The Hoyle State

4 February, 2021

Nuclear physics is complicated compared to atomic physics, because the strong force is complicated compared to the electromagnetic force, and nucleons—protons and neutrons—are bag-like groupings of quarks and gluons held together by the strong force. They resemble elastic bags that attract each other. They jostle each other in the nucleus… governed by the rules of quantum mechanics.

To begin to understand such a complex thing as a nucleus, people started with approximate models. In 1930 George Gamow introduced the ‘liquid drop model’, which was further developed by Niels Bohr, John Archibald Wheeler and Carl F. von Weizsäcker. The idea is to treat the nucleus as a droplet of an incompressible fluid with some surface tension—but again, quantum-mechanically.

Another model, more reminscent of atomic physics, is the shell model. Here neutrons are protons are treated as moving in a potential well (which is actually created by their interaction with each other). Since protons and neutrons each separately obey the Pauli exclusion principle, there are—approximately—separate shells for each kind of particle, which fill up when their reaches a so-called magic number

2, 8, 20, 28, 50, 82, 126, …

Thus, nuclei with a magic number of either protons or neutrons are especially stable, and ‘doubly magic’ nuclei, with a magic number of protons and a a magic number of neutrons, are even more stable: think of helium-4, oxygen-16, calcium-40, nickel-56, or lead-208 (with 82 protons and 126 neutrons).

Yet another interesting approximation is to think of a nucleus as made of smaller nuclei, especially these four:

• the deuteron: a deuteron, the nucleus of deuterium or 2H, is a proton and neutron stuck together.

• the triton: a triton, the nucleus of tritium or 3H, is a proton and two neutrons stuck together.

• the helion: a helion, the nucleus of 3He, is two protons and a neutron stuck together.

• the alpha particle: an alpha particle, the nucleus of 4He, is two protons and two neutrons stuck together.

Of these, all but the triton is stable on its own, and the triton has a half-life of 4500 days, which is essentially forever on the timescale at which particles in the nucleus do things. Compare a truly unstable nucleus like 4H, consisting of one proton and three neutrons: this has a half-life of 1.39 × 10-22 seconds.

Here’s a great example of how it can sometimes be useful to think of atomic nuclei as assemblages of deuterons, tritons, helions and alpha particles. The nucleus of ordinary carbon, 12C, consists of 6 protons and 6 neutrons. But it has an excited state—a state of higher energy—in which it acts like 3 alpha particles orbiting each other! This is called the Hoyle state.

The Hoyle state has energy 7.65 MeV more than the lowest-energy state of 12C. To get a feeling for how much that is, it helps to know that when a 12C is in its lowest-energy state—called its ground state—its energy is 11,177.93 MeV. So, the Hoyle state has just a tiny bit more energy than the ground state.

But here’s a better way to think about it. The ground state energy of 12C is 7.27 MeV less than that of 3 alpha particles. So, the Hoyle state of 12C has

7.65 MeV – 7.27 MeV = 0.38 MeV

more energy than 3 alpha particles. This means that carbon in its Hoyle state can break apart into 3 alpha particles! It can also decay back to the ground state of 12C. But it’s not a bound state: it’s not held together, it can fall apart into alpha particles.

But here’s the really interesting thing about the energy of the Hoyle state: it’s almost the same as the energy of a beryllium nucleus plus an alpha particle! An ordinary beryllium nucleus, 8Be, is made of 4 protons and 4 neutrons. Thus, it can be made from two alpha particles. 12C, as we’ve seen, can be made from three. But the rest energy of 8Be plus that of an alpha particle exceeds that of 12C, thanks to binding energy. In fact, that sum is closer to the energy of the Hoyle state!

Let’s see how it works. The ground state energy of 8Be is 7456.89 MeV. The ground state energy of an alpha particle is 3727.38 MeV. Summing them up, we get

7456.89 MeV + 3727.38 MeV = 11184.27 MeV

This is more than the ground state energy of 12C, which—I said a while back—is 11,177.93 MeV. How much more?

11184.27 MeV – 11,177.93 MeV = 6.34 MeV

On the other hand, we’ve seen the energy of the Hoyle state is 7.64 MeV more than that of 12C. These numbers are pretty close.

This coincidence is important, and it has a romantic history. The astrophysicist Fred Hoyle predicted its existence based on stellar evolution. Without a state of this sort, it’s unlikely that carbon would be formed when alpha particles smack into beryllium nuclei in a star! And that would be a serious roadblock to the formation of carbon.

This is sometimes counted a success of the anthropic principle, since without carbon there would be no life…

…. well, no life containing carbon anyway.

As far as I’m concerned, the anthropic twist is wholly unnecessary and distracting. You see carbon in stars, you know it must have gotten there somehow, and you guess there must be an excited state of carbon to explain this. It’s a perfectly fine piece of detective work. Why spoil it by tacking on the observation “and if there were no carbon, there would be no intelligent life!”

In fact Hoyle didn’t mention the anthropic business in his original argument in 1953: he was focused on the observed appearance of elements in stars. Only in 1965 did he add a remark that had the energy levels been different, “it is likely that living creatures would never have developed”. And only later, in 1979, did Carr and Rees claim that the prediction of the Hoyle state was, or could have been, a triumph of the anthropic principle.

