Of Particular Significance

Article 4 on Zero-Point Energy: Mass, Fermions, and a Good Wrong Idea

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 02/19/2024

I have posted my fourth article discussing zero-point energy. (Here are the firstthe second, and the third, which covered respectively the zero-point energy of a ball on a spring, a guitar string, and a bosonic field whose particles have zero mass, such as the electromagnetic field.) Today’s article looks at fields whose particles have non-zero mass, such as the Higgs field, and fermionic fields, such as the electron field and quark fields. It presents some simple formulas, and in its final section, shows how one can obtain them using math.

Along the way we’ll encounter the idea of “supersymmetry” and its failed role in the cosmological constant problem. This is a word which (for some good historical reasons) generates a lot of heat. But stay calm; I’m neither promoting it nor bashing it. Supersymmetry is an idea which proves useful as a conceptual tool, whether it is true in nature or not.

So that you know where I’m headed: after this article, we’ll now be in a position to understand (using only simple formulas) where the hierarchy puzzle comes from and why it is tied up with the concept of zero-point energy. Then, finally, we can grasp what’s puzzling about the hierarchy, and look at various proposed solutions to it, ranging from fancy math to historical drama, or even denying that it’s puzzling at all.

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27 Responses

  1. Matt, OT so feel free to delete this comment when you’ve read it. On Amazon.com the hard edition of your book comes under Books/Science & Math/Mathematics while on Amazon.co.uk it’s under Science, Nature & Maths/Engineering & Technology/Civil Engineering. I doubt this is what you had in mind.

      1. Actually the publisher can’t do anything; it’s all internal to Amazon. Don’t know if there’s a way to raise the issue there…? Most unfortunate, but on the other hand, most people search by title and author, not topic…

        1. You might check that the Library of Congress classification(s) for the book are correct. From

          It has “QA8.4”: Mathematics: Philosophy: General works, treatises, and textbooks

          I haven’t read the book, but
          * QC6 Physics: Philosophy, Methodology: General works, treatises, and textbooks
          seems closer and
          * QC174.45.A6-Z (a range of terminal characters) Physics: Atomic physics: Quantum Theory: Quantum Field theory: General works, treatises, and textbooks
          seems like it might be very close.

        2. A read of: “Amazon Book Categories : What Authors Need to Know” suggests to me that the publisher gets to choose. There may be subtle marketing reasons behind publishers choosing odd categories for a book, yet Brian Greene’s ‘The Elegant Universe’ in paper back is under ‘Crafts, Hobbies & Home/Crafts/Colouring Books for Grown-Ups’ for Amazon.co.uk (I’m not joking!)

          You may find it annoying if your book is highlighted on the Amazon-UK page as ‘number one best seller in civil engineering’ when it’s published.

  2. Is the cosmic microwave background (CMB) radiation the zero-point energy (ZPE) of this cycle* of the universe?

    If the temperature is 2.725 K, is that because matter won over antimatter at the Big Bang & created a positive offset?

    Did matter win because of the spatial expansion?

  3. https://profmattstrassler.com/2024/02/19/article-4-on-zero-point-energy-mass-fermions-and-a-good-wrong-idea/#comment-449935. Matt Strassler: “Something about this subject has you quite confused. I am afraid I haven’t yet figured out what it is.”

    I think Terry’s difficulty with accepting zero point energy is very similar to the 4/3 problem of the electron in the early 20th century: the electromagnetic energy of an electron didn’t transform via E = gamma m c^2. A solution was realizing that a pressure had to be added in the form of Poincare stresses to keep the electron together and subtracting its -1/3 contribution to give the relativistic expected answer. I recall that Einstein was involved in this, writing many papers on the transformation of physical stresses, so that he had an expert’s view on the physics involved and didn’t get lead astray by just the maths.

    Matt wrote about this, linking some papers of Rothman:

  4. Dr.Strassler:
    Would like to ask you about your response to another post, for clarification. You wrote:

    “But all particles of any sort, Higgs mechanism or no, violate special relativity — even photons, whose energy depends on your frame of reference”

    I thought one of the main tenets of Special Relativity was that there are no privileged reference frames. The energy and momentum of an object / particle is not reference frame invariant. In your response above, what do you mean “violate special relativity”
    I thought that energy / momentum dependence on the reference frame was in accordance with special relativity. Unless you are talking about the rest energy, of massive particles, which is reference frame invariant.

