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@narfwhals7843 I think most of your intuition is correct, but I will add a couple of comments to hopefully clarify some things. > If I understand correctly, when we have a local symmetry, we need the gauge field to accommodate for the local transformation. > To me, this indicates that every particle that has this local symmetry must couple to the gauge field, because that's the point of the gauge field. > This would mean that the charge of the theory is the same property as having the local symmetry. You're exactly correct that any field that transforms under the local symmetry must couple to the gauge field. Keeping in mind that the gauge field is really the connection between two points which are now allowed to transform differently under the symmetry transformation, this is very easy to see: suppose I don't couple such a field to the gauge connection. Then, I am not accounting for this extra local transformation as I compare two different points, so two field configurations which are related by the local symmetry transformation will actually "look" different according to the theory and the symmetry is explicitly broken (two fields which are related by a symmetry transformation should always look the same). The charge is essentially how much the field transforms under the symmetry, and is always going to be proportional to the generators of the symmetry group. In QED (and any U(1) theory in general), the generators are any real numbers, so fields can have any charge under a U(1) symmetry. For a more complicated symmetry group like SU(3), fields are restricted to only transform as representations under the group (this is true for U(1) as well, it's just that any real number corresponds to a valid representation). > In QCD this seems to be the case as SU(3) is just the symmetry of the color space. Can we have particles that have global symmetry in this space but not local? Is that what confinement is? Not quite. When we make a symmetry transformation, the transformation has a set number of transformation parameters (the same as the number of generators), i.e. "how much" you are transforming by. These transformation parameters can vary on spacetime, but they will be the same for any field we are transforming. A global symmetry would then need to be completely disconnected from the local one since it would have totally different transformation parameters, so it would probably make more sense to just consider them as separate symmetries. Confinement just has to do with the behavior of couplings/potentials of a theory. However, you can do the opposite, where you have a large global symmetry and only gauge a sub-group of it. For example, I could have a global U(2) symmetry where I gauge the U(1) sub-group of the U(2). > In QED we usually talk of the charge separately. Like we can have particles that have local phase invariance but do not need the gauge field to make up for it. Is that the case? Do neutral particles have local phase invariance? > In terms of the covariant derivative, with the gauge as the connection term, we have this term multiplying the charge, which is zero for neutral particles. But that seems like its just a convenient way to write it. Any time that you have a theory which will depend on *differences* (i.e. derivatives) of fields at spacetime points where the fields transform under a local symmetry, you will need a gauge field. Again, this just has to do with the fact that we are really connecting different points of fields that could be transformed differently, and the gauge field automatically accounts for this. It absolutely isn't wrong to say that a neutral field transforms with "charge" zero. This is just because the "zero" representation is always going to be a representation in any group, since this corresponds to the identity operator and by definition, a group must have an identity element. In this case, yes, the gauge term in the covariant derivative vanishes. I don't think that this is just a convenient way to write it, but it is the correct way to think about it. > In your QED video, (if I understand correctly) you write this term as psibar*psi times the gauge derivative. Will this turn out to be zero, if the field is not locally symmetric? I'm not sure what you mean here by "not locally symmetric." If you mean that the theory truly isn't invariant under local transformations of this field, then the symmetry is explicitly broken, and you can no longer talk about having a local symmetry. If you mean that it doesn't transform under the symmetry (or I guess more properly that it transforms as a 0 representation of the group), then the part of the covariant derivative containing the gauge field will vanish when acting on this field, but the standard derivative will still hang around. However, it will only vanish in the coupling to that field, not to any fields which actually transform non-trivially under the symmetry. Hope that helps!

​@@zapphysics Wow, thank you for always giving such extensive answers! So having a non-zero charge tells us how much the gauge transformation affects the field. A zero charge means the field "transforms trivially", so no change at all. I think my confusion was exactly the difference between your last two paragraphs. A trivial transformation is still a local symmetry. So the "amount" of charge is a distinct property of the fields, which is restricted to representations of the gauge group. And that is why we (sometimes) need to talk about it separately. How is that related to the psibar*psi term in the derivative here czcams.com/video/qtf6U3FfDNQ/video.html ? Or am I misunderstanding which derivative that is?

