Waves in an Impossible Sea

Going Beyond the Book

If you’re reading Waves in an Impossible Sea, and you have a question about a related subject that the book doesn’t directly cover, please ask it here!

79 Responses

  1. In chapter 16, you write that a photon can be spread out across a room and then absorbed by “a single atom, located at one microsopic spot on the room’s walls.” Suppose the wall is a box of gas. Are atoms of the gas wavicles, too. Would such an atom still be located at a small spot when it absorbs the photon, or would it be spread throughout the box? If the former, how big a spot, and what kept it from being spread out more?

    1. So, this is indeed “beyond the book”, but worse, probably beyond what I can explain clearly in a few paragraphs. Really, this needs a whole lot of careful buildup. Quantum physics is very, very strange, and to explain it well takes more pedagogical experience than I currently feel I have. But I’ll try.

      Let’s keep it as simple as possible: there’s just one atom in the room, not a wall, and there’s one photon crossing the room. Chances are the photon will miss the atom, but the probability of interaction isn’t zero, so let’s look at it.

      An atom is not a simple wavicle, because it is made of many objects, but it does have wavelike behaviors that it inherits from the elementary wavicles that it contains. For this reason it can indeed spread out. [This kind of thing is actually observed, most clearly as reported (poorly) in https://www.space.com/atom-transforms-into-quantum-wave-schrodinger … at some point maybe I’ll have time to explain the paper and the physics behind this.]

      How spread out is the atom? This will depend in part on when and how the atom was inserted in the room; the more mass an object has, the longer it takes to spread out across a macroscopic space. (See https://profmattstrassler.com/2024/03/19/yes-standing-waves-can-exist-without-walls/ ) But let’s assume it has had enough time to spread out across the room.

      Now we have a photon spread out across the room, and an atom spread out across the room. What happens when they interact? Do they interact everywhere at once?

      In a sense yes — they could interact anywhere — and in a sense no — when they interact, the interaction occurs in a small region… atomic sized, because that’s the scale of the electromagnetic dipole moment of the atom, which is what interacts with a photon. The fact that both atom and photon are spread out doesn’t change that.

      The fact that they are spread out has only the following effect: the region where they interact could have been anywhere in the room, and could not otherwise have been predicted.

      Or at least, that’s one consistent way to interpret the math. It’s not the only one, but I don’t think I could correctly go through all the others.

      [If we have multiple atoms, then we have to deal with the fact that the atoms are all correlated with one another, which means that the location of one could in principle depend on all the others, complicating things infinitely. Only in a Bose-Einstein condensate, where the correlation is perfect and simple, would things become easier, but I haven’t thought through how a Bose-Einstein condensate interacts with a photon.]

      Obviously this raises all sorts of questions about where information, objects, and relationships among objects are actually stored. Correlations between objects matter; in fact, interactions and measurements always create new correlations between objects or between an object and an observer. Reality contains these correlations, and can’t simply be stored in the way that seems intuitive to us, where there is space that pre-exists, and there are definite objects at definite locations in that space that we can measure without significantly affecting them. Trying to picture this, or follow the logical flow where it needs to be taken, goes beyond what is easy for the human brain, and physicists and philosophers are still struggling for easy-to-explain ways to answer questions like yours.

  2. Thank you for your book. Not only did I enjoy it, as a high school physics and math teacher, it taught me a lot, and has helped me give better answers to my curious students when asked about the Higgs field.
    I also appreciate your insightful comments about fields and relativity. I feel I can use it when I talk about relativity.

    As a physics teacher, I want to give my students a conceptual model of quantum mechanics, to help introduce it in a way that makes some sense. I think I have succeeded in the following, and since intuitive models of physics seem to be your thing, I am hopeful you will be generous enough to give me your thoughts on it.

    My conceptual model is: (1) Quantum objects can be thought of as instantly reformable wave packets, all part of one thing. An electron, say, is an infinite bunch of waves of the electron field, summed up to produce a wave packet. All the constituent waves are part of one thing, the quantum object, the electron. Upon measurement (or a proper interaction with other quantum objects) it instantly reforms (collapses), and you get a new wave packet. (2) These wave packets instantly reform when interacting with other quantum objects, in a random way obeying the Born rule. I tell my students (because it makes more sense) that instant reformation (collapse) probably happens when things interact, but we only know for sure that it happens upon measurement. The size of the new wave packet depends upon the wavelength of the quantum object that caused the collapse. I also go over the case (to illustrate what the Born Rule implies) where half of a quantum wave packet hits a detector and the other half hits nothing. The “particle” is detected half the time, half the time it isn’t.

    The reason I like this conceptual model is that enables one to reason out the results of the experiment where you send electrons or photons one at a time though a double slit, even if you’ve never heard of the experiment before. Before it encounters the double slit, the electron is a large wave packet, bigger than the double slit, so you can reason that if it makes it through it will interfere with itself, like any wave will do after passing through a double slit. Most of the time the quantum object collapses on the slit material, because most of it is interacting with the material, following the Born rule. Sometimes it makes it through, because of the holes (not all of it is interacting), and interferes with itself as it travels to the detector. Where will it collapse on the detector? It will never collapse at the points where it is cancelling itself out, and has the greatest probability to collapse in the middle, where the squared amplitude is the greatest (the Born rule). When it instantly reforms, it gives its energy to a detector electron, which is much smaller than it, so upon collapse, it shrinks a great deal, becoming a new wave packet. Thus we see small “particles” on the detector screen.

