These give Hamiltonian densities differing from each other only by a constant, so the physics is the same, but the first choice is more convenient, as we will see below. Using this Hamiltonian, the time translation operator to second order was given in Eq. (9.7),

We now reduce this expression for the time translation operator using Wick's theorem. First, note that the instantaneous term can be written ij dt JÏinst(î) = -i]^ y^J\n,t)J\r2,t)

where the insertion of the T product in the second line has no effect but produces a formula more easily compared with the radiation part. The radiation part of the time translation operator, to second order, includes terms where the photon field is contracted. As we have just seen, these reduce to

-i J dixld4x2T[J\xl)J^x2)] (0|T (j4i(x1)v4J (x2)) |0)

Now, if either the incoming or outgoing electron is virtual, it is no longer true that = 0, but it can be proved, to any given order in the electric charge e, that all k^J^ terms cancel, so that we may make the replacement klJl

0 rO


Fig. 11.1 Example of disconnected Feynman diagrams generated by the first term in Eq. (11.30).

The first term, the fully normal-ordered term, has no contractions, and hence the integration over and diX2 can be carried out independently. This generates two independent energy-momentum conserving delta functions, the form of which depends on the process under consideration. Since the term is fully normal ordered, only matrix elements involving a combined presence of six particles in both the initial and final states can contribute. An example of such a process, shown in Fig. 11.1, is e~ + e~ e~ + e~ + 2j. Since there are no contractions, the interactions at x\ and 12 are independent of each other, and the delta functions in this case are

They are zero because a physical electron cannot decay into a real photon and another physical electron. Examination of other processes generated by this interaction shows that the above analysis holds in every case. The first term makes no contribution to any process.

annihilation exchange

Fig. 11.2 Annihilation and exchange diagrams generated by the second term in Eq. (11.30).

annihilation exchange




Note that the M now involves a non-trivial integral over the internal four-momentum of the virtual photon, k, and corresponds to the Feynman diagram shown in Fig 11.4. In this process, the four-momentum of the virtual photon is not constrained, even though energy-momentum is conserved at every vertex. The process illustrates a new Feynman rule:

Rule 5: integrate over each internal four-momentum k not fixed by energy-momentum conservation with a weight equation (11.34) therefore gives, to second order in the electron charge, the following result for the electron self-energy:

We will postpone discussion of how to evaluate such integrals until Sec. 11.6. Now we will discuss the physical significance of the self-energy.

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