For a careful dissection of how the mythology surrounding Hoyle’s prediction grew and grew over time, read this:

• Helge Kragh, Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology, Section 9.2: Anthropic Reasonings, Oxford U. Press, Oxford, 2011.

This is a truly wonderful book, though I find the failed theories of the 1800s inspiring, and the recent ones merely depressing, since for decades I’ve had to watch seemingly intelligent scientists cling to these recent theories despite their lack of success.

Anyway, what fascinates me about the Hoyle state is not the anthropic baloney, but the idea of a carbon nucleus as three alpha particles engaged in a complicated quantum dance!

Of course this is just a simplified picture, an approximation. For more, see these:

• Natalie Wolchover, A primordial nucleus behind the elements of life, Quanta, 4 December 2012.

• David Jenkins and Oliver Kirsebom, The secret of life, Physics World, 7 February 2013.

While these authors are unable to resist retelling the anthropic just-so story, their articles contain interesting details, simply explained, of how physicists are trying to get a better understanding of the Hoyle state.

Recent supercomputer calculations show that even carbon in its ground state is approximately described by three alpha particles, orbiting each other in a compact triangle! Carbon in its Hoyle state, on the other hand, is approximately a ‘bent arm’ configuration of three alpha particles:

This image is a modified version of one in the Physics World article by Jenkins and Kirsebom.

For something about experiments rather than computations, try this:

• Hans O. U. Fynbo and Martin Freer, Rotations of the Hoyle state in carbon-12, Physics, 14 November 2011.


2 February, 2021

Having an even number of neutrons and/or an even number of protons tends to make a nucleus more stable against radioactive decay:

• Wikipedia, Even and odd nuclei.

I just learned there are only 5 stable nuclei with an odd number of neutrons and an odd number of protons:

• deuterium (hydrogen-2), with 1 proton and 1 neutron.

• lithium-6, with 3 protons and 3 neutrons.

• boron-10, with 5 protons and 5 neutrons.

• nitrogen-14, with 7 protons and 7 neutrons.

• tantalum-180, with 73 protons and 107 neutrons.

Deuterium is rare compared to hydrogen and helium-4. Lithium-6 is rare compared to lithium-7.

Tantalum-180 is rare compared to tantalum-181, though I was lying slightly when I said it was stable: theoretically it’s predicted to decay, though with such a long half-life—over 1016 years—that it’s never actually been seen to decay. It’s also weird because it’s the only nuclear isomer found naturally in nature: that is, a nucleus that’s in an excited state, not its ground state. To add to the weirdness, the ground state of tantalum-180 is less stable than the excited state: it decays into tungsten or hafnium with a half-life of 8 hours!

Anyway: in these cases, one gets the feeling that odd-odd nuclei are hard for nature to produce. But 20% of boron is boron-10 (the rest being boron-11), and 99.6% of nitrogen is nitrogen-14 (the rest being nitrogen-15).

What’s up with nitrogen-14? Why is it the most abundant isotope of an element despite it being doubly odd? Or more precisely, why is it the only isotope with this property?

This Week’s Finds (1–50)

12 January, 2021

Take a copy of this!

This Week’s Finds in Mathematical Physics (1-50), 242 pages.

These are the first 50 issues of This Week’s Finds of Mathematical Physics. This series has sometimes been called the world’s first blog, though it was originally posted on a “usenet newsgroup” called sci.physics.research — a form of communication that predated the world-wide web. I began writing this series as a way to talk about papers I was reading and writing, and in the first 50 issues I stuck closely to this format. These issues focus rather tightly on quantum gravity, topological quantum field theory, knot theory, and applications of n-categories to these subjects. There are, however, digressions into elliptic curves, Lie algebras, linear logic and various other topics.

Tim Hosgood kindly typeset all 300 issues of This Week’s Finds in 2020. They will be released in six installments of 50 issues each, for a total of about 2610 pages. I have edited the issues here to make the style a bit more uniform and also to change some references to preprints, technical reports, etc. into more useful arXiv links. This accounts for some anachronisms where I discuss a paper that only appeared on the arXiv later.

The process of editing could have gone on much longer; there are undoubtedly many mistakes remaining. If you find some, please contact me and I will try to fix them.

By the way, sci.physics.research is still alive and well, and you can use it on Google. But I can’t find the first issue of This Week’s Finds there — if you can find it, I’ll be grateful. I can only get back to the sixth issue. Take a look if you’re curious about usenet newsgroups! They were low-tech compared to what we have now, but they felt futuristic at the time, and we had some good conversations.

CP Violation

5 January, 2021

Here are two more open questions about physics. I have a question of my own at the end!

Why are the laws of physics not symmetrical when we switch left and right, or future and past, or matter and antimatter? Why do the laws of nature even violate “CP symmetry”? That is: why are the laws not symmetrical under the operation where we simultaneously switch matter and antimatter and switch left and right?

Violation of P symmetry, meaning the symmetry between left and right, is strongly visible in the Standard Model: for example, all directly observed neutrinos are “left-handed”. But violation of CP symmetry is subtler: in the Standard Model it appears solely in interactions between the Higgs boson and quarks or leptons. Technically, it occurs because the numbers in the Cabibbo–Kobayashi–Maskawa matrix and Pontecorvo–Maki–Nakagawa–Sakata matrix (discussed in the previous question) are not all real numbers. Interestingly, this is only possible when there are 3 or more generations of quarks and/or leptons: with 2 or fewer generations the matrix can always be made real.

