Planet Musings

This is the color of something infinitely hot. Of course you’d instantly be fried by gamma rays of arbitrarily high frequency, but this would be its spectrum in the visible range.

This is also the color of a typical neutron star. They’re so hot they look the same.

It’s also the color of the early Universe!

This was worked out by David Madore.

As a blackbody gets hotter and hotter, its spectrum approaches the classical Rayleigh–Jeans law. That is, its true spectrum as given by the Planck law approaches the classical prediction over a larger and larger range of frequencies.

So, for an extremely hot blackbody, the spectrum of light we can actually see with our eyes is governed by the Rayleigh–Jeans law. This law says the color doesn’t depend on the temperature: only the brightness does!

And this color is shown above.

This involves human perception, not just straight physics. So David Madore needed to work out the response of the human eye to the Rayleigh–Jeans spectrum — “by integrating the spectrum against the CIE XYZ matching functions and using the definition of the sRGB color space.”

The color he got is sRGB(148,177,255). And according to the experts who sip latte all day and make up names for colors, this color is called ‘Perano’.

Here is some background material Madore wrote on colors and visual perception. It doesn’t include the whole calculation that leads to this particular color, so somebody should check it, but it should help you understand how to convert the blackbody spectrum at a particular temperature into an sRGB color:

• David Madore, Colors and colorimetry.

In the comments you can see that Thomas Mansencal redid the calculation and got a slightly different color: sRGB(154,181,255). It looks quite similar to me:

I was driving home from picking up sushi the other night, and another car was tailgating me. I was really annoyed. I was on a curvy road, it was icy out, and I was going the speed limit, 25 – and this guy was riding my bumper, with those new really bright halogen headlights shining right into my rear-view mirror. I was not going to speed up to satisfy him, and anyway I was just going a couple more blocks. But when I turned onto my block, the tailgater turned with me, and when I pulled into my driveway, he parked next to my house. Now I was kind of freaked out. Was the guy going to get out of his car and scream at me for slowing him down? He did get out of his car. No chance of avoiding a conversation. He came up to me and asked where a certain address on my street was. He was a DoorDash delivery guy. Tailgating me because his ability to make enough money to live on depends on getting a certain number of deliveries done per hour, and that means that it’s an economic necessity for him to drive too fast on icy roads.

There have been dramatic articles in the news media suggesting that a Nobel Prize has essentially already been awarded for the amazing discovery of a “fifth force.” I thought I’d better throw some cold water on that fire; it’s fine for it to smoulder, but we shouldn’t let it overheat.

There could certainly be as-yet unknown forces waiting to be discovered — dozens of them, perhaps.   So far, there are four well-studied forces: gravity, electricity/magnetism, the strong nuclear force, and the weak nuclear force.  Moreover, scientists are already fully confident there is a fifth force, predicted but not yet measured, that is generated by the Higgs field. So the current story would really be about a sixth force.

Roughly speaking, any new force comes with at least one new particle.  That’s because

• every force arises from a type of field (for instance, the electric force comes from the electromagnetic field, and the predicted Higgs force comes from the Higgs field)
• and ripples in that type of field are a type of particle (for instance, a minimal ripple in the electromagnetic field is a photon — a particle of light — and a minimal ripple in the Higgs field is the particle known as the Higgs boson.)

The current excitement, such as it is, arises because someone claims to have evidence for a new particle, whose properties would imply a previously unknown force exists in nature.  The force itself has not been looked for, much less discovered.

The new particle, if it really exists, would have a rest mass about 34 times larger than that of an electron — about 1/50th of a proton’s rest mass. In technical terms that means its E=mc² energy is about 17 million electron volts (MeV), and that’s why physicists are referring to it as the X17.  But the question is whether the two experiments that find evidence for it are correct.

In the first experiment, whose results appeared in 2015, an experimental team mainly based in Debrecen, Hungary studied large numbers of nuclei of beryllium-8 atoms, which had been raised to an “excited state” (that is, with more energy than usual).  An excited nucleus inevitably disintegrates, and the experimenters studied the debris.  On rare occasions they observed electrons and positrons [a.k.a. anti-electrons], and these behaved in a surprising way, as though they were produced in the decay of a previously unknown particle.

In the newly reported experiment, whose results just appeared, the same team observed  the disintegration of excited nuclei of helium.  They again found evidence for what they hope is the X17, and therefore claim confirmation of their original experiments on beryllium.

When two qualitatively different experiments claim the same thing, they are less likely to be wrong, because it’s not likely that any mistakes in the two experiments would create fake evidence of the same type.  On the face of it, it does seem unlikely that both measurements, carried out on two different nuclei, could fake an X17 particle.

However, we should remain cautious, because both experiments were carried out by the same scientists. They, of course, are hoping for their Nobel Prize (which, if their experiments are correct, they will surely win) and it’s possible they could suffer from unconscious bias. It’s very common for individual scientists to see what they want to see; scientists are human, and hidden biases can lead even the best scientists astray.  Only collectively, through the process of checking, reproducing, and using each other’s work, do scientists create trustworthy knowledge.

So it is prudent to await efforts by other groups of experimenters to search for this proposed X17 particle.  If the X17 is observed by other experiments, then we’ll become confident that it’s real. But we probably won’t know until then.  I don’t currently know whether the wait will be months or a few years.

Why I am so skeptical? There are two distinct reasons.

First, there’s a conceptual, mathematical issue. It’s not easy to construct reasonable equations that allow the X17 to co-exist with all of the known types of elementary particles. That it has a smaller mass than a proton is not a problem per se.  But the X17 needs to have some unique and odd properties in order to (1)  be seen in these experiments, yet (2) not be seen in certain other previous experiments, some of which were explicitly looking for something similar.   To make equations that are consistent with these properties requires some complicated and not entirely plausible trickery.  Is it impossible? No.  But a number of the methods that scientists suggested were flawed, and the ones that remain are, to my eye, a bit contrived.

Of course, physics is an experimental science, and what theorists like me think doesn’t, in the end, matter.  If the experiments are confirmed, theorists will accept the facts and try to understand why something that seems so strange might be true.  But we’ve learned an enormous amount from mathematical thinking about nature in the last century — for instance, it was math that told us that the Higgs particle couldn’t be heavier than 1000 protons, and it was on the basis of that `advice’ that the Large Hadron Collider was built to look for it (and it found it, in 2012.) Similar math led to the discoveries of the W and Z particles roughly where they were expected. So when the math tells you the X17 story doesn’t look good, it’s not reason enough for giving up, but it is reason for some pessimism.

Second, there are many cautionary tales in experimental physics. For instance, back in 2003 there were claims of evidence of a particle called a pentaquark with a rest mass about 1.6 times a proton’s mass — an exotic particle, made from quarks and gluons, that’s both like and unlike a proton.  Its existence was confirmed by multiple experimental groups!  Others, however, didn’t see it. It took several years for the community to come to the conclusion that this pentaquark, which looked quite promising initially, did not in fact exist.

The point is that mistakes do get made in particle hunts, sometimes in multiple experiments, and it can take some time to track them down. It’s far too early to talk about Nobel Prizes.

[Note that the Higgs boson’s discovery was accepted more quickly than most.  It was discovered simultaneously by two distinct experiments using two methods each, and confirmed by additional methods and in larger data sets soon thereafter.  Furthermore,  there were already straightforward equations that happily accommodated it, so it was much more plausible than the X17.] 

And just for fun, here’s a third reason I’m skeptical. It has to do with the number 17. I mean, come on, guys, seriously — 17 million electron volts? This just isn’t auspicious.  Back when I was a student, in the late 1980s and early 90s, there was a set of experiments, by a well-regarded experimentalist, which showed considerable evidence for an additional neutrino with a E=mc² energy of 17 thousand electron volts. Other experiments tried to find it, but couldn’t. Yet no one could find a mistake in the experimenter’s apparatus or technique, and he had good arguments that the competing experiments had their own problems. Well, after several years, the original experimenter discovered that there was a piece of his equipment which unexpectedly could absorb about 17 keV of energy, faking a neutrino signal. It was a very subtle problem, and most people didn’t fault him since no one else had thought of it either. But that was the end of the 17 keV neutrino, and with it went hundreds of research papers by both experimental and theoretical physicists, along with one scientist’s dreams of a place in history.

In short, history is cruel to most scientists who claim important discoveries, and teaches us to be skeptical and patient. If there is a fifth sixth force, we’ll know within a few years. Don’t expect to be sure anytime soon. The knowledge cycle in science runs much, much slower than the twittery news cycle, and that’s no accident; if you want to avoid serious errors that could confuse you for a long time to come, don’t rush to judgment.

Last Thursday, an experiment reported that the magnetic properties of the muon, the electron’s middleweight cousin, are a tiny bit different from what particle physics equations say they should be. All around the world, the headlines screamed: PHYSICS IS BROKEN!!! And indeed, it’s been pretty shocking to physicists everywhere. For instance, my equations are working erratically; many of the calculations I tried this weekend came out upside-down or backwards. Even worse, my stove froze my coffee instead of heating it, I just barely prevented my car from floating out of my garage into the trees, and my desk clock broke and spilled time all over the floor. What a mess!

—

Broken, eh? When we say a coffee machine or a computer is broken, it means it doesn’t work. It’s unavailable until it’s fixed. When a glass is broken, it’s shattered into pieces. We need a new one. I know it’s cute to say that so-and-so’s video “broke the internet.” But aren’t we going a little too far now? Nothing’s broken about physics; it works just as well today as it did a month ago.

More reasonable headlines have suggested that “the laws of physics have been broken”. That’s better; I know what it means to break a law. (Though the metaphor is imperfect, since if I were to break a state law, I’d be punished, whereas if an object were to break a fundamental law of physics, that law would have to be revised!) But as is true in the legal system, not all physics laws, and not all violations of law, are equally significant.

There’s a plot afoot. It’s a plot that involves a grid of earthquake locations, under the island of La Palma.

Conspiracy theory would be hysterically funny if it weren’t so widespread and so incredibly dangerous. Today it threatens democracy, human health, and world peace, among many other things. In the internet age, scientists and rational bloggers will have no choice but to take up arms against it on a regular basis.

The latest conspiracy theory involves the ongoing eruption of the Cumbre Vieja volcanic system on the island of La Palma. This eruption, unlike the recent one in Iceland, is no fun and no joke; it is occurring above a populated area. Over the past month, thousands of homes have been destroyed by incessant lava flows, and many more are threatened. The only good news is that, because the eruption is relatively predictable and not overly explosive, no one has yet been injured.

The source of the latest conspiracy theory is a graph of earthquakes associated with the eruption. You can check this yourself by going to www.emsc-csem.org and zooming in on the island of La Palma. You’ll see something like the plot below, which claims to show earthquake locations. You can see something is strange about it: the earthquakes are shown as occurring on a grid.

Clearly there’s something profoundly unnatural about this. That is exactly what thousands and thousands of people are concluding around the world. They are absolutely correct. There’s no way this could be natural.

When faced with something unnatural like this, there are two possible conclusions that a human can draw.

1. The earthquakes are natural, but their positions appear on a grid because of something unnatural about the way the data is plotted.
2. The data is plotted correctly, and the earthquakes really are happening on a grid — which suggests that the earthquakes can’t be made by nature, and must be human-made.

Now, when faced with these two options, what does a reasonable person guess is more likely? Option 2 requires a spectacular technology that can set off huge explosions five to twenty miles (10-30 km) underground without anyone noticing, by a group of people who are evil enough to want to set off earthquakes miles underground and clever enough to keep their super-high-tech methods secret, but dumb enough to set off the earthquakes in a grid so that a simple look at the earthquake’s locations by non-experts perusing the internet reveals their dastardly plot. Option 1 requires a tiny amount of human error or computer error.