@narfwhals7843 ah, this term comes exactly from trying to take the derivative of the gauge-transformed field. I apologize in advance for the formatting, but when I take psi-> exp(i*e*theta(x))*psi and psibar -> psibar*exp(-i*e*theta(x)), where e is the charge of the field and theta(x) is the spacetime-dependent transformation parameter, the impertant terms with the derivative of psi transform as psibar*dpsi/dx -> psibar*(dpsi/dx + i*e*psi*dtheta/dx). The first term isn't a problem, since this is the term we started with, but the second term is an issue, since it adds a term we didn't have before, which is clearly an issue for a symmetry. However, when we have a gauge-covariant derivative, we also have a coupling like psibar*i*e*A*psi where A is the gauge field. Under a gauge transformation, the gauge field also transforms like A -> A - dtheta/dx. So, when we add this all together, the combined transformation of the gauge field as well as the local transformation of the charged fields exactly cancel, leaving the theory invariant under the combined transformation. Note that this is exactly related to the fact that the charge in the transformation is the same as the charge in the gauge-covariant derivative: the transformation of the gauge field makes no reference to the particular charge of any one field, so in order for the cancellation to happen correctly, the gauge field needs to paired with the appropriate charge in the covariant derivative acting on the field. Now, this story gets slightly more complicated for a non-abelian group, but the general idea is the same. Just replace the charge with the appropriate generators of the group corresponding to the representation of the given field.

@@zapphysics Thank you, this was very helpful!

Energy is dual to mass -- Einstein. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps synthesize stereo graphic projection. Time dilation is dual to length contraction -- Einstein, special relativity. Duality:- two equivalent descriptions of the same thing -- Leonard Susskind, physicist. Symmetry is dual to conservation -- the duality of Noether's theorem. "Always two there are" -- Yoda.

Energy is dual to mass -- Einstein. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps synthesize stereo graphic projection. Time dilation is dual to length contraction -- Einstein, special relativity. Duality:- two equivalent descriptions of the same thing -- Leonard Susskind, physicist. Symmetry is dual to conservation -- the duality of Noether's theorem. "Always two there are" -- Yoda.

@@hyperduality2838 Thanks for the reply which is clearly very relevant to what I wrote.

@@RabbitInAHumanWoild There are new laws of physics that you may not be aware of:- Making predictions to track targets, goals and objectives is a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Information is dual:- Average information (entropy) is dual to co or mutual information (syntropy). Potential or imaginary information is dual to real or kinetic information. Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Vectors (contravariant) are dual to co vectors (covariant) -- dual bases or Riemann geometry is dual. Converting potential information into real information is a syntropic process. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Your mind integrates information to form predictions (syntropy) -- integrated information theory. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! "Always two there are" -- Yoda. Integrating information is a syntropic process, teleological -- hence there is a 4th law of thermodynamics. "Entropy is a measure of randomness" -- Roger Penrose. Syntropy is a measure of order.

@@RabbitInAHumanWoild Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual! The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- language or communication. If mathematics is a language then it is dual. Electro is dual to magnetic -- photons, light or pure energy is dual -- bi-vectors are dual. Vectors (contravariant) are dual to co or dual vectors (covariant) -- dual bases or Riemann geometry is dual. Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum or wave/particle duality. "Always two there are" -- Yoda. Energy is duality, duality is energy. The conservation of duality (energy) will be know as the 5th law of thermodynamics -- Generalized Duality! Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Synchronic points/lines (positive singularities) are dual to enchronic points/lines (negative singularities). Points are dual to lines -- the principle of duality in geometry. Everything in physics is made out of energy (duality). Time contraction is dual to length dilation -- the twin paradox. Time dilation is dual to length contraction -- Einstein, special relativity. Space is dual to time -- Einstein.

​@@hyperduality2838 that's why we should watch Jean-Pierre Petit and his "janus"theory. There was poincarré, who elaborate this, einstein, who made it popular, and Jean-Pierre Petit, who turn it right.