    Now of course, I tell my students about other ways to think about quantum mechanics, and discuss particles and waves and so on, and I tell them that what I have just described is one way to look at what the math tells us, but it just seems like a conceptually easier way to think about it all, seems to give one more insight.

    In this model, the only sense that a quantum object is a “particle” is that all the waves that make up the wave packet are part of “one thing”. If you split a quantum object in two, the two parts are still part of “one thing” and if one of the parts instantly reforms upon measurement, the other part disappears. The “whole thing” reacts to part of it successfully interacting, which is a “particle” like trait. However, that seems to be it. There is no particle in this model, as there is no “little ball” anywhere.

    This conceptual model is also helpful in talking about electrons about an atom, which are trapped wave packets that form standing waves in a spherical sort of enclosure. The wave packets are not accelerating, but are vibrating in place, so they don’t lose photons. Of course, the wave packets are now of a more uniform frequency, and they still cancel out in certain areas, so you can never detect them there, according to the Born rule. I go over this and show how you can get the shape of electron “orbitals” through the standing waves vibrating in place and having a non zero squared amplitude in certain spots.

    It’s also useful for talking about the uncertainty principle, but I’m writing too much…

    What I really want to ask you about is the way this model deals with entanglement. The natural model for entanglement with this picture is that if you have two entangled electrons, A and B, the waves of A “occupy” B and the waves of B “occupy” A. Upon measurement of, say A, the waves of B are instantly kicked out of A, and vice versa. You can also think of it as, say, the spin of A exists in B (and A) and the spin of B exists in A (and B). Upon measurement, they instantly reform and kick the other spin out. It makes some sense that whatever you measure in A will be related to what you measure in B (like an opposite spin). This makes sense to me, but I haven’t been able to talk about it much with anyone more knowledgeable, like you. How does this sound to you?

    Anyways, I don’t want to make too long of a post. When I have presented this to students (with all the necessary caveats – I hope!) they say they really appreciate it and it makes sense to them, much better than chemistry class. Of course when I first came up with it, I was nervous that I was leading them down a wrong path, so I showed this to a few local physicists, and they gave me some approval, which I hope you will as well.

    So my question for now (I hope you will indulge me by answering a few more) is: How does this sound to you? Also, very much hoping to get your thoughts on the entanglement model.

  3. Prof. Strassler:

    First and foremost, thank you. Your book and other explanations on this site have been wonderful to my superficial understanding of the concepts behind modern physics. I’ve re-read the book many times, and enjoyed it every iteration.

    I hope my question isn’t too naive, I’m a physician and have long been an enthusiast of grasping the Standard Model, though I lack the needed math to go deeper.

    My concrete question: Could light speed be infinite, and not c? I mean could the cosmic speed limit be just an artifact that we experience as a consequence of our perspective as beings made from massive particles?

    If light, and gravitational waves move at c speed, but no matter how fast we go we can’t overtake them, and even measure their speed as the same no matter what, could it be possible that their speed is actually infinite, and not “just” very fast?

    Since from the perspective of a photon (assuming it as a being) it would not experience neither time nor space, because of extreme time slowing and extreme length contraction, could it be possible that our perception of a slower time and a physical space be “artifacts” of us being massive, or made out of massive particles?

    Again, hope I’m not speaking absolute nonsense, and thanks again.

    1. Thanks for the question, and I am glad you enjoyed the book.

      The answer to your question is no, their speed could not be infinite. We directly can measure the time that it takes for light to go from point A to point B, because we have materials that emit light, and screens that absorb light, whose timing is precise to better than a nanosecond (a billionth of a second), which is the time it takes light to travel just 30 centimeters. The travel time observed is always the same: it is the travel distance divided by the cosmic speed limit c.

      Another way we can be sure of this is to study not light or gravitational waves but ordinary matter. If we send an electron from point A to point B, we can measure its time of travel, also. Now the travel time is the travel distance divided by the speed of the electron, v. No matter how much motion-energy we give the electron, we find that its speed v is always less than the cosmic speed limit c — that is, the travel time is never shorter than the travel distance divided by c.

      Furthermore, there are many details in the relationship between energy and speed that can be measured, and they are in exact accord with Einstein’s relativity equations, which are essential in preserving Galileo’s relativity principle. If light and gravitational waves traveled at infinite speed, they would violate either Galileo’s relativity principle or Einstein’s relativity principles, in conflict with the results of experiment.

      Indeed, the entire Large Hadron Collider — the timing for the accelerator, and the timing for the particle detectors — relies on knowing the travel times of photons and other particles to better than one nanosecond. If our understanding of particle speeds were not precise and reliable, the Large Hadron Collider and its particle detectors ATLAS, CMS, LHCb, and ALICE simply would not work.

  4. Dr.Strassler:
    I actually posted this question on a different thread, but realized it is more appropriate here. It is concerning fields, specifically the magnetic field. If I have two magnets, a north pole and a south pole facing each other, and pull them apart, doing work on the system. after pulling them apart, I would say the work I did appears as an increase of mass of the system.

    I then, heat up one magnet. Or both, until they lose their magnetization. What happens to the energy I put into the system? Does the collapsing magnetic field transfer the energy to something else?

