From Special Relativity to Feynman Diagrams.pdf

24 actually the treatment of the aforementioned

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24 Actually the treatment of the aforementioned divergences was given for matrix elements out | S ( 2 ) | in of S ( 2 ) between initial and final single-particle states obeying 24 Note that the problem of the electron self-energy already exists in the classical theory of the electron. Indeed, either one assumes the electron to be a point particle without structure, in which case the total energy of the electron together its associated field is infinite; or one assumes a finite electron radius, in which case it should explode as a consequence of the internal charge distribution.
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508 12 Fields in Interaction the equations of motion of the free theory (on-shell particles), namely 25 ( p m ) u ( p , s ) = 0 , (12.255) k 2 = 0 ; ε · k = 0 . (12.256) Photon and electron self-energies are just an example of diagrams containing loops . As already mentioned, the presence of loops in a Feynman diagram entails an inte- gration over the momentum k of the virtual particle circulating in the loop, and this, in general, implies an ultraviolet divergence of the integral when the k → ∞ . Thus, when we consider higher order terms in the perturbative expansion many more ultraviolet divergences (actually infinitely many) show up in the computation of the S -matrix. This tells us that the Feynman rules for the computation of the amplitudes are in some sense incomplete since they do not tell us what to do with divergent inte- grals when computing amplitudes beyond the lowest tree-level. It turns out, however, that if we express amplitudes in terms of the physical measurable parameters of the theory, namely the mass and the coupling constant, the amplitudes become finite. Letusmakethisstatementmoreprecise.Itmustbeobservedthatwhenweconsider higher order terms in perturbation theory, the parameters m and e appearing in the Lagrangian do not represent the experimental values of mass and coupling constant, as it was anticipated in the introduction. For example the electron experimental mass is defined as the expectation value of the Hamiltonian (the energy operator) when the one-particle electron state has zero three-momentum. This has to be computed, to the order of precision required, using m exp = p , s | ˆ H | p , s p , s | p , s p = 0 , | p , s and ˆ H being the states and Hamiltonian operator of the complete interacting theory, the former being perturbatively expressed in terms of free states in ( 12.3 ). Notice that in no situation an electron state can be identified with a free state in the Fock space, of the kind we have been using so far in our analysis: the higher order terms in the expansion ( 12.3 ) are always present. The reason for this is that an electron is never isolated since it always interacts at least with its own electromagnetic field, and its self interaction contributes to the perturbative expansion ( 12.3 ).
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