Does the strong force violate CP symmetry? In the Standard Model it would be very natural to add a CP-violating term to the equations describing the strong force, proportional to a constant called the “θ angle”. But experiments say the magnitude of the θ angle is less than 2 × 10-10. Is this angle zero or not? Nobody knows. Why is it so small? This is called the “strong CP problem”. One possible solution, called the Peccei–Quinn mechanism, involves positing a new very light particle called the axion, which might also be a form of dark matter. But despite searches, nobody has found any axions.

• Wikipedia, CP Violation.

• Wikpedia, Strong CP Problem.

• Michael Beyer, editor, CP Violation in Particle, Nuclear, and Astrophysics, Springer, Berlin, 2008.

• I. Bigi, CP Violation — An Essential Mystery in Nature’s Grand Design.

It’s a theorem that quantum field theories are symmetrical under CPT: the combination of switching matter and antimatter, left and right, and future and past. Thus, a violation of CP implies a violation of time reversal symmetry. For more on this, see:

• R. G. Sachs, The Physics of Time Reversal, University of Chicago Press, Chicago, 1987.

What are the electric dipole moments of the electron and the neutron?

As of 2020, experiments show the electric dipole moment of the electron is less than 1.1 × 10-29 electron charge centimeters. According to the Standard Model it should have a very small nonzero value due to CP violation by virtual quarks, but various extensions of the Standard Model predict a larger dipole moment.

Also as of 2020, experiments show the neutron’s electric dipole is less than 1.8 × 10-26 e·cm. The Standard Model predicts a moment of about 10-31 e·cm, again due to CP violation by
virtual quarks, and again various other theories predict a larger moment.

Measuring these moments could give new information on physics beyond the Standard Model.

• Wikipedia, Electron Electric Dipole Moment.

• Wikipedia, Neutron Electric Dipole Moment.

• Maxim Pospelov and Adam Ritz, Electric Dipole Moments as Probes of New Physics.

Here’s my question. Do you know papers that actually calculate what the Standard Model predicts for the electric dipole moments of the electron and neutron?

Cosmic Censorship

31 December, 2020

I seem to be getting pulled into the project of updating this FAQ:

Open questions in physics.

The more I look at it, the bigger the job gets. I started out rewriting the section on neutrinos, and now I’m doing the part on cosmic censorship. There are even bigger jobs to come. But it’s fun as long as I don’t try to do it all in one go!

Here’s the new section on cosmic censorship. If you have any questions or have other good resources to suggest, let me know.

Does Cosmic Censorship hold?  Roughly, is general relativity a deterministic theory—and when an object collapses under its own gravity, are the singularities that might develop guaranteed to be hidden behind an event horizon?

Proving a version of Cosmic Censorship is a matter of mathematical physics rather than physics per se, but doing so would increase our understanding of general relativity. There are actually at least two versions: Penrose formulated the “Strong Cosmic Censorship Conjecture” in 1986 and the “Weak Cosmic Censorship Hypothesis” in 1988. Very roughly, strong cosmic censorship asserts that under reasonable conditions general relativity is a deterministic theory, while weak cosmic censorship asserts that that any singularity produced by gravitational collapse is hidden behind an event horizon. Despite their names, strong cosmic censorship does not imply weak cosmic censorship.

In 1991, Preskill and Thorne made a bet against Hawking in which they claimed that weak cosmic censorship was false. Hawking conceded this bet in 1997 when a counterexample was found by Matthew Choptuik. This features finely-tuned infalling matter poised right on the brink of forming a black hole. It almost creates a region from which light cannot escape—but not quite. Instead, it creates a naked singularity!

Given the delicate nature of this construction, Hawking did not give up. Instead he made a new bet, which says that weak cosmic censorship holds “generically”—that is, except for very unusual conditions that require infinitely careful fine-tuning to set up. For an overview see:

• Robert Wald, Gravitational Collapse and Cosmic Censorship.

In 1999, Christodoulou proved that for spherically symmetric solutions of Einstein’s equation coupled to a massless scalar field, weak cosmic censorship holds generically. For a review of this and also Choptuik’s work, see:

• Carsten Gundlach, Critical Phenomena in Gravitational Collapse.

While spherical symmetry is a very restrictive assumption, this result is a good example of how, with plenty of work, we can make progress in rigorously settling the questions raised by general relativity.

What about strong cosmic censorship? In general relativity, for each choice of initial data—that is, each choice of the gravitational field and other fields at “time zero”—there is a region of spacetime whose properties are completely determined by this choice. The question is whether this region is always the whole universe. That is: does the present determine the whole future?

The answer is: not always! By carefully choosing the fields at time zero you can manufacture counterexamples. But Penrose, knowing this, claimed only that generically the fields at time zero determine the whole future of the universe.

In 2017, Mihalis Dafermos and Jonathan Luk showed that even this is false if you don’t demand that the fields stay smooth. But perhaps the conjecture can be saved if we require that:

• Kevin Hartnett, Mathematicians Disprove Conjecture Made to Save Black Holes.

• Oscar J.C. Dias, Harvey S. Reall and Jorge E. Santos, Strong Cosmic Censorship: Taking the Rough with the Smooth.