    1. The first paragraph and second paragraph are logically separate.

      First paragraph: the work you did does not appear as an increase of the mass *of the system*, unless you tell me what the system is. In particular, if you are part of the system, the mass of you and the magnets has not changed; in fact you will radiate some heat and the mass of the whole system will slightly decrease. If you are not part of the system, then we can’t measure the mass until you let go of the magnets, at which point the question is incomplete. Is something holding the magnets in place? If there is nothing holding them at the moment you let go, they will move, and so the energetics of the system has to be followed carefully. For instance, if they now slam into each other, this will create some heat and electromagnetic radiation, which will carry energy away. In the end the mass of the two magnets, now stuck together, may in fact be reduced rather than increased; one has to think about it carefully.

      Second question: what happens if a magnet is heated and loses its magnetization? Well, heating them puts energy in, and increases the mass. The loss of the magnetic field (and any effects thereof) will be smaller than the heat-energy put into the magnet, which will partially be inside the magnetic (in vibration motion of its atoms) and partly radiated away in photons. The magnetic field energy gets swallowed up in this complicated system in a way that can be tracked if necessary, but it’s certainly not simple.

      Of course, if you now allow the magnet to cool again, then the energy will be radiated away, although some of it may be used to remagnetize the magnet, either as a whole or locally in magnetic “domains”.

      All of this is to say that when you want to do precise bookkeeping and make sure you know where energy came from and went to, and how much an object’s mass may have changed during a process, it requires careful tracking of all the details, and real effort.

  5. Dear Matt,

    Thank you for taking the time and effort to write such a wonderful book. You’ve truly exemplified the famous saying often attributed to Albert Einstein: “If you can’t explain it to a six-year-old, you don’t understand it yourself.” As a musician, I often feel like a six-year-old when it comes to the complex concepts you’ve so skillfully broken down into small, easily grasped notions and metaphors. You’ve opened up a new world of understanding for me.

    I have a question: Did you play the clarinet? I inferred this from your remarks about memories of cold mornings. Or was this just a metaphor?

    For more than 15 years, I’ve struggled to understand the physics of how a clarinet works. While I can’t claim complete understanding yet, your book and explanations have brought a lot of clarity. I can confidently say I now perceive much more about the phenomena behind it.

    Many kind regards and thanks, and I eagerly look forward to your next books.

    With fond appreciation,

    1. Thanks for your kind words about the book! And yes, I played the clarinet for about a decade, and have played the piano for five decades. It is absolutely true that in music camp chamber music rehearsals, I was often out of tune before 9:30 am.

      While the following website is about flutes, some (not all) of the issues are exactly the same as for a clarinet. Maybe you’ll find it enlightening.


  6. Matt, your book is super awesome! I’ll try to come to Lenox, if stars aligned, to see you in person.
    Book gives many good analogies, and I wonder if you have good analogy for energy and field values itself?
    E.g. for example electron has that amount of energy because at certain location in electron field value is vibrating (changing back and forth?) with frequency f. What amplitude of such vibration could represent? I mean, if it’s in fermionic field, then it’s bounded by some value, but this value is not part of E=hf formula. So is this property of the field and not the quant then? How can we think of the value that vibrates?

  7. I don’t think I have a question, but perhaps something worthy of an endnote. (Aside: I much prefer footnotes to endnotes, but, whatever works…)

    In figure 27, you lay out the geometric optics of rainbow formation. I recall an article 10-20 years ago that explained a small adjustment to that picture that gives the actually observed rainbow angle. The idea was that the total internal reflection induces a surface wave/evanescent wave/plasmon wave/something (this part of my recollection is fuzzy) that travels for about degree, then internally emits the photons from a slightly different point on the surface. Then the remainder of the path in the diagram is adjusted by about a degree. I think I recall that CAM (complex angular momentum) calculations contributed. More recent papers about Regge pole contributions to atomic rainbows swamp my Google searches, so I’m not able to find a relevant reference. Sorry.

    Why this is relevant to the book: My mental model of the fundamental fields frequently leans on my understanding of coupled pendula. (Random example: https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/08%3A_Oscillations/8.04%3A_Coupled_Oscillators ) The pedula stick out in various directions (in a more-than-3-dimensional space) and are coupled by springs. (If you try to embed this picture in Euclidean 3-space, there’s nowhere to put all the springs.) This is my go-to get-the-brain-rolling model for QFT, providing a mental model for energy transfer between fields. This seems to be an important idea in the book. It’s also happening in the rainbow water droplet — the external to the droplet photon modes are coupling to the internal to the droplet photon modes are coupling to the surface (something) modes — the energy is sloshing around among these various oscillators.

    1. (I prefer footnotes too — this was my publisher’s decision.)

      I haven’t ever looked at this subtlety in rainbows; thanks for pointing it out. If you do find a reference to it, let me know.

      Your mental model for fields is indeed the one I use.

  8. Are zero-point fluctuations of a fermionic field subjected in some way to the Pauli exclusion principle? My question is motivated by the following reasoning. Quantum theory predicts immense energy density for the lowest possible scale for these fluctuations. If (1) the exclusion principle is applicable, they must seemingly repel each other and expand as a whole at breakneck speed. Then if (2) this expansion is decoupled from spacetime dynamics, this might explain both cosmological constant and hierarchy problems, in a unified setting.

