A central issue in discussions of particle physics’ present and future is known as **the hierarchy puzzle**. Although I discuss the hierarchy — its confusing nature and the debates that it generates — in my upcoming book, I do so rather briefly, and so, I’ll be putting up some new pages on this website with supplemental information. The same information is relevant for the cosmological constant problem. (Older pages already giving various perspectives on these issues can be found here, here and here.)

I have just posted the first new page, on “zero-point motion” and “zero-point energy.” It begins with a verbal, non-technical description of zero-point motion and zero-point energy. There follows a sketch of the details using pre-university math. Future pages will apply these ideas to quantum fields, addressing notions of “vacuum energy density” and the “cosmological constant”, and then turning to “Higgs feedback” and the core of the hierarchy puzzle.

A quick description of the hierarchy in question: it is a hierarchy of energy scales, or of mass scales. One way it can be described is in terms of particle masses:

- The masses
*[meaning “rest masses”]*of all elementary particles are absurdly small compared to the Planck mass, the lowest possible mass for a black hole

The masses of the known elementary particles lie between 200 times larger or 2000 smaller than a proton’s mass, excepting neutrinos’ masses, which are even smaller, and the zero rest mass of photons, gluons, and (presumed) gravitons. But if our understanding of gravity is taken at face value *(and maybe it should not be)*, the smallest possible black hole would have a mass of at least a **billion billion** protons.

Alternatively, we may rephrase the hierarchy in terms of forces:

- For all known elementary particles, gravity is astonishingly weak compared to the other elementary forces.

The strong nuclear force, acting on the quarks inside of protons, is about as strong as a force can possibly be; electromagnetism’s strength is about 1% of that; but gravity’s effects on a proton are a **billion billion** times smaller than that of the strong nuclear force.

The origin of the hierarchy is unknown. What makes the hierarchy * puzzling* is that when we look at the particles we know, organized in what is called the Standard Model of Particle Physics, it seems highly non-generic. If we consider the Standard Model not in isolation but as a representative of a large class of similar possible universes, it is unusual, in that almost all universes of similar type would have no such hierarchy. In most such theories, masses would either be zero or huge. To be precise, some particles might have no mass at all, while all others would have masses near the Planck scale — and for the latter, gravity would be as strong as the other forces.

Of course, we only live in one universe, and it doesn’t have to be generic. But in the past, when we’ve found an aspect of the universe that’s highly non-generic, there has usually been a story behind it — and so, on heuristic grounds, one might suspect that this gigantic and non-generic hierarchy might point toward some important facts about the universe that we still don’t know. The questions of how much weight to give that suspicion, and of whether the missing facts might lie within close experimental reach, is one we can return to after I’ve fully laid out the underlying scientific issues.

## 24 Responses

So, how small is the gravitional force the absolute scenario, i.e. you could say in a unified theory, if it exist?

Gravitional Force ~ 1 / r^2 … so, when going down the scale to the Planck’s scale the radii get pretty small so you would expect the gravitional force to be huge.

Yes the masses drop to a scale that we may, thus far, have missed the gravitional (quantum) force relationship wrt to the other three (at least) forces. The point being that may be gravity is the fundament force of nature opun which all other forces are derived from. Assuming this theory, then there could many, many more forces at or even below Planck’s scale that we have yet to conceptualize and certainly don’t have the experiment set up to discover, yet.

This concept of quantum gravity was triggered when a saw the graphical proof that pi is an irrational number. You could find it on the internet. It has two arms rotating, one about a fix axis and the second about the end point of the first arm, two degrees of freedom. As they rotate to complete one cycle the outer arm will never concide with the starting point thus proving that pi is irrational.

But, it you look closely to what the end graphically representation shows, what you’ll see is Bohr’s atom!!!

So, how do you get to quantum gravity from this graphical representation? Again, Fg ~ M1*M2/r^2, the masses are essentially the energy densities and these energy density could be thought as fluxes rotating about the nucleus at near the speed of light, c. So, the density at the nucleus must be maximum possible at such a small radii, and yes, so big that keep the proton intact, the strong force.

The strong force is a function of quantum gravity?

It is certainly true that gravity becomes very strong at short distances, but not because of the 1/r^2. All the different forces have the same 1/r^2.

Instead, the strengthening of gravity compared to the other forces arises because (in contrast to Newton’s law) Einstein’s laws of gravity contain not M (mass) but E (energy). Thanks to quantum physics, distances of interaction at high energy are proportional to hbar/E (Planck’s constant divided by energy, and so Einstein’s law has extra short-distance growth compared to the other forces, which do not have energy directly appearing in their laws. Therefore the force becomes stronger than just the 1/r^2 effect that we’re so familiar with from Newton’s law.

Ok, so, I’m assume “gravity at short distance” will also refer to quantum gravity. If that’s what you saying, is it possible that quantum gravity is, indeed, the fundamental force, coupling, between energy packets, (density fluxes), and all the other forces can be derived from quantum gravity?

Have you seen the graphical representation mentioned above? “Bohr atom”. Do you want me to send it it you on your twitter account? I think you will be intrigued/inspired/interested, maybe insulted, :).