    1. It’s a good question. The difference is between (a) the laws of physics themselves and (b) the specifics of the universe you are in.

      The fact that there are no preferred reference frames in the laws of physics implies that a particle (relative to you) can have any speed.

      But once your universe has a specific particle with a specific motion, then there is only one frame in which it is stationary. In that sense, relativity is broken (but not ruined) by the presence of that particle.

      The Sun sets a preferred frame too: there’s only one frame in which it is stationary.

      It’s the same with the universe as a whole. There is a preferred frame in our universe — the one in which the temperature of the universe is the same in all directions.

      But relativity still holds in the laws of physics themselves. The Sun could have been moving with any speed and direction (and indeed there are stars that move with every speed and direction imaginable.)

      This kind of subtlety comes up a lot. The fact that the laws of physics themselves (and the equations that follow from them) have a symmetry does not imply that the solutions to those equations will have that symmetry. For instance, the laws of physics are rotationally invariant, but that does not mean that all solutions to those laws are spheres. Any one asteroid breaks rotational symmetry — but it could have been oriented in any direction, reflecting the invariance of the underlying laws.

      I probably should have been more precise in answering the question you were referring to.

      1. Dr. Strassler:
        Got it, I understand, thank you. I have always looked at it from this perspective:
        The laws of physics are the same in all reference frames, two observers viewing the same “event” from different reference frames, will disagree on the “components” of the event (total energy, total momentum, time, distance…..etc) however, all observers will agree on the results. Any issues with that perspective ?

      2. Dr.Strassler:
        Trying to further my understanding. I had always thought the main tenet of relativity, either basic Galileo relativity or special relativity, is that the laws of physics are reference frame invariant. You have affirmed this. However, could you expand on as to why the rest frame, of let’s say the Sun, is the preferred frame?

        1. It’s not “the” preferred frame. It’s a preferred frame if you are sitting in the presence of the Sun; it’s the only frame in which the Sun and all its effects are stationary.

          Take the Earth, it’s more obvious. No human ever experiences the invariance of reference frames, because all our experiences are affected dramatically by the Earth’s presence. In that sense, the Earth-at-rest frame is special for physics as we experience it on the Earth.

          This does not, of course, mean that the laws of nature aren’t reference frame invariant. It just means that it is very far from obvious, on Earth, that they are reference frame invariant. And that’s exactly why we humans required geniuses like Kepler, Galileo, Newton and Einstein to figure it out!

  5. “… and $\ell_p$ is the ridiculously …” You probably intend to replace that inline TeX with a tine rendered picture.

    “Since all known elementary particles’ masses are less than a thousand times a proton’s mass, … extremely tiny …” Having just introduced m_{pl} <= 10^{18} m_0, maybe just actually write "m_0/m_{pl} \leq 10^{-15}, which is extremely tiny." "Perhaps, if the universe had exactly the same number of fermionic fields as the number of its bosonic fields, the former contributing negative zero-point energy and the latter positive zero-point energy" -- why do there have to be equal numbers of fields? Why not just these fields cancel those fields however many of each there are? (I know, supersymmetry assumes equality, but it's strange to jump to equal numbers of fields without saying *something* about non-equal numbers of fields.) "they would have to have exactly the same resonance frequencies" -- We live in a universe with a positive Hubble constant, so *exact* cancellation doesn't match observation. Absurdly precise, but not exact, cancellation is what we see looking out the window. (One can then tie this back to approximate supersymmetry -- we don't have to have partner by partner cancellation, just aggregate cancellation.) (Random aside: I've occasionally wondered is the universe is too small for the de Broglie wavelength of some particles, maybe neutrinos, and after enough expansion, neutrinos pick up mass because their waves finally fit in the universe.)

    1. Thanks for the excellent suggestions: I’ve taken the first two. I’ll think about the last two; I’m not sure your approach is clearer to the average reader.

      As for your random aside,, I don’t see how one could make sense of it; do you really mean de Broglie wavelength? That wavelength would be different for each individual particle. By constrast, rest mass can’t be different particle-to-particle without making the particles distinguishable, in which case they would not act as identical fermions, and there would be no Fermi pressure, rendering calculations of supernova physics wrong.

