Despite the fact that the intermediate states in ofpt

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Despite the fact that the intermediate states in OFPT are on-shell, we saw that it washelpful to write the answer in terms of a Lorentz 4-vectorkμwithk20representing themomentum of an unphysical, off-shell photon. We were led tokμby combining two diagramswith different temporal orderings, which we called advanced and retarded. It would be nice if wecould getkμwith just one diagram, where 4-momentum is conserved at vertices and so propaga-tors can be Lorentz invariant from the start. In fact we can! That’s what we’ll be doing for therest of the course. As we will see, there is just one propagator in this approach, the Feynmanpropagator, which combines the advanced and retarded propagators into one in a beautifullyefficient way. So we won’t have to keep track of what happens first. This new formalism willgive us a much more cleanly organized framework to address the confusing infinities whichplague quantum field theory calculations. Before finishing OFPT, as additional motivation andfor its important historical relevance, we will heuristically review one such infinity.3InfinitiesHistorically, one of the first confusions about the second-quantized photon field was that theHamiltonianH=integraldisplayd3k(2π)3ωkparenleftbiggakak+12parenrightbigg(32)withωk=vextendsinglevextendsinglekvextendsinglevextendsingleseemed to imply that the vacuum has infinite energyE0=(0|H|0)=12integraldisplayd3k(2π)3vextendsinglevextendsinglekvextendsinglevextendsingle=(33)Fortunately, there is an easy way out of this paradoxical infinity: how do you measure theenergy of the vacuum? You don’t! Only energy differences are measurable, and in these differ-ences thezero-point energy, the energy of the ground state, drops out. This is the basic ideabehind renormalization – infinities can appear in intermediate calculations, but they must dropout of physical observables. This zero-point energy does have consequences, such as the Casimireffect (Lecture III-1) which comes from the difference in zero point energies in different sizeboxes, and the cosmological constant problem, which comes from the fact that energy gravitates.We will come to understand these two examples in detail later in the course, but it makes moresense to start with some less exotic physics.In 1930, Oppenheimer thought to use perturbation theory to compute the shift of the energyof the Hydrogen atom due to the photons. (R. Oppenheimer,Phys Rev.35461, 1930). He gotinfinity and concluded that quantum electrodynamics was wrong.In fact the result is notinfinite but a finite calculable quantity known as the Lamb shift which agrees perfectly withdata. However, it is instructive to understand Oppenheimer’s argument.3.1Oppenheimer and the Lamb shiftUsing old-fashioned perturbation theory we would calculate the energy shift usingΔEn=(ψn|Hint|ψn)+summationdisplaymn|(ψn|Hint|ψm)|2EnEm(34)This is the standard formula from time-independent perturbation theory. The basic problem isthat we have to sum over all possible intermediate states|ψ

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