By Nayak C.

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The sum over k really terminates when ωk is the Debye energy. For the most part, we will not be interested in this energy (see, however, the problem set), so we will drop it. This can be done by introducing the notion of a normal-ordered product, which will be useful later. The normal-ordered product of a set of a†ki ,si ’s and akj ,sj ’s is the product with all of the a†ki ,si ’s to the left and all of the aki ,si ’s to the right. It is denoted by a pair of colons. 39) Since creation operators commute with one another and annihilation operators do as well, we do not need to speficy their orderings.

The expansion in loops is actually an expansion in powers of Planck’s constant, since, as you will show in the problem set each propagator comes with a factor of Chapter 5: Feynman Diagrams and Green Functions (a) (b) (c) + ... 1). 47 Chapter 5: Feynman Diagrams and Green Functions 48 h ¯ and each vertex comes with a factor of 1/¯ h. An L-loop diagram comes with a coefficient of h ¯ L−1 . Turning now to the evaluation of multi-loop diagrams, we find the following trick (due to - you guessed it - Feynman) very useful.

19) Of course, we can rarely evaluate UI (t, t ) exactly, so we must often expand the exponential. 21) Squaring this, we recover Fermi’s Golden Rule. There is a slight subtlety in that the t integral leads to an amplitude proportional to δ(Ei − Ef ). This appears to lead to a transition probability which is proportional to the square of a δ-function. We understand, however, that this is a result of taking the limits of integration to infinity carelessly: the square of the δ-function is actually a single δ-function multiplied by the difference between the initial and final times.