  6. Matt, despite its support from Bohr, I’ve completely given up zero-point energy as a viable physics concept In a world where special relativity rules to more decimal places than we can measure. That’s relevant because I haven’t seen a way to talk about zero-point energy without subtly invoking some version of an aether.

    I’ll check back through your articles, but can you point to any well-proven experimental evidence for the existence of zero-point energy? I’ve seen multiple debatable examples, but certainly nothing like the experimental proofs of special relativity through high-energy gamma-ray analysis.

    1. I’m hiding some very subtle points in these articles. When a theory has strong interactions among its fields, even defining what you mean by “zero-point energy” as opposed to other sources of energy-density becomes ambiguous. So if you want to claim that zero-point energy is zero, then you run into problems when you try to make sense of that claim in a strong interacting theory. I can’t get into this now, but this is just to warn you that you are heading onto thin ice.

      There is ample evidence of zero-point energy in material systems, such as crystals. It affects their melting temperatures, for instance. Of course, there was also ample evidence that speeds of sound are not the same for all observers in material systems, and yet the speed of light in empty space turns out to be a completely different matter. Clearly, space (more generally, spacetime) is different from material systems, and what is true for the latter may sometimes not be true for the former.

      But you don’t solve the problem by just declaring that some calculation which is large should be set to zero for no good reason. First, you hit ambiguities when you consider interactions, as I just mentioned. And second, that’s not an explanation; it’s burying your head in the sand. Instead, it’s by facing the issue head-on, carefully, from all its aspects, that you have a chance of making progress… and that’s exactly what Einstein did, better than anyone, when it came to the speed of light.

      The experimental evidence for zero-point energy of empty space is at best subject to interpretation. However, we do know that the energy density of empty space is not *always* zero. If it were, the Higgs boson would be massless (and there would be a similar massless composite particle similar to a pion.) So the energy density can definitely be non-zero, and it depends on the values of certain fields. The question of why that energy density — which includes but is not limited to zero-point energy — is so close to zero in the universe’s current vacuum state (or why it seems to be so close to zero, even if it actually isn’t) is not going to be resolved by just setting various things willy-nilly to zero.

      1. Matt/Terry, see The Foundation of General Relativity ( https://einsteinpapers.press.princeton.edu/vol6-trans/197?highlightText=gravitatively) where in 1916 Einstein said “the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy”. This means the energy density of space is greater near a massive body such as a star than it is several light years away. By 1920 in his Leyden Address he was referring to space as the aether of general relativity. Also see the Robert B Laughlin quote here: https://en.wikipedia.org/wiki/Aether_theories#Quantum_vacuum.

      2. Matt Strassler, thank you, that’s a great reply with excellent info. I especially liked your point about the Higgs mechanism requiring non-zero vacuum energy and empty space. I have no reason whatsoever to doubt that, but the interpretations that come to mind for me differ a bit from the standard ones.

        The high-energy concerns stemming from deep elaboration of quantum field theory are closely akin to the vacuum density problem, are they not? What is the status of that these days?

        My special relativity concern is simple: Any mechanism that that creates experimentally detectable energy associated with a small, well-observed region of space also ties that energy to that observer frame; it is no longer a relativistically invariant region of space, and has become intimately associated with the matter creating that effect.

        All variants of Casimir experiments I’ve seen lock the region of space to an observer and associate that region closely and profoundly with condensed matter. Also, the energy levels involved are always very low and easily compatible with atomic-scale van der Waals attractions due to (classical analogy) very short-range charge reshuffling.

        Bohr… well, even Casimir never fully endorsed his claim that his experiments demonstrated energy from the vacuum, versus mundane and fully energy-conserving condensed matter mechanisms.

        That’s telling, I think, given that without Bohr’s enthusiasm, Casimir would now be a barely known name in condensed matter physics. Yet even with so much to lose, Casimir deferred to Bohr for the defense of the vacuum energy explanation of his experimental result.

        That, again, is why your point about the Higgs mechanism is so intriguing. Yet even there, is that truly the same universal frame of special relativity? The Higgs mechanism is the very definition of particle-field intimacy, and the very existence of that massive particle defines membership in an inertial reference frame.

