Are photons or other quanta at least somewhat localized in a quantum field?

My limited understanding of quantum field theory is that photons or other fundamental particles (quanta) are excitations of quantum fields. The underlying properties of the quantum field are not known, but they are sometimes modeled as an array of quantum oscillators. And the "particles" are really waves moving through the field.

Matrix of oscillators

PBS Space Time host Matt O'Dowd gave a nice presentation of this here (pictured above).

Particles in a field

And YouTuber Arvin Ash gives a similar description here, though notice that this visualization seems to show some notion of "location."

In another post, helpful commenter Jagerber48 gave a link to an interesting article by author Art Hobson titled "There are no particles, there are only fields." Ref (full article here). The included math is beyond my level of study, but I was able to follow at least a part of the narrative discussion. Much of what Hobson explains rings true, but some of it is confusing to me. He states (section IIIA, p10) (emphasis added by me):

Quanta that are superpositions of different frequencies can be more spatially bunched and in this sense more localized, but they are always of infinite extent. So it's hard to see how photons could be particles.

... photons never have positions--position is not an observable and photons cannot be said to be "at" or "found at" any particular point.

Hobson then references Gerhard Hegerfeldt on p19 (emphasis added by me).

Hegerfeldt shows that any free (i.e. not constrained by boundary conditions or forces to remain for all time within some finite region) relativistic quantum "particle" must, if it's localized to a finite region to begin with, instantly have a positive probability of being found an arbitrarily large distance away. But this turns out to violate Einstein causality (no superluminal signaling). The conclusion is then that an individual free quantum can never--not even for a single instant--be localized to any finite region.

...It's remarkable that even localizability in an arbitrarily large finite region can be so difficult for a relativistic quantum particle--its probability amplitude spreads instantly to infinity. Now here is the contradiction: Consider a particle that is localized within Vo at t0 . At any t > t0 , there is then a nonzero probability that it will be found at any arbitrarily large distance away from V o . This is not a problem for a non-relativistic theory, and in fact such instantaneous spreading of wavefunctions is easy to show in NRQP. 66 But in a relativistic theory, such instantaneous spreading contradicts relativity's prohibition on superluminal transport and communication, because it implies that a particle localized on Earth at t 0 could, with nonzero probability, be found on the moon an arbitrarily short time later. We conclude that "particles" cannot ever be localized. To call a thing a "particle" when it cannot ever be localized is surely a gross misuse of that word.

It seems that the word "local" has very specific meanings when considered in quantum mechanics. In this post, user udrv gives a good description of locality (I added emphasis).

Basically, the general "principle of locality" (Wikipedia ref.) requires that "for an action at one point to have an influence at another point, something in the space between the points, such as a field, must mediate the action". In view of the theory of relativity, the speed at which such an action, interaction, or influence can be transmitted between distant points in space cannot exceed the speed of light. This formulation is also known as "Einstein locality" or "local relativistic causality". It is often stated as "nothing can propagate faster than light, be it energy or merely information" or simply "no spooky action-at-a-distance", as Einstein himself put it. For the past 20 years or so it has been referred to also as the "no-signaling" condition.

It seems that original quantum mechanics (QM) states that there is a wave function $\psi$ independent of any underlying structure, and $|\psi|^2$ gives the probability of detection during measurement. Quantum field theory (QFT) seems to give an underlying structure and rationale for the wave function, such that a particular part of the field is influencing its neighbor to pass along the wave. This would seem to me to indicate that the theory depends on "locality", and there is no "spooky action at a distance." And yet, Hobson states (p22) (emphasis added by me):

Nonlocality is pervasive, arguably the characteristic quantum phenomenon. It would be surprising, then, if it were merely an "emergent" property possessed by two or more quanta but not by a single quantum.

The real-life example that I hear the most supporting non-locality is the separation of entangled particles. E.g. two entangled photons can be separated a great distance before measurement. And by detecting a property (say spin) of a first particle, instantly the spin of the second particle becomes fixed. No information is passed faster than light, but yet it is said the wave equation $\psi$ describing the pair collapses, affecting both over distances which would seem to imply faster-than-light coordination. The easiest (and reportedly wrong) explanation of this would be that the state of each particle was determined before separation, and measurement merely unmasked the situation. But for this to be true, there would have to be a hidden local variable, with the state (e.g. spin) being carried by the particle. And Bell's inequality has proven (though I admit I don't understand the proof) that local hidden variables are disallowed. Here, my understanding of the local qualifier is to mean that the particle holds the variable in itself somehow, not the sense of the word quoted from user udrv above whereby one element influences another.

