6 - Composite operators

Summary

In Chapter 2 we met a number of equations involving products of field operators at the same point. Examples are given by the equations of motion (2.1.10) and the Ward identities (2.7.6). These products we will call composite operators. When computed directly they have ultra-violet divergences: the product ø(x)ø(y) makes unambiguous sense if x is not equal to y, but if x equals y then we have ø(x)², which diverges. Since the equations of motion and the Ward identities express fundamental properties of the theory, it is useful to construct finite, renormalized composite operators with which to express these same properties.

It could be argued that there is no need to have renormalized equations of motion. One could say that one only actually needs the equations of motion in the regulated theory, where they are finite. A situation of practical importance where we actually do need renormalized composite operators is the operator-product expansion, to be discussed in Chapter 10. This is used in a phenomenological situation such as deep-inelastic scattering (Chapter 14) where we wish to compute the behavior of a Green's function when some of its external momenta get large. Equivalently, we need to know how a product of operators, like ø(x)ø(y), behaves as x → y.