        1. I have little to say about the Casimir effect; I do not view it as unambiguous.

          As for the Higgs mechanism: perhaps the single most important aspect of the Higgs field is that giving it a non-zero constant value leaves special relativity untouched. This can only be true for a spin-zero cosmic field.

          In this regard, I strongly disagree with the statement that “The Higgs mechanism is the very definition of particle-field intimacy, and the very existence of that massive particle defines membership in an inertial reference frame.” No, no and no; you’re confused about how it works.

          The Higgs mechanism is a completely relativistic means by which one field can change properties of other fields. No special reference frame required.

          The particles of those changed fields — their minimal ripples — can then have rest mass as a result of this relativistically-invariant change. But all particles of any sort, Higgs mechanism or no, violate special relativity — even photons, whose energy depends on your frame of reference. Don’t confuse that ubiquitous violation of special relativity with what the Higgs field is doing.

          I do explain this in the book, though I don’t use math to do it. Some of the corresponding math is in the following two sets of articles; have you read them?



          1. Hi Dr. Strassler,

            Thanks for the links, I’ll take a look.

            What would be your best reference, particularly experimental, on why you think the Casimir effect is unambiguous?

            Casimir was not that sure, but those are old discussions. The recent literature seems pretty murky, from what I can see, with more than one pretty good author saying it’s never anything more than van der Waals force.

            You might be surprised that I disagree with that. I’m mostly convinced there’s something very interesting going on in Casmer effect experiments, but I’m also very firmly convinced that the energies involved are always nothing more than conversations of existing mass-energy in a local system. Absolute mass-energy conservation: I have to stick 100% with Einstein on that one. It is interesting to wonder if Einstein and Bohr ever addressed zero-point energy in any of their never-ending, never-concluding conversations.

            I also wonder what Feynman had to say about it. I suspect that he was not impressed with the idea, since he was openly skeptical about the implication of infinite computation in one cubic centimeter of space that his own QED method seem to imply. I don’t think I know of any other case of a Nobel Laureate openly mocking his theory in that fashion, and getting a good laugh from a room full of mathematicians in the process.

            I do not understand what you mean by saying “all particles … violate special relativity.” I know darned well you don’t mean that literally since you are extremely adept at navigating the Poincaré symmetries. I still chuckle at how nicely you caught my error one time when I slipped up on collision perspectives! But those symmetries don’t mean anything if you don’t have particles and energy to apply them to!

            What I meant about zero point radiation violating special relativity is that it’s impossible to define a _finite_ emission spectrum that does not lock the effect down into a specific inertial frame. Thus I’m fine with some frame-dependent version of zero-point energy — whatever that means! — but any version of that just makes it a condensed matter effect that has nothing to do with energy-free space that all inertial frames see as identical in special relativity.

            Dr. Strassler, thanks again, and I’ll look up your references and any more you may add about the Casimir effect. But I’m going to drop out of this conversation and let others talk. It’s been fun, though!

            1. My fault for using a double negative: I wrote that “I do not view it as unambiguous”, but I would have been clearer if I’d written “I view it as ambiguous.” I would rather not rely on the Casimir effect.

              There is no frame-dependent notion of zero-point energy. It is not like a condensed matter effect. The entire effect is relativistically invariant, precisely because the zero-point energy is combined with negative zero-point pressure.

              The presence of any one particle breaks special relativity “spontaneously”, just as the Earth does. The Earth defines a frame of reference; there is only one frame, at any moment, in which the Earth is stationary. The same is true of any particle. Like the Earth, though, the presence of a particle does not change the fact that the elementary laws of nature are still relativistically invariant.

              The Higgs boson does not, in any way, do anything worse to relativity than any other particle.

              The Higgs field, even when non-zero, does not define a reference frame in any sense. The fact that it gives mass to particles does not violate relativity any more than the presence of the particle itself does; without a particle present, there is no violation of any sort; and with a particle present, the particle’s energy does define a frame, but so does the existence of that particle. As far as relativity is concerned, there is *zero* difference between a particle that got its mass from the Higgs field and a particle which got its mass from somewhere else.

              Something about this subject has you quite confused. I am afraid I haven’t yet figured out what it is.

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