Hi Narf! Thank you for the insightful questions as always. The first thing to note is that, before electroweak symmetry breaking, even though the particles themselves are massless, they still interact with the Higgs, and this interaction is what ends up giving them their mass after EWSB. So, all the flavors of both the leptons and quarks are always distinguished from each other by their interactions with the Higgs. However, for pretty much all of the particles aside from the top (really, the left-handed quark doublet and right-handed up-like singlet of the third generation), these interactions are very small, so there is actually quite a large approximate flavor symmetry of the standard model: U(2)^2 x U(3)^3 (the U(2)^2 comes from the first two generation quark doublets and up singlets while the U(3)^3 comes from the down singlets, the lepton doublets and the charged lepton singlets). The smaller U(2)^5 symmetry disregarding all third-generation particles is actually an even better approximation. These approximate symmetries allow us to essentially build flavor multiplets exactly like you suggest, and can be very useful for model-building in new physics scenarios. The fundamental difference between the quarks and the leptons is that in the standard model, the neutrinos are massless, while both the up- and down-like quarks are massive. The key feature of this is that when one of the two particles in the weak doublets is massless after EWSB, the particle is always a physically propagating state with definite-energy, no matter how one rotates it with a unitary transformation (this is just because there is no mass matrix that needs to be diagonalized to give definite masses to the particles). So, one can always rotate the massless particles (neutrinos in this case) in the opposite direction of the massive particles (charged leptons) so that the weak eigenstates are aligned with the mass eigenstates and no off-diagonal interactions arise. However, since we now know that neutrinos are massive, we need to include this mixing effect into the neutrino sector as well which is accomplished using an analog of the CKM matrix, known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, and is the reason we end up with neutrino oscillations. The short answer to the question of more quarks between the bottom and top masses is that no, this is not really possible. The reason is actually quite interesting: we know that the coupling constant of QCD runs with the energy scale, and this running can both be calculated as well as measured (of course, "measured" has a loose meaning here since couplings aren't totally physical objects. However, the parametric dependence of physical observables on these couplings can be determined by the theory, allowing one to extract them from data). This running has a strict dependence on the number of quarks with masses below the energy scale at which one is determining the coupling constant. And it turns out that, below the top mass scale, a five-quark-flavor running agrees with the experimental determination remarkably well (see Fig. 9.1 in pdg.lbl.gov/2006/reviews/qcdrpp.pdf, where the theory prediction is the Spectroscopy (Lattice) point). If there were additional quarks in nature, we would have to see an additional effect here since all quarks interact with via QCD with the same strength. Hope that answers your questions!

@@zapphysics Hi Zap! :) Thank you, for the interesting answers! The neutrino part and another one of your replies here gave me a thought. If we think of lepton, and quark generations as doublets, should we not expect the neutrinos to be electrically charged, just like the quarks? Or is this just about flavor? Is the charge of the neutrino predicted, or is it a measured value?(of course we needed it to be neutral to match observation in the first place) Is there a relationship between the massiveness and the neutral charge of the neutrino? The fact that we see no charged particles without mass seems "suggestive". There is nothing that forbids this, right? And an unrelated question. I have been (slowly...)working my way down to the fundamentals of General Relativity and Differential Geometry and I am *struggling* with the concept of a connection, and how to rigorously define one. Then I remembered that Sean Carroll said that, in QFT, the boson fields are connection fields, just like the ones in GR, which tell us how to parallel transport the charged vectors. What is the bundle in this formulation? And is the "Photon Connection" just given by Maxwell's Equations? Fun anecdote: I had just happily accepted that "connection", "parallel transport", and "covariant derivative" mean the same thing. Today I started a lecture that opened with "in general relativity these are often used synonymously. You need to forget all that." ... Math is hard.

