From Special Relativity to Feynman Diagrams.pdf

We have two sources of infinities in the above

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We have two sources of infinities in the above expression: One given by the δ ( 4 ) ( 0 ) factor due to the absence of external lines in the diagram, implying that the matrix element is proportional to the four-dimensional volume in space–time; The other infinity shows up in the double integral which is ultraviolet divergent in p as well as in q . Actually we may simply ignore this diagram along with all vacuum-vacuum transition amplitudes, of any order in the perturbative expansion. For example at fourth order we may have the vacuum diagrams in Fig. 12.12 a. To show that the sum of all these diagram is physically irrelevant, we recall that the S -matrix elements describe the evolution of a state vector from t = −∞ to t = +∞ in the inter- action picture (it is a mapping between asymptotic free-particle states). Under this transformation the vacuum state must remain invariant. Let us denote by C = 0 | S | 0 , the sum of all the vacuum–vacuum transitions to all orders in perturbation theory. Conservation of the four-momentum p μ implies that the S -matrix can only map the vacuum state, which has p μ = 0 , into itself. Therefore S | 0 = C | 0 = 0 | S | 0 | 0 .
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12.7 Divergent Diagrams 507 Fig.12.12 a Fourth order vacuum–vacuum transition; b disconnected fourth order graph for the Compton scattering However unitarity of the S -matrix implies 0 | SS | 0 = 0 | CC | 0 = 0 | 0 = 1 | C | 2 = 1 . The conclusion is that C is just a phase factor and can be disregarded. Note that each Feynman diagram can be accompanied by a set of vacuum graphs. For example at fourth order we may have the disconnected graph in 12.12b and so on at any order in perturbation theory. Since in any disconnected diagram the S -matrix element is the product of the matrix elements of the disconnected parts, we conclude that the constant C appears as an overall multiplicative phase factor in the S -matrix. If S is the S -matrix with all the disconnected diagrams omitted CS is the full S -matrix differing from S by a trivial phase factor. It follows that all the disconnected Feynman diagrams can be omitted in studying the perturbative expansion. 12.8 A Pedagogical Introduction to Renormalization In Sect.12.7 we have shown that the last three diagrams of Fig. 12.10 are expressed in terms of divergent integrals. Aside from the divergences associated with vacuum– vacuum transitions (which, as we have seen, can be disregarded because their effect is of multiplying any S -matrix element by a same phase factor), the divergence associated with the photon self-energy transitions (vacuum polarization) was shown to vanish on the grounds of gauge invariance. On the other hand, the divergence associated with the electron self-energy graph was found to be somewhat “serious” in that there seems to be no simple and consistent way to eliminate it.
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