atyy said:
Even in this theory, would it be fair to say, that there are intermediate steps where we don't even have a picture of an electron and a photon as distinct particles. However, we know that the electron and photon ultimately emerge from the intermediate steps, and we have means of calculating those ultimate results?
Yes and no. But in order to even begin to explain that, you should understand a little bit how quantum field theory "works". It is so very different from any classically-looking description that it would be pointless to say anything about it before you get at least a gross picture of how it does work - because it will be totally unexpected.
I'll give it a try, but the only true answer would be to learn it thoroughly, which will occupy you for a few years at least.
In quantum field theory, there are, well, quantum fields. There are no "particles" or forces or whatever, there are quantum objects which we call quantum fields. There's such a single quantum field for each KIND of "particle" that we know of. Now, the only quantum fields which we understand more or less correctly, are FREE quantum fields, that is, quantum fields which don't interact. These are boring things, but at least, we can deal with them. Boring as they are, they are nevertheless extremely strange.
Let's think about just electrons. There is ONE quantum field, the "electron quantum field" if you want, that deals with them. As any quantum system, it can be in its ground state, which corresponds to "there are no electrons", or it can be in several thinkable excited states, and each excited state corresponds to "a certain number of electrons (and positrons) are present, and have certain states of motion". So, in this picture, all thinkable configurations of electrons and positrons are nothing else but stable excited states of our quantum field for the electron. As it is a free field, these states are stable, that is, the only thing that happens is the motion of electrons, but there are no electrons that change direction, or disappear, or whatever.
A "quantum jump" of this quantum field would mean that we change the state of it, in other words, we arrive in a state with more or less electrons and positrons, and different states of motion (different speeds and directions). But a free system doesn't undergo quantum jumps.
Now, we can introduce several quantum fields. One for the electrons (and positrons), another one for the photons (also sometimes called: the quantum electrodynamical field), eventually some for the quarks etc...
But as long as they are all free, they act independently of one another. Nothing changes, nothing happens.
And then we introduce "interactions" between them. And that's where things get difficult. We run in fundamental mathematical problems when we do that. The whole system doesn't seem to be even internally consistent... But we can do a thing: we can consider "incoming free fields in the remote past", "turn on interactions", "turn them off again", and consider "outgoing free fields in the remote future". Although we have to do mathematical operations which are not entirely sound in order to do so, we CAN calculate the changes between the "free fields of the past" and the "free fields of the future" when we turn on certain interactions.
And then it looks like as if, say, the electron field "de-excited" from a "positron/electron" excited state to the ground state, and the electromagnetic field "got excited" from the ground state to the "two photon state".
If you want to look in between you have a problem, because first of all it is mathematically not even sound, but on top of that, we don't know how to interpret the 'states' of an interacting field. We only know what mean the states of free fields, in terms of particles. We don't know what eventual intermediate field states mean.
