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I've watched a few PBS Space Time videos on YouTube, but from them I can't get a picture of quantum fields that makes sense. My QFT understanding is of course very limited.

Let's take the electron field as an example. On one hand it's a quantum field, so it's values are multiples of some base unit. So if say the value in a given point is 0 there are no electrons there, if the value is 1 there is 1 electron, if the value is 2 - 2 eletrons, more than 2 elecrons are not possible because of Pauli exclusion principle.

On the other hand an electron exists in a state of superposition, so in one point there may be 80% probability to find an electron there, further away point 70%, etc. What are the values of the electron field in these points? Is it like 99 units - very likely there, 50 units - somewhat likely, 1 - barely possible? And in what units is the field measured.

Also beacause of Heisenberg uncertainty principle even in vacuum free of eletrons, the electron field has (most of the time?) value different than 0. What is this value?

AquaRacer
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    When something is "quantum", that means it is operator valued. The fact that some operators have a quantized spectrum is purely incidental. – Connor Behan Oct 15 '21 at 03:34
  • What does "operator valued" mean? When Plank came with the first quantum theory, didn't quantum just mean having integer values? Did the definition of quantum change later? – AquaRacer Oct 15 '21 at 04:42
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    "On one hand it's a quantum field, so it's values are multiples of some base unit. So if say the value in a given point is 0 there are no electrons there, if the value is 1 there is 1 electron, if the value is 2 - 2 eletrons" Unfortunately this is not how it works at all. As much as I like PBS spacetime, you cannot understand the mathematics of QFT from these videos. – Koschi Oct 15 '21 at 06:59
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    If you have understanding of non-relativistic quantum mechanics, most importantly the quantum harmonic oscillator, then "Quantum Field Theory for the Gifted Amateur" might be a good book to start. – Koschi Oct 15 '21 at 07:07
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    see my answers here https://physics.stackexchange.com/questions/343237/how-does-the-wave-particle-duality-fit-with-quantum-field-theory/343345#343345 and here https://physics.stackexchange.com/questions/256663/from-quantum-mechanics-to-quantum-field-theory-to-string-theory/263597#263597 – anna v Oct 15 '21 at 12:59
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    @AquaRacer Yes, the word quantum gained a new meaning (among physicists) that is independent of its colloquial meaning as a synonym of "discrete". This kind of language-evolution often happens when our understanding of the physics improves: we keep using the old words but with new meanings that reflect our new understanding. In modern terminology, we call a theory "quantum" if its observables are not all compatible with each other (more technically: if they are represented by operators that don't all commute with each other). – Chiral Anomaly Oct 15 '21 at 13:34

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