Solar Neutrinos

29 December, 2020

Over on the Category Theory Community Server, John van de Wetering asked me how many times a typical solar neutrino oscillates on its flight from the Sun to the Earth. I didn’t know, and I thought it would be fun to estimate this.

So let’s do it! Let’s do a rough calculation, and worry about details later. For those too lazy to even jump to the end, here are the results:

• A neutrino takes about 500 seconds to travel from the Sun to the Earth.

• Because a typical solar neutrino moving is moving close to the speed of light, time dilation affects it dramatically, and the time of travel from the Sun to the Earth experienced by the neutrino is much less: very roughly, 1/6 of a millisecond.

• There are different kinds of oscillation. If we keep track only of its slower oscillations, a typical solar neutrino oscillates roughly once for each 1250 meters of its flight through space.

• As it travels from Sun to Earth, this typical neutrino does about 120 million oscillations.

Let’s start at the beginning.

The Sun emits a lot of electron neutrinos. Most are produced from a reaction where two protons collide and one turns into a neutron, emitting a positron and an electron neutrino. The proton and neutron then stick together forming a ‘deuteron’, but let’s not worry about that.

More importantly, the energy of the neutrinos produced from these so-called pp reaction is at most 400 keV. That means 400,000 eV, where an eV or ‘electron volt’ is the energy an electron picks up as it falls through a potential of one volt. If you look at this chart:

you’ll see most solar neutrinos have a somewhat lower energy. Let’s say 300 keV.

By comparison to the rest mass of a neutrino, this is huge. Nobody knows neutrino masses very accurately—as we’ll see, people know more about differences of squares of the three neutrino masses. But a very rough estimate for the rest mass of the lightest neutrino might be 0.1 eV/c2. Here like particle physicists I’m measuring mass in units of energy divided by the speed of light squared. An eV, or electron volt, is the change in energy of an electron as it undergoes a one-volt change in potential.

This mass could be way off, say by a factor of 10 or more. But it’s good enough to show this: solar neutrinos are moving very close to the speed of light!

Remember, the energy of a moving particle, divided by its ‘mass energy’, the energy due to its mass, is

\displaystyle{ \frac{1}{\sqrt{1 - v^2/c^2}} }

Our solar neutrino, using our very rough guess about its mass, has

\displaystyle{ \frac{1}{\sqrt{1 - v^2/c^2}} \approx \frac{300 \textrm{keV}}{0.1 \textrm{eV}} = 3 \cdot 10^6 }

It has an energy 3 million times its rest energy! That gives

\displaystyle{  1 - v^2/c^2 \approx \frac{1}{9 \cdot 10^{12}} }


\displaystyle{  v^2/c^2 \approx 1 - \frac{1}{9 \cdot 10^{12}} }

or using a Taylor series trick

\displaystyle{  v/c \approx 1 - \frac{1}{18 \cdot 10^{12}} }

or if I didn’t push the wrong button on my calculator

v  \approx  0.99999999999994 \; c

This is ridiculously close to the speed of light.

It’s more useful to remember that our neutrino’s energy is roughly 3 million times what it would be at rest. And relativity says that due to time dilation, the passage of time experienced by this neutrino is slowed down by the same factor!

It takes 500 seconds for light to go from the Sun to the Earth. Our neutrino will take a tiny bit longer—the difference is not worth worrying about. But because of time dilation, the travel time ‘experienced by the neutrino’ will be

\displaystyle{ \frac{500 \; \textrm{sec}}{3 \cdot 10^6} \approx 1.67 \cdot 10^{-4} \; \textrm{sec} }

This figure is very rough, due to how poorly we know the neutrino’s mass, but it’s about a 1/6 of a millisecond.

Now let’s think about how the neutrino oscillates.

To keep things simple, let’s assume our electron neutrino gets out of the Sun without anything happening to it. What happens next?

There are three flavors of neutrino—and as it shoots through space, what started as an electron neutrino will ‘oscillate’ between all three flavors, like this:

Here black means electron neutrino, blue means muon neutrino and red means tau neutrino.

You’ll notice that both high-frequency and low-frequency oscillations are going on. This is because the three flavors of neutrino are nontrivial linear combinations of three ‘mass eigenstates’, each of which has a phase that oscillates at a different rate. Two of the mass eigenstates are very close in mass, and this small mass difference causes a small energy difference which causes the slower oscillation. The third mass eigenstate is farther away from the other two, so we also get a more rapid oscillation. As you can see, this is especially noticeable in how the neutrino flickers back and forth between being a muon and a tau neutrino.

But all this is a bit complicated, so let’s just focus on the slower oscillations. How many of those oscillations happen as our friend the neutrino wings its way from Sun to Earth?

To estimate this, let’s pretend there are only the two mass eigenstates that are very close in mass, and ignore the third. The two masses m_1 and m_2 are not actually known very accurately. What we know is

m_2^2 - m_1^2 \approx 0.000074 \; \textrm{eV}^2/c^4

The reason we know these differences in squares of mass is actually by doing measurements of neutrino oscillations: these differences actually determine the frequency of the neutrino oscillations! Let’s see why.