No research is needed to conclude Option 1 is more plausible, but five minutes’ research confirms it’s true. First, other websites plotting the same earthquakes do not show the grid pattern. Second, as this video pointed out and as you yourself can check, the same website, plotting earthquakes in other locations such as Hawaii, again shows the grid pattern — so it’s a fact of the emsc website, not of the La Palma earthquakes. Third, as pointed out by the excellent Volcano Discovery website, looking at the actual data that the emsc website uses, one sees that the latitude and longitude are rounded off to the nearest 1/100th, and thus north-south and east-west locations on the map are rounded off to (roughly) the nearest kilometer. This “rounding off” moves each earthquake location to the nearest point on a grid. That’s the cause. No conspiracy, no magical technology, just a plotting issue. There’s nothing more here than nature doing its thing: making earthquakes, just as it does with every volcanic eruption on Earth.

This effect, where writing numbers to a particular choice of significant figures leads to a plot with a grid pattern, is well known to every scientist. Here’s an example of how it works. Below are thirty points chosen at random in a small region, shown at left. I plotted them using a wide range at the top, and then zoomed in to make the lower plot.

Next, the points are rounded to one significant figure after the decimal point, using the same methods we are all taught in school, and the points are replotted. Instant grid.

In short, we’re not looking at a plot to destroy La Palma and set off a tsunami. We’re looking at a plot of rounded-off locations. I agree that’s not nearly as exciting; but as any scientist with some experience will tell you, boring explanations are usually true and conspiracies, especially wild ones, are usually not.

What’s the point of this post? Well, aside from being a source that you can send to any friends, relatives or acquaintances who are falling for this ridiculous conspiracy theory, it’s an apolitical context in which to contemplate the real problem.

The real problem is that we face an increasing flood of half-reasoned badly-researched pseudo-science, combined with irrational knee-jerk conspiratorialism, the whole thing driven by an unholy mixture of fear, maliciousness, narcissism and greed. It’s a war between calm reason and emotional darkness, a war in which people are actually dying, and in which nations are actually at risk. At this rate, the voices of rationality may soon be drowned. So perhaps we might consider this question: how can an apolitical conspiracy such as this one be used as an example, one from which we can learn lessons that we can apply more broadly, in territory that’s much more complex and dangerous?

—

p.s., predictably, someone questioned whether the statement about Hawaiian earthquakes on the emsc website appearing on a grid was true. Well, here’s the plot below — the earthquakes aren’t so many, so the grid isn’t full, but you can see every plotted earthquake lies on a grid point. And the same is true for earthquakes on the island of Crete, as shown in the second plot. All of it data that has been rounded off to the nearest 1/100th of latitude and longitude.

 Since a few days, I am following the hype and play wordle. I think I got lucky the first days but I had already put in some strategy as in starting with words where the possible results are most telling. I was thinking that getting the vowels right early is a good idea so I tend to start with "HOUSE" (continuing three vowels and an S) possibly followed by "FAINT" (containing the remaining vowels plus important N and T).

With this start it never took me more than four guesses so far and twice I managed to find the solution in three guesses.

Of course, over time you start thinking how to optimise this. I knew that Donald Knuth had written a paper solving the original Mastermind showing that five moves are sufficient to always find the answer. So today, I sat down and wrote a perl script to help. It does not do the full minimax (but that shouldn't be too hard from where I am) but at least tells you which of your possible next guesses leaves the best worst case in terms of number of remaining words after knowing the result of your guess. 

In that metric, it turns out "ARISE" is the optional first guess (leaving at most 168 out of the possible 2314 words on this list after knowing the result). In any case, here is the source: 

NB: Since i started playing, there was no word that contained the same letter more than once, so I am not 100% sure how those cases are handled (like what color do the two 'E' in "AGREE" receive if the solution is "AISLE" (in mastermind logic, the second would be green the other grey, not yellow) and what when the solution were "EARLY"? So my script does not handle those cases correct probably (for EARLY it would color both yellow).

#!/usr/local/bin/perl -w

use strict;

# Load the word list of possible answers
my @words = ();
open (IN, "answers.txt") || die "Cannot open answers: $!\n";
while(<IN>) {
  chomp;
  push @words, uc($_);
}
close IN;

my %letters = ();
my @appears = ();

# Positions at which letter $l can still appear
foreach my $c (0..25)  {
  my $l = chr(65 + $c);
  $letters{$l} = [1,1,1,1,1];
}


# Running without an initial guess shows that ARISE is the best guess at it leaves 168 words.

&filter("ARISE", &bewerten("ARISE", "SOLAR"));
#&filter("SMART", &bewerten("SMART", "SOLAR"));

# Find the remaining words
my @remain = @words;
# Only keep words containing the letters in @appeads
foreach my $a(@appears) {
  @remain = grep {/$a/} @remain;
}
my $re = &makeregex;

# Apply positional constraints
@remain = grep {/$re/} @remain;


my $min = @remain;
my $best = '';

# Loop over all possible guesses and targets and count how ofter a potential result appears for a guess
foreach my $g(@remain) {
  my %results = ();
  foreach my $t(@remain) {
    ++$results{&bewerten($g, $t)}
  }
  my $max = 0;
  foreach my $res(keys %results) {
    $max = $results{$res} if $results{$res} > $max;
  }
  #print "$g leaves at most $max.\n";
  if ($min > $max) {
    $min = $max;
    $best = $g;
  }
}

print "Best guess: $best leaves at most $min.\n";

# Assemble a regex for the postional informatiokn
sub makeregex {
  my $rem = '';
  foreach my $p (0..4) {
    $rem .= '[';
    foreach my $l (sort keys %letters) {
      $rem .= $l if $letters{$l}->[$p];
    }
    $rem .= ']';
  }
  return $rem;
}

# Find new constraints arising from the result of a guess
sub filter {
  my ($guess, $result) = @_;

  my @a = split //, $result;
  my @w = split //, uc($guess);
  foreach my $p (0..4) {
    my $l = $w[$p];
    if ($a[$p] == 0) {
      $letters{$l} = [0,0,0,0,0];
    } elsif ($a[$p] == 1) {
      &setletter($l, $p, 0);
      push @appears, $l;
    } else {
      foreach my $o (sort keys %letters) {
	&setletter($o, $p, 0);
      }
      &setletter($l, $p, 1);
    }
  }
}

# Update the positional information for letter $l at position $p with value $v
sub setletter {
  my ($l, $p, $v) = @_;
  my @a = @{$letters{$l}};
  $a[$p] = $v;
  $letters{$l} = \@a;
}

# Find the result for $guess given the $target
sub bewerten {
  my ($guess, $target) = @_;
  my @g = split //, $guess;
  my @t = split //, $target;

  my @result = (0,0,0,0,0);
  foreach my $p(0..4) {
    if($g[$p] eq $t[$p]) {
      $result[$p] = 2;
      $t[$p] = '';
      $g[$p] = 'x';
    }
  }
  $target = join('', @t);
  foreach my $p(0..4) {
    if($target =~ /$g[$p]/) {
      $result[$p] = 1;
    }
  }
  return join('', @result);
}

I took a short break from other projects this weekend. Poking around, I found three particularly lovely science stories, which I thought some readers might enjoy as well. They all involve mapping, but the distances involved are amazingly different.

Starbirth Near to Home

The first one has been widely covered in the media (although a lot of the articles were confused as to what is new news and what is old news.) Our galaxy is full of stars, but also of gas (mainly hydrogen and helium) and dust (tiny grains of material made mainly from heavier elements forged in stars, such as silicon). That gas and dust, referred to as the “interstellar medium”, is by no means uniform; it is particularly thin in certain regions which are roughly in the shape of bubbles. These bubbles were presumably “blown” by the force of large stellar explosions, i.e. supernovas, whose blast waves cleared out the gas and dust nearby.

It’s been known for several decades that the Sun sits near the middle of such a bubble. The bubble and the Sun are moving relative to one another, so the Sun’s probably only been inside the bubble for a few million years; since we’re just passing through, it’s an accident that right now we’re near its center. Called simply the “Local Bubble”, it’s an irregularly shaped region where the density of gas and dust is 1% of its average across the galaxy. If you orient the galaxy in your mind so that its disk, where most of the stars lie, is horizontal, then the bubble stretches several hundred light-years across in the horizontal direction, and is elongated vertically. [For scale: a light-second is 186,000 miles or 300,000 km; the Sun is 8 light-minutes from Earth; the next-nearest star is 4 light-years away; and our Milky Way galaxy is about 100,000 light years across.] It’s been thought for some time that this bubble was created some ten to twenty million years ago by the explosion of one or more stars, probably siblings that were born close by in time and in space, and which at their deaths were hundreds of light-years from the Sun, far enough away to do no harm to Earthly life. [For scale: recall the Sun and Earth are about 5 billion years old.]

Meanwhile, it’s long been suggested that explosion debris from such supernovas can sweep up gas and dust like a snowplow, and that the compression of the gas can lead it to start forming stars. It’s a beautiful story; large stars live fast and hot, and die young, but perhaps as they expire they create the conditions for the next generation. [A star with 40 times the raw material as the Sun has will burn so hot that it will last only a million years, and most such stars die by explosion, unlike Sun-like stars.]

If this is true, then in the region near the Sun, most if not all the gas clouds where stars are currently forming should lie on the current edge of this bubble, and moreover, all relatively young stars, less than 10 million years old, should have formed on the past edge of the bubble. Unfortunately, this has been hard to prove, because measuring the locations, motions and ages of all the stars and clouds isn’t easy. But the extraordinary Gaia satellite has made this possible. Using Gaia’s data as well as other observations, a team of researchers (Catherine Zucker, Alyssa A. Goodman, João Alves, Shmuel Bialy, Michael Foley, Joshua S. Speagle, Josefa Groβschedl, Douglas P. Finkbeiner, Andreas Burkert, Diana Khimey & Cameren Swiggum) has claimed here that indeed the star-forming regions lying within a few hundred light-years of the Sun all lie on the Local Bubble’s surface, and that nearby stars younger than ten million years were born on the then-smaller shell of the expanding bubble. Moreover they claim that the bubble probably formed 14-15 million years ago, at a time when the Sun was about 500 light-years distant.

The one exception to this rule appears to be a star-forming region on the backside of a second bubble, adjoining to the Local Bubble. The discovery of this “Per-Tau superbubble,” and the fact that two star-forming regions lie on opposite sides of it, was announced by the (nearly) same team just a few months ago; in fact this earlier paper gives the first direct evidence of the explosion-snowplow-to-star-formation effect. (You can see a 3d image of that adjoining bubble, created by the research team, here. and you can see its relation to the Local Bubble in the right-hand-side of this image, where the Local Bubble is marked in purple and the adjoining “Per-Tau” bubble is marked in green. Other visuals and captions are at this site.) The supernova or supernovas that may have created this bubble also probably went off between 5 and 20 million years ago, based on the size of the bubble and the slow motion of the bubble walls.

That stars can reproduce by generating new ones from the debris of the old is reminiscent of a similar effect in thunderstorms; when a thunderstorm complex is mature and loses its powerful updrafts, its cold air drops downward and then flows outward, and the resulting outflow can generate new storms. (See this blog post for some detailed discussion.) Perhaps there are other examples of natural engines generating offspring in a similar way; do you know of any? I have long wondered whether life could have gotten its start through a mechanism like this one, a rudimentary form of imperfect physical reproduction on which evolution might then have begun to act.

Galaxies Across the Cosmos

Meanwhile the Dark Energy Spectroscopic Instrument (DESI), seeking to understand the history of the universe with precision, has been engaged in a similar mission to Gaia but on a different scale and with different methods. It is mapping distances not to nearby stars in our own galaxy but to other galaxies, covering a very substantial fraction of the visible universe. Its data has now allowed the creation of the largest ever 3D map of a region of the cosmos, extending across part of the sky out to distances reaching five billion light years from Earth. (I’m particularly fond of this type of map, because my first scientific paper was an attempt, led by Jerry Ostriker, to make sense of the structures seen in the first 2D slice map of the universe, made by Valerie de Lapparent, Margaret Geller and John Huchra, scarcely reaching out 200 million light-years.) The 3D map, seen as a movie that sweeps through 2D slices of the map, can be found here. Each dot on the map is a galaxy; our own galaxy is located at the lower left corner. Within the map (ignoring dark wedges which presumably arise from regions that are blocked from view by nearby objects in the sky) you can see emptier regions threaded by dense filaments and other features.

