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?

Applied Category Theory 2021 — Adjoint School

2 January, 2021

Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you:

Applied Category Theory 2021 — Adjoint School. Applications due Friday 29 January 2021. Organized by David Jaz Myers, Sophie Libkind, and Brendan Fong.

There are four projects to work on with great mentors. You can see descriptions of them below!

By the way, it’s not yet clear if there will be an in-person component to this school —but if there is, it will happen at the University of Cambridge. ACT2021 is being organized by Jamie Vicary, who teaches in the computer science department there.

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

School overview

Participants are divided into four-person project teams. Each project is guided by a mentor and a TA. The Adjoint School has two main components: an Online Seminar that meets regularly between February and June, and an in-person Research Week in Cambridge, UK on July 5–9.

During the online seminar, we will read, discuss, and respond to papers chosen by the project mentors. Every other week, a pair of participants will present a paper which will be followed by a group discussion. Leading up to this presentation, study groups will meet to digest the reading in progress, and students will submit reading responses. After the presentation, the presenters will summarize the paper into a blog post for The n-Category Cafe.

The in-person research week will be held the week prior to the International Conference on Applied Category Theory and in the same location. During the week, participants work intensively with their research group under the guidance of their mentor. Projects from the Adjoint School will be presented during this conference. Both components of the school aim to develop a sense of belonging and camaraderie in students so that they can fully participate in the conference, for example by attending talks and chatting with other conference goers.

Projects to choose from

Here are the projects.

Topic: Categorical and computational aspects of C-sets

Mentors: James Fairbanks and Evan Patterson

Description: Applied category theory includes major threads of inquiry into monoidal categories and hypergraph categories for describing systems in terms of processes or networks of interacting components. Structured cospans are an important class of hypergraph categories. For example, Petri net-structured cospans are models of concurrent processes in chemistry, epidemiology, and computer science. When the structured cospans are given by C-sets (also known as co-presheaves), generic software can be implemented using the mathematics of functor categories. We will study mathematical and computational aspects of these categorical constructions, as well as applications to scientific computing.


Structured cospans, Baez and Courser.

An algebra of open dynamical systems on the operad of wiring diagrams, Vagner, Spivak, and Lerman.

Topic: The ubiquity of enriched profunctor nuclei

Mentor: Simon Willerton

Description: In 1964, Isbell developed a nice universal embedding for metric spaces: the tight span. In 1966, Isbell developed a duality for presheaves. These are both closely related to enriched profunctor nuclei, but the connection wasn’t spotted for 40 years. Since then, many constructions in mathematics have been observed to be enriched profunctor nuclei too, such as the fuzzy/formal concept lattice, tropical convex hull, and the Legendre–Fenchel transform. We’ll explore the world of enriched profunctor nuclei, perhaps seeking out further useful examples.


The Legendre–Fenchel transform from a category theoretic perspective, Willerton.

On the fuzzy concept complex (chapters 2-3), Elliot.

Topic: Double categories in applied category theory

Mentor: Simona Paoli

Description: Bicategories and double categories (and their symmetric monoidal versions) have recently featured in applied category theory: for instance, structured cospans and decorated cospans have been used to model several examples, such as electric circuits, Petri nets and chemical reaction networks.

An approach to bicategories and double categories is available in higher category theory through models that do not require a direct checking of the coherence axioms, such as the Segal-type models. We aim to revisit the structures used in applications in the light of these approaches, in the hope to facilitate the construction of new examples of interest in applications.


Structured cospans, Baez and Courser.

A double categorical model of weak 2-categories, Paoli and Pronk.

and introductory chapters of:

Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories, Paoli.

Topic: Extensions of coalgebraic dynamic logic

Mentors: Helle Hvid Hansen and Clemens Kupke

Description: Coalgebra is a branch of category theory in which different types of state-based systems are studied in a uniform framework, parametric in an endofunctor F:C → C that specifies the system type. Many of the systems that arise in computer science, including deterministic/nondeterministic/weighted/probabilistic automata, labelled transition systems, Markov chains, Kripke models and neighbourhood structures, can be modeled as F-coalgebras. Once we recognise that a class of systems are coalgebras, we obtain general coalgebraic notions of morphism, bisimulation, coinduction and observable behaviour.