If something has energy E, quantum mechanics says its phase will oscillate over time like this:

\exp(-i t E / \hbar)

where \hbar is Planck’s constant and the minus sign is just an unfortunate convention. But all we detect is the absolute value of this, which is just 1: that doesn’t change. So to actually see oscillations we should think about something that can have two different energies E_1 and E_2. Then we need to think about things like

\exp(-i t E_1 / \hbar) - \exp(-i t E_2 / \hbar)

or other linear combinations of these two functions. But their difference illustrates the point nicely: we have

\exp(-i t E_1 / \hbar) - \exp(-i t E_2 / \hbar) =

\exp(-i t E_1) (1 - \exp(-it (E_2 - E_1)/\hbar)

and the absolute value of this changes with time! It’s

| 1 - \exp(-it (E_2 - E_1)/\hbar)|

and the takeaway message here is that it oscillates at a frequency depending on the energy difference,

\omega = (E_2 - E_1) / \hbar

So, if we have two kinds of neutrino, it’s the energy difference of the two mass eigenstates that determines how fast a superposition of these two will oscillate. It’s very similar to how when two piano strings are oscillating at almost but not quite the same frequency, you’ll hear ‘beats’ as they go in and out of phase—and the frequency of these beats depends on the difference of their piano strings’ frequencies.

So energy differences are what we care about. But how is energy related to mass? In units where the speed of light is 1, special relativity tells us this:

E^2 = m^2 + p^2

where m is mass and p is momentum. One of the mind-blowing moments of my early physics education was watching someone do a Taylor expansion for low momenta and getting this:

\displaystyle{ E = \sqrt{m^2 + p^2} \approx m + \frac{p^2}{2m} + \cdots }

It looks more impressive if you don’t set the speed of light c equal to 1:

\displaystyle{ E = \sqrt{m^2c^4 + p^2c^2} \approx mc^2 + \frac{p^2}{2m} + \cdots }

So we see that at low momenta the energy is Einstein’s famous E = mc^2 plus the kinetic energy p^2/2m famous from classical mechanics before relativity!

But all this is useless for our solar neutrino, which is ‘ultra-relativistic’: it’s moving almost at the speed of light! Now p^2 is much bigger than m^2, not smaller, in units where c = 1. So we should do a different Taylor expansion, where we treat m^2 as the small perturbation:

\displaystyle{ E = \sqrt{p^2 + m^2} \approx p + \frac{m^2}{2p} + \cdots }

Cute, eh? Everything is backwards from what I learned in school: we just switch m and p.

This shows us that if we have a neutrino with some large momentum p and it’s a linear combination of two different mass eigenstates with masses m_1 and m_2, it’ll be a blend of two energies:

\displaystyle{ E_1 = \sqrt{p^2 + m_1^2} \approx p + \frac{m_1^2}{2p} + \cdots }


\displaystyle{ E_2 = \sqrt{p^2 + m_2^2} \approx p + \frac{m_2^2}{2p} + \cdots }

So, the energy difference is

\displaystyle{E_2 - E_1 = \frac{1}{2 p} (m_2^2 - m_1^2) }

and this is what determines the rate at which the neutrino oscillates.

If we stop working in units where c = 1 we get

\displaystyle{E_2 - E_1 = \frac{c^3}{2 p} (m_2^2 - m_1^2) }

So, the frequency of oscillations is

\displaystyle{\omega = (E_2 - E_1) / \hbar = \frac{c^3}{2 \hbar p}  (m_2^2 - m_1^2) }

This frequency says how the relative phase rotates around in radians per second. But it’s more useful to think about radians per distance traveled; let’s call that k. Since our neutrino is moving at almost the speed of light, to get this we just divide by c.

\displaystyle{k = \frac{c^2}{2 \hbar p}  (m_2^2 - m_1^2) }

And because the neutrino is ultra-relativistic, its momentum almost obeys E = p c. Here E could be either E_1 or E_2; they’re so close the difference doesn’t matter here. So we get

\displaystyle{k = \frac{c^3}{2 \hbar E}  (m_2^2 - m_1^2) }

This is why people doing experiments with neutrino oscillations measure differences of squares of neutrino masses, not neutrino masses.

For our solar neutrino we’re assuming

E = 300 \; \mathrm{keV}

and remember

m_2^2 - m_1^2 \approx 0.000074 \; \textrm{eV}^2/c^4

Plugging these in we get

\displaystyle{k = \frac{1}{2 \hbar c} \frac{0.000074 \; \textrm{eV}}{300,000}  }

Now it gets annoying, and this is where I usually make mistakes. We use

c = 3 \cdot 10^8 \; \textrm{meter} / \textrm{second}

\hbar =  1.05 \cdot 10^{-34} \; \textrm{kilogram} \, \textrm{meter}^2 / \textrm{second}

\textrm{eV} = 1.60 \cdot 10^{-19} \; \textrm{kilogram} \, \textrm{meter}^2 / \textrm{second}^2

and get

\displaystyle{ k \approx \frac{1}{1600 \; \textrm{meter}} }

It’s funny how multiplying and dividing all these large and tiny numbers leaves us with something at the human scale!