And DESI has much more to tell us; it has measured far more galaxies than appear in this map; it has measured the distance to many galaxies extending out to 10 billion light years, most of the distance across the visible universe; and it’s just getting started, with many more years of mapping left to do. By the way, the technology behind DESI is pretty amazing; I leave it to you to read about it on the DESI website.

The Ocean Deep

Finally I bring us closer to home, to another frontier where humans still have remote mapping to do: the bottom of the sea. The story begins with partial failure in an experiment to explore the deepest region of the ocean, the Challenger Deep, which extends below sea level 25% further than Mount Everest extends above it. In 2014 a research team put two thick glass spheres full of scientific instruments into the water above the Challenger Deep, expecting one to reach the very bottom and the other to remain just above it. Unfortunately the first one must have had a small but fatal flaw in its construction, because when still roughly 2 km (1.2 miles) above the sea floor, it imploded under the immense pressure. One can only begin to imagine the disappointment of those who built the instrument and were expecting reams of data from it. At least the other sphere survived.

But now comes a story of scientists at their most creative, making lemonade from this lemon. The implosion that destroyed the first instrument created a shock wave which reverberated all across the ocean floor, back to the ocean surface, yet again back to the floor, and so on; and those reverberations were detected by the surviving instrument. By carefully investigating all these echoes from the disaster, a team of three scientists has recently claimed to have obtained the most precise measurement ever of the depth of the Challenger Deep: 10,983 meters (36033 feet) with an uncertainty of only 6 meters (20 feet) up or down, one fourth the uncertainty of any previous measurement. While I’m in no position to evaluate the validity of this claim, it represents an inspiring effort to make the very best of bad news, and to treat an experiment’s failure as an experiment in its own right.

Now if someone could just boldly go and map the internet…

 It's a very busy time, so no lengthy content, but here are a few neat things I came across this week.

•  A new PRL came out this week that seems to have a possible* analytic solution to Hilbert's 18th problem, about the density of random close-packed spheres in 2D and 3D.  This is a physics problem because it's closely related to the idea of jamming and the onset of mechanical rigidity of a collection of solid objects.  (*I say possible only because I don't know any details about any subtle constraints in the statement of the problem.)
• The Kasevich group at Stanford has an atom interferometric experiment that they claim is a gravitational analog of the Aharonov-Bohm effect.  This is a cool experiment, where there is a shift in the quantum phase of propagating atomic clouds due to the local gravitational potential caused by a nearby massive object.  (Phase goes like the argument of \(\exp(-i S(x(t))/\hbar)\), where the action can include a term related to the gravitational potential, \(m \times \Phi_{G}(x(t)\).)  At a quick read, though, I don't see how this is really analogous to the AB effect.  In the AB case, there is a relative phase due to magnetic flux enclosed by the interfering paths even when the magnetic field is arbitrarily small at the actual location of the path.  I need to read this more closely, or perhaps someone can explain in the comments.
• A colleague pointed out to me this great review article all about charge shot noise in mesoscopic electronic systems.  
• Speaking of gravity, there has been interest in recent years about "warp drives", geometries of space-time allowed by general relativity that seem to permit superluminal travel for an observer in some particular region.  One main objection to these has been that past proposed incarnations violate various energy conditions in GR - requiring enormous quantities "negative matter", for example, which does not seem to exist.  Interestingly, people have been working on normal-matter-only ideas for these, and making some progress as in this preprint.  Exercises like this can be really important for illuminating subtle issues with GR, just like worrying about "fast light" experiments can make us refine arguments about causality and signaling.  
• Thomas Wong from Creighton University has a free textbook (link on that page) to teach about quantum computing, where the assumed starting math knowledge is trig.  It looks very accessible!
• People recommended two other books to me recently that I have not yet had time to read.  The Alchemy of Us is a materials-and-people focused book from Ainissa Ramirez, and Sticky: The Secret Science of Surfaces is all about surfaces and friction, by Laurie Winkless.  Gotta make time for these once the semester craziness is better in hand....

Lily Zhao (Flatiron) has been looking at the variability of the SDSS-IV BOSS spectrograph calibration frames, in the thought that if we can understand them well enough, we might be able to substantially reduce the observing overheads for the robotic phase of SDSS-V. To start, we are doing simple cross-correlations between calibration images. After all, the dominant calibration changes from exposure to exposure appear to be shifts. But these cross-correlations have weird features in them that I don't understand: In general if you are cross-correlating two images that are very similar, the cross-correlation function should look very symmetric, I think? But Zhao's cross-correlation functions look asymmetric in weird ways. We ended our session puzzled.

As the saying goes, it is better not to see laws or sausages being made. You’d prefer to see the clean package on the outside than the mess behind the scenes.

The same is true of science. A good paper tells a nice, clean story: a logical argument from beginning to end, with no extra baggage to slow it down. That story isn’t a lie: for any decent paper in theoretical physics, the conclusions will follow from the premises. Most of the time, though, it isn’t how the physicist actually did it.

The way we actually make discoveries is messy. It involves looking for inspiration in all the wrong places: pieces of old computer code and old problems, trying to reproduce this or that calculation with this or that method. In the end, once we find something interesting enough, we can reconstruct a clearer, cleaner, story, something actually fit to publish. We hide the original mess partly for career reasons (easier to get hired if you tell a clean, heroic story), partly to be understood (a paper that embraced the mess of discovery would be a mess to read), and partly just due to that deep human instinct to not let others see us that way.

The trouble is, some of that “mess” is useful, even essential. And because it’s never published or put into textbooks, the only way to learn it is word of mouth.

A lot of these messy tricks involve numerics. Many theoretical physics papers derive things analytically, writing out equations in symbols. It’s easy to make a mistake in that kind of calculation, either writing something wrong on paper or as a bug in computer code. To correct mistakes, many things are checked numerically: we plug in numbers to make sure everything still works. Sometimes this means using an approximation, trying to make sure two things cancel to some large enough number of decimal places. Sometimes instead it’s exact: we plug in prime numbers, and can much more easily see if two things are equal, or if something is rational or contains a square root. Sometimes numerics aren’t just used to check something, but to find a solution: exploring many options in an easier numerical calculation, finding one that works, and doing it again analytically.

“Ansatze” are also common: our fancy word for an educated guess. These we sometimes admit, when they’re at the core of a new scientific idea. But the more minor examples go un-mentioned. If a paper shows a nice clean formula and proves it’s correct, but doesn’t explain how the authors got it…probably, they used an ansatz. This trick can go hand-in-hand with numerics as well: make a guess, check it matches the right numbers, then try to see why it’s true.

The messy tricks can also involve the code itself. In my field we often use “computer algebra” systems, programs to do our calculations for us. These systems are programming languages in their own right, and we need to write computer code for them. That code gets passed around informally, but almost never standardized. Mathematical concepts that come up again and again can be implemented very differently by different people, some much more efficiently than others.

I don’t think it’s unreasonable that we leave “the mess” out of our papers. They would certainly be hard to understand otherwise! But it’s a shame we don’t publish our dirty tricks somewhere, even in special “dirty tricks” papers. Students often start out assuming everything is done the clean way, and start doubting themselves when they notice it’s much too slow to make progress. Learning the tricks is a big part of learning to be a physicist. We should find a better way to teach them.

I spent much of the day understanding Other People's Code (tm). One piece of code is legacy code from Aaron Dotter (Harvard) that computes magnitudes from flux densities. I was looking at it to confirm my frantic writing of yesterday. Another piece of code is the code by Weichi Yao (NYU) that implemented our group-equivariant machine learning. I was looking at it to see if we can modify it to impose units equivariance (dimensional scaling symmetries). I think we can, and I confirmed that it Just Works (tm).

A couple of weeks ago I had intense arguments with Belokurov (Cambridge) and Farr (Stony Brook) about the definition of a photometric bandpass and a photometric magnitude. And a few months ago I had long conversations with Breivik (Flatiron) about the bolometric correction. For some reason these things took over my mind today and I couldn't stop myself from starting a short pedagogical note about how magnitudes are related to spectral energy distributions. It's not trivial! Indeed, the integral isn't the integral you naively think it might be, because most photometric systems count photons (they don't integrate energy). People often say that a magnitude is negative-2.5 times the log of a flux. But that's not right! It is negative-2.5 times the log of a ratio of signals measured in two experiments.

(Hopefully no one has taken taken that title yet!)

I waste a large fraction of my existence just reading about what’s happening in the world, or discussion and analysis thereof, in an unending scroll of paralysis and depression. On the first anniversary of the January 6 attack, I read the recent revelations about just how close the seditionists actually came to overturning the election outcome (e.g., by pressuring just one Republican state legislature to “decertify” its electors, after which the others would likely follow in a domino effect), and how hard it now is to see a path by which democracy in the United States will survive beyond 2024. Or I read about Joe Manchin, who’s already entered the annals of history as the man who could’ve halted the slide to the abyss and decided not to. Of course, I also read about the wokeists, who correctly see the swing of civilization getting pushed terrifyingly far out of equilibrium to the right, so their solution is to push the swing terrifyingly far out of equilibrium to the left, and then they act shocked when their own action, having added all this potential energy to the swing, causes it to swing back even further to the right, as swings tend to do. (And also there’s a global pandemic killing millions, and the correct response to it—to authorize and distribute new vaccines as quickly as the virus mutates—is completely outside the Overton Window between Obey the Experts and Disobey the Experts, advocated by no one but a few nerds. When I first wrote this post, I forgot all about the global pandemic.) And I see all this and I am powerless to stop it.

In such a dark time, it’s easy to forget that I’m a theoretical computer scientist, mainly focused on quantum computing. It’s easy to forget that people come to this blog because they want to read about quantum computing. It’s like, who gives a crap about that anymore? What doth it profit a man, if he gaineth a few thousand fault-tolerant qubits with which to calculateth chemical reaction rates or discrete logarithms, and he loseth civilization?

Nevertheless, in the rest of this post I’m going to share some quantum-related debunking updates—not because that’s what’s at the top of my mind, but in an attempt to find my way back to sanity. Picture that: quantum mechanics (and specifically, the refutation of outlandish claims related to quantum mechanics) as the part of one’s life that’s comforting, normal, and sane.

There’s been lots of online debate about the claim to have entangled a tardigrade (i.e., water bear) with a superconducting qubit; see also this paper by Vlatko Vedral, this from CNET, this from Ben Brubaker on Twitter. So, do we now have Schrödinger’s Tardigrade: a living, “macroscopic” organism maintained coherently in a quantum superposition of two states? How could such a thing be possible with the technology of the early 21^st century? Hasn’t it been a huge challenge to demonstrate even Schrödinger’s Virus or Schrödinger’s Bacterium? So then how did this experiment leapfrog (or leaptardigrade) over those vastly easier goals?

Short answer: it didn’t. The experimenters couldn’t directly measure the degree of freedom in the tardigrade that’s claimed to be entangled with the qubit. But it’s consistent with everything they report that whatever entanglement is there, it’s between the superconducting qubit and a microscopic part of the tardigrade. It’s also consistent with everything they report that there’s no entanglement at all between the qubit and any part of the tardigrade, just boring classical correlation. (Or rather that, if there’s “entanglement,” then it’s the Everett kind, involving not merely the qubit and the tardigrade but the whole environment—the same as we’d get by just measuring the qubit!) Further work would be needed to distinguish these possibilities. In any case, it’s of course cool that they were able to cool a tardigrade to near absolute zero and then revive it afterwards.

I thank the authors of the tardigrade paper, who clarified a few of these points in correspondence with me. Obviously the comments section is open for whatever I’ve misunderstood.

People also asked me to respond to Sabine Hossenfelder’s recent video about superdeterminism, a theory that holds that quantum entanglement doesn’t actually exist, but the universe’s initial conditions were fine-tuned to stop us from choosing to measure qubits in ways that would make its nonexistence apparent: even when we think we’re applying the right measurements, we’re not, because the initial conditions messed with our brains or our computers’ random number generators. (See, I tried to be as non-prejudicial as possible in that summary, and it still came out sounding like a parody. Sorry!)