Modal logics are well-known formalisms for specifying properties of state-based systems, and one of the central contributions of coalgebra has been to show that modal logics for coalgebras can be developed in the general parametric setting, and many results can be proved at the abstract level of coalgebras. This area is called coalgebraic modal logic.

In this project, we will focus on coalgebraic dynamic logic, a coalgebraic framework that encompasses Propositional Dynamic Logic (PDL) and Parikh’s Game Logic. The aim is to extend coalgebraic dynamic logic to system types with probabilities. As a concrete starting point, we aim to give a coalgebraic account of stochastic game logic, and apply the coalgebraic framework to prove new expressiveness and completeness results.

Participants in this project would ideally have some prior knowledge of modal logic and PDL, as well as some familiarity with monads.


Parts of these:

Universal coalgebra: a theory of systems, Rutten.

Coalgebraic semantics of modal logics: an overview, Kupke and Pattinson.

Strong completeness of iteration-free coalgebraic dynamic logics, Hansen, Kupke, and Leale.

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?

Theoretical Physics in the 21st Century

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 Zurich 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.

Consolidated Appropriations Act, 2021

22 December, 2020

You may not have noticed, but the US Congress just passed the biggest climate-related bill in long time, with measures to help save the ozone layer and speed progress on solar, wind and nuclear energy, battery storage and carbon capture. It’s big news, though it’s been overshadowed by the pandemic and resulting economic disaster. Everyone is focused on another portion of the 5593-page Consolidated Appropriations Act: namely, Division M, the Coronavirus Response and Relief Supplemental Appropriations Act.

This is important. We’ll get through this pandemic, though the US at least has been doing a bad job so far. Global warming will be a much tougher test of our resolve. So it’s good to see this step toward recognizing its gravity.

• Sarah Kaplan and Dino Grandoni, Stimulus deal includes raft of provisions to fight climate change, Washington Post, 21 December 2020.

In one of the biggest victories for U.S. climate action in a decade, Congress has moved to phase out a class of potent planet-warming chemicals and provide billions of dollars for renewable energy and efforts to suck carbon from the atmosphere as part of the $900 billion coronavirus relief package.

The legislation […] wraps together several bills with bipartisan backing and support from an unusual coalition of environmentalists and industry groups.

It will cut the use of hydrofluorocarbons (HFCs), chemicals used in air conditioners and refrigerators that are hundreds of times worse for the climate than carbon dioxide. It authorizes a sweeping set of new renewable energy measures, including tax credit extensions and new research and development programs for solar, wind and energy storage; funding for energy efficiency projects; upgrades to the electric grid and a new commitment to research on removing carbon from the atmosphere. And it reauthorizes an Environmental Protection Agency program to curb emissions from diesel engines.

The legislation also includes key language on the “sense of Congress” that the Energy Department must prioritize funding for research to power the United States with 100 percent “clean, renewable, or zero-emission energy sources” — a rare declaration that the nation should be striving toward net-zero carbon emissions.

“This is perhaps the most significant climate legislation Congress has ever passed,” said Grant Carlisle, a senior policy adviser at the Natural Resources Defense Council.

The HFC measure, which empowers the EPA to cut the production and use of HFCs by 85 percent over the next 15 years, is expected to save as much as half a degree Celsius of warming by the end of the century. Scientists say the world needs to constrain the increase in the average global temperature to less than 2 degrees Celsius compared with preindustrial times to avoid catastrophic, irreversible damage to the planet. Some places around the globe are already experiencing an average temperature rise beyond that threshold.

Advocates say the $35 billion of new funding for renewable technology and energy efficiency in the legislation will also help reduce pollution that is driving global warming and provide a much-needed boost to federal energy programs that haven’t been updated since 2007.

“It doesn’t have regulations or mandates in it,” Sasha Mackler, director of the energy project at the Bipartisan Policy Center, said of the energy package. “But from the bottom up it’s advancing the technology that’s needed. … This is definitely a bill that creates the enabling conditions for decarbonization.”

Support among lawmakers for the package suggests that tax incentives and research funding may be a rare area of common ground between two parties that have been divided on climate change.

