The simple energy momentum analysis is a basic technique that can be used to give a sanity check on any proposed physical process.

The technique works. It is a valid way to show when descriptions of a process are inadequate. If before and after states of an alleged interaction don't balance, then the process is not physically possible.

"Defences" that appeal to subtle properties of an alleged process are red herrings. Those details are immaterial; what is required is a plain description of where energy and momentum ends up.

"Defences" that appeal to quantum principles in which brief conservation violations can occur are also not responsive. They can be useful for calculating details of how a process occurs, like rates or cross sections or durations; but any "violation" of conservation laws is virtual, or transitory, and does not remove the responsibility for finding a balance of energy and momentum as part of the complete electron-photon interaction.

Lyndon has said repeatedly that my analysis assumes longitudinal waves. This is emphatically false. No such assumptions were used or required. Simply add up energy and momentum based on final states as described in Lyndon's paper, and this analysis remains totally unaffected by details of how we get from the initial state to the final state. The problem is simply that energy and momentum don't add up.

If someone proposes a new theory in which photons perform triple Mossbauer backflip with reverse M-brane pike and two virtual gluons to manage the absorption, the argument I have given will still apply. A new theory could propose a photon transforms briefly into a bowl of petunias, within the time constraints of Heisenberg's uncertainty principle; and that the electron reacts to beauty or charm for the instant of their existence. My argument is unchanged. Someone may well note -- correctly -- that classical wave analysis requires a transverse push to the electron. This makes no difference to the simple observation that initial and final states don't balance.

From the transverse nature of the electromagnetic wave, it follows that absorption by an electron is impossible; because the photon must be scattered to balance the sideways push on the electron. But we know it is impossible already by the energy and momentum analysis made independently of the nature of the wave.

To do an energy momentum analysis, just add up energy terms and momentum terms, before and after the interaction. This is what Lyndon still needs to do, but he's stuck because the interaction is necessarily imbalanced.

The three electrons and two nails was not such an analysis. It neglected to consider momentum. Lyndon gave the electron a kinetic energy of about 4e-19 J to balance energy, but this required momentum 612 times greater than the original momentum, and in the wrong direction. No other momentum terms were identified. It is not the responsibility of readers to guess such things. Lyndon needs to account for momentum explicitly himself.

Lyndon's paper does not give such an analysis; it fails to balance energy. The paper uses standard formulae based on momentum conservation to give the electron kinetic energy of about 1e-24 J, leaving 99.99976% of the energy unaccounted for.

Lyndon has spoken vaguely of other particles. A real energy momentum analysis must quantify where the momentum and energy ends up. Potential energy involves displacing particles in the field to new locations. Kinetic energy involves a particle velocity. Both require forces that can be calculated. Oscillation is not a separate energy term; it merely refers to cyclical exchanges of kinetic and potential energy with two or more particles and restoring forces.

Recently, we've had mention of Feynman diagrams, suggesting that intermediate states of the diagram are unbalanced. So they are; but they don't describe the end state of an interaction. A related red herring is the uncertainty principle. These quantum based ideas are sophisticated ways to calculate the details of a process; but they don't give a license to ignore energy momentum conservation for the conclusion. The good old high school level energy momentum analysis continues to be valid when applied to the consequences of an interaction, even when the full power of quantum mechanics is applied to spell out how the interaction proceeds.

Cheers -- Sylas

PS. Thanks Grey. I did indeed say "longitude" in one place where I meant "transverse", but the rest of the context makes clear what I meant. I spoke of a sideways push. The matter of "escape velocity" for two electrons is poor wording, but the number still reflects correctly the magnitude of the forces. In the case of repulsive forces between like charges, it is the velocity attained in the limit by a particle starting from rest and driven to a distance where it escapes further influence. If Lyndon is unwilling to deal with the substantive issues from someone who occasionally makes such slips, I don't mind in the least. But it's a bit hypocritical to make a big deal about this, and just keep mousy quiet about his own acceleration provided for an electron by another electron at one meter difference, which was out by a factor of ten. These kinds of minor defects are not significant, for either side.