Sabine sets up the usual dichotomy that people argue against superdeterminism only because they’re attached to a belief in free will. She rejects Bell’s statistical independence assumption, which she sees as a mere dogma rather than a prerequisite for doing science. Toward the end of the video, Sabine mentions the objection that, without statistical independence, a demon could destroy any randomized controlled trial, by tampering with the random number generator that decides who’s in the control group and who isn’t. But she then reassures the viewer that it’s no problem: superdeterministic conspiracies will only appear when quantum mechanics would’ve predicted a Bell inequality violation or the like. Crucially, she never explains the mechanism by which superdeterminism, once allowed into the universe (including into macroscopic devices like computers and random number generators), will stay confined to reproducing the specific predictions that quantum mechanics already told us were true, rather than enabling ESP or telepathy or other mischief. This is stipulated, never explained or derived.

To say I’m not a fan of superdeterminism would be a super-understatement. And yet, nothing I’ve written previously on this blog—about superdeterminism’s gobsmacking lack of explanatory power, or about how trivial it would be to cook up a superdeterministic “mechanism” for, e.g., faster-than-light signaling—none of it seems to have made a dent. It’s all come across as obvious to the majority of physicists and computer scientists who think as I do, and it’s all fallen on deaf ears to superdeterminism’s fans.

So in desperation, let me now try another tack: going meta. It strikes me that no one who saw quantum mechanics as a profound clue about the nature of reality could ever, in a trillion years, think that superdeterminism looked like a promising route forward given our current knowledge. The only way you could think that, it seems to me, is if you saw quantum mechanics as an anti-clue: a red herring, actively misleading us about how the world really is. To be a superdeterminist is to say:

OK, fine, there’s the Bell experiment, which looks like Nature screaming the reality of ‘genuine indeterminism, as predicted by QM,’ louder than you might’ve thought it even logically possible for that to be screamed. But don’t listen to Nature, listen to us! If you just drop what you thought were foundational assumptions of science, we can explain this away! Not explain it, of course, but explain it away. What more could you ask from us?

Here’s my challenge to the superdeterminists: when, in 400 years from Galileo to the present, has such a gambit ever worked? Maxwell’s equations were a clue to special relativity. The Hamiltonian and Lagrangian formulations of classical mechanics were clues to quantum mechanics. When has a great theory in physics ever been grudgingly accommodated by its successor theory in a horrifyingly ad-hoc way, rather than gloriously explained and derived?

Update: Oh right, and the QIP’2022 list of accepted talks is out! And I was on the program committee! And they’re still planning to hold QIP in person, in March at Caltech, will you fancy that! actually I have no idea—but if they’re going to move to virtual, I’m awaiting an announcement just like everyone else.

I got stuck today on a problem that seems trivial but is in fact not at all trivial: Given N inputs, how to make all possible dimensionless monomials of those N inputs, at (or less than) degree d. For our purposes (which are the units-equivariant machine-learning projects I have with Soledad Villar, JHU), a dimensionless monomial of degree d is a product of integer powers of the inputs, in which those powers can be positive or negative, such that the dimensions cancel out completely, and for which the the max (or sum or some norm) of the absolute values of the powers is &leqd. We have a complete basis of dimensionless monomials, such that any valid dimensionless monomial can be expressed as a monomial of the basis monomials. Because of this, the dimensionless monomials can be seen as the vertices of a Bravais lattice, technically. The problem is just to traverse the entire lattice within some kind of ball. Why is this hard? Or am I just dull? I feel like there are fill algorithms that should do this correctly.

Christina Eilers (MIT) and I discussed our referee report today, on our paper on re-calibrating abundance ratios as measured by APOGEE to remove log-g-dependent systematics. The referee report came quickly! And it was very useful: The referee found an assumption that we are making that we had not explicitly stated in the paper. And this is important: As my loyal reader knows, I believe that a data-analysis paper is correct only insofar as it is consistent with its explicitly stated (and hopefully tested and justified) assumptions. So if a paper is missing an assumption, it is wrong!

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It’s always nice to wake up to good news, and it is not necessarily a common experience as we reach the end of the second year of a pandemic. But for astronomers, two decades of anxiety were laid to rest over the weekend as the last mirror segments of the James Webb Space Telescope, or JWST, were locked into place.

Part of me always responds a little cynically when NASA’s public relations machine hypes the difficulty of forthcoming space-based feats, whether it is the 7 Minutes of Terror as a rover lands on Mars or the 344 single points of failure that could each spell doom for this wonderful telescope. There is some butt-covering in there if things do go wrong, but it is also the patter of an old-school magician as he prepares to saw his assistant in half. NASA is, at heart, a bureaucracy and does not set out to do things it does not believe it can accomplish. 

But things do go wrong in space. The probe that drilled a hundred million hole in the side of Mars because metric and imperial units were accidentally combined when computing its trajectory. Or the Hubble telescope itself, whose mirror was mis-configured in a way that an old-school lens-grinder polishing a glass blank would be embarrassed to reveal. And rockets do sometimes fail. So there is always room for worry. 

In fact, many failures result from organisational complexity – miscommunications that plant the seeds for a catastrophe which only become apparent when everything is fitted together and hurled into the heavens. And the JWST had a remarkably baroque bureaucratic journey.  Its origin is often traced to a 1989 workshop on successors to the Hubble Space Telescope – held before Hubble itself was in space. The green light for NASA to build it followed a decade later, with a US$1.6 billion budget and 10 year roadmap to launch. 

By 2010 it was clear that this budget and timeline had been somewhere between a shared, consensual hallucination and an active untruth. Launch was still years away and the budget had swelled to something like $10 billion; at one point the joke was the JWST slipped a year every year. Cynics argued that this reflected a belief that it is easier to ask for forgiveness than permission, trusting Congress to chase sunk-cost rather than scuttling a program with billions already spent. 

The telescope was the subject of Congressional hearings and became something close to a public scandal, suffering a “near death experience” when its future budget was briefly erased during the wrangling on Capitol Hill. Likewise, the management of the JWST itself was the subject of a blunt report that drove changes in its leadership and process. 

The following decade saw the project stay (mostly) inside the new budget cap but more schedule slippages during testing meant that it finally made it to space on Christmas Day 2021. By then, the project’s many travails had ratcheted up the anxiety-levels of everyone in the field to the point that you could almost hear your colleagues gently buzzing from stress. 

So it was a massive relief and perhaps something of a surprise that a pitch-perfect Christmas-day flight was followed by an almost flawless deployment of the massive sunshade and the unfolding of the telescope mirror.  The next few months will be spent aligning the individual segments of the mirror and cooling the instruments to their operating temperature, but the JWST now seems to promise two decade’s worth of unprecedented observations of the universe. 

So I’m happy.

Not just happy in fact, as watching a group of people come together to do something this hard and this unprecedentedly complex – often in the face of administrative inertia – reminds me that it is not only possible to reach for our dreams but that sometimes we manage to take hold of them.

I still have a lot of text files from when I was in college and even high school, sequentially copied from floppy to floppy to hard drive to hard drive over the decades. I used to write poems and they were not good and neither is this one, but to my surprise it had some lines in it that I remembered but did not remember that I wrote myself. What was I doing with the line breaks though? I am pretty sure this would have been written in my junior year of college, maybe spring of 1992. Around this same time I submitted a short story to a magazine and the editor wrote back to me saying “free-floating anxiety cannot be what drives a narrative,” but I disagreed, obviously.

To a Crackpot

He eschews the shoulders
of giants. He chooses instead
the company of thin men, coffee-stained,
stooped with knowledge. They huddle
on the sidewalk, nodding, like crows
or rabbis. He speaks:
the world is hollow and we live
on the inside. (Murmurs of assent.) There
is a hole at the top where the water runs in. The sun
is smaller than my hand, and the stars
are smaller than the sun.

A woman walks by, drawing
his eye. She has no idea. Beneath their feet,
out in the dark, secret engines. The Earth turns like milk.

I had an interesting conversation with a colleague last week about the challenges of writing a broadly appealing, popular book about condensed matter.  This is a topic I've been mulling for (too many) years - see this post from the heady days of 2010.

He made a case that condensed matter is inherently less wondrous to the typical science-interested person than, e.g., "the God Particle" (blech) or black holes.  This is basically my first point in the old post linked above.  He was arguing that people have a hard time ever seeing something that captures the imagination in items or objects that they have around them all the time.  The smartphone is an incredible piece of technology and physics, but what people care about is how to get better download speeds, not how or why any of it works.  

I'm curious:  Do readers think this is on-target?  Is "lack of wonder" the main issue, or one of many?  

I’m teaching Quantum Field Theory this year. One of the things I’ve been trying to emphasize is the usefulness of spinor-helicity variables in dealing with massless particles. This is well-known to the “Amplitudes” crowd, but hasn’t really trickled down to the textbooks yet. Mark Srednicki’s book comes close, but doesn’t (IMHO) quite do a satisfactory job of it.

Herewith are some notes.

The first step in constructing perturbation theory is to quantize the free fields. Following Weinberg and Srednicki, I’m using the “mostly-plus” signature convention (my 2-component spinor conventions are those of Dreiner et al if you define the macro \def\signofmetric{1} in the LaTeX file). For $k^2=0$, we can define helicity spinors

which allow us to straightforwardly canonically-quantize.

Spin-1/2

For a Weyl fermion, $$\mathcal{L}= i\psi^\dagger\overline{\sigma}\cdot\partial \psi$$ the general solution to the equations of motion is $$\begin{aligned} \psi_\alpha(x)&=\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\lambda_\alpha \left(\xi^\dagger_{\vec{k}}e^{-ik\cdot x}+\eta_{\vec{k}}e^{ik\cdot x}\right)\\ \psi^\dagger_{\dot\alpha}(x)&=\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\lambda^\dagger_{\dot\alpha}\left(\eta^\dagger_{\vec{k}}e^{-ik\cdot x}+\xi_{\vec{k}}e^{ik\cdot x}\right) \end{aligned}$$ The Equal-Time Anti-Commutation Relations $$\{\psi_\alpha(\vec{x},0),\psi^\dagger_{\dot\beta}(\vec{x}',0)\}=\sigma^0_{\alpha\dot\beta}\delta^{(3)}(\vec{x}-\vec{x}')$$ become the canonical anti-commutation relations $$\begin{aligned} \{\xi_{\vec{k}},\xi^\dagger_{\vec{k}'}\}&= {(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \{\eta_{\vec{k}},\eta^\dagger_{\vec{k}'}\}&= {(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \end{aligned}$$ for creation and annihilation operators for fermions of definite helicity.

The upshot, after tracking this through the LSZ reduction formula, is that external fermion lines are contracted with the corresponding helicity spinor ($\lambda_i$ or $\lambda^\dagger_i$) depending on the helicity of the $i^{\text{th}}$ incoming/outgoing particle. When we take the absolute square of the amplitude, we use (1) to rewrite $\lambda_i\lambda^\dagger_i=-k_i\cdot\sigma$, etc.

Spin-1

There’s a certain amount of hand-wringing associated to quantizing the free Maxwell Lagrangian, $$\mathcal{L} = -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ If we take the canonical variables to be $A^\mu$ and $\pi_\mu =\frac{\delta \mathcal{L}}{\delta\partial_0 A^\mu}$, then the gauge-invariance entails that the symplectic structure is degenerate ($\pi_0$ vanishes identically). The usual approach is to fix a gauge (Weinberg and Srednicki use Coulomb gauge) and then work very hard (replacing Poisson brackets with Dirac brackets, because the constraints are 2^nd class, …).

On the other hand, if we

1. realize that the phase space is the space of classical solutions and
2. introduce spinor helicity variables, as before,

it’s easy to write down the general solution to the equations of motion

The (non-degenerate) symplectic structure on the space of classical solutions leads to the Equal-Time Commutation Relations

which, in turn, give the canonical commutation relations $$\begin{aligned} [\varepsilon_+(\vec{k}),\varepsilon^\dagger_+(\vec{k}')]&={(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ [\varepsilon_-(\vec{k}),\varepsilon^\dagger_-(\vec{k}')]&={(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \end{aligned}$$ of the creation and annihilation operators for photons of definite helicity.