Despite President Trump’s numerous efforts to roll back climate regulations, leading Republicans backed the package, which has been a top priority for Sen. Lisa Murkowski (R-Alaska) for years. Senators John Barrasso (R-Wyo.) and John Neely Kennedy (R-La.) helped craft the bipartisan agreement to scale down polluting refrigerants.

“These measures will protect our air while keeping costs down for the American people,” Barrasso, chair of the Senate Environment and Public Works Committee, said in a statement Monday.

Sen. Thomas R. Carper (D-Del.), an ally of President-elect Joe Biden and co-sponsor of the HFC provision, called it “a watershed moment” that bodes well for lawmakers interested in working with the incoming administration on climate change.

“The debate on whether climate change is real is over. It is real. It’s not getting better,” Carper said in a recent interview. “Our Republican colleagues, they get it, for the most part.”

The agreement comes on the heels of a major United Nations climate report, which found that nations’ current plans to reduce greenhouse gasses are just one-fifth of what’s needed to avoid catastrophic warming.

If leaders invest heavily in green infrastructure and renewable energy as part of coronavirus stimulus spending, the world could trim as much as 25 percent from its expected 2030 emissions, the U.N. report said.

Democrats and environmental groups say the legislation is not quite the sweeping “green stimulus” that’s needed. Though it meets Biden’s call to extend tax incentives for solar and wind generation and provide more money for clean energy research, it falls short of his requests for aggressive subsidies for electric vehicles and new requirements that utilities eliminate their contributions to global warming by 2035.

It also excludes a provision from earlier versions of the bill that would have set voluntary standards for energy efficiency in buildings — something that could significantly curb emissions from cities.

“Let’s be clear: Are these provisions enough to meet the demands of the science? No,” said Senate Minority Leader Charles E. Schumer (D-N.Y). “But are they a significant step in the right direction? Yes.”

The HFC rule lays the groundwork for the United States to sign onto the Kigali Amendment, an international agreement in which more than 100 nations committed to replacing the chemicals with refrigerants that have a smaller climate impact. Signed in the final days of the Obama administration, the treaty was never submitted by Trump for ratification by the Senate. By voting to curb the climate pollutant now, Congress has eased the path for approval once Biden takes office.

Included in the energy package are roughly $4 billion for solar, wind, hydropower and geothermal research and development; $1.7 billion to help low-income families install renewable energy sources in their homes; $2.6 billion for the Energy Department’s sustainable transportation program; and $500 million for research on reducing industrial emissions.

It also authorizes $2.9 billion for the Advanced Research Projects Agency-Energy, a program that funds high-risk, high-reward research and that Trump has sought to eliminate multiple times.

The increased funding is expected to make emerging clean-energy technology cheaper and more widespread. This is especially significant for ideas that have proved effective but are struggling to make the jump to commercial viability.

“This is an opportunity to not only make significant advances in climate action and reducing HFCs, but to help maintain leadership of U.S. technology and our competitiveness in that global market,” said Marty Durbin, an energy lobbyist at the U.S. Chamber of Commerce, the largest corporate lobbying group in Washington.

In a boon for renewable energy companies, Congress extended tax credits for wind and solar and introduced a new credit for offshore wind projects, which Heather Zichal, chief executive of the American Clean Power Association, called “America’s largest untapped clean energy source.” One Department of Energy analysis suggested that developing just 4 percent of the total U.S. offshore wind capacity could power some 25 million homes and reduce the nation’s greenhouse gas emissions by almost 2 percent.

But many green groups were critical of provisions dedicating more than $6 billion to efforts to remove carbon from the air and store it, as well as funding for enhanced oil recovery projects, which reuse carbon dioxide to flush residual oil from existing wells.

“It just perpetuates the fossil fuel system,” said Jean Su, an attorney and director of the energy justice program at the Center for Biological Diversity. “If you pass something like this, you’re not doing the best we can do in terms of transforming our energy system.”

Others see carbon capture as a necessary tool for mitigating emissions from sources that aren’t easily decarbonized, such as air travel. The bill directs the energy secretary to estimate “the magnitude of excess carbon dioxide” that needs to be removed from the air to stabilize the climate.

The legislation includes more than $11 billion for nuclear energy [….]

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.