Existential problem on Electromagnetism and the combination of Relativity and Quantum Mechanics

In Feynman's famous Physics book, in a discussion of the generality of Maxwell's equations in the static case, in which he addresses the problem of whether they are an approximation of a deeper mechanism that follows other equations or not, he says:

Strange enough, it happens that the combination of relativity and quantum mechanics, in the present form of these theories, seems to prohibit one from finding an equation that is fundamentally different from the Poisson equation and that does not at the same time lead to some kind of contradiction. Not a simple disagreement with experience, but an internal contradiction.

I was wondering first of all if this was a personal observation of Feynman's, or if it was a known thing that I will find in the future while studying and will somehow be pointed out to me in some Physics course. Then I was wondering if you could understand qualitatively what you are talking about, i.e., how does one theory ensure that there cannot be another different and more precise theory for the phenomena it describes...

Hak said:

In Feynman's famous Physics book

Which one?

Vanadium 50 said:

Which one?

"The Feynman Lectures on Physics". Thanks.

Jeez...this is going to be like pulling teeth. We've isolated it down to somewhere in three books.

Where does he say this? Seriously, how would you like to answer a question "Feynman says X...somewhere...explain it to me."

Vanadium 50 said:

Where does he say this?

In the last paragraph of section 12-7 (The "underlying unity" of nature) of volume II.

Hak said:

I was wondering first of all if this was a personal observation of Feynman's, or if it was a known thing that I will find in the future while studying and will somehow be pointed out to me in some Physics course.

It is well known that quantum electrodynamics is not mathematically satisfactory. It is seen as an "effective" theory that is a very good approximation to some deeper theory.

Hak said:

Then I was wondering if you could understand qualitatively what you are talking about, i.e., how does one theory ensure that there cannot be another different and more precise theory for the phenomena it describes...

I think this is an overstatement. There can be no "proof" that a more precise theory cannot exist. There are many no-go theorems, but the proofs always depend on some innocent-looking assumption (which may turn out not to be satisfied in the real world). Nobody has yet constructed a mathematically clean quantum field theory in 3+1 dimensions for electrons with non-zero charge. It seems we are attacking the problem from a wrong angle.

WernerQH said:

In the last paragraph of section 12-7 (The "underlying unity" of nature) of volume II.It is well known that quantum electrodynamics is not mathematically satisfactory. It is seen as an "effective" theory that is a very good approximation to some deeper theory.I think this is an overstatement. There can be no "proof" that a more precise theory cannot exist. There are many no-go theorems, but the proofs always depend on some innocent-looking assumption (which may turn out not to be satisfied in the real world). Nobody has yet constructed a mathematically clean quantum field theory in 3+1 dimensions for electrons with non-zero charge. It seems we are attacking the problem from a wrong angle.

Combining Quantum Mechanics and Relativity (especially General Relativity) I think we get to very, very, very complicated things. At this point, given your answer, I think Feynman's point is to be understood like this: people have tried for a long time to build theories that combine them and in all cases Poisson-style equations popped up somewhere, so there's no theory on the horizon that seems to work but doesn't have that kind of structure. What do you think?

Feynman's statement is tautologically true. What does it mean to be "fundamentally different". If it is consistent, one can say, "well, it's different, but not fundamentally different."

The question is really A-level, but it appears from the context that Feynman is commenting that one cannot construct a gauge invariant theory of massive photons. Such a theory would not satisfy the Poisson equation. However, shortly after the lectures, it was shown how to do this by Peter Higgs and others, for which he and Francois Englert won the Nobel Prize.

The inconsistency that is mentioned above is called a "Landau Pole". It does not happen physically, as the unification of electromagnetism and the weak nuclear force happens well before that happens. Further, even if that did not happen, the energy scale is trillions of times larger than all the energy in the visible universe.

Finally, Feynman is not talking about GR.

Hak said:

Combining Quantum Mechanics and Relativity (especially General Relativity) I think we get to very, very, very complicated things.

In this context, Feynman is not concerned with gravity. We don't even have a consistent quantum electrodynamics in flat spacetime.

Hak said:

people have tried for a long time to build theories that combine them and in all cases Poisson-style equations popped up somewhere, so there's no theory on the horizon that seems to work but doesn't have that kind of structure.

Such equations emerge under very diverse circumstances, as Feynman demonstrates in that chapter. There's no problem with that. I believe the more "precise" (mathematically well-defined) quantum electrodynamics of the future will produce the same equations, but permit a better understanding of the theory. Feynman is concerned with apparent "internal contradictions" of the theory. Already the classical electron theory had the problem that a point charge would have infinite energy, and for a smeared out charge no consistent description could be found.

Maxwell's theory of the aether (the original electrodynamics) had similar "internal contradictions". The aether had to be solid (allowing transverse waves) and fluid (for vortex lines to form) at the same time. For years, physicists tried to construct mechanical models for such a medium. For Maxwell and his contemporaries it must have been self-evident that light waves cannot propagate without a medium carrying them. And that time is absolute and the same for all observers in the universe. After 1905 Maxwell's equations remained unchanged, but were seen in a new light. I believe there will be a similar change with quantum field theory, after we've jettisoned some "self-evident" metaphysical assumptions on what the theory is about.

Thank you very much.