Unfortunately, to couple to charged matter fields, we need an expression for $A^\mu$, not just $F^{\mu\nu}$, so (2) does not quite suffice for our purposes. But, again, helicity spinors come to the rescue.

Introduce a fixed fiducial null vector $\check{k}^2=0$ and the corresponding helicity spinors $$(\check{k}\cdot\sigma)_{\alpha\dot\beta}= -\mu_\alpha\mu^\dagger_{\dot\beta},\qquad (\check{k}\cdot\overline{\sigma})^{\dot\alpha\beta} = -\mu^{\dagger\dot\alpha}\mu^\beta$$ We then can write

which satisfies $\partial\cdot A=0$ and (exercise for the reader) $$\begin{aligned} \partial^\mu A^\nu-\partial^\nu A^\mu&= \frac{1}{\sqrt{2}}\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon^\dagger_{-}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon^\dagger_{+}(\vec{k})\right)e^{-ik\cdot x}+\\ &\qquad+\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon_{+}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon_{-}(\vec{k})\right)e^{ik\cdot x}\\ &=F^{\mu\nu}(x) \end{aligned}$$ as before. Together, these ensure that changing the reference momentum $\check{k}$ changes $A^\mu(x)$ by a harmonic gauge transformation^†.

To completely justify (4), we choose R-$\xi$ gauge, and use BV-BRST quantization, but that’s the subject for another blog post.

Here, it suffices to say that the Feynman rules contract every external photon line with a $\frac{\mu\sigma^\mu\lambda^\dagger}{\mu\lambda}$ or a $\frac{\mu^\dagger\overline{\sigma}^\mu\lambda}{\mu^\dagger\lambda^\dagger}$, depending on the helicity of the incoming/outgoing photon. We’re free to make any choice of reference momentum $\check{k}$ that we want, but verifying that the final answer is independent of $\check{k}$ is a nice check on our calculations.

^† Notoriously, Lorentz gauge $\partial\cdot A = 0$ does not completely fix the gauge: we can still shift $A_\mu\to A_\mu+\partial_\mu f$, where $f$ is any solution to the scalar wave equation, $\square f = 0$.

Fellow science communicators, think you can explain everything that goes on in your field? If so, I have a challenge for you. Pick a day, and go through all the new papers on arXiv.org in a single area. For each one, try to give a general-audience explanation of what the paper is about. To make it easier, you can ignore cross-listed papers. If your field doesn’t use arXiv, consider if you can do the challenge with another appropriate site.

I’ll start. I’m looking at papers in the “High Energy Physics – Theory” area, announced 6 Jan, 2022. I’ll warn you in advance that I haven’t read these papers, just their abstracts, so apologies if I get your paper wrong!

arXiv:2201.01303 : Holographic State Complexity from Group Cohomology

This paper says it is a contribution to a Proceedings. That means it is based on a talk given at a conference. In my field, a talk like this usually won’t be presenting new results, but instead summarizes results in a previous paper. So keep that in mind.

There is an idea in physics called holography, where two theories are secretly the same even though they describe the world with different numbers of dimensions. Usually this involves a gravitational theory in a “box”, and a theory without gravity that describes the sides of the box. The sides turn out to fully describe the inside of the box, much like a hologram looks 3D but can be printed on a flat sheet of paper. Using this idea, physicists have connected some properties of gravity to properties of the theory on the sides of the box. One of those properties is complexity: the complexity of the theory on the sides of the box says something about gravity inside the box, in particular about the size of wormholes. The trouble is, “complexity” is a bit subjective: it’s not clear how to give a good definition for it for this type of theory. In this paper, the author studies a theory with a precise mathematical definition, called a topological theory. This theory turns out to have mathematical properties that suggest a well-defined notion of complexity for it.

arXiv:2201.01393 : Nonrelativistic effective field theories with enhanced symmetries and soft behavior

We sometimes describe quantum field theory as quantum mechanics plus relativity. That’s not quite true though, because it is possible to define a quantum field theory that doesn’t obey special relativity, a non-relativistic theory. Physicists do this if they want to describe a system moving much slower than the speed of light: it gets used sometimes for nuclear physics, and sometimes for modeling colliding black holes.

In particle physics, a “soft” particle is one with almost no momentum. We can classify theories based on how they behave when a particle becomes more and more soft. In normal quantum field theories, if they have special behavior when a particle becomes soft it’s often due to a symmetry of the theory, where the theory looks the same even if something changes. This paper shows that this is not true for non-relativistic theories: they have more requirements to have special soft behavior, not just symmetry. They “bootstrap” a few theories, using some general restrictions to find them without first knowing how they work (“pulling them up by their own bootstraps”), and show that the theories they find are in a certain sense unique, the only theories of that kind.

arXiv:2201.01552 : Transmutation operators and expansions for 1-loop Feynman integrands

In recent years, physicists in my sub-field have found new ways to calculate the probability that particles collide. One of these methods describes ordinary particles in a way resembling string theory, and from this discovered a whole “web” of theories that were linked together by small modifications of the method. This method originally worked only for the simplest Feynman diagrams, the “tree” diagrams that correspond to classical physics, but was extended to the next-simplest diagrams, diagrams with one “loop” that start incorporating quantum effects.

This paper concerns a particular spinoff of this method, that can find relationships between certain one-loop calculations in a particularly efficient way. It lets you express calculations of particle collisions in a variety of theories in terms of collisions in a very simple theory. Unlike the original method, it doesn’t rely on any particular picture of how these collisions work, either Feynman diagrams or strings.

arXiv:2201.01624 : Moduli and Hidden Matter in Heterotic M-Theory with an Anomalous U(1) Hidden Sector

In string theory (and its more sophisticated cousin M theory), our four-dimensional world is described as a world with more dimensions, where the extra dimensions are twisted up so that they cannot be detected. The shape of the extra dimensions influences the kinds of particles we can observe in our world. That shape is described by variables called “moduli”. If those moduli are stable, then the properties of particles we observe would be fixed, otherwise they would not be. In general it is a challenge in string theory to stabilize these moduli and get a world like what we observe.

This paper discusses shapes that give rise to a “hidden sector”, a set of particles that are disconnected from the particles we know so that they are hard to observe. Such particles are often proposed as a possible explanation for dark matter. This paper calculates, for a particular kind of shape, what the masses of different particles are, as well as how different kinds of particles can decay into each other. For example, a particle that causes inflation (the accelerating expansion of the universe) can decay into effects on the moduli and dark matter. The paper also shows how some of the moduli are made stable in this picture.

arXiv:2201.01630 : Chaos in Celestial CFT

One variant of the holography idea I mentioned earlier is called “celestial” holography. In this picture, the sides of the box are an infinite distance away: a “celestial sphere” depicting the angles particles go after they collide, in the same way a star chart depicts the angles between stars. Recent work has shown that there is something like a sensible theory that describes physics on this celestial sphere, that contains all the information about what happens inside.

This paper shows that the celestial theory has a property called quantum chaos. In physics, a theory is said to be chaotic if it depends very precisely on its initial conditions, so that even a small change will result in a large change later (the usual metaphor is a butterfly flapping its wings and causing a hurricane). This kind of behavior appears to be present in this theory.

arXiv:2201.01657 : Calculations of Delbrück scattering to all orders in αZ

Delbrück scattering is an effect where the nuclei of heavy elements like lead can deflect high-energy photons, as a consequence of quantum field theory. This effect is apparently tricky to calculate, and previous calculations have involved approximations. This paper finds a way to calculate the effect without those approximations, which should let it match better with experiments.

(As an aside, I’m a little confused by the claim that they’re going to all orders in αZ when it looks like they just consider one-loop diagrams…but this is probably just my ignorance, this is a corner of the field quite distant from my own.)

arXiv:2201.01674 : On Unfolded Approach To Off-Shell Supersymmetric Models

Supersymmetry is a relationship between two types of particles: fermions, which typically make up matter, and bosons, which are usually associated with forces. In realistic theories this relationship is “broken” and the two types of particles have different properties, but theoretical physicists often study models where supersymmetry is “unbroken” and the two types of particles have the same mass and charge. This paper finds a new way of describing some theories of this kind that reorganizes them in an interesting way, using an “unfolded” approach in which aspects of the particles that would normally be combined are given their own separate variables.

(This is another one I don’t know much about, this is the first time I’d heard of the unfolded approach.)

arXiv:2201.01679 : Geometric Flow of Bubbles

String theorists have conjectured that only some types of theories can be consistently combined with a full theory of quantum gravity, others live in a “swampland” of non-viable theories. One set of conjectures characterizes this swampland in terms of “flows” in which theories with different geometry can flow in to each other. The properties of these flows are supposed to be related to which theories are or are not in the swampland.

This paper writes down equations describing these flows, and applies them to some toy model “bubble” universes.

arXiv:2201.01697 : Graviton scattering amplitudes in first quantisation

This paper is a pedagogical one, introducing graduate students to a topic rather than presenting new research.

Usually in quantum field theory we do something called “second quantization”, thinking about the world not in terms of particles but in terms of fields that fill all of space and time. However, sometimes one can instead use “first quantization”, which is much more similar to ordinary quantum mechanics. There you think of a single particle traveling along a “world-line”, and calculate the probability it interacts with other particles in particular ways. This approach has recently been used to calculate interactions of gravitons, particles related to the gravitational field in the same way photons are related to the electromagnetic field. The approach has some advantages in terms of simplifying the results, which are described in this paper.

Last time we were thinking about categories enriched over $\bar{\mathbb{R}}_+$, the extended non-negative reals; such enriched categories are sometimes called generalized or Lawvere metric spaces. In the context of optimal transport with cost matrix $k$, thought of as a $\bar{\mathbb{R}}_+$-profunctor $k\colon \mathcal{S}\rightsquigarrow\mathcal{R}$ between suppliers and receivers, we were interested in the centre of the ‘Kan-type adjunction’ between enriched functor categories, which is the following:

In this post I want to give some examples of the Kan-type centre in low dimension to try to give a sense of what they look like over $\bar{\mathbb{R}}_+$. Here’s the simplest kind of example we will see.

Quick Recap

It’s been a while since my last post on this and I don’t want to a big recap but certainly a small recap is in order! I’m interested in transporting goods from a set $\{S_i\}_{i=1}^s$ of suppliers to a set $\{R_j\}_{j=1}^r$ of receivers. (Think of transporting loaves of bread from bakeries to cafés.) We have a supply of $\sigma_i$ units of good at supplier $S_i$ and a demand of $\rho_j$ at receiver $R_j$ and the cost of transporting one unit of goods from $S_i$ to $R_j$ is $k_{ij}\in [0,\infty)$.

We saw that in the dual problem we want to find prices $v=(v_1,\dots,v_s)$ and $u=(u_1,\dots, u_r)$ at the suppliers and receivers respectively that maximize the total revenue $$\sum_j u_j\rho_j -\sum_i v_i\sigma_i$$ subject to the competitivity constraint $k_{ij}\ge u_j-v_i$ for all $i,j$.

I’ll model this is the language of $\bar{\mathbb{R}}_+$-categories. Recall that an $\bar{\mathbb{R}}_+$-category is a generalization of a metric space: in an $\bar{\mathbb{R}}_+$-category $X$ we have a ‘hom-object’ $X(a,b)\in [0,\infty]$ between each pair of objects $a,b\in X$. This hom-object should be thought of as a distance which does not have to be finite, nor symmetric, and you can have distinct objects having zero distance between them. The right notion of ‘functor’ here is that of ‘distance non-increasing map’; the term ‘short map’ is used for short.

I model the situation in the following way.

Note that I didn’t mention how to model the supply and demand. That is not of immediate interest to us, so I won’t say anything about that now.

We also saw that in finding prices which maximize total revenue we can restrict our attention to ‘tight price plans’, that is pairs of prices $(v; u)$ which lie in the centre $Z_{\mathrm{K}}(k)$ of the following Kan-type adjunction.