But actually my computation was sloppy at one point. I warned you! I think it’s actually off by a factor of two. Wikipedia says right answer is

\displaystyle{k = \frac{c^3}{4 \hbar E}  (m_2^2 - m_1^2) }

and this gives

\displaystyle{ k \approx \frac{1}{3200 \; \textrm{meter}} }

So, the neutrino oscillates at a rate of about one radian every 3200 meters! And to get the wavelength of the oscillation we need to multiply by 2 \pi. So our solar neutrino makes a complete oscillation about once every 20 kilometers!

And the distance from the Earth to the Sun is 150 million kilometers. So, our neutrino oscillates about 7.5 million times on its trip here.

You should take all this with a grain of salt since I easily could have made some mistakes. If you find errors please let me know! I leave you with a puzzle:

Puzzle. Where does the missing factor of 2 come from?

I don’t think you need to know fancy physics to solve this. I think the mistake is visible in my calculations.

Neutrino Puzzles (Part 2)

26 December, 2020

Okay, I’ve drafted an update to my list of open questions in physics.

I eliminated a bunch of questions that seem to have been answered. It’s really remarkable how accelerator experiments in the last decade or so have settled questions in particle physics without discovering any new mysterious phenomena. The really big mysteries remain.

I have not gotten around to adding the new questions about black holes raised by LIGO. I have not gotten around to updating the sections on ultra-high energy cosmic rays or gamma ray bursters, both of which sorely need it. But I have updated the section on neutrinos!

Here’s the new version. I still need some more good new general reviews of neutrino experiments and theoretical questions. Do you know some?

What’s going on with neutrinos?  Why are all the 3 flavors of neutrino—called the electron neutrino, the muon neutrino and the tau neutrino—so much lighter than their partners, the electron, muon, and tau?  Why are the 3 flavors of neutrino so different from the 3 neutrino states that have a definite mass?  Could any of the observed neutrinos be their own antiparticles?  Do there exist right-handed neutrinos—that is, neutrinos that spin counterclockwise along their axis of motion even when moving very near the speed of light?  Do there exist other kinds of neutrinos, such as “sterile” neutrinos—that is, neutrinos that don’t interact directly with other particles via the weak (or electromagnetic or strong) force?

Starting in the 1990s, our understanding of neutrinos has dramatically improved, and the puzzle of why we see about 1/3 as many electron neutrinos coming from the sun as naively expected has pretty much been answered: the three different flavors of neutrino—electron, muon and tau—turn into each other, because these flavors are not the same as the three “mass eigenstates”, which have a definite mass.  But the wide variety of neutrino experiments over the last thirty years have opened up other puzzles.

For example, we don’t know the origin of neutrinos’ masses.  Do the observed left-handed neutrinos get their mass by coupling to the Higgs and a right-handed partner, the way the other quarks and leptons do?  This would require the existence of so-far-unseen right-handed neutrinos.  Do they get their mass by coupling to themselves?  This could happen if they are “Majorana fermions“: that is, their own antiparticles.  They could also get a mass in other, even more exciting ways, like the “seesaw mechanism“. This requires them to couple to a very massive right-handed particle, and could explain their very light masses.

Even what we’ve actually observed raises puzzles.  With many experiments going on, there are often “anomalies”, but many of these go away after more careful study.  Here’s a challenge that won’t just go away with better data: the 3×3 matrix relating the 3 flavors of neutrino to the 3 neutrino mass eigenstates, called the Pontecorvo–Maki–Nakagawa–Sakata matrix, is much further from the identity matrix than the analogous matrix for quarks, called the Cabibbo–Kobayashi–Maskawa matrix.  In simple terms, this means that each of the three flavors of neutrino is a big mix of different masses.  Nobody knows why these matrices take the values they do, or why they’re so different from each other.

For details, try:

The Neutrino Oscillation Industry.

• John Baez, Neutrinos and the Mysterious Pontevorco–Maki–Nakagawa–Sakata Matrix.

• Paul Langacker, Implications of Neutrino Mass.

• A. Baha Balantekin and Boris Kayser, On the Properties of Neutrinos.

• Salvador Centelles Chuliá, Rahul Srivastava and José W. F. Valle, Seesaw Roadmap to Neutrino Mass and Dark Matter.

The first of these has lots of links to the web pages of research groups doing experiments on neutrinos.  It’s indeed a big industry!

Neutrino Puzzles (Part 1)

24 December, 2020

Merry Xmas, Ymas, and Zmas—and a variable New Year!

For a long time I’ve been meaning to update this list of open questions on the Physics FAQ:

Open questions in physics, Physics FAQ.

Here’s what it said about neutrinos as of 2012:

• What is the correct theory of neutrinos?  Why are they almost but not quite massless?  Do all three known neutrinos—electron, muon, and tau—all have a mass?  Could any neutrinos be Majorana spinors?  Is there a fourth kind of neutrino, such as a “sterile” neutrino?

Starting in the 1990s, our understanding of neutrinos has dramatically improved, and the puzzle of why we see about 1/3 as many electron neutrinos coming from the sun as naively expected has pretty much been answered: the different neutrinos can turn into each other via a process called “oscillation”. But, there are still lots of loose ends.

It’s held up fairly well: all of those questions are still things people wonder about. But I should add a question like this, because it’s nice and concrete, and physicists are fascinated by it:

• Is the tau neutrino heavier than the mu and electron neutrinos, or lighter?

