You are lost in maths. Displacement current is a feature of light. It's a time-varying electric field. And it does what it says on the can. It isn't called displacement current for nothing.
Argument by etymology. New one!
Because that displacement current is alternating. That's why we have vacuum impedance, impedance being resistance to alternating current. The two waves "ride over one another", like two ocean waves ride over one another. There's a displacement up, then down. Then each continues on its way. So it looks like there was no interaction. But when you increase the energy the displacement is so drastic that each wave is displaced into itself. And ends up displacing itself into a closed path. Simples.
Awesome.
Your proof that Maxwell's Equations imply photon-photon scattering is ... um ... a restatement of your unproven, unpublished crackpot theory of photons twisted into knots.
Dude, one of the reasons I disagree with your photon-knot theory is because they're so obviously not a solution to Maxwell's Equations. As I'm sure I've said repeatedly. You've never derived photon-knots from Maxwell's Equations; if you tried to do so you'd fail. I repeat: Maxwell's Equations are linear. They do not do what you have daydreamed they do.
Your walkthrough of "displacement current" is gibberish and corresponds to nothing ever seen in Maxwell's Equations, nor in E&M experiments, nor to anything in theoretical or experimental quantum electrodynamics. It's just pouring out of your head and yours alone.
I don't make mistakes. You just don't know much physics, that's all. I think it's because your physics degree was all maths. That's the usual thing. People spend three years doing maths, not physics.
Actually, my undergraduate degree consisted entirely of drawing arrows on a ping-pong-ball with a marker. My master's thesis was on "staring at spirals" and my postdoc work consisted entirely of misquoting late-19th-century E&M lectures. I later joined the faculty at the Harvard Department of Numerology, where I've been combining physical constants, in various unit systems, into 1/137, ever since. It's a tough life but I'm good at what I do.
Huh? It's a photon-photon collider. We collide photons with photons. So then we know that photons interact with photons to create electrons and positrons. Duh!
I'm sorry, I have been talking at you as though you'd seen a Feynman diagram at some point in your life. I guess I was wrong; it is impossible for someone who is even recognizes Feynman-diagram-language to write the sentence Farsight just wrote.
How low do we have to go? You don't know unit systems, you don't know what a vector is, you don't know what a linear equation is, now you don't know what a vertex is?
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