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By Stetz A.W.

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The first-order term has eight fields each with two terms. If you were to simply multiply them out there would be 28 = 256 terms, so here’s where Mr. G. C. Wick earns his keep. First notice that the vacuum expectation value of any normal-ordered product of fields vanishes. We need only be concerned with terms in which all eight fields are contracted. Because of the normal-ordered Hamiltonian, we don’t contract the four ϕ(x)’s with one another. The only surviving terms are those in which each of ϕ(x1 ), ϕ(x2 ), ϕ(x3 ), and ϕ(x4 ) is contracted with one of the ϕ(x)’s.

I have also labeled the loop momentum in such as way as to acknowledge this rule. 3) There are three points to be made about this. The first is that the integral over q is ubiquitous whenever there are loops. This is in fact the last of the Feynman rules for this simple scalar field theory. 63 64 CHAPTER 4. 1: The one-loop correction to the propagator. 2: The truncated one-loop diagram. • For every continuous loop there will be one undetermined momentum and an integral over it of the form d4 q .

If there are no loops all the internal momenta are determined uniquely by the external momenta. Diagrams without loops are called tree diagrams. Life is simple in the trees. For every loop, however, there is an integral over an undetermined momentum. These integrals usually diverge and are generally deep trouble. We’ll get to them in the next chapter. 6. THE PROBLEM OF SELF INTERACTIONS 49 • For each internal line with momentum q there is a corresponding factor of i∆F (q) in the scattering amplitude.

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