    Similar to the cosmic inflation which produces flat fluctuation spectrum due to superhorizon fluctuation “freezing”, effective (observable) zero-point fluctuations spectrum might be flat as well. The cosmological constant and hierarchy then look like the right picture scaled down by tenths of orders of magnitude. The grand cosmic cycle starts at a hot big bang with the zero average Higgs field value. From then the feedback mechanism works in a quenched regime, i.e. currently Higgs field value just hasn’t had enough time to reach higher values. After eons of eons, the feedback mechanism will have pushed the Higgs field value to such height that wall of doom starts, creating hot and dense plasma while relaxing the field value back to zero in a way similar to the reheating during inflation, and renewing the grand cosmic cycle.

    Recently (2019) the quantum optoelectronics group at ETH Zurich has performed some measurement the zero-point fluctuation spectrum, what is interestingly they observed excess at the lower end and deficit at the higher end of the probed frequency range. Could it be a hint to the flatness of the spectrum? Fig.3d of https://www.nature.com/articles/s41586-019-1083-9

    1. The Pauli exclusion principle does apply, but not in the way that you are suggesting. It’s the exclusion principle [or rather, the same relative minus sign that leads to the exclusion principle] that makes the zero-point energies negative. More precisely, it makes the energy density negative and the pressure positive — i.e., a negative cosmological constant. The positive pressure actually causes the universe to collapse, not expand, because of the way that gravity reacts — just as gravity would collapse a gas with relativistically-large positive pressure.

      These equations aren’t so hard, you can learn them in places such as Kolb and Turner’s book, or indeed any book on cosmology. Possibly even wikipedia. You’ll see that they don’t accord with your speculations.

      1. Dr.Strassler:
        In stars, the positive outward pressure, due to thermal motion, balances gravity trying to collapse the star. However, positive pressure…energy density…contributes to curvature and hence gravity. Obviously, there must be a break even point where the increased outward force, due to thermal motion, outweighs the pressure contribution to curvature due to energy density…..and the star is in equilibrium. Is there an established break even point? It seems that in most “normal” stars the increased outward force…due to pressure….overwhelms the pressure contribution to gravity. In extreme density situations, like neutron stars, would the energy density contribution to gravity start becoming dominant?

        1. Until the system becomes extremely relativistic, with all speeds at or near the cosmic speed limit, the energy density is always larger than the pressure, and that results in usual gravity. In fact, even most relativistic systems still are dominated by energy density. It takes special situations for gravity to be significantly impacted by the effects of pressure, and even more special for those to result in expansionary gravity, such as are seen when one has vacuum energy density and comparable negative pressure.

          1. Dr.Strassler:
            I may be convoluting some terms here, hopefully you can clarify for me. I was associating “pressure” with the definition from gases where pressure is described as energy per unit volume, isn’t that the same as energy density? So isn’t pressure a statement of energy density? Or, is pressure being used differently in General Relativity?

            1. Even for gases, this isn’t true; you’re thinking only of the kinetic and associated potential energy of the atoms, and forgetting the energy stored in their masses. Gravity knows all about that energy density; if it didn’t, then Einstein’s laws of gravity wouldn’t agree with Newton’s in the appropriate regimes where they ought to do so.

              For example, in an ordinary gas, the ideal gas law does indeed say that p = n k T, where n is the number density of atoms or molecules, and k T is proprtional (not necessarily equal!) to the kinetic energy plus internal potential energy per atom or molecule. Typically that kinetic+potential energy equals x k T, where x might be 3/2, 5/2, or 7/2. If we have a gas of atoms only, then x = 3/2, and 3/2 k T = 1/2 m v^2 = the average kinetic energy per atom, where v is the average speed (really, root-mean-squared speed) of the atoms.

              That’s what Newton’s followers thought was the entirety of the energy density. But Einstein discovered that the true energy density is n (m c^2 + x k T), and in any normal gas, m c^2 is far larger than the kinetic + potential energy. That’s what gravity then responds to. And you can see that this is always true for a monatomic gas as long as 3/2 k T = 1/2 m v^2 << m c^2 -- i.e., if the gas is made of atoms moving at non-relativistic speeds, with v << c. Only if the speeds of the atoms are of order c --- relativistic motion, far outside what one learns in first-year physics --- is the pressure per atom is of order m c^2. Moreover, in this case both the energy density and the pressure are positive, so the effect of gravity is still attractive. This is true also for a gas of photons, for which the pressure and total energy density really are equal. But no gas of particles can have negative pressure. That has to come from something more exotic. And for it to have large negative pressure, enough to counter the total energy density, it must be both exotic and relativistic. This cannot happen in Newtonian physics.

              1. Dr.Strassler:
                Thank you, I see where my error was made now. So, when they are talking about energy density, in General Relativity, they are talking about how tightly the masses are squeezed together….is that correct?

                1. That’s too limited a vision. A photon gas at high temperature has high energy density, but the photons have no mc^2 rest-mass energy. Zero-point energy of a field leads to huge energy-density (and pressure) but that energy-density has nothing to do with masses of particles, and it could be positive (for bosonic fields) or negative (for fermionic fields).

                  For a non-relativistic gas of particles with non-zero rest mass, the energy density is potentially dominated by the mc^2 energies of the particles in the gas. But if the gas is sufficiently cold, then various quantum effects can contribute additionally to the energy density; this is critical in neutron stars and in white dwarf stars.

                  So the answer to your question is that you’ve given one correct example, but there are many others.

                2. Dr.Strassler:
                  Ahh ok, I see where my conceptual understanding was flawed, basically it’s just “energy” including all forms of energy, contributing to energy density. My background is in Aeronautics, and when we think about energy density, it’s how energy is stored in the degrees of freedom of molecules….but we generally don’t consider the energy locked up in the masses of molecules, and we aren’t generally dealing with a photon gas.

