I reckon it's turtles all the way down

One of those, "what's it all about, I mean really, when you get right down to it ?" articles.

Do electrons feel forces from their own electromagnetic fields? Either answer leads to trouble. First, suppose the answer is yes. The electromagnetic field of an electron gets stronger as you get closer to the electron. If you think of the electron as a little ball, each piece of that ball would feel an enormous outward force from the very strong electromagnetic field at its location. It should explode. Henri Poincaré conjectured that there might be some other forces resisting this self-repulsion and holding the electron together – now called ‘Poincaré stresses’. If you think of the electron as point-size, the problem is worse. The field and the force would be infinite at the electron’s location.

But would that mean the particle itself can feel anything ? If it's infinitely small (along any dimension), how can it be said to have any substance that can be affected ? Then again, interactions between particles of opposite charge would involve infinite energy, and that's not nice. And if it's not infinitely small, what the heck is it made of ? Perhaps instead it has extension, but it's an indivisible substance that simply cannot be ripped apart further. If so, how is it possible that it can be so readily converted into energy in the right circumstances ? That matter-energy duality seems weirder by far than any philosophical mind-body dualism...

Nope, I reckon it's all magical turtles. Solves the whole thing at a stroke. But for the sake of it, let's continue.

So, let us instead suppose that the electron does not feel the field it produces. The problem here is that there is evidence that the electron is aware of its field. Charged particles such as electrons produce electromagnetic waves when they are accelerated. That takes energy. Indeed, we can observe electrons lose energy as they produce these waves. If electrons interact with their own fields, we can correctly calculate the rate at which they lose energy by examining the way these waves interact with the electron as they pass through it. But, if electrons don’t interact with their own fields, then it’s not clear why they would lose any energy at all.

Faraday asked: ‘What real reason, then, is there for supposing that there is any such nucleus in a particle of matter?’ That is, why should we think that there is a hard core at the centre of a particle’s electromagnetic field? In modern terms, Faraday has been interpreted as proposing that we eliminate the particles and keep only the electromagnetic fields.

 But what actually is a field ? What gives it "substance", for lack of a better word ? My head hurts.

In a 1938 paper, Dirac proposed a modification to the laws of electrodynamics, changing the way that fields exert forces on particles. For a point-size particle, his new equation eliminates any interaction of the particle with its own electromagnetic field, and includes a new term to mimic the kind of self-interaction that we actually observe – the kind that causes a particle to lose energy when it makes waves. However, the equation that Dirac proposed has some strange features. One oddity is ‘pre-acceleration’: a particle that you’re going to hit with a force might start moving before you hit it.

In the 1930s and ’40s, a different strategy was pursued by four notable physicists [who] proposed ways of changing the laws that specify how particles produce electromagnetic fields so that the fields produced by point particles never become infinitely strong. When you change these laws, you change a lot. As Hubert explained in his presentation, we don’t fully understand the consequences of these changes. In particular, it is not yet clear whether the Born-Infeld and Bopp-Podolsky proposals will be able to solve the self-interaction problem and make accurate predictions about the motions of particles.

So on the smallest of scales it looks like things are inevitably weird. I think I'll stick to the largest scales, where everything is well-behaved and there are no controversies of any kind.

Wheeler and Feynman – like Ritz – do away with the electromagnetic field and keep only the particles. As I mentioned earlier, Ritz’s field-free theory has particles interact across gaps in space and time so that each particle responds to the past states of the others. In the Wheeler-Feynman theory, particles respond to both the past and the future behaviour of one another. As in a time-travel movie, the future can influence the past. That’s a wild idea, but it seems to work. In appropriate circumstances, this revision yields accurate predictions about the motions of particles without any true self-interaction.

In the action-at-a-distance theories put forward by these physicists, you can’t tell what a particle will do at a particular moment just by looking at what the other particles are doing at that moment. You also need to look at what they were doing in the past (and perhaps what they will do in the future). Lazarovici argued that the electromagnetic field is merely a useful mathematical bookkeeping device that encodes this information about the past and future, not a real thing out there in the world.

If you think of electrons as particles, you’ll have to think of photons differently – either eliminating them (Lazarovici’s story) or treating them as a field (Hubert’s story). On the other hand, if you think of electrons as a field, then you can think of photons the same way. I see this consistency as a virtue of the all-fields picture.

But how does this help with the double-slit experiment ? It's all magical turtles, I'm sure. At least they make sense. How dare the Universe be so bloody complicated...

Is everything made of particles, fields or both combined? - Charles Sebens | Aeon Essays

Long before philosophy and physics split into separate career paths, the natural philosophers of Ancient Greece speculated about the basic components from which all else is made. Plato entertained a theory on which everything on Earth is made from four fundamental particles.