The Kan centre is the full subcategory of $\bar{\mathbb{R}}_+^\mathcal{S}\times \bar{\mathbb{R}}_+^\mathcal{R}$ consisting of objects $(v,u)$ such that $k\circ v = u$ and $v=k\triangleright u$; in the notation of previous posts I would write $\hat{v} =u$ and $v=\tilde{u}$.

Note that the Kan-type adjunction is not the same as the Isbell-type (*) adjunction I’ve discussed in previous posts on profunctor nuclei and Isbell tight-spans. I will discuss a relationship between them in a future post.

(*) I don’t like using the terms ‘Kan’ and ‘Isbell’ for these adjunctions, I’d much rather use more descriptive adjectives. It anyone has any ideas…

In this post I want to visualize the Kan centre $Z_{\mathrm{K}}(k)$ for you, and as the Kan centre is a subset of $\bar{\mathbb{R}}_+^\mathcal{S}\times \bar{\mathbb{R}}_+^\mathcal{R}\cong \bar{\mathbb{R}}_+^{\mathcal{S}\sqcup\mathcal{R}}$, it would be wise to start in a case with a very small number of suppliers and receivers, namely where $\left | \mathcal{S}\sqcup\mathcal{R}\right|\le 3$. We will see that we can in fact visualize, in a certain way, higher dimensional cases, but we will start with a slightly silly example — from an optimal transport perspective — to illustrate some basic features.

A first example

Let’s start with two suppliers and one receivers, $|\mathcal{S}|=2$ and $|\mathcal{R}|=1$. I will think of these as bakeries and cafés and I’ll pick some simpler numbers than I did in the old café example I used. The numbers on the arrows represent the cost of transporting one loaf of bread. The numbers in circles represent the supply and demand in loaves of bread.

The profunctor in this situation can be represented by the matrix $k=\left(\begin{smallmatrix}3\\2\end{smallmatrix}\right)$. (I might be regretting my choice of conventions and I might have preferred to have the transpose of this matrix.)

It is clear what the solution to the optimal transport problem is in this case as there’s only one feasible solution, namely both bakeries send all their bread to the one café. But this should be a useful example none-the-less.

We have the adjunction

with

$$k\circ(v_1, v_2)=(\min\{3+v_1, 2+v_2\}) \qquad k\triangleright(u_1) = (u_1\,\dot{{}-{}}\,3, u_1\,\dot{{}-{}}\,2).$$

You can check that the centre is

$$\{((\lambda\,\dot{{}-{}}\,1, \lambda), (2+\lambda))\mid \lambda\in [0,\infty]\} \subset \bar{\mathbb{R}}_+^2\times \bar{\mathbb{R}}_+^1$$

and this (or at least a finite portion of it) can be drawn as follows.

You can observe that this is the union of two closed line segments, or, as might be more useful, that this is the union of two open line segments (one bounded, one unbounded) and two points. In fact, in general the Kan centre is a cell complex of open, piece-wise affine cells.

Another thing to observe here is that if $(v,u)$ is a pair in the Kan centre then $v$ determines $u$ and $u$ determines $v$, so we can project onto $\bar{\mathbb{R}}_+^2$ or $\bar{\mathbb{R}}_+^1$ without losing any information. Here are the projections, I’ve drawn the boundary of the view box that I’ve picked, but the picture should be unbounded.

Here they are, drawn in a more standard way.

Some generalities

Projecting in this way gives us two views of the Kan centre and will allow us to visualize some higher dimensional examples. You need to bear in mind, however, that neither picture is the ‘true’ centre as this naturally sits in the higher dimensional space. For instance, the total revenue function can be restricted to the tight price plans, i.e. the centre, this is the restriction of a linear function when the centre is considered as a subset of $(v,u)$-space, but it is not the restriction of a linear function when the centre is considered as a subset of $v$-space or $u$-space. Remember that we are hoping to maximize the total revenue function on the Kan centre.

There’s a very general statement about adjunctions when enriching over commutative quantales like $\bar{\mathbb{R}}_+$. I won’t go into the details, many of which can be found in my Legendre-Fenchel transform paper. Anyway, roughly speaking, for an adjunction $L\dashv R$ between $\bar{\mathbb{R}}_+$-categories we have a counit $L R\Rightarrow\mathrm{id}$ and a unit $\mathrm{id}\Rightarrow R L$, thus we have $R L R\Rightarrow R$ and $R\Rightarrow R L R$ (which means $0\ge \mathcal{C}(R L R(d), R(d))$ and $0\ge \mathcal{C}(R(d), R L R(d))$ for all $d\in \mathcal{D}$) and so $R \cong R L R$. Similarly $L \cong L R L$. We saw that manifested earlier as $\tilde{\hat{\tilde{v}}}= \tilde{v}$. This gives us that any monad on an $\bar{\mathbb{R}}_+$-category is idempotent and also the following.

Lemma 1. For an adjunction $L\dashv R$ between $\bar{\mathbb{R}}_+$-categories $\mathcal{C}$ and $\mathcal{D}$ there is the following identification of categories where $\mathrm{Im}$ denotes the image of a functor and $\mathrm{Fix}$ denotes the fixed category of an endofunctor.

So in our case this means that the left-hand view or the space of tight price plans on the suppliers can be thought of as the image of the functor $k\triangleright{-}$, or as the fixed space of the monad $k\triangleright(k\circ{-})$.

Examples

Here are some more examples. In each case I’ll give the profunctor and then pictures of the two views of the Kan-type centre, or, if you prefer, the tight price plans.

I computed these (as cell complexes) using some rather inefficient SageMath code; however, having seen what they look like I now realise there will be better algorithms out there. I’ll maybe say more on this next time.

$\begin{pmatrix}2&1\\3&5\end{pmatrix}$

$\begin{pmatrix}2&7\\3&5\end{pmatrix}$

$\begin{pmatrix} 4 & 6 \\ 10 & 5 \\ 11 & 11 \\ 8 & 14 \end{pmatrix}$

In the above example I couldn’t actually show you the picture of the left-hand view as I’m not good at putting 4d objects on the the screen. However in the next example, by the magic of three.js, I can (hopefully!) put some fairly basic 3d objects on the screen which you can navigate and move with your mouse/touchpad/fingers. By clicking on the border you should be able to open models in fullscreen if you desire.

Due to laziness on my part, I’ll only show the one-cells; however, the two-cells and three-cells fill in the obvious holes (or at least I think they are obvious).

$\begin{pmatrix} 1 & 4 & 6 \\ 2 & 7 & 1 \\ 8 & 7 & 8 \end{pmatrix}$

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So there’s some examples to get one thinking.

Next time

I hope that next time I will explain some of the structure of these examples and relate these to the notion of the semi-tropical convex hull. (The tropically aware amongst you might have noticed something convex hully going on.) And later on I would like to compare and contrast these with Isbell-type centres.

The Topos Institute has a new seminar:

• Intercats: seminar on categorical interaction. Every other Tuesday at 17:00 UTC, beginning with a talk by David Spivak on the 25th of January, 2022.

The talks will be streamed and also recorded on YouTube.

It’s a new seminar series on the mathematics of interacting systems, their composition, and their behavior. Split in equal parts theory and applications, we are particularly interested in category-theoretic tools to make sense of information-processing or adaptive systems, or those that stand in a ‘bidirectional’ relationship to some environment. We aim to bring together researchers from different communities, who may already be using similar-but-different tools, in order to improve our own interaction.

INTERCATS SCOPE

Although by no means an exhaustive or prescriptive list, Intercats seminar topics are likely to be related to the following mathematical topics:

• Generalized lenses, optics, and related structures, such as Chu spaces or Dialectica categories;

• polynomial functors, and their associated ecosystem;

• applications of the foregoing to:

  • bidirectional processes,

  • dynamical systems,

  • wiring diagrams,

  • open games,

  • database theory,

  • automatic differentiation,

  • computational statistics,

  • machine learning.

ORGANIZERS

Jules Hedges, University of Strathclyde

David Spivak, Topos Institute

Toby St Clere Smithe, Topos Institute

Scott’s foreword

One week ago, E. O. Wilson—the legendary naturalist and conservationist, and man who was universally acknowledged to know more about ants than anyone else in human history—passed away at age 92. A mere three days later, Scientific American—or more precisely, the zombie clickbait rag that now flaunts that name—published a shameful hit-piece, smearing Wilson for his “racist ideas” without, incredibly, so much as a single quote from Wilson, or any other attempt to substantiate its libel (see also this response by Jerry Coyne). SciAm‘s Pravda-like attack included the following extraordinary sentence, which I thought worthy of Alan Sokal’s Social Text hoax:

The so-called normal distribution of statistics assumes that there are default humans who serve as the standard that the rest of us can be accurately measured against.

There are intellectually honest people who don’t know what the normal distribution is. There are no intellectually honest people who, not knowing what it is, figure that it must be something racist.

On Twitter, Laura Helmuth, the editor-in-chief now running SciAm into the ground, described her magazine’s calumny against Wilson as “insightful” (the replies, including from Richard Dawkins, are fun to read). I suppose it was as “insightful” as SciAm‘s disgraceful attack last year on Eric Lander, President Biden’s ultra-competent science advisor and a leader in the war on COVID, for … being a white male, which appears to have been E. O. Wilson’s crime as well. (Think I must be misrepresenting the “critique” of Lander? Read it!)

Anyway, in response to Scientific American‘s libel of Wilson, I wrote on my Facebook that I’ll no longer agree to write for or be interviewed by them (you can read my old stuff free of charge here or here), unless and until there’s a complete change of editorial direction. I encourage all other scientists to commit likewise, thereby making it common knowledge that the entity that now calls itself “Scientific American” bears the same relation to the legendary home of Martin Gardner as does a corpse to a living being. Fortunately, there are high-quality online venues (e.g., Quanta) that partly fill the role that Scientific American abdicated.

After reading my Facebook post, my friend Ashutosh Jogalekar was inspired to post an essay of his own. Ashutosh used to write regularly for Scientific American, until he was fired seven years ago over a column in which he advocated acknowledging Richard Feynman’s flaws, including his arrogance and casual sexism, but also understanding those flaws within the context of Feynman’s whole life, including the tragic death of his first wife Arlene. (Yes, that was really it! Read the piece!) Below, I’m sharing Ashutosh’s moving essay about E. O. Wilson with Ashutosh’s very generous permission. —Scott Aaronson

Guest Post by Ashutosh Jogalekar

As some know, I was “fired” from Scientific American in 2014 for three “controversial” posts (among 200 that I had written for the magazine). When I parted from the magazine I chalked up my departure to an unfortunate misunderstanding more than anything else. I still respected some of the writers at the publication, and while I wore my separation as a badge of honor and in retrospect realized its liberating utility in enabling me to greatly expand my topical range, I occasionally still felt bad and wished things had gone differently.

No more. Now the magazine has done me a great favor by allowing me to wipe the slate of my conscience clean. What happened seven years ago was not just a misunderstanding but clearly one of many first warning signs of a calamitous slide into a decidedly unscientific, irrational and ideology-ridden universe of woke extremism. Its logical culmination two days ago was an absolutely shameless, confused, fact-free and purely ideological hit job on someone who wasn’t just a great childhood hero of mine but a leading light of science, literary achievement, humanism and biodiversity. While Ed (E. O.) Wilson’s memory was barely getting cemented only days after his death, the magazine published an op-ed calling him a racist, a hit job endorsed and cited by the editor-in-chief as “insightful”. One of the first things I did after reading the piece was buy a few Wilson books that weren’t part of my collection.