Dear prof, you said here about Higgs feedback, then I ask you a big request: I know that you just wrote a book about the Higgs Boson, but there are many doubts that even physicist doctorate level has them. This doubt, I believe that you did not approach it in your book, and if was included, I ask you to clarify us here. Then I ask you, but at the real level, in practice, on the tactile world, what really is MASS and how it is transmissed? How HB do it, because on formulas, equations, in a mathematical explanation, it is easy for an expert to demonstrate it, but I challenge this technician to explain it in the form and through, the real world, at tactile world, concrete, scaping from formulas, math. That is, everyone knows mathematically what the sum of 2 plus 2, being easy to explain in the real world, until using apples, but what about the transmission of mass, would it be possible to explain using the real world? Or tp explain what the mass really is? Could you give us an article about it? regards. thanks..

The purpose of my book was to explain what mass is and how the Higgs field (not the boson) causes elementary particles to obtain mass — to the extent we know the answer. So I think you will find the book answers many of your questions. Three more weeks. If it doesn’t answer them, then please do ask me again.

I am not sure I know what you mean by “transmission” of mass. This is not a term that I use and so I’m not sure what you are really asking.

Dr.Strasler:

A check of my understanding, if I could. The issue is that the Planck mass is fairly large, 10^-9 kilograms I believe. The Planck volume is really small, 10^-105. So is the issue that you have to stuff a relatively large amount of mass….10^-9 into a volume very much smaller to get a black hole? And the reason why you still need 10^-9 kg, in such a small volume, is that gravity is so incredibly weak, and we don’t know why it’s so much weaker than the other fundamental forces?

That’s right, yes.

Matt, around a decade ago after the discovery of the 125Gev Higgs in 2012 you gave a lecture, which can be found on Youtube, ‘A Metacrisis in Particle Physics Naturalness On the Brink by Matt Strassler’.

How has the hierarchy puzzle changed for you since then?

In the video you mention the various experiments trying to discover the ether drift, physicists coming up with hacked physical models to explain null observations; until Einstein came along. I think another interesting historical example around the same time was the anomalous precession of Mercury’s perihelion, which astronomers proposed was a consequence of another planet, Vulcan, orbiting the Sun inside Mercury’s orbit. I find it remarkable that within just a decade, Einstein turned upside down how physicists viewed space, time, energy and momentum so that they could confidently abandon Newton’s model of gravity, despite it appearing far simpler. In our era we have black holes and ideas about information conservation etc increasingly taking center stage.

The main changes: Experimentally, the LHC has continued not to find any other particles or other phenomena that might bear on the hierarchy puzzle, despite much more data. Meanwhile, no theorist has introduced a really great idea that might resolve the hierarchy puzzle without easily-observed LHC phenomena (while perhaps predicting other phenomena). There are ideas, for sure, but none of them is really great the way relativity was a great idea. Nor has a theorist given a convincing argument that everyone is wrong and there is no hierarchy puzzle in the first place. (Although certain prominent science popularizers seem to think they have done so, they haven’t convinced a single major theorist.)

During this past decade, I have worked on some of the more subtle possible solutions to the hierarchy puzzle, for example the approach summarized in https://arxiv.org/abs/1501.05310. But none of them ring true; they all have significant problems. I doubt the right idea has been found.

The example of Vulcan and Mercury is not really as crisp as that of the ether, although I understand why you might see a parallel. Historically, there were other proposals (a ring of dust near the Sun, for instance) that were alternatives to a planet like Vulcan. Non-observation of Vulcan was a puzzle but definitely not a crisis. Moreover, the perihelion of Mercury was an observed small unexpected effect, not the absence of an expected effect. It’s different.

The null experiment that revealed no sign of the ether was really a crisis, because it flew in the face of basic understanding of waves, and as there was no sensible explanation — ether drift was a pretty awful solution, and one that was very difficult to make sense of. Similarly, the hierarchy puzzle flies in the face of basic understanding of quantum fields, and the straightforward solutions to the hierarchy puzzle that are still consistent with LHC data are also mostly pretty awful, except for some small loopholes.

So I think the current situation — a null experiment that has no really good explanation — it is a little more like the ether crisis than it is like the case of Mercury’s perihelion. It’s not identical by any means, but there are parallels worth thinking about. The Mercury discrepancy was slowly on its way to becoming a crisis, but really hadn’t reached that point yet. Einstein came up with an explanation before it really became impossible to think of alternatives. In this regard, the “Hubble tension” strikes me as more similar to the case of Mercury.

“But then again, an electron is not a dot.” I usually get at this with “Where is a brick?” Physics-y people tend to want to assign the CoM as its position (good reaction for Mechanics and Dynamics, questionable for other settings), but that’s not the answer that an “is there a brick there” machine would give — it says “yes” over an extended volume…

The hierarchical problem is not as complex as it might appear. The gravitational and electrostatic forces are fundamentally the same, except for the fact that the gravitational force is distributed over three dimensions, whereas the electrostatic force is one-dimensional. This is expressed perfectly by the fundamental equation E=mc^2 . In the cgs system, c is very nearly 3 x 10^10 cm/sec. and c^2 is therefore 9 x 10^20 cm/sec. The ratio between the two forces is thus 9 x 10^20, mass being the three-dimensional equivalent of energy (a one-dimensional entity), as the equation clearly indicates.