But the part that confuses me the most is why the quantum field itself can't act as a global (as opposed to something self-contained) hidden variable that carries state of entangled particles. This would seemingly brush away a lot of "spookiness".

Summary of question(s):

  1. Although a photon quanta is not a discrete thing, i.e. not a little ball flying through empty space, it seems that it has at least a general location in the quantum field. I would think that most agree that energy put into the field from a distant sun initially has location near that sun, and that after many years the energy travels to us as visible starlight. Why then do the above quotes seem to indicate that a photon can never be said to localized in some arbitrary finite region?

  2. The fields, as illustrated by YouTubers Matt O'Dowd and Arvin Ash seem to have some real existence, with behavior at least somewhat analogous to the behavior of other fields (materials) at human-sized scales (e.g. water, air etc). They seem to show completely normal action without any spookiness. Although analogies are never perfect, are they at least mostly correct and on the right path? Or are they off base?

  3. Hobson's article, There are no particles, there are only fields, seems to be a push to very much downplay the concept of particles. When particles are sometimes conceived of as little billiard balls, as I did when starting college physics, I can understand Hobson drive to emphasize that waves in a quantum field don't really have a location. But isn't there at least a semi-location? Can't we at least say which side of the galaxy a quanta of field excitation exists in?

If possible, please use math as simply as possible in any explainations. When an answer is given in the form of "Element A can't be true because the equation for theory B says otherwise", I often can't follow the logic. If I don't already understand the equations and how a theory ties into reality, then it seems a bit like circular reasoning to me. This is limitation on my part and I continue to try to extend my knowledge as best I can.

I have reviewed these other posts but didn't find an answer to my question:

  • 1
    $\begingroup$ This question is very long, but at its end I'm not sure what it is really asking: You already link to this answer which clearly states that particles can never be exactly localized but that it is possible to have approximately localized states for all practical purposes, and it even specifies the shape of that approximate localization (an exponentially falling-off profile). Why does that not already answer this question? The question reads as if people were denying the possibility of any approximate localization at all, which just isn't true. $\endgroup$
    – ACuriousMind
    Nov 25, 2022 at 19:11
  • $\begingroup$ Also, see physics.stackexchange.com/q/13157/50583 and physics.stackexchange.com/q/154391/50583 before putting too much effort into arguments that rely heavily on the analogy between quantum particles and excitations ("ripples") in a classical field that is visible in the images in the question. $\endgroup$
    – ACuriousMind
    Nov 25, 2022 at 19:16
  • $\begingroup$ @ACuriousMind The link you give starts off: "In a strictly nonrelativistic QFT, a field operator 𝜑(𝑥) may indeed create a particle that is strictly localized at 𝑥, with the same caveats in nonrelativistic single-particle quantum mechanics: those states don't really belong to the Hilbert space because they're not normalizable." At the end of my question, I asked for a simpler answer, as this one is above my understanding. And the length of my question is giving documentation for my points of confusion. Did you read the question, or was it too long? $\endgroup$
    – kdtop
    Nov 25, 2022 at 21:23
  • $\begingroup$ If the problem is just that you can't follow the mathematical jargon in the answer, then just saying that would have been much clearer to me than this question that cites a wealth of other materials. In any case: Even if you may not understand the mathematical details prior to it, I still don't see how the third-to-last paragraph of that answer is not an answer to this question - again, it says very explicitly (even quantitatively!) in what sense both massive and massless particles can be localized in QFT. $\endgroup$
    – ACuriousMind
    Nov 25, 2022 at 21:40
  • $\begingroup$ @ACuriousMind Thank you for your replies. Looking at the third-to-last paragraph, it does indeed state that semi-localization is allowed. The references I gave seemed to say the contrary. It is difficult for me to understand some things when different sources seem to vary. And it is difficult to determine when people just disagree, vs. when they are saying the same thing a different way. Thanks again. $\endgroup$
    – kdtop
    Nov 25, 2022 at 22:07


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