Ed Wilson was one of the gentlest, most eloquent, most brilliant and most determined advocates for both human and natural preservation you could find. Under Southern charm lay hidden unyielding doggedness and immense stamina combined with a missionary zeal to communicate the wonders of science to both his fellow biologists and the general public. His autobiography, “Naturalist”, is perhaps the finest, most literary statement of the scientific life I have read; it was one of a half dozen books that completely transported me when I read it in college. In book after book of wide-ranging intellectual treats threading through a stunning diversity of disciplines, he sent out clarion calls for saving the planet, for enabling dialogue between the natural and the social sciences, for understanding each other better. In the face of unprecedented challenges to our fragile environment and continued barriers to interdisciplinary communication, this is work that likely will make him go down in history as one of the most important human beings who ever lived, easily of the same caliber and achievement as John Muir or Thoreau. Even in terms of achievement strictly defined by accolades – the National Medal of Science, the Crafoord Prize which recognizes fields excluded by the Nobel Prize, and not just one but two Pulitzer Prizes – few scientists from any field in the 20th century can hold a candle to Ed Wilson. My friend Richard Rhodes who knew Wilson for decades as a close and much-admired friend said that there wasn’t a racist bone in his body; Dick should know since he just came out with a first-rate biography of Wilson weeks before his passing.

The writer who wrote that train wreck is a professor of nursing at UCSF named Monica McLemore. That itself is a frightening fact and should tell everyone how much ignorance has spread itself in our highest institutions. She not only maligned and completely misrepresented Wilson but did not say a word about his decades-long, heroic effort to preserve the planet and our relationship with it; it was clear that she had little acquaintance with Wilson’s words since she did not cite any. It’s also worth noting the gaping moral blindness in her article which completely misses the most moral thing Wilson did – spend decades advocating for saving our planet and averting a catastrophe of extinction, climate change and divisiveness – and instead focuses completely on his non-existent immorality. This is a pattern that is consistently found among those urging “social justice” or “equity” or whatever else: somehow they seem to spend all their time talking about fictional, imagined immorality while missing the real, flesh-and-bones morality that is often the basis of someone’s entire life’s work.

In the end, the simple fact is that McLemore didn’t care about any of this. She didn’t care because she had a political agenda and the facts did not matter to her, even facts as basic as the definition of the normal distribution in statistics. For her, Wilson was some obscure white male scientist who was venerated, and that was reason enough for a supposed “takedown”. And the editor of Scientific American supported and lauded this ignorant, ideology-driven tirade.

Ironically, Wilson would have found this ideological hit job all too familiar. After he wrote his famous book Sociobiology in the 1970s, a volume in which, in a single chapter about human beings, he had the temerity to suggest that maybe, just maybe, human beings operate with the same mix of genes that other creatures do, the book was met by a disgraceful, below-the-belt, ideological response from Wilson’s far left colleagues Richard Lewontin and Stephen Jay Gould who hysterically compared his arguments to thinking that was well on its way down the slippery slope to that dark world where lay the Nazi gas chambers. The gas chamber analogy is about the only thing that’s missing from the recent hit job, but the depressing thing is that we are fighting the same battles in 2021 that Wilson fought forty years before, although turbocharged this time by armies of faithful zombies on social media. The sad thing is that Wilson is no longer around to defend himself, although I am not sure he would have bothered with a piece as shoddy as this one.

The complete intellectual destruction of a once-great science magazine is now clear as day. No more should Scientific American be regarded as a vehicle for sober scientific views and liberal causes but as a political magazine with clearly stated ideological biases and an aversion to facts, an instrument of a blinkered woke political worldview that brooks no dissent. Scott Aaronson has taken a principled stand and said that after this proverbial last straw on the camel’s back, he will no longer write for the magazine or do interviews for them. I applaud Scott’s decision, and with his expertise it’s a decision that actually matters. As far as I am concerned, I now mix smoldering fury at the article with immense relief: the last seven years have clearly shown that leaving Scientific American in 2014 was akin to leaving the Soviet Union in the 1930s just before Stalin appointed Lysenko head biologist. I could not have asked for a happier expulsion and now feel completely vindicated and free of any modicum of regret I might have felt.

To my few friends and colleagues who still write for the magazine and whose opinions I continue to respect, I really wish to ask: Why? Is writing for a magazine which has sacrificed facts and the liberal voice of real science at the altar of political ideology and make believe still worth it? What would it take for you to say no more? As Oscar Wilde would say, one mistake like this is a mistake, two seems more like carelessness; in the roster of the last few years, this is “mistake” 100+, signaling that it’s now officially approved policy. Do you think that being an insider will allow you to salvage the reputation of the magazine? If you think that way, you are no different from the one or two moderate Republicans who think they can still salvage the once-great party of Lincoln and Eisenhower. Both the GOP and Scientific American are beyond redemption from where I stand. Get out, start your own magazine or join another, one which actually respects liberal, diverse voices and scientific facts; let us applaud you for it. You deserve better, the world deserves better. And Ed Wilson’s memory sure as hell deserves better.

Update (from Scott): See here for the Hacker News thread about this post. I was amused by the conjunction of two themes: (1) people who were uncomfortable with my and Ashutosh’s expression of strong emotions, and (2) people who actually clicked through to the SciAm hit-piece, and then reported back to the others that the strong emotions were completely, 100% justified in this case.

I felt like a gum ball trying to squeeze my way out of a gum-ball machine. 

I was one of 50-ish physicists crammed into the lobby—and in the doorway, down the stairs, and onto the sidewalk—of a Manhattan hotel last December. Everyone had received a COVID vaccine, and the omicron variant hadn’t yet begun chewing up North America. Everyone had arrived on the same bus that evening, feeding on the neon-bright views of Fifth Avenue through dinnertime. Everyone wanted to check in and offload suitcases before experiencing firsthand the reason for the nickname “the city that never sleeps.” So everyone was jumbled together in what passed for a line.

We’d just passed the halfway point of the week during which I was pretending to be a string theorist. I do that whenever my research butts up against black holes, chaos, quantum gravity (the attempt to unify quantum physics with Einstein’s general theory of relativity), and alternative space-times. These topics fall under the heading “It from Qubit,” which calls for understanding puzzling physics (“It”) by analyzing how quantum systems process information (“Qubit”). The “It from Qubit” crowd convenes for one week each December, to share progress and collaborate.¹ The group spends Monday through Wednesday at Princeton’s Institute for Advanced Study (IAS), dogged by photographs of Einstein, busts of Einstein, and roads named after Einstein. A bus ride later, the group spends Thursday and Friday at the Simons Foundation in New York City.

I don’t usually attend “It from Qubit” gatherings, as I’m actually a quantum information theorist and quantum thermodynamicist. Having admitted as much during the talk I presented at the IAS, I failed at pretending to be a string theorist. Happily, I adore being the most ignorant person in a roomful of experts, as the experience teaches me oodles. At lunch and dinner, I’d plunk down next to people I hadn’t spoken to and ask what they see as trending in the “It from Qubit” community. 

One buzzword, I’d first picked up on shortly before the pandemic had begun (replicas). Having lived a frenetic life, that trend seemed to be declining. Rising buzzwords (factorization and islands), I hadn’t heard in black-hole contexts before. People were still tossing around terms from when I’d first forayed into “It from Qubit” (scrambling and out-of-time-ordered correlator), but differently from then. Five years ago, the terms identified the latest craze. Now, they sounded entrenched, as though everyone expected everyone else to know and accept their significance.

One buzzword labeled my excuse for joining the workshops: complexity. Complexity wears as many meanings as the stereotypical New Yorker wears items of black clothing. Last month, guest blogger Logan Hillberry wrote about complexity that emerges in networks such as brains and social media. To “It from Qubit,” complexity quantifies the difficulty of preparing a quantum system in a desired state. Physicists have conjectured that a certain quantum state’s complexity parallels properties of gravitational systems, such as the length of a wormhole that connects two black holes. The wormhole’s length grows steadily for a time exponentially large in the gravitational system’s size. So, to support the conjecture, researchers have been trying to prove that complexity typically grows similarly. Collaborators and I proved that it does, as I explained in my talk and as I’ll explain in a future blog post. Other speakers discussed experimental complexities, as well as the relationship between complexity and a simplified version of Einstein’s equations for general relativity.

I learned a bushel of physics, moonlighting as a string theorist that week. The gum-ball-machine lobby, though, retaught me something I’d learned long before the pandemic. Around the time I squeezed inside the hotel, a postdoc struck up a conversation with the others of us who were clogging the doorway. We had a decent fraction of an hour to fill; so we chatted about quantum thermodynamics, grant applications, and black holes. I asked what the postdoc was working on, he explained a property of black holes, and it reminded me of a property of thermodynamics. I’d nearly reached the front desk when I realized that, out of the sheer pleasure of jawing about physics with physicists in person, I no longer wanted to reach the front desk. The moment dangles in my memory like a crystal ornament from the lobby’s tree—pendant from the pandemic, a few inches from the vaccines suspended on one side and from omicron on the other. For that moment, in a lobby buoyed by holiday lights, wrapped in enough warmth that I’d forgotten the December chill outside, I belonged to the “It from Qubit” community as I hadn’t belonged to any community in 22 months.

Happy new year.

¹In person or virtually, pandemic-dependently.

Thanks to the organizers of the IAS workshop—Ahmed Almheiri, Adam Bouland, Brian Swingle—for the invitation to present and to the organizers of the Simons Foundation workshop—Patrick Hayden and Matt Headrick—for the invitation to attend.

Happy New Year, everyone!

It was exactly two years ago that it first became publicly knowable—though most of us wouldn’t know for at least two more months—just how freakishly horrible is the branch of the wavefunction we’re on. I.e., that our branch wouldn’t just include Donald Trump as the US president, but simultaneously a global pandemic far worse than any in living memory, and a world-historically bungled response to that pandemic.

So it’s appropriate that I just finished reading Viral: The Search for the Origin of COVID-19, by Broad Institute genetics postdoc Alina Chan and science writer Matt Ridley. Briefly, I think that this is one of the most important books so far of the twenty-first century.

Of course, speculation and argument about the origin of COVID goes back all the way to that fateful January of 2020, and most of this book’s information was already available in fragmentary form elsewhere. And by their own judgment, Chan and Ridley don’t end their search with a smoking-gun: no Patient Zero, no Bat Zero, no security-cam footage of the beaker dropped on the Wuhan Institute of Virology floor. Nevertheless, as far as I’ve seen, this is the first analysis of COVID’s origin to treat the question with the full depth, gravity, and perspective that it deserves.

Viral is essentially a 300-page plea to follow every lead as if we actually wanted to get to the bottom of things, and in particular, yes, to take the possibility of a lab leak a hell of a lot more seriously than was publicly permitted in 2020. (Fortuitously, much of this shift already happened as the authors were writing the book, but in June 2021 I was still sneered at for discussing the lab leak hypothesis on this blog.) Viral is simultaneously a model of lucid, non-dumbed-down popular science writing and of cogent argumentation. The authors never once come across like tinfoil-hat-wearing conspiracy theorists, railing against the sheeple with their conventional wisdom: they’re simply investigators carefully laying out what they’re confident should become conventional wisdom, with the many uncertainties and error bars explicitly noted. If you read the book and your mind works anything like mine, be forewarned that you might come out agreeing with a lot of it.

I would say that Viral proves the following propositions beyond reasonable doubt:

• Virologists, including at Shi Zhengli’s group at WIV and at Peter Daszak’s EcoHealth Alliance, were engaged in unbelievably risky work, including collecting virus-laden fecal samples from thousands of bats in remote caves, transporting them to the dense population center of Wuhan, and modifying them to be more dangerous, e.g., through serial passage through human cells and the insertion of furin cleavage sites. Years before the COVID-19 outbreak, there were experts remarking on how risky this research was and trying to stop it. Had they known just how lax the biosecurity was in Wuhan—dangerous pathogens experimented on in BSL-2 labs, etc. etc.—they would have been louder.
  
• Even if it didn’t cause the pandemic, the massive effort to collect and enhance bat coronaviruses now appears to have been of dubious value. It did not lead to an actionable early warning about how bad COVID-19 was going to be, nor did it lead to useful treatments, vaccines, or mitigation measures, all of which came from other sources.
  
• There are multiple routes by which SARS-CoV2, or its progenitor, could’ve made its way, otherwise undetected, from the remote bat caves of Yunnan province or some other southern location to the city of Wuhan a thousand miles away, as it has to do in any plausible origin theory. Having said that, the regular Yunnan→Wuhan traffic in scientific samples of precisely these kinds of viruses, sustained over a decade, does stand out a bit! On the infamous coincidence of the pandemic starting practically next door to the world’s center for studying SARS-like coronaviruses, rather than near where the horseshoe bats live in the wild, Chan and Ridley memorably quote Humphrey Bogart’s line from Casablanca: “Of all the gin joints in all the towns in all the world, she walks into mine.”
  
