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I'm trying to understand Weisskopf's 1939 paper 'On the Self-Energy and the Electromagnetic Field of the Electron'. Here Weisskopf demonstrates that in Dirac's positron theory, the self-energy of an electron diverges only logarithmically (Eq. 26), because the quadratically divergent energy due to vacuum fluctuations (equation below Eq. 25) cancels with the quadratically divergent term of energy due to the spin (Eqs. 18, 20).

My question is about the integral in Eq. 18. It seems to suggest that the following holds

$$\int_0^P \frac{p^2}{\sqrt{m^2c^4+p^2c^2}} dp = \frac{1}{c}\left(P\sqrt{P^2+m^2c^2}-\frac{m^2c^2}{2} \ln \frac{P+\sqrt{P^2+m^2c^2}}{mc}\right).$$

However, it seems to me that in the limit of large $p$, the integrand can be approximated as $\frac{p}{c}$, whose integral would yield $\frac{P^2}{2c}$, i.e. a factor two off from Weisskopf's quadratic term. When I ask WolframAlpha or ChatGPT to calculate this integral, they also seem to divide the quadratic term by $2c$ instead of $c$. This factor of two is significant, because it determines whether the quadratic divergences cancel out or not.

So my question is: am I making a careless mistake when I expect that the quadratic term should be $\frac{P^2}{2c}$ instead of $\frac{P^2}{c}$? If not, then how do we still arrive at the result that the quadratic divergences cancel?

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You are correct, that Equation (18) seems to just be wrong.

The usual thing to do is to straight up evaluate this integral. The integral identities to use is rapidity. i.e. Let $p=mc\sinh\chi$ and $Y=\arg\!\sinh\frac P{mc}$ so that $$ \begin{align} \tag1&\hphantom{=\ }\int_0^P\frac{p^2}{\sqrt{m^2c^4+p^2c^2}}\mathrm dp\\ \tag2&=\int_0^Y\frac{(mc\sinh\chi)^2}{mc^2\cosh\chi}mc\cosh\chi\,\mathrm d\chi\\ \tag3&=m^2c\int_0^Y\sinh^2\chi\,\mathrm d\chi\\ \tag4&=m^2c\int_0^Y\frac{\cosh2\chi-1}2\mathrm d\chi\\ \tag5&=\frac12m^2c\left[\frac12\sinh2\chi-\chi\right]_0^Y\\ \tag6&=\frac12m^2c\left[\sinh\chi\cosh\chi-\chi\right]_0^Y\\ \tag7&=\frac12m^2c\left[\frac P{mc}\sqrt{1+\left(\frac P{mc}\right)^2}-\arg\!\sinh\frac P{mc}\right]\\ \tag8&=\frac1{2c}\left[P\sqrt{P^2+m^2c^2}-m^2c^2\ln\left(\frac P{mc}+\sqrt{\left(\frac P{mc}\right)^2+1}\right)\right]\\ \tag9&=\frac1{2c}\left[P\sqrt{P^2+m^2c^2}-m^2c^2\ln\frac{P+\sqrt{P^2+m^2c^2}}{mc}\right] \end {align} $$ It thus seems like Equation (18) of the paper is just wrong. But maybe the argument follows through, being consistently wrong in such a way that the cancellation proceeds just fine, and the error cancels out in the end.

Also, unless you are trying to understand the history, you should simply ignore the early papers. The pioneers were plentifully confused, and it would be rather easy to be infected by their confusion. You are much better off reading and making sense of modern treatments, after they have finally made sense of the general ideas and have a unified treatment suitable for comparison with experiment. After all, only when one needs to compare with experiment does one need to dot all the i and cross all the t.

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    $\begingroup$ Nice answer. But strong disagreement from me about ignoring early papers in general. When I have discovered disagreements between "confused pioneers" compared to modern practice, tracking down the origins of those disagreements has been extremely educational. $\endgroup$ Commented Dec 2 at 0:39
  • $\begingroup$ @rob agreed. But that is a rather special case, isn't it? There are plenty of scenarios in both quantum theory and in Maxwell's original treatises whereby confusion abounds and threatens to sink newcomers along with them, whereas a modern treatment is far more understandable because we have found ways to ring-fence those scary tar-pits. Otherwise, yes, consulting the original papers can often be insightful. $\endgroup$ Commented Dec 2 at 0:42
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    $\begingroup$ I think we agree with each other. Historical papers were written for an audience of experts whose expertise rested on a different curriculum than we know today. Making sense of them is a skill separate from the skill of reading a modern introduction. A fabulous example of this is in Tomonaga's book "The Story of Spin," where he reconstructs two alternate conventions for spin quantum numbers that eventually vanished from the literature. I think one of them would have been fine, but one had a fatal inconsistency. I understand spin much better for having thought about it — but not as a freshman. $\endgroup$ Commented Dec 2 at 0:53
  • $\begingroup$ @rob Yikes, that would have been seriously scary. It is very probable that the initial people might make a mistake in the manipulations and define the spinors wrongly. After all, they argued phenomenologically with sloppy arguments that spin should have this or that behaviour, that might well be inconsistent with actual experimentation. In the end, the dreaded gruppenpest is simply the correct way to deal with this. $\endgroup$ Commented Dec 2 at 1:07
  • $\begingroup$ Many thanks for this helpful answer. I would still be grateful if somebody could clear up the historical aspect. The book 'QED and the men who made it' mentions on p.124 that Weisskopf made a mistake in his 1934 calculation, which he corrected after Furry pointed it out, and then Heisenberg repeated the calculation. It then says: 'The most detailed analysis of the self energy problem [...] was given by Weisskopf in an important paper published in 1939.' So I would not expect any careless yet significant errors. A reference to the same but correct calculation would be much appreciated. $\endgroup$ Commented 2 days ago

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