                3. Yes, this is unfamilar territory even for most professional scientists. But it’s crucial. Newton’s law of gravity creates a pull proportional to the masses of objects. Einstein’s, however, creates a pull proportional to the *energies* of objects, if all velocities are slow — and proportional to a more complicated combination of energy and momentum (the latter of which contributes to pressure) when relativistic speeds are involved. It’s only because *all* the energy creates gravity, including the mc^2 energy stored in all particles with rest mass, that Einstein’s gravity law agrees with Newton’s gravity law in the regime of slow velocities (and sufficiently low densities.)

                  In fact, Newton’s law for the gravity of a proton would be wrong. Most of a proton’s mass comes from the kinetic and potential energy of the particles inside it, not from the mass of the particles inside it. This is discussed in Chapters 6-9 of my book.

                4. Could you please give some ways of thinking about negative energy density of fermionic zero-point fluctuations, as for me it’s really confusing. Is it like “borrowed” energy, or like future potential energy?

                5. You already know that binding energy in atoms or planetary systems can be negative. It is no more confusing than that… It is a statement that if you have a field whose particles have very large mass, but with an interaction that makes them massless inside a box, then if the field is fermionic, you will have to add energy if you want the box to expand, while if the field is bosonic, the box will tend to expand on its own accord. [I should make this more precise; probably there’s a way to turn this into an explicit example…]

                  The question of why a fermionic harmonic oscillator’s zero-point energy is negative should have an intuitive explanation. (For a bosonic harmonic oscillator, it is easy.) I do not have a good one yet, but it’s on my mind.

  9. Hi Matt,
    Regarding a first possible wave in that impossible sea (smallest oscillating thing?), you got me wondering! In your book, do you explore how Planck’s quantum of angular momentum might act on that condensate of weak hypercharge (Higgs-type field)?

    1. Sorry, but I don’t understand the question. Angular momentum doesn’t “act” on condensates, or more specifically on the Higgs field. Maybe ask again, more precisely?

      1. Hi Matt,

        Thanks for the prod! Thinking of that smallest possible wave in your impossible sea, mediated by Planck’s constant of action, implied a quantum of curvature (for those keen to quantize GR).

        Being irreducible, such a quantum of angular momentum would persist, a “first measurable thing”; effectively a quantum of localized energy, hence density. Hence curvature.

        So what I had in mind was: what if this Planck scale oscillator, this smallest spinning thing, had a rest frame, and it dragged that local frame as it spun? If this quantum of curved action is acting in/on a superfluid tachyonic nonzero scalar field (say, a condensate of weak hypercharge), would we have a quantized vortex of Lenny Susskind’s zilch?

        Which suggested a new twist on the old axion idea. Meanwhile, could such a “cold” axion, a classical quantum of curved action, actually define Planck’s constant? Such thoughts sprang to mind when considering minimal waves in your marvelous sea!


  10. Thanks a lot for the book, it inspired me and made me think (I’ve recently had several intense half-nighters). I don’t want to spam your blog, so ask for an advice of how and with whom better to share:
    – Short fictional story illustrating the equivalence principle of GR (in English, characters excluding spaces – 4438)
    – Simple geometric derivation of Lorentz transformations, relativistic energy-momentum relation, invariant mass, for boson-like wavicles.

  11. Several questions about cosmic walls.
    – Does the wall of doom you described in The Wizardry of Quantum Fields run at the cosmic speed limit? I think so as you mentioned the “unsuspecting” universe, and we humans will just cede to exist if the wall reaches us, like unfortunate passengers of the Titan submersible for whom its catastrophic implosion went unnoticed.
    – The Hubble constant of 70 km/s/Mpc means that at ~4Gpc the expansion is superluminal and nothing which is father away now, including a wall of doom, could reach us. Therefore, we only need to worry about the local walls which have much lower probability to appear due to the restricted volume, right?
    – I’m thinking of a more benign version of the domain wall resembling the iron’s magnetic domains and their walls. Suppose the Higgs field acquired some constant non-zero value but with different potential phase across the universe (considering the Mexican hat shape). If so, are the different vacua “compatible” with each other, i.e. can a wavicle traverse the wall between phase domains? And how such a wall might look like?

    1. Excellent questions.

      1) In most models it approaches the cosmic speed limit closely after a short time, and yes, we don’t get much if any warning.

      2) Yes, what happens outside our cosmic horizon cannot hurt us, but there are some regions that are currently outside our horizon which might someday come within it. This depends in part on what “dark energy” really is and how it evolves.

      3) Domain walls of this sort are not possible with our Higgs field, because the different phases you are imagining are physically equivalent (“gauge equivalent” in the language of physicists.) This is different from a magnet where the different directions of pointing are physically inequivalent, making domains and domain walls possible. [Buzzwords: global symmetries can have inequivalent domains related by a symmetry; gauge symmetries are fakes — they are symmetries that act on the math used but not on the physics — so they cannot.] Other scalars fields could in principle have domains; but indeed, such domain walls have big effects on cosmology and so are sharply constrained by observation.

  12. Matt, the electron field interacts with the electromagnetic/photon field to produce an electron; but also an anti-electron, aka positron, to maintain charge conservation. The fact that the two charges are related by a sign change is something I find fascinating and, to my mind, appears to be saying something about (a) some property of the photon that needs to be conserved in the interaction or (b) to maintain a symmetry such as parity: Is this vaguely correct?