This is a bit sloppy because the neutrinos of definite mass are linear combinations of the neutrinos of definite flavor (the electron, mu and tau neutrinos). The neutrinos of definite mass are called mass eigenstates and the neutrinos of definite flavor are called flavor eigenstates. This picture by Xavier Sarazin makes the two competing scenarios clearer:

In the normal hierarchy the mass eigenstate that’s mainly made of tau neutrino is the heaviest. In the inverted hierarchy it’s the lightest.

We don’t know which of these scenarios is correct. The problem is that we can’t easily measure neutrino masses! The rate at which neutrinos oscillate from flavor to flavor gives us information about absolute values of differences of squared masses! Currently we’re pretty sure the three masses obey

|m_1^2 - m_2^2| \approx 0.00008\; \mathrm{eV}^2


|m_2^2 - m_3^2| \approx 0.003 \;\mathrm{eV}^2

So, m_1 and m_2 are close and m_3 is farther, but we don’t know if m_3 is bigger than the other two (normal hierarchy) or smaller (inverted hierarchy).

We also don’t know which is bigger, m_1 or m_2. And as the FAQ points out, we’re not even sure all three masses are nonzero!

By the way, I will bet that we’ve got the normal hierarchy, with m_1 < m_2 < m_3. My reason is just that this seems to match the behavior of the other leptons. The electron is lighter than the muon which is lighter than the tau. So it seems to vaguely make sense that the electron neutrino should be lighter than the mu neutrino which in turn is lighter than the tau neutrino. But this ‘seems to vaguely make sense’ is not based on any theoretical reason! We haven’t the foggiest clue why any of these masses are what they are—and that’s another question on the list.

I also want to change this question to something less technical, so people realize what a big deal it is:

Could any neutrinos be Majorana spinors?

A less technical formulation would be:

• Is any kind of neutrino its own antiparticle?

On the one hand it’s amazing that we don’t know if neutrinos are their own antiparticles! But on the other hand, it’s really hard to tell if a particle is its own antiparticle if its very hard to detect and when you make them they’re almost always whizzing along near the speed of light.

We’d know at least some neutrinos are their own antiparticles if we saw neutrinoless double beta decay. That’s a not-yet-seen form of radioactive decay where two neutrons turn into two protons and two electrons without emitting two antineutrinos, basically because the antineutrinos annihilate each other:

Physicists have looked for neutrinoless double beta decay. If it happens, it’s quite rare.

Why in the world should we suspect that neutrinos are their own antiparticles? The main reason is that this would provide another mechanism for them to have a mass—a so-called ‘Majorana mass’, as opposed to the more conventional ‘Dirac mass’ that explains the mass of the electron (for example) in the Standard Model.

I will bet against the observed neutrinos being their own antiparticles, because this would violate conservation of lepton number and an even more sacred conservation law: conservation of baryon number minus lepton number. On the other hand, if some so-far-unobserved right-handed neutrinos are very heavy and have a Majorana mass, we could explain the very light masses of the observed neutrinos using a trick called the seesaw mechanism. And by the way: even the more conventional ‘Dirac mass’ requires that the observed left-handed neutrinos have right-handed partners, which have so far not been seen! So here’s another interesting open question:

• Are there right-handed neutrinos: that is, neutrinos that spin counterclockwise along their direction of motion when moving at high speeds?

So many unanswered questions about neutrinos!

My list of references hasn’t held up as well:

For details, try:

The Neutrino Oscillation Industry.

• John Baez, Neutrinos and the nysterious Maki-Nakagawa-Sakata Matrix.

• Paul Langacker, Implications of neutrino mass.

• Boris Kayser, Neutrino mass: where do we stand, and where are we going?.

The first of these has lots of links to the web pages of research groups doing experiments on neutrinos. It’s indeed a big industry!

In fact the first page is now full of silly random posts, but oddly still titled Paul Langacker’s page is missing. Boris Kayser’s review uses an old link to the arXiv, back when it was at His review is still on the arXiv, and it’s nice—but it dates to 1998, so I should find something newer!

What are the best places to read a lot of clearly explained information about neutrino puzzles? Are there other big neutrino puzzles I should include?

Sustainability Week

22 December, 2020

In 2021, March 8–13 will be “Sustainability Week” in Switzerland. During this week, students at all Swiss universities will come together to present their current work, promote a sustainable lifestyle and draw extra attention to changes that must be made at the institutional level. Anna Knörr, a third year Physics Bachelor student at ETH Zürich, is president of the Student Sustainability Commission. She and Professor Niklas Beisert invited me to give the Zürich Theoretical Physics Colloquium on Monday the 8th of March.

She proposed the modest title “Theoretical Physics in the 21st Century”. I like this idea because it would give me a chance to think about the ways in which theoretical physics is stuck, the ways it’s not, and the ways theoretical physics can help us adapt to the Anthropocene. So, I could blend ideas from these two talks:

Unsolved mysteries in fundamental physics, Cambridge University Physics Society, October 3, 2018.

Energy and the environment—what physicists can do, Perimeter Institute, April 17, 2013.

but update and improve the second one. I think it’ll be pretty easy for me to explain that the Anthropocene is about much more than global warming. The hard part is giving suggestions for “what physicists can do”.

