P P = 1 , which is the definition of unitarity. There are many more representations of the Poincare group than the unitary ones, but the unitary ones are the only ones we’ll be able to compute Poincare invariant matrix elements from. As an aside, it is worth pointing out here that there is an even stronger requirement on phys- ical theories: the S matrix must be unitary. Recall that the S-matrix is defined by | f ) = S | i ) (5) 1
Say we start with some normalized state ( i | i ) = 1 , meaning that the probability for finding any- thing at time t = − ∞ is P = ( i | i ) 2 = 1 . Then ( f | f ) = (big i | S S | i )big . So we better have S S = 1 , or else we end up at t = + with less than we started! Thus unitarity of the S -matrix is equiva- lent to conservation of probability, which seems to be a property of our universe as far as anyone can tell. We’ll see consequences of unitary of S