“As I understand it, an electron isn’t an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc.”
It is hard to tell from the brief description, but it seems to me that you are talking about localized electrons and Wikipedia is talking about delocalized electrons. To describe particles in quantum field theory you have some field in spacetime. In the simplest case of a scalar field it is described by some function f(x,y,z,t). Note that f(x,y,z,t) is not a quantum wavefunction, it is just a classical field. Quantum mechanically, there is an amplitude corresponding to each possible configuration of this field. (Thus the wavefunction is technically a “functional”). Different configurations have different energies. The Langrangian tells you what energy corresponds to what configuration. The Lagrangian for a single field not interacting with anything looks sort of like the Lagrangian for material that can vibrate. (This is just an analogy, it has nothing to do with the aether.)
By a change of basis, we can write the Lagrangian in terms of normal modes, which each behave like harmonic oscillators, and which are decoupled from each other. As a one dimensional example, the normal modes for a violin string are the sine waves whose wavelengths are the length of the string, half the length of the string, 1⁄3 the length of the string, etc. These modes thus correspond to sinusoidal variation of the field. (This has nothing to do with string theory. The violin string is just a handy example of a vibrating system.) We know how to “quantize” the Harmonic oscillator. It turns out that the allowed energies are (n+1/2)h*omega, where n=1,2,3,..., and omega is the resonant frequency of the mode and h is planck’s constant. If the mode of frequency omega is excited to n=1 and the mode of frequency omega’ is excited to n=5 that corresponds to a six electron state with one electron of frequency omega and five electrons of frequency omega’. (Similarly for photons, or any other particles. For photons these frequencies correspond to colors.)
We can have superpositions of different such states. For example we could have quantum amplitude 1/sqrt(2) for mode omega to have n=1 and quantum amplitude 1/sqrt(2) for mode omega to have n=2. If we just have quantum amplitude 1 for a given mode omega to be in the n=1 state, and amplitude zero for all other configurations of the field, then this is a one electron state, where the electron is completely delocalized. What state corresponds to an electron in a particular region? A localized electron does not correspond to the field being nonzero in only a small region (e.g. the violin string has a localized bump in it like this ---^---). That would be a multi-electron state, because it decomposes into a classical superposition of many different sine waves, so we would have n>0 in multiple modes. Instead we can build a localized state of an electron by making a quantum superposition over different modes being occupied. It is important not to get the wavefunction confused with the field f(x,y,z,t). (If you have heard about the Dirac and Klein-Gordon equations, the solutions are analogous to f(x,y,z,t), not analogous to Schrodinger wavefunctions. Historically, there was some confusion on this point.)
Everything I have described so far is the quantum field theory of non-interacting particles. Although I may not have explained that well, it is actually not too complicated. However, if the particles interact, then the normal modes are coupled. Nobody knows how to treat this directly, so you need to use perturbation theory. This is where the complicated stuff about Feynman diagrams and so forth comes in.
“As I understand it, an electron isn’t an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc.”
It is hard to tell from the brief description, but it seems to me that you are talking about localized electrons and Wikipedia is talking about delocalized electrons. To describe particles in quantum field theory you have some field in spacetime. In the simplest case of a scalar field it is described by some function f(x,y,z,t). Note that f(x,y,z,t) is not a quantum wavefunction, it is just a classical field. Quantum mechanically, there is an amplitude corresponding to each possible configuration of this field. (Thus the wavefunction is technically a “functional”). Different configurations have different energies. The Langrangian tells you what energy corresponds to what configuration. The Lagrangian for a single field not interacting with anything looks sort of like the Lagrangian for material that can vibrate. (This is just an analogy, it has nothing to do with the aether.)
By a change of basis, we can write the Lagrangian in terms of normal modes, which each behave like harmonic oscillators, and which are decoupled from each other. As a one dimensional example, the normal modes for a violin string are the sine waves whose wavelengths are the length of the string, half the length of the string, 1⁄3 the length of the string, etc. These modes thus correspond to sinusoidal variation of the field. (This has nothing to do with string theory. The violin string is just a handy example of a vibrating system.) We know how to “quantize” the Harmonic oscillator. It turns out that the allowed energies are (n+1/2)h*omega, where n=1,2,3,..., and omega is the resonant frequency of the mode and h is planck’s constant. If the mode of frequency omega is excited to n=1 and the mode of frequency omega’ is excited to n=5 that corresponds to a six electron state with one electron of frequency omega and five electrons of frequency omega’. (Similarly for photons, or any other particles. For photons these frequencies correspond to colors.)
We can have superpositions of different such states. For example we could have quantum amplitude 1/sqrt(2) for mode omega to have n=1 and quantum amplitude 1/sqrt(2) for mode omega to have n=2. If we just have quantum amplitude 1 for a given mode omega to be in the n=1 state, and amplitude zero for all other configurations of the field, then this is a one electron state, where the electron is completely delocalized. What state corresponds to an electron in a particular region? A localized electron does not correspond to the field being nonzero in only a small region (e.g. the violin string has a localized bump in it like this ---^---). That would be a multi-electron state, because it decomposes into a classical superposition of many different sine waves, so we would have n>0 in multiple modes. Instead we can build a localized state of an electron by making a quantum superposition over different modes being occupied. It is important not to get the wavefunction confused with the field f(x,y,z,t). (If you have heard about the Dirac and Klein-Gordon equations, the solutions are analogous to f(x,y,z,t), not analogous to Schrodinger wavefunctions. Historically, there was some confusion on this point.)
Everything I have described so far is the quantum field theory of non-interacting particles. Although I may not have explained that well, it is actually not too complicated. However, if the particles interact, then the normal modes are coupled. Nobody knows how to treat this directly, so you need to use perturbation theory. This is where the complicated stuff about Feynman diagrams and so forth comes in.
I hope this is helpful.