@@narfwhals7843 I'll start with your first question, mainly because it is much easier to answer. So the main thing to remember is that electric charge is really a combination of a particle's weak isospin (T3) and its hypercharge (Y) after EWSB as e = T3 + Y (sometimes there will be some extra factors of 1/2 thrown in, but this is just how you normalize the generators of the group). Just like regular spin, for doublets under SU(2), the isospin values take either +1/2 or -1/2. Also remember that, since these are doublets, both elements of the doublet must have the same weak hypercharge. So, we know experimentally that the up quark has electric charge +2/3 while the down quark has electric charge -1/3. The only way to put these into a doublet in a consistent way is to have T3u = +1/2, T3d = -1/2 and Yquark = +1/6. For the case of the leptons, if we measure an electron with electric charge -1, we have two options: T3e = +1/2, Ylepton = -3/2 or T3e = -1/2, Ylepton = -1/2. The first choice then includes a particle with T3=-1/2 which would then have electric charge -2 and the other choice has T3=+1/2 and zero electric charge. The distinction is then made by the fact that we observe electrically neutral neutrinos. However, this isn't all experimental, there is a theoretical aspect to it, namely in terms of the relative choices of hypercharge of the quarks and leptons. This has to do with the fact that, whenever you have a gauge theory which talks to left- and right-handed fermions differently, quantum corrections to the theory can actually induce a non-conservation of the corresponding current and therefore break the gauge symmetry. This is known as an anomaly, and it is really important that for any chiral gauge theory, these anomalies cancel. As it turns out, this is actually quite restrictive in terms of the allowed charges your fermions can take in the theory. In the case of the standard model, once you include all of the lepton and quark doublets as well as the singlets, it actually pretty much completely fixes the allowed choices of the hypercharges of the particles in order to preserve the symmetries of the SM. Weinberg actually has a really good discussion of this in Chapter 22 of his second QFT volume. I might be able to help a little bit with the topic of gauge fields as connections, but I will warn you that this topic has always confused me as well, so my help will be limited. I think that the best way to get a more intuitive grasp on what these things mean is to just look at the flat, Euclidean plane described in polar coordinates. Something that I find is not necessarily emphasized enough is that this choice of coordinates *depends on where you are in the plane*, or in other words, if you start at one point in the plane and move to another, your basis vectors have changed their orientation. This is obviously going to be an issue when we try to compare two points using e.g. a partial derivative, which takes the difference of some object evaluated at two neighboring points. The problem is that how I describe that object actually changes between the two points because the basis vectors are not aligned (think of defining the vector (1, 0) where the first component is radial and the second is azimuthal at the two separate points: the vector looks different at each point even though it should be the "same" object!). So, in order to actually reasonably take the difference of this object evaluated at two points, I need some way of consistently accounting for this additional change coming from the rotation of the basis vectors at the two points. The way to do this is to introduce some new object that "connects" the two points by "transporting" the basis vector information at one point to the other point. As this would suggest, this object is exactly the connection, while the action of this connection is parallel transport. I can then define an operation which consistently compares two neighboring points by introducing this connection into the derivative to account for the rotation of the basis vectors. This combination, of course, is the covariant derivative. Now, in spacetime, this has a fairly concrete meaning: the choice of coordinates can be spacetime dependent, so we have to account for the way that the components of spacetime tensors change as the basis vectors are varied from point to point. Remember that each component of a tensor is contracted with a set of basis vectors to define the true, coordinate-independent tensor object, so the number of indices that we need to describe these components will change the way that we need to compensate the variation of the basis vectors. This is why Christoffel symbols have three indices: they eat one spacetime index (how one piece of the tensor components change with the basis vectors), they have to replace this index for consistency between the right and left sides of the equation, and it has to have an additional spacetime index that tells us along which spacetime basis vector we are changing. When we are working with gauge theories in QFT, the story is quite similar, except the variations no longer happen in spacetime (we always work in flat spacetime, so we can simply use Cartesian coordinates), but instead some internal space which depends on the symmetry group of the theory. When we make this symmetry local (i.e. spacetime dependent), the field's orientation in the internal space is going to change as we move in spacetime. So again, when we try to compare the field at two points in spacetime, the field has a different internal orientation at these two points, and we are trying to compare two inequivalent objects. So we do the same thing: we introduce an object which transports us from spacetime point to another in a consistent way in the *internal* space. So by analogy to the Christoffel symbols, this connection will need two internal indices to "eat" and replace the internal space information of the field at the two points, and a spacetime index to give the information of the path through spacetime. So, we always have a spacetime vector which serves as a connection, though its internal structure depends on the representation of the field under the symmetry group (this turns out to be exactly proportional to the generators of the representation). This connection is the spacetime vector gauge field that we introduce to preserve local symmetries. Of course, when we add this connection into the derivative to properly compare the spacetime points, we end up with a gauge-covariant derivative. Unfortunately, I don't think I'll be much help with the bundles...this is usually the point where my eyes glaze over a bit seeing the mathematical definitions, theorems, etc. (I agree that math is hard.) However, maybe this review of the subject will be more useful to you: arxiv.org/pdf/1607.03089

@@zapphysics Thank you _very_ much! So the vector we are transporting is not the state vector in Hilbert space, but a vector in this "internal space", given by the symmetry group. This should make it clear what the bundle is. I'll check that paper! The first paragraph is already fascinating.