Of course we can all resolve to fly less, etc.—but none of those suggestions take advantage of special skills that physicists have. Anna Knörr correctly noted that many theoretical physicists have trouble seeing what they can do to help our civilization adapt to the Anthropocene, since many of them are not good at designing better batteries, solar cells, fission or fusion reactors comes easily. To the extent that I’m a theoretical physicist I fit into this unhappy class. But I think there are more theoretical activities that can still be helpful! And I have more to say about this now than in 2013.

One lesson I may offer is this:

If something is not working, try something different.

This applies to the Anthropocene as a whole, all the social problems that afflict us, and also fundamental physics. I just ran into a talk that the famous particle physicist Sheldon Glashow gave 40 years ago, called “The New Frontier”. He said:

Important discoveries await the next generation of accelerators. QCD and the electroweak theory need further confirmation. We need to know how b quarks decay. The weak interaction intermediaries must be seen to be believed. The top quark (or the perversions needed by topless theories) lurks just out of range. Higgs may wait to be found. There could well be a fourth family of quarks and leptons. There may even be unanticipated surprises. We need the new machines.

That was in 1980. The ‘weak interaction intermediaries’—the W and Z—were found three years later, in 1982. The top quark was found in 1995. The Higgs boson was found in 2012. No fourth generation of quarks and leptons, and we now have good evidence that none exists. To the great sorrow of all physcists, particle accelerators have found no unanticipated surprises!

On the other hand, we have for the first time an apparently correct theory of elementary particle physics. It may be, in a sense, phenomenologically complete. It suggests the possibility that there are no more surprises at higher energies, at least at energies that are remotely accessible.

He’s proved right on this, so far.

Proton decay, if it is found, will reinforce belief in the great desert extending from 100 GeV to the unification mass of 1014 GeV. Perhaps the desert is a blessing in disguise. Ever larger and more costly machines conflict with dwindling finances and energy reserves. All frontiers come to an end.

You may like this scenario or not; it may be true or false. But, it is neither impossible, implausible, nor unlikely. And, do not despair nor prematurely lament the death of particle physics. We have a ways to go to reach the desert, with exotic fauna along the way, and even the desolation of a desert can be interesting.

Proton decay has not been found despite a huge amount of effort. So, that piece of evidence for grand unified theories is missing, and with it a strong piece of evidence that there should be a “desert” of new phenomena between the electroweak unification energy scale and the GUT energy scale.

But, we’re not seeing anything beyond the Standard Model: no “exotic fauna”.

Glashow’s “new frontier” was the “passive frontier”: non-accelerator experiments like neutrino measurements, and this is indeed where the progress came since 1980: we now know neutrinos are massive and oscillate, and there is still some mystery here and room for surprises—though frankly I suspect that neutrino masses will work very much like quark masses, via coupling to the Higgs. (This is in a sense the most conservative, least truly exciting scenario.)

So, very little dramatic progress has happened in particle physics since 1980—except for a profusion of new theories that haven’t made any verified predictions. I’ll argue that physicists should turn elsewhere! There are other things for them to do, that are much more exciting.

Theories of Aether and Electricity (Part 1)

19 December, 2020

I’ve been reading an amazing book, a little bit every night in bed:

• Edmund Whittaker, A History of the Theories of Aether and Electricity, Two Volumes Bound As One. Volume I: The Classical Theories. Vol. II: The Modern Theories, 1900-1926. Dover, 1989, 753 pages.

How in the world did our species figure out the laws governing the electric field, magnetic field, and charged particles? A lot started with pure luck. Two unusual stones played a key role: amber and lodestone.

The first, really fossilized tree sap, easily acquires an electric charge if you rub it against wool or silk. This was one of human’s introductions to the electric field, and electrons. Indeed, the ancient Greek word for amber was ēlektron. The second, called magnetite, is naturally magnetic.

How odd that of all the minerals in nature, there were two with peculiar abilities to attract and repel! This duality foreshadowed the duality between electric and magnetic fields, now understood mathematically using the Hodge star operator. Who could have guessed that a pair of stones would eventually lead to such deep discoveries?

Isaac Newton caught a glimpse of it. In the early 1700s he commented about both amber and lodestones in the third book of his Opticks, called simply The Queries. He was imagining challenging someone skeptical of the existence of aether:

Let him also tell me, how an electrick Body can by Friction emit an Exhalation so rare and subtile, and yet so potent, as by its Emission to cause no sensible Diminution of the weight of the electrick Body, and to be expanded through a Sphere, whose Diameter is above two Feet, and yet to be able to agitate and carry up Leaf Copper, or Leaf Gold, at the distance of above a Foot from the electrick Body? And how the Effluvia of a Magnet can be so rare and subtile, as to pass through a Plate of Glass without any Resistance or Diminution of their Force, and yet so potent as to turn a magnetick Needle beyond the Glass?

While these are brilliant questions, he and some later thinkers had to struggle for a long time to sort out the relation between what we’d later call electrons and the electric field. It’s easy to see why, since they’re so intimately related.

As it turns out, electrons are not emitted but absorbed by amber when it rubs against wool. Later there were long arguments about whether there were two kinds of ‘electrical fluid’, positively and negatively charged, or just one. But maybe the ‘exhalation’ he mentions is really the electric field, just as the ‘effluvia’ of a magnet are the magnetic field.

There is a lot more to say about all this, but I think I’ll do it in short bits, to avoid writing a 753-page tome like Whittaker’s.