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Page 83 of Bruce Schumm’s book Deep Down Things: The Breathtaking Beauty of Particle Physics says that the bare charge $ e_{0} $ of an electron is infinite. However, Page 348 of Robert Klauber’s book Student Friendly Quantum Field Theory says that $ e_{0} = 0 $. Klauber’s argument goes like this:

The renormalization of electron charge takes the form of the equation $$ e = \sqrt{Z_{3}} e_{0}, $$ where $ e $ is the physical electron charge and $ Z_{3} $ is a renormalization constant depending on some parameter (which is determined by the regularization scheme being used). We have $ Z_{3} \to \infty $ as this parameter tends to some limit, and in the end, as $ e $ is finite, we obtain $ e_{0} \to 0 $.


My question is thus:

Question. How do we reconcile these two contradictory statements?

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    $\begingroup$ Not sure, myself. I know that the claim of divergent electric charge is related to loop calculations where the polarizability of the vacuum screens the charge of the point particle (that is, loops where charged particle anti-particle pairs are produced). I've never encountered the other claim, so I can't comment. $\endgroup$ – Sean E. Lake Sep 15 '16 at 1:05
  • $\begingroup$ There is usually more than one way to renormaluzate a divergent quantity in QFT and different schemes will have different definitions of the "bare" quantity. Hence, it is not unusual for these schemes to differ in the details as long as the residual finite results agree. $\endgroup$ – Lewis Miller Sep 15 '16 at 2:15
  • $\begingroup$ This difference should be covered by one of the answers already present on this site. Perhaps the basis for the difference goes back to the statement which is determined by the regularization scheme being used ( not renormalization) more on this distinction can be found at en.m.wikipedia.org/wiki/Regularization_(physics) $\endgroup$ – user108787 Sep 15 '16 at 3:53
  • $\begingroup$ Thank you for your comments, everyone. After some more reading and lots of reflection, it appears to me that the determination of the bare electron charge $ e_{0} $ is a meaningless task. One can never measure it, as no physical experiment can access a bare electron by getting past its screen of virtual interactions. One should thus view $ e_{0} $ and $ Z_{3} $ as formal power series in $ e $, with coefficients depending on a parameter that is provided by a regularization scheme. $\endgroup$ – Transcendental Oct 6 '16 at 0:05
  • $\begingroup$ One then computes $ \sqrt{Z_{3}} $ as a formal power series and forms the product power series $ \sqrt{Z_{3}} e_{0} $, which must be identical to $ e $ (viewed as a power series with coefficient in the first power equal to $ 1 $ and coefficients in all other powers equal to $ 0 $). $\endgroup$ – Transcendental Oct 6 '16 at 0:13

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