Hi, these are very great questions, and looking back, I think I definitely could have done a better job addressing them in the video itself. I think the main points to answer your questions are around 6:36 and 9:58 in the video, respectively, but I'll try to answer both of your questions in some more depth, because they definitely deserve to be expanded upon. (1) This is a bit tricky mainly due to the fact that the same "intuition" for a ball on a spring doesn't really apply for a quantum harmonic oscillator. Probably a better way to think of a quantum harmonic oscillator is something like a piggy bank that only accepts dimes: it can only hold an exactly integer number of dimes, and since the hole is only large enough to fit dimes if you try to put in a different amount of money, it won't fit. In the case of a quantum harmonic oscillator, the dimes are the quanta of energy: you can only add or take away integer multiples of the exact value of energy accepted by the oscillator. Now, when we couple together multiple identical oscillators, any one oscillator can still only accept a discrete number of "dimes" at a time, but they are allowed to pass them back and forth between each other. When we increase the number of oscillators in the system, it gets easier and easier for neighbors to pass these quanta between each other, just like the neighboring masses respond more quickly to pushes and pulls in the the classical case, but you have to handle it a bit carefully to avoid things from blowing up (just like replacing the discrete masses and spring constants with a continuous mass density and tension). The end result is a line of continuous sites where these quanta of energy can live, but it is free to move from one to the next. The other key point is that, due to the uncertainty principle, we can't know exactly where these quanta of energy are living; we have to describe it by a probability distribution for them to be at any one site. The "spreading" effect that you are looking for an analogy for from the classical case is the spreading of this probability of the quantum to be at each site. However, it's very important to not get this probability mixed up with the location of the quantum: the unit of energy lives *at a single site* at any one time (remember, the oscillators can only accept/give up an integer multiple of this energy at a time, so if there is only one unit of energy in the system, it cannot be split up between sites), we just can't know exactly which site it is living at at any one time, so the best we can do is a probability distribution. (2) You're completely right that when we go to higher dimensions, the probability distribution will no longer "look" particle-like. In fact, in d-dimensions, instead of points, you would see the probability of locating the quantum in the system as a (d-1)-spherical shell expanding outward at the speed of sound. This is because we initially put the quantum of energy in a single site, so we have to have infinite uncertainty in momentum. Since the particle is massless, though, it has to travel at the speed of sound (or light in the case of a vacuum theory), and that is why you see a shell instead of a more "dispersive" effect where the probability spreads out over the full space. What you are really seeing is the natural Lorentz-invariance of the field theory! Again, I need to reiterate that the particle-like behavior isn't coming from the behavior of the probability distribution, and in fact this distribution looks very classical in higher dimensions: it is essentially a wave expanding at the speed of sound. The particle-like behavior is coming from the discrete nature of the quantum harmonic oscillators that make up the system. Consider we surround the site where we place the initial displacement (classical) or unit of energy (quantum) with a spherical detector. In both cases, the perturbation travels outward at the speed of sound until it hits our detector. In the classical case, the spherical wave originating from the displacement hits the *entire* detector, i.e. the full detector sees an excitation at once. On the other hand, in the quantum case, the chunk of energy travels outward at the speed of sound, but since there is only a single unit of energy in the system and this unit of energy can only live at one site at a time, the detector will only see a "hit" at *one, specific point*. Of course, we can't predict where that point will be, since all points have equal probability (the spherical probability distribution), but the main takeaway is that there is a fundamentally different behavior between the classical and quantum cases, and this "single-hit" behavior that we see in the quantum case is exactly what we expect from a particle! Hopefully that clarifies your questions a bit more!

@@zapphysics those are fantastic answers, and I so appreciate the time you took to respond! I am much clearer on the first question, and your answer on the second question covers my question well. The remaining questions I have is less about your video and more general questions about quantum field theory. I know it is a mistake to take the metaphors used in the visual depiction of a theory and overextend them, and I know slogans like "the electron is an excitation in the electron field" are gross simplifications. So the many CZcams videos of multiple interacting two dimensional fields with high (or low) points representing particles can only be taken so far. But they all rely on there being "particle-like" localization of the probability density field that endures for some duration of time, and for an unconstrained excitation, I don't understand how that is possible. Everything gets smeared out at the speed of sound. (The other question I have is about how you START with a localized distribution, but that is just the preparation end of the measurement problem, so also not a question directly related to your video!) I am also confused about the actual quantum field - is there THE electron field for the universe, or separate ones for each system/preparation? How many dimensions does it have in multiple-particle systems? I guess I don't understand what entanglement looks like in the field picture. Don't feel obligated to respond, but if any future videos come down the pipeline addressing these kinds of issues... put me down as excited!

