What are “vibrations” in a field in Quantum Field Theory (QFT)?

For example, in a 2013 article for NOVA Don Lincoln writes:

Everywhere in the universe there is a field called the electron field. A physical electron isn’t the field, but rather a localized vibration in the field. In fact, every electron in the universe is a similar localized vibration of that single field.

https://www.pbs.org/wgbh/nova/article/the-good-vibrations-of-quantum-field-theories/

He doesn’t really talk about where he came up with the word “vibration,” but uses it as if it’s a normal term. Is this a pop-sci metaphor, or is this a real thing? What solid intro to QFT can I read for a reliable translation of the science to a college grad who took two semesters of physics (basic Newtonian physics, and electromagnetism) ages ago?

Will such an intro likely use the word “vibration” and explain why we use that instead of “particle,” or is “vibration” a terrible approximation for the real concept, and a reliable intro would never use that term?

• You are right the term "vibration" as used here is a self-declared folksy synonym for the technical "normal mode" or "phonon". You may, or may not, be invited to think of a specific note on a string and analogize its mathematics to that of quantum field theory, but your remonstrance obviates the pedagogical tack attempted. – Cosmas Zachos May 18 at 23:06
• "A physical electron isn’t the field, but rather a localized vibration in the field." - this type of description has perplexed me for some time. In QFT, there are the operator fields (operators indexed by their coordinates in spacetime) that create and destroy quanta. But a single electron state with definite momentum isn't a 'vibration' in this operator field is it? In other words, what precisely is the entity with states in the Fock space? – Alfred Centauri May 19 at 0:05
• @Afred Centauri A single electron state of momentum p is the normal mode of that field with momentum index p, in its first excited state. We prefer to work in Fourier space because that's where the normal modes decouple, and the Fock space description is practical, but not ineluctable. Spacetime localization in QFT is always approximate and, basically, flakey. – Cosmas Zachos May 19 at 0:26
• You would need some Quantum Mechanics to really get going but this is the most accessible and yet not "populist" resource I can think of: damtp.cam.ac.uk/user/tong/qft.html – Dvij Mankad May 19 at 2:06
• @CosmasZachos, I understand that the definite momentum state $p$ is the normal mode with momentum index $p$ - that's not the question I posed. As I understand it, there is at least (1) an operator valued field and (2) something else with states $|0\rangle, |...,1,...\rangle$, etc., correct? When 'we' say that a particle is an excitation of 'the quantum field', what is it precisely that is excited? – Alfred Centauri May 19 at 2:24

The word vibration has more of a historical sense, and we use excitation more often. In QFT, real particles are excitation of the underlying field.

Now the reason why we use vibration is because historically, these fields were modeled through waves, mathematically, and if you imagine a guitar string, and create a vibration, that will create a sound, a real thing, and so historically we use the expression vibration because it is very similar to a water surface where if it gets disturbed (excitation), it will create a wave, and these waves are modeling real particles in our currently accepted theories, the SM, GR, and QM, together with QFT.

• Not in GR though. GR isn’t a QFT. Please edit – Prof. Legolasov May 19 at 7:03
• @SolenodonParadoxus GR, at energies well below the Planck scale, is very much like any other standard effective field theory. – Avantgarde May 19 at 13:25
• @Avantgarde that is a very tricky statement. GR is not an effective theory in Minkowski space. The Poincare group doesn’t act on GR states (it is a symmetry of one solution of GR, not the full theory). If you disagree, you’re welcome to tell me what the result of boosting the FLRW metric is. – Prof. Legolasov May 19 at 13:50
• @SolenodonParadoxus I'm not sure how to answer that. GR as an EFT, however, has been in (active) study since the mid 90s, eg. arxiv.org/abs/gr-qc/9405057. There are many recent papers too. – Avantgarde May 19 at 14:10
• That’s the perturbative expansion around the flat Minkowski space. I completely agree that it’s a QFT. However, the original formulation of GR by Einstein predicts objects that can never be understood perturbatively (black holes, cosmological expansion, etc — note that these objects live far from the Planck scale), so the perturbative version is not the full story. – Prof. Legolasov May 19 at 14:13