My final point before an extended radio silence:

The magnetostatic force, being two-dimensional, conforms to the equation M=mc, where M is the magnetostatic force, m is mass and c is the speed of light. For the reasons given above, it is 3 x 10^10 weaker than the electrostatic force and 3 x 10^10 stronger than the gravitational force.

Matt, this is a great quick summary of the hierarchy problem, stated in an interestingly different way from what I mostly seen. Alas, the enthusiasm I’ve had for most of my life for Planck scale logic has disappeared in the last couple of years. These days, it just just feels like math noise.

I think you’ll understand the hierarchy puzzle much better before I am done. Remember, you don’t have to think about it in terms of the Planck scale. Just ask: “why is gravity so weak?” And when you realize that the answer is: “because the Higgs field’s average value is so small”, then you see that this is a feature of nature that inevitably, in some way, links gravity and forefront particle physics. That’s what makes it interesting, independent of anyone’s math.

Nice, thank you! Your point about the Higgs field average value being so small particularly hits home. As you say, that is a very solid point that is independent of the issue of how best to represent the situation in math.

John Duffield, thank you for that great reference. I recalled some of the discussions from early papers about whether there was or was not the point hidden within the wave function, but I’ve lost track of where I’d read it. That is very relevant, so again, thanks.

Interesting stuff Matt. Personally I think a lot hinges on something you said on your Zero-Point Motion web page: “But then again, an electron is not a dot”. In the 1920s there was a tussle between the “realists” like Einstein, Schrodinger, de Broglie, and Darwin etc, and the “shut up and calculate” Copenhagen School consisting of Bohr, Heisenberg, Pauli, and Dirac etc. The realists wrote papers talking about the electron as a wave in a closed path. The Copenhagen school adopted Frenkel’s point-particle electron. (See his 1926 paper here: http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/frenkel_-_electrodynamics_of_rotating_electrons.pdf). The Copenhagen School won, the point-particle electron led to the “problem of infinities”, which then led to the need for renormalization, which then lead to the hierarchy problem. Read the old papers, and IMHO you get a different slant on things.

I don’t really agree with this, John. Yes, you can learn a lot about scientific history by reading papers from 1926. But quantum field theory of the 1940s, 1950s and 1960s changed the story; the Frenkel-like point-particle electron does not exist in quantum field theory (or, if you prefer, it is an auxiliary construction.) Real interacting electrons in quantum field theory are far more wavelike than particle-like, and that explains a lot… e.g., why they are identical.

As for the hierarchy problem, again, historically what you say is true, but scientifically I would argue that it is not. Renormalization is not about infinities (although, if you don’t understand it and you ignore it, some of your calculations will appear infinite). It is about doing physically meaningful calculations instead of physically meaningless ones. And it does not so much “lead to” the hierarchy problem as make it clear what the problem is. So I think you’re drawing a

historicalline between concepts here, and somewhat misinterpreting it as alogicalline between those concepts.Noted Matt, but IMHO I’m not drawing a historical line and misinterpreting it as a logical line. There’s too many examples for that. See for example Einstein talking about the speed of light here: https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22 . Now remember c = 1/√(ε₀μ₀). This means gravity is related to electromagnetism. From that you could assert that it’s weak because it’s a residual force.

I’ve stayed away from the discussion for some months following Matt’s admonition not to promote rival theories on his site. I hope he will tolerate this brief comment, in which I promote nothing, but merely point out how the strong force originated. Not only is this force “about as strong as a force could possibly be” (Matt’s words), it is exactly as strong as physicists *need it to be* in order for it to perform the task it was designed to perform, namely to counter the positive electric charge of the protons allegedly residing at close quarters within the Rutherfordian atomic nucleus and thus keep the latter from instantly disintegrating. It was invented in the 1930s for this very purpose. The positive charge of the protons postulated to exist as such within the atomic nucleus would cause them to repel one another with great force. Therefore the countervailing force needed to stabilize such a nucleus had to be equally strong.

Ok, that was the polite version of my argument. Here is the rude one that will probably get me banned: “Your atomic nucleus doesn’t hold together? Here’s some crazy glue. A couple of drops should do it!”

🙂

Again, I think you are confusing history with physics. The force was proposed to explain why the nucleus doesn’t disintegrate. But today the strong nuclear force is used as the basis for computations that can compute the difference between the proton and the neutron mass, https://arxiv.org/abs/2202.01613, or the whole spectrum of hadrons, none of which were known in the 1930s, see the last figure in https://webific.ific.uv.es/web/en/content/lattice-qcd-numerical-approach-strong-force .

This is the way history works. Some crazy idea is proposed, to solve a problem; maybe it’s wrong. But if decades go by and that same idea is used over and over again to solve a wide range of problems, including ones that weren’t even imagined when it was first proposed, well… maybe it’s right.

Thanks for your tolerance, Matt. I will keep lurking again for a while, not to muddy the waters too much..