Not always, actually. (At least, not in the sense you probably think of waves.)
That's not what happens in quantum physics. In QM the electron's momentum modes are all linear de Broglie waves, just like the photon's.
It is, at least as far as mainstream models are concerned, in the sense that it has point-like interactions. It's a quantum point particle (as opposed to, say, a quantum string). I've seen you go around "correcting" people telling them the electron is not like a classical billiard ball. That's pointless: nobody in physics thinks the electron is like a classical billiard ball. That's not what we mean when we say the electron is a point particle.
Have you ever wondered maybe why papers like those don't get much attention? Maybe their models don't actually accomplish what they say on the tin.
I understand QED just fine, thank you, and none of this contradicts what I told you: in quantum physics in general you can't model the electron as a photon bound state, and in QED this is not what happens. As AlphaNumeric explained to you (if you bothered to read his post) electrons are treated as fundamental particles in QED. In QED, the photons literally disappear and electrons appear in their place. The photon is an excitation of the quantum vector field usually noted $$A_{\mu}$$. The electron is an excitation of a quantum fermionic field usually noted $$\psi$$ (not to be confused with quantum wavefunctions, which are also often noted $$\psi$$ or $$\Psi$$).
This isn't completely accurate. In QED the two photons don't interact directly - there's no such fundamental interaction. You see this on the Feynman diagram for the pair production process (from your own Wikipedia page):
What's depicted here is one photon splitting into an electron-positron pair, and the other photon being later absorbed by either the electron or the positron. The photons never interact directly.
None of this is true. For starters, not all wave equations are Lorentz invariant. For example, the non-relativistic Schrödinger equation, as its name would suggest, is not Lorentz invariant. Incidentally, the equation itself is:
$$
- \, \frac{\hbar^{2}}{2m} \, \frac{\partial^{2} \psi}{\partial x^{2}} \,=\, i \hbar \frac{\partial \psi}{\partial t} \,.
$$
If you're at all experienced with relativity, the easy way of seeing this is that the equation doesn't treat space and time the same way: it contains a second derivative with respect to x, but only a first derivative with respect to t. This is the same sort of argument I've used agains the idea of electromagnetism being curved space: the idea separates space and time and says the curvature is only happening in the space part. That immediately breaks Lorentz invariance.- \, \frac{\hbar^{2}}{2m} \, \frac{\partial^{2} \psi}{\partial x^{2}} \,=\, i \hbar \frac{\partial \psi}{\partial t} \,.
$$
(Also, though it's not really relevant here, Robert Close is wrong about another thing: the massless relativistic wave equation is not the only Lorentz invariant equation. He's way behind the times here. We've known how to easily contruct Lorentz invariant equations since Minkowski's geometrical formulation of special relativity around 1907 or so. Basically, any equation constructed out of four-vectors and tensors and using certain operations (addition, subraction, or comparison of equal rank tensors; tensor contraction) is automatically Lorentz invariant.)
What are you talking about? There is no evidence that electrons are actually made out of photons. There is only evidence that there are processes in which photons go in and electrons come out. That doesn't necessarily imply one particle is made out of the other. In fact this is specifically not the case, as I explained above and as AlphaNumeric has explained, in the best models we have for those interactions.
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