@andrewmilne9535 Yes, I certainly took the easy way out in this video by doing a non-interacting QFT :) I think one must always be extremely wary of anyone trying to present a "true" visual representation of an interacting quantum field theory. The main issue is that, outside of extremely special theories (e.g. superconformal field theories), it is not known how to find exact solutions for these probability distributions, whereas non-interacting QFTs are actually quite straightforward to solve, though they aren't really all that interesting for describing nature. In fact, it isn't even known whether or not interacting quantum fields in general are consistent mathematical objects (if you're interested in learning more, this is a good place to start: en.wikipedia.org/wiki/Quantum_field_theory#Mathematical_rigor). What's typically done in particle physics is that one starts with the non-interacting case, which we can solve, and then add in small perturbations caused by interactions. In this case, one can get reasonable answers and predictions (though the convergence of the resulting series at extremely high orders is questionable), but the mathematics behind it quickly becomes substantially more difficult than the free case, so visual representations of these on CZcams, particularly of realistic scenarios like QED, are immediately going to be a bit suspicious. Not saying that you shouldn't ever trust these, but I'm just trying to get the point across that it becomes significantly more difficult to get the answer, especially beyond first order in perturbation theory. Going beyond, if someone is saying that the theory they are visualizing is non-perturbative, they would need some sort of lattice calculations, which are unbelievably computationally expensive, or an unrealistic theory with extremely high degrees of symmetry that is going to be somewhat difficult to extrapolate to reality. I have seen some videos of the sort that you are describing and some are honestly a bit horrifying that they are being passed off as true. One that jumps to mind essentially just took Feynman diagrams, spread out the lines and called them excitations of the corresponding fields. So they ended up with things like virtual photon field excitations in their visualization, which is just nonsensical: we know that everything must be described in terms of probability densities of some observation (in this case, it would be a position observation), and we cannot observe "virtual particles" because they are unphysical. In some sense, these virtual particles never even exist and are really an intermediate tool we use to do perturbation theory; if you were able to solve the theory exactly, you would never need to talk about virtual particles. I think that your intuition is good on the matter: if something is showing a completely non-dispersive, but point-like probability density in any spatial dimension higher than one (and even in one spatial dimension, you should at least see left-right dispersion like in the video), it certainly calls into question the validity of what they are showing. One could always start with some Gaussian distribution which minimizes both spatial and momentum uncertainty, but at the end of the day, there is no getting around the uncertainty principle if you have a quantum theory. Calling something e.g. the electron field only makes sense in this perturbative picture. Really, only in the free picture, but in perturbation theory, we assume that the fields are close to their non-interacting counterparts. In this case, yes, single units of energy that you add to the field will always result in particles of that type (really, they will be slightly different than the free cases, altered by the interactions). The fields themselves are infinitely expansive and fill the whole universe. That is why all electrons have the same mass, charge, etc. As soon as we go away from this nice, perturbative case, though, everything goes downhill: for example, if I try to add a single unit of energy to the Yang-Mills field, I won't end up with a gluon, even though the fundamental degrees of freedom that describe this field are the gluons. (I can't tell you what you would actually get, since again, nobody knows how to solve this problem, but the likely answer is that you would end up with some *massive* glueball which is uncharged under the Yang-Mills interactions.) If I'm honest, I'm not exactly sure how to answer all of your other questions unfortunately. I do, however, want to make a follow up to this video that addresses some of these points eventually (along with some other misconceptions that I am seeing a lot in the comments). I am even toying with the idea of actually trying to do a similar setup with some (perturbative) interacting theory, but like I said, it is a lot of work to make sure that it is done properly, so we will see.

@@zapphysics I'm sorry I didn't see this earlier, just to make sure I thanked you! You have just answered more questions in a few paragraphs and in a substantive way than the 2 QFT text books I am struggling through! You are a fantastic explainer, and as a teacher myself I really, really appreciate it! I avidly await anything you take the time to produce!