    1. It’s mainly just electric charge that’s at stake. The photon’s interaction with the electron field requires that whatever the former does to the latter, the total electric charge can’t change. Therefore, no matter what happens, a photon can only be transformed into n electrons and n positrons (where n is most commonly 1).

      1. Is there a version of the Standard Model that is mathematically consistent in allowing like charges to attract rather than repel?
        If so, how does this influence the need for positrons within it?

        I’m thinking that at the point of creation, electron/positron wavicles will attract one another because of quantum gravitational effects from their interacting rest-energy/mass; decreasing with adequate separation. Likewise an additional electric attraction is needed to ensure work is done in separating two charged wavicles created at the same instant.

  13. Here’re a couple of ideas for the page’s name to discuss the book’s pedagogical choices. First is “Final Draft” and inspired by a recent addition to the Alan Wake computer game series. The game protagonist is an eponymous writer drawn to some kind of a parallel world by an obscure, powerful and evil entity called Dark Presence. And by reading your “strange space-suffusing entity, an enigmatic presence known as the Higgs’ field” passage it immediately stroked me how similar they are. Another is “Yoctosecond Edition” which plays with the double meaning of “second” and refers to the smallest timescale currently accessible.

  14. Hi Matt,
    Great book (and blog)! You clarify some major issues that, it is true, are left unanswered, or maybe are taken for granted, in most sources (eg. the fact that “particles” are wavicles, etc)!
    My question is this: you say, on the one hand, that mass is intransigence while, on the other, that mass doesn’t slow things down. How can these two statements be compatible with each other? And, in the end, what is it exactly that doesn’t allow objects to reach the cosmic speed limit? Mass or inertia? (I recently heard about an, as yet unverified, idea about “quantized inertia” that attempts to explain this fact)

  15. In 12 What Ears Can’ t Hear and Eyes Can’ t See you discuss the question of why a rainbow is narrow. Your explanation implies that dispersion spreads initially bright narrow arc to the spectrum and we see just a small portion of it. That’s true but not the whole story. The question is why that bright narrow arc appears in the first place. The sun has a finite angular diameter of half a degree, and its rays need to be redirected three times to reach an observer. Any initial misalignment would be amplified during the redirection within the constrained drop’s geometry, and we’ll end up seeing a wide band of light. But a thorough analysis shows that the deflection function has an extremum near the rainbow’s angle which acts like a lens focusing part of incoming light into the narrow band. There’s even another extremum having more complex optical path, which gives the outer, less bright arch with inverted dispersion.

  16. You were such a courageous boy jumping while flying in a jet. You might have expected that if you’d jumped high enough, you would have been tackled by the jet’s wall rushing at 500 mph! Here’s a fun exercise also playing with relativity principle, which I came up with many years ago during my long subway commutes. In a subway car take a seat for you to look sideways. Suppose the train goes to the right of you, and you’re in a long steady hop between two stations. Close your eyes. Without any visual clues, feeling just monotonous bumps and shaking, it’s rather easy to trick you mind thinking that the train goes in opposite direction, to the left of you, because by relativity principle they’re indistinguishable. And when the train starts braking, you’ll feel – quite opposite – rather intense speeding up! The feeling lasts until you realize that instead of intensifying, bumps and shaking subside. *Initially I posted it on “Got a Question” page, but now can’t find it there, so repost here.

  17. Can’t stop thinking about Lorentz-invariance. Here’s another toy/wild idea inspired by your latest posts on zero-point energy and standing waves. Starting points:
    – Cosmic fields (vacuum) energy density depends on the smallest scale of its constituent parts
    – Non-interacting wave excitations (particles) gradually spread
    – We have one example when the cosmic field(s) property – curvature – are influenced by matter-energy
    Let’s assume that a) the cosmic fields’ constituent parts are like non-interacting particles. Then in absence of other disturbances they will spread, overlap, and drive the smallest scale up suppressing the zero-point excitations. Then assume that b) in presence of any form of matter-energy these constituent particles interact with it and localize, driving the zero-point excitations to their usual level. Then it’s plausible that localization interaction is Lorentz-invariant because it happens due matter-energy presence, i.e. within its frame of reference. Another advantage that it might alleviate the vacuum energy problem as only induced zero-point excitations count.

    1. Concerning the vacuum energy and induced vs suppressed zero-point excitations. On intergalactic scale the suppressed zero-point excitations might still dominate as the space there is mostly empty. Then this model predicts the vacuum equation of state value as close to but still larger than minus one (due to dilution of the induced excitations during the expansion), which is vaguely consistent with the observed value.

      1. Ok, realized that the assumption for constituent particles to be non-interacting contradicts the need to provide a sort of stiffness in response to gravity.

  18. Dear Matt. I have tried to order your book using the discount code but I live in Spain and it only allows me to put a USA address. How can I get the book shipped to Spain?

    Kind regards,

    Julian Collins

    1. I’ll have to ask the publisher; the publication and shipping of books remains a black box to me. It may be that you have to go through their UK office and I’m not sure they are offering the discount there.

      1. Indeed, you can’t get the US discount, but the publisher says that Amazon in Spain is offering a comparable discount, and that is probably your best bet for the moment. I hope that works for you!

  19. It appears that the reason that 2 atoms cannot overlap is the same as the reason that we have white dwarves or neutron/quark stars. Could you expand on this topic in a blog post on this topic ?

