Discussion between QuantumBrick and Keshav Srinivasan

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05:48

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A: Does the $\frac{4}{3}$ problem of classical electromagnetism remain in quantum mechanics?

Problem outline In the cited reference, Feynman starts his argument by stating that the energy for an inertial sphere of radius $a$ and uniform charge $q$ is given by $$ U_{elec} = \int_{\mathbb{R}^3} dV \, u_{elec} = \int_{\mathbb{R}^3} dV \, \left( \frac{\epsilon_0 E}{2} \right) = \frac{1}{2...

Keshav Srinivasan

"Do we have the right of invoking SR in a derivation's single step? Although we used Einstein's famous formula to arrive at the 43 problem, notice that we have not used SR earlier: (∗) and (∗∗) are not even Lorentz invariant!" Actually Feynman fully takes deformation issue into account but it doesn't resolve the 4/3 problem. Scroll down to where he says "Let’s pursue our electromagnetic theory of mass. Our calculation was for v≪c; what happens if we go to high velocities?"feynmanlectures.caltech.edu/II_28.html You get a Lorentz invariant form but it still has a $\frac{2}{3}$ factor.

QuantumBrick

First, Feynman's formula 28.7 is for the relativistic change of the wrong mass, $m_e$, which does not arise in SR: The correct term is $m_e'$. Second, fully accounting for electron deformation involves not neglecting an integral that is only null inside a bare electron. I urge you to check Rohrlich's paper, where everything is extremely well presented, especially the last section where he solves for the stress-energy tensor on the inside and outside of an ellipsoidal electron.

Keshav Srinivasan

I don't know what you mean by "Feynman's formula 28.7 is for the relativistic change of the wrong mass, $m_e$, which does not arise in SR". Formula 28.7 is just calculating the relativistic momentum, it's not about electromagnetic mass yet.

QuantumBrick

Check closely: Feynman defines the electromagnetic mass $m_e$ in 28.4, with the 2/3 factor in front of it. Then he later uses it in 28.7 to define relativistic momentum. However, this is not right, and Feynman is only working out the consequences of a poor choice. The proper, correct mass comes from SR and has no 2/3 factor in front of it, since it is given by $m_e' = U_{elec}/c^2$. This mass, then, is later used to obtain the relativistically correct momentum $\vec{p}' = \gamma m_e' \vec{v}$. The mass in 28.4 is wrong, since it comes from an expression that is not correct in SR.

Keshav Srinivasan

He does not derive 28.7 from 28.4. He first derives 28.4 from 28.3, then he provides 28.7 as the relativistic analogue of the calculation done to get 28.3, in order to show that 28.4 comes from 28.7 in the same way that it comes from 28.3.

QuantumBrick

05:48

I'll reinforce what I said before: The proper mass that pops up when using the correct relativistic momentum is $m_e'$, not $m_e$. This is explicitly done in my calculations. Feynman's formula 28.7 is wrong.

Keshav Srinivasan

That cannot possibly be a problem with formula 28.7, because formula 28.7 is not about electromagnetic mass at all. It is about electromagnetic momentum, and then 28.7 is used to obtain the electromagnetic mass.

QuantumBrick

Exercise 1: Arrive at formula 28.7 without starting from an initial momentum equal to $m_e \vec{v}$. Hint: it's impossible. Exercise 2: define the mass using $m_e' = U_{elec}/c^2$, then redo the calculations I displayed and arrive at the correct momentum in the small velocity regime $m_e' \vec{v}$.

Keshav Srinivasan

Feynman describes in outline how it's done, and it's not by using formula 28.4. It's by integrating the momentum density taking relativity into account: "Early attempts led to a certain amount of confusion, but Lorentz realized that the charged sphere would contract into a ellipsoid at high velocities and that the fields would change in accordance with the formulas (26.6) and (26.7) we derived for the relativistic case in Chapter 26. If you carry through the integrals for p in that case..." I highly doubt Feynman was bluffing, he presumably did the relevant calculation.

QuantumBrick

I will let you do the exercises I proposed. Then, please, find the error in: for $\vec{p}=0$, $m_e' = U_{elec}/c^2 \Rightarrow \vec{p}' = \gamma U_{elec} \vec{v}/c^2 = \gamma m_e' \vec{v}$. This calculation was done by Schwinger, Fermi, Rohrlich, and dozens of others. Without understanding my answer properly or reading the references I provided, there is no point to continue this discussion.

Keshav Srinivasan

Why don't I do this: I can post a new question on the site asking how the integral for formula 28.7 is done. As you said Feynman is "one of the most brilliant physicists in history", so we shouldn't dismiss what he's saying about integrating to find $p$ out of hand.

QuantumBrick

05:48

You can ask this, but it's trivial: to arrive at 28.7 Feynman started from 28.3, and later used the transformation law $\vec{p}' = \gamma \vec{p}$ to start from initial momentum $m_e \vec{v}$ and map it to $\gamma m_e \vec{v}$. Equation 28.7 is clearly not relativistically correct, since the $2/3$ factor in front of it prevents the associated 4-momentum to transform as a 4-vector. Thus, equation 28.7 uses the right transformation law for the wrong momentum, obtained from a relativistically wrong equation. The correct one is obtained from $\vec{p} = m_e' \vec{v}$.

Keshav Srinivasan

"to arrive at 28.7 Feynman started from 28.3, and later used the transformation law $p′=γp$ to start from initial momentum $m_ev$ and map it to $γm_ev$." That is not what he says he did. What he says he did is integrate the momentum density, using the field transformations from formulas 26.6 and 26.7.

QuantumBrick

If have to be careful to understand what you are talking about. Equations in chapter 26 only deal with the transformational properties of fields, they have absolutely nothing to do with momentum. If you use those formulas to calculate $p$ as he did in 28.3, it would still be wrong, because integrating the poyting vector as he did is not relativistically correct. This is the point mentioned by mostly everyone. Wikipedia itself says that

"Another solution was found by authors such as Enrico Fermi (1922),[32] Paul Dirac (1938)[33] Fritz Rohrlich (1960),[34] or Julian Schwinger (1983),[35] who…

The correct procedure, relavistically speaking, is to use the stress-energy tensor to perform all integrations. This is what Rohrlich does. The point is that this is not needed, because with a very simple argument I have showed that the definition of electromagnetic mass provided by 28.4 and used in 28.7 is not correct in SR.

If you do want madly to perform all integrations using proper objects, please check Rohrlich's last section on deformed ellipsoids. If not, than please think about the definition of electromagnetic mass as simply U/c^2. This is the correct object to be associated with momentum, since no factors arise, only 4-vectors are dealt with, and no paradox emerges. The 4/3 problem exists only because a wrong definition of electromagnetic mass is chosen.

Check also what was said by Fermi:

" It's known that simple electrodynamic considerations[1] lead to the value 4/3 U c^2 for the electromagnetic mass of a spherical electricity-distribution of electrostatic energy U, when c denotes the speed of light. On the other hand, it's known that relativistic considerations for the mass of a system containing the energy U give the value U c^2. Thus we stand before a contradiction between the two views, whose solution seems not unimportant to me, especially with respect to the great importance of the electromagnetic mass for general physics, as the foundation of the electron theory of m…

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Discussion between QuantumBrick and K…