  20. Hello Dr. Strassler, I’ve been a reader of your blog for several years (your series of posts culminating in measuring the distance to the Sun via meteor showers is fabulous) and I really enjoyed Waves in an Impossible Sea, so thank you for your writing!

    I have a question about confinement. In your post “A Half Century Since the Birth of QCD” from November 2023, you describe how confinement is the result of the dual Meissner effect. My question is, does this effect occur in classical Yang-Mills field theory, or is it strictly a quantum effect?

    1. Thanks for your kind comments about the blog and the book!

      Confinement is a quantum effect. In fact, the classical theory has a symmetry, known as a “scaling symmetry”, as does electromagnetism; that symmetry implies directly that the force between two particles at a distance r must be proportional to 1/r^2. Both quantum electromagnetism and quantum Yang-Mills theory violate that symmetry; in the former case the force is a little weaker than 1/r^2 at long distance, while in the latter case it is stronger at long distance. This part is easy to calculate in the quantum theory. What is hard to prove is that in the quantum theory the force eventually becomes constant — independent of r. It is this constancy of the force that represents true confinement. It is not only a quantum effect but a non-perturbative effect (i.e. not calculable using the simple techniques of Feynman diagrams or their equivalents.) It can only be caculated using computers — the numerical methods of lattice gauge theory. If someone can prove it using pure mathematics, they win a million dollars from the Clay foundation. Such motivation has not been enough to generate a solution, however.

  21. Dr.Strassler:
    I have a question inspired by one of your posts on one of your other sites, about the two slit experiment

    its my understanding, that in order to see this characteristic of wavicles, the two slits have to be very close together…..like less than a millimeter, does that sound correct?

    also, in the two slit experiment, say for a single photon, how is momentum conserved? in other words, if I fire my “photon gun” one photon, and the photon gun recoils in the negative X direction, the photon should, have momentum in the positive X direction. After it passes thru both slits, and ends up as a mark on the wall…..absorbed by an atom not directly behind the slits, what did it exchange momentum with? does it exchange momentum with the slits, as it passes thru both of them? If the photon gets absorbed by an atom, way off to the side, not directly behind the slits, it must have had sideways momentum given to it at one point. I watched a lecture by Dr. Susskind on this, and he seemed to indicate momentum was conserved, but didn’t say how…where was the momentum exchange?

    1. There is an interplay between the slit spacing, the wavelength of the light, and the size of the interference pattern, so your question doesn’t have an answer unless you are a lot more specific. Not also the interference pattern itself is not a sign of quantum physics; that happens with classical waves. It’s only when you see the interference pattern built up photon by photon that you are seeing quantum physics. [There are many wrong YouTube videos about this.]

      The photon can exchange momentum with the wall, just as a classical light wave can.

  22. Thank you so much for your book, which I just finished reading with great interest. Your use of the wavicle concept makes so much more sense to me than the Copenhagen interpretation. One question (of many):
    If a single photon is released that could be seen by two observers an equal distance away from the source, will only one of them see it, or both? Is this hypothetical similar to the two-slit experiment or different? Many thanks.

    1. Great question., I wish I could tell you that thinking in terms of wavicles resolves some of the puzzles in quantum physics, but it does not. It rephrases them, to some degree, and I think it prevents one from getting caught up in wrong ways of thinking (while the Copenhagen interpretation of “particle-wave duality” easily leads to unnecessary confusions.) Nevertheless, the puzzles are all still there. I carefully skirted them in the book because the story of wavicles and their rest masses does not require resolving them, and they would have been an enormous distraction from that story.

      As for your question: Only one observer (at most) will see your photon. And yes, it is very similar to the double slit experiment. The photon goes through both slits; it interferes with itself, creating a spread-out, complex pattern. Yet only one atom on the screen will grab it.

      However, this is not because a photon is a dot-like object when it is absorbed, as the Copenhagen interpretation would want you to imagine. The photon is still a wave, with a frequency. The absorbing atom, too, is not a dot; it has a radius of 1/3 of a billionth of a meter, which is not small on the scale of subatomic objects. The absorption process is an interaction among waves — or more precisely, an interaction among wavicles — that allow the photon to be absorbed by the electron wavicles that make up the outskirts of the atom.

      But the process by which the atom absorbs the spread-out photon, thereby making it impossible for any other atom to absorb it — and how this process is described in terms of probabilities rather than certainties — remains confusing. Either you use the many-world picture, in which the universe proliferates into a gazillion branches of possibilities, or you assert that the equations of quantum field theory leave something out, or… or like me, you sit back and hope someone smarter than you will come up with a better way to think about it.

      The many-worlds interpretation has the merit of being consistent with the equations (which the Copenhagen interpretation is not) but that does not make it intellectually or emotionally satisfying to most practitioners. It is popular, though, since it seems to be the best we have for now.

      1. Many thanks. This is very helpful. As between the many-worlds interpretation and confusion, I prefer the latter…

  23. Matt,
    Swimming with you through the wavey sea was great! No re-reading, except a paragraph here and there, was needed (although I am). You done good!
    As a non-physicist and non-mathematician, I have been reading to understand quantum mechanics for six decades. Ruth Kastner’s Transactional Interpretation finally makes sense to me (after four readings) so I would like to see if it is consistent with yours.
    While you offer many demurrals, I understand you to be describing that wavicles traverse trajectories in a pre-existing space-time container to translocate energy. Alternatively, Kastner’s (not Cramers’s) Transactional Interpretation proposes that instead, waves interact outside of space-time to translocate energy thereby creating events and their space-time separation. Note that this interpretation replaces particles, trajectories, and a space-time box with events, pre-space-time waves, and an evolving space-time.
    I wonder if your “demurrals” would make yours and Kastner’s interpretations compatible?
    Many thanks,
    Tony Way

    1. I’m afraid I can’t comment on that, since I don’t know Kastner’s interpretation well enough. But I am not taking a position on the interpretation of quantum physics in the book, just trying to give readers a useful picture without claiming that it is complete. I did make the remark that we don’t know if space exists or should be thought of as fundamental, and so for myself, as a scientist, I’m not assuming a space-time container in my research. But the picture I provided to readers does assume it, and might need for that very reason to be replaced someday — a point that I tried to make clear at the end of Chapter 14.

  24. Where can I find the Mathematica programs that you’ve used to make your blog animations, esp. the ones from “Fields and Their Particles: With Math?”

        1. I meant to ask where I can find your animation codes. I would like to try to reproduce you plot animations.

          1. Ah. This is all stuff I write myself and it’s kind of a mess — not commented or anything. And I have to think about what I do and don’t want to release. (Believe it or not, despite 10 years of animations, you’re the first one to ask.) Lemme think about it. There are more animations coming and you can remind me about it.

  25. Dr. Strassler:
    finished the book, absolutely loved it. I have placed it in my library right next to The Feynman Lectures on Physics. I would like, if possible, sometime in the future, a more detailed discussion of how a wavicle starts to spread out, but then collapses back to “point like” when it collides (exchanges momentum / energy) with another wavicle, and then starts spreading again.

    1. Great to hear you enjoyed the book!! Please leave a review on Amazon or GoodReads if you have the time.

      As for how to think about what happens when a wavicle collides with another wavicle — ah yes, I’d like a description of that too. This lies at the heart of perhaps the most difficult conceptual issue in physics. It’s a great question, but I’ll have to build up a whole infrastructure to even define and illustrate the question, so this will be something I probably won’t return to for months or even longer.

      1. I’m eager to get my hand on the book now, blame the recast site – the notion of science as “spectator sports” or since I rarely watch sports I don’t perform myself perhaps “spectator architecture” seems fitting. So seeing we have to wait for months or even longer perhaps some preliminaries to expand on the question or to have preliminary partial answers?

        In the following I may make the mistake of confusing the wavicle with the wavefunction but unless given hints I have to assume it is essentially the same interaction collapse. FWIW then here is a related question: Is the work “Answering Mermin’s challenge with conservation per no preferred reference frame” published in the less cited Nature Communications useful (does it survive basic criticism)? [Stuckey, W.M., Silberstein, M., McDevitt, T. et al. Answering Mermin’s challenge with conservation per no preferred reference frame. Sci Rep 10, 15771 (2020). https://doi.org/10.1038/s41598-020-72817-7%5D The conceptual issue does not seem testable but the work reinterprets “wavefunction collapse” as a relativistic effect. My naive view is that it adds to the wavepacket time dilation and length contraction the interaction collapse in order to conserve the wavefunction spin for the observer (here: interaction partner?). Then if we can familiarize ourselves with the first two effects, we could do the same with the potential third candidate.

        1. Well, this is not the type of question the book attempts to address; in fact, I deliberately skirt these issues.

          You are indeed at risk of confusing wavicle with wavefunction. But worse, it’s far from clear that wavefunctions collapse, or what it would mean for them to do so in a relativistic theory, or what would cause collapse, or how that collapse could be describable and predictable. This is why many of my colleagues either subscribe to Everett’s many-world view (as do Sean Carroll and Max Tegmark) or throw up their hands in dismay and confusion (as I do.) Someday I will try to explain the problem carefully; I have no solution.

          Regarding that specific paper (DOI number is wrong, but it is also here: https://www.nature.com/articles/s41598-020-72817-7 ), I would have to study it. Any serious papers trying to clarify how quantum physics really works have to be studied with care; otherwise one is likely to come away with the wrong impression. But real progress on a problem that has troubled us for a century is likely to require either a new experimental discovery or a profound new theoretical idea. This paper sounds potentially interesting, but not likely to be significant. I would expect bigger ideas to arise potentially from the interplay of quantum computing and quantum gravity research.

          1. Matt, thanks for the response! Yes, the paper discuss a relativistic consistent cause for collapse/Born postulate, which else is seen as a problem in search for bigger (likely testable) ideas.

    1. Thanks!! The book’s correct, but we have a missing redirect on the site. I’ve put in a temporary redirect and will make it automatic shortly. Try it again!

      1. Thanks. Just tried this and the redirect works great.

        [P.S. Just a quick note to tell how much I’m enjoying —and learning from— WiaIS.

        I’ve been (highly) recommending it to all my friends who find these things interesting.

        Finally, I very much enjoyed your appearance on Sean Carroll’s podcast.
        And I want to thank you for one specific point.

        That is, I’ve struggled for a long time to understand the precise mechanism as to how “the Higgs mechanism imparts mass to certain elementary particles”.

        During the episode you briefly mentioned, almost in passing, that the Higgs field affects (say) the electron field in such a way as to alter the form of the waves in the field, consequently changing its frequency, which in turn “determines the mass of the particle”.

        I can only tell you that when I heard this, simple as it may have been, lightbulbs went off all over the place. Thank you once again]

        Nichael Cramer
        Guilford VT

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