• The seafood market was probably “just” an early superspreader site, rather than the site of the original spillover event. No bats or pangolins at all, and relatively few mammals of any kind, appear to have been sold at that market, and no sign of SARS-CoV2 was ever found in any of the animals despite searching.
  
• Most remarkably, Shi and Daszak have increasingly stonewalled, refusing to answer 100% reasonable questions from fellow virologists. They’ve acted more and more like defendants exercising their right to remain silent than like participants in a joint search for the truth. That might be understandable if they’d already answered ad nauseam and wearied of repeating themselves, but with many crucial questions, they haven’t answered even once. They’ve refused to make available a key database of all the viruses WIV had collected, which WIV inexplicably took offline in September 2019. When, in January 2020, Shi disclosed to the world that WIV had collected a virus called RaTG13, which was 96% identical to SARS-CoV2, she didn’t mention that it was collected from a mine in Mojiang, which the WIV had sampled from over and over because six workers had gotten a SARS-like pneumonia there in 2012 and three had died from it. She didn’t let on that her group had been studying RaTG13 for years—giving, instead, the false impression that they’d just noticed it recently, when searching WIV’s records for cousins of SARS-CoV2. And she didn’t see fit to mention that WIV had collected eight other coronaviruses resembling SARS-CoV2 from the same mine (!). Shi’s original papers on SARS-CoV2 also passed in silence over the virus’s furin cleavage site—even though SARS-CoV2 was the first sarbecoronavirus with that feature, and Shi herself had recently demonstrated adding furin cleavage sites to other viruses to make them more transmissible, and the cleavage site would’ve leapt out immediately to any coronavirus researcher as the most interesting feature of SARS-CoV2 and as key to its transmissibility. Some of these points had to be uncovered by Internet sleuths, poring over doctoral theses and the like, after which Shi would glancingly acknowledge the points in talks without ever explaining her earlier silences. Shi and Daszak refused to cooperate with Chan and Ridley’s book, and have stopped answering questions more generally. When people politely ask Daszak about these matters on Twitter, he blocks them.
  
• The Chinese regime has been every bit as obstructionist as you might expect: destroying samples, blocking credible investigations, censoring researchers, and preventing journalists from accessing the Mojiang mine. So Shi at least has the excuse that, even if she’d wanted to come clean with everything relevant she knows about WIV’s bat coronavirus work, she might not be able to do so without endangering herself or loved ones. Daszak has no such excuse.

It’s important to understand that, even in the worst case—that (1) there was a lab leak, and (2) Shi and Daszak are knowingly withholding information relevant to it—they’re far from monsters. Even in Viral‘s relentlessly unsparing account, they come across as genuine believers in their mission to protect the world from the next pandemic.

And it’s like: imagine devoting your life to that mission, having most of the world refuse to take you seriously, and then the calamity happens exactly like you said … except that, not only did your efforts fail to prevent it, but there’s a live possibility that they caused it. It’s conceivable that your life’s work managed to save minus 15 million lives and create minus $50 trillion in economic value.

Very few scientists in history have faced that sort of psychic burden, perhaps not even the ones who built the atomic bomb. I hope I’d maintain my scientific integrity under such an astronomical weight, but I’m doubtful that I would. Would you?

Viral very wisely never tries to psychoanalyze Shi and Daszak. I fear that one might need a lot of conceptual space between “knowing” and “not knowing,” “suspecting” and “not suspecting,” to do justice to the planet-sized enormity of what’s at stake here. Suppose, for example, that an initial investigation in January 2020 reassured you that SARS-CoV2 probably hadn’t come from your lab: would you continue trying to get to the bottom of things, or would you thereafter decide the matter was closed?

For all that, I agree with Chan and Ridley that COVID-19 might well have had a zoonotic origin after all. And one point Viral makes abundantly clear is that, if our goal is to prevent the next pandemic, then resolving the mystery of COVID-19 actually matters less than one might think. This is because, whichever possibility—zoonotic spillover or lab leak—turns out to be the truth of this case, the other possibility would remain absolutely terrifying and would demand urgent action as well. Read the book and see for yourself.

Searching my inbox, I found an email from April 16, 2020 where I told someone who’d asked me that the lab-leak hypothesis seemed perfectly plausible to me (albeit no more than plausible), that I couldn’t understand why it wasn’t being investigated more, but that I was hesitant to blog about these matters. As I wrote seven months ago, I now see my lack of courage on this as having been a personal failing. Obviously, I’m just a quantum computing theorist, not a biologist, so I don’t have to have any thoughts whatsoever about the origin of COVID-19 … but I did have some, and I didn’t share them here only because of the likelihood that I’d be called an idiot on social media. Having now read Chan and Ridley, though, I think I’d take being called an idiot for this book review more as a positive signal about my courage than as a negative signal about my reasoning skills!

At one level, Viral stands alongside, I dunno, Eichmann in Jerusalem among the saddest books I’ve ever read. It’s 300 pages of one of the great human tragedies of our lifetime balancing on a hinge between happening and not happening, and we all know how it turns out. On another level, though, Viral is optimistic. Like with Richard Feynman’s famous “personal appendix” about the Space Shuttle Challenger explosion, the very act of writing such a book reflects a view that you’re still allowed to ask questions; that one or two people armed with nothing but arguments can run rings around governments, newspapers, and international organizations; that we don’t yet live in a post-truth world.

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

• 2022 Adjoint School: application.

Read on for more about how this works!

The school will be run online from February to June, 2022, and then — coronavirus permitting — there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11–15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

• About the Adjoint School.

There are four topics to work on, and you can see descriptions of them below.

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.

Also check out our inclusivity statement.

Topic 1: Compositional Thermodynamics

Mentors: Spencer Breiner and Joe Moeller

TA: Owen Lynch

Description: Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.

Readings:

• John C. Baez, Owen Lynch, and Joe Moeller, Compositional thermostatics.

• F. William Lawvere, State categories, closed categories and the existence of semi-continuous entropy functions.

Topic 2: Fuzzy Type Theory for Opinion Dynamics

Mentor: Paige North

TA: Hans Reiss

Description: When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.

Readings:

• Daniel R. Grayson, An introduction to univalent foundations for mathematicians.

• Jakob Hansen and Robert Ghrist, Opinion dynamics on discourse sheaves.

Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

Mentor: Nicoletta Sabadini

TA: Mario Román

Description coming soon!

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description coming soon!

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

• 2022 Adjoint School: application.

The school will be run online from February to June, 2022, and then—coronavirus permitting—there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11 – 15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

• About the Adjoint School.

There are four topics to work on, and you can see descriptions of them below.

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.

Also check out our inclusivity statement.

Topic 1: Compositional Thermodynamics

Mentors: Spencer Breiner and Joe Moeller

TA: Owen Lynch

Description: Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.

Readings:

• John C. Baez, Owen Lynch, and Joe Moeller, Compositional thermostatics.

• F. William Lawvere, State categories, closed categories and the existence of semi-continuous entropy functions.

Topic 2: Fuzzy Type Theory for Opinion Dynamics

Mentor: Paige North

TA: Hans Reiss

Description: When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.

Readings:

• Daniel R. Grayson, An introduction to univalent foundations for mathematicians.

• Jakob Hansen and Robert Ghrist, Opinion dynamics on discourse sheaves.

Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

Mentor: Nicoletta Sabadini

TA: Mario Román

Description coming soon!

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description coming soon!

read more

This history seamlessly provides context for the portrait of Anderson, a brilliant, intuitive theorist who prized profound, essential models over computational virtuosity, and who had a litany of achievements that is difficult to list in its entirety.  The person described in the book gibes perfectly with my limited direct interactions with him and the stories I heard from my thesis advisor and other Bell Labs folks.  Some lines ring particularly true (with all that says about the culture of our field):  "Anderson never took very long to decide if a physicist he had just met was worth his time and respect."

On a separate note:  Thanks for reading, and I wish you a very happy new year!  I hope that you and yours have a safe, healthy, and fulfilling 2022.

• 30 Dec 2021:  Project Hail Mary, by Andy Weir.
• 5 Dec 2021: The Green Futures of Tycho, by William Sleator.
• 30 Nov 2021:  The Cup of Fury, by Upton Sinclair.
• 28 Nov 2021: Horse Walks Into A Bar, by David Grossman (Jessica Cohen, trans.)
• 5 Nov 2021: All Of The Marvels, by Douglas Wolk.
• 13 Oct 2021: Great Days, by Donald Barthelme.
• 30 Sep 2021:  Beautiful World, Where Are You? by Sally Rooney.
• 18 Sep 2021:  Because Internet, by Gretchen McCulloch.
• 10 Sep 2021:  Hidden Valley Road, by Robert Kolker.
• 26 Aug 2021:  The Hairdresser of Harare, by Tendai Huchu.
• 7 Aug 2021:  Max Beckmann at the St. Louis Art Museum:  The Paintings, by Lynette Roth.
• 2 Aug 2021:  To Live, by Yu Hua. (Michael Berry, trans.)
• 1 Aug 2021:  Blackman’s Burden, by Mack Reynolds.
• 29 Jul 2021:  Highly Irregular, by Arika Okrent.
• 21 Jul 2021: Sleeping Beauties, by Stephen King and Owen King.
• 12 Jul 2021:  Giovanni’s Room, by James Baldwin.
• 7 Jul 2021:  Daisy Miller, by Henry James.
• 29 Jun 2021:  Parable of the Sower, by Octavia Butler.
• 6 Jun 2021:  Walkman, by Michael Robbins.
• 4 Jun 2021:  Darryl, by Jackie Ess.
• 25 May 2021:  Likes, by Sarah Shun-Lien Bynum.
• 18 May 2021:  Subdivision, by J. Robert Lennon.
• 20 Apr 2021:  Big Trouble, by J. Anthony Lukas (not finished yet, will I finish it?)
• 15 Apr 2021:  No More Parades, by Ford Madox Ford (not finished yet, will I finish it? (Finished Sep 2021)
• 26 Mar 2021:  Some Do Not…, by Ford Madox Ford.
• 12 Jan 2021:  Metropolitan Life, by Fran Lebowitz.

My plan was that 2021 was going to be the year of reading long books. Why not? When the year started, there was no vaccine and I was teaching on the screen and so I was at home a lot of the time. Perfect time to really sink into a long book. I had gotten through and really valued some long ones in recent years — Bleak House, Vanity Fair, 1Q84, Hermione Lee’s Wharton biography — so why not Ford Madox Ford’s four-book Parade’s End, which I have wanted to read ever since I read and fell for The Good Soldier (aka the next book you have to read if you think Gatsby is the perfect novel and want “something else like that” but don’t think there could be something else like that — there is and this is it.)

But I didn’t read Parade’s End in 2021. I’ve read most of it! Two of the novels and half of the third. I will probably finish it. But — too high-modernist, too stream-of-consciousness, too hard under the circumstances. I have read a lot of it but I have never really been in it. It has made an impression but I could tell you only fragments about what happened in it. So my plan to read this, and The Man Without Qualities, and USA by Dos Passos, fell away. Sorry, early 20th century high modernism. And sorry too to Kevin Young’s Bunk and the 800-page history of Maryland and The Rest Is Noise and all the other I’ll-get-to-this bricks that didn’t even get taken off the shelf, as I’d imagined they might.

I read what I usually read. Stuff that caught my eye on the shelf. New books by people I know. Paperbacks small enough to fit in the pocket of my cargo shorts, for long walks — in 2021, we tried to spend a lot of time outside. (The small ones were: Blackman’s Burden and Great Days.)

I found I didn’t really read for pleasure this year. There were a lot of these books that I liked and admired; I don’t think there was one that gave me the sensation of “the thing I feel most like doing right now is reading this book.” But I’m not sure when a book (or a TV show, or movie) last made me feel that way, so it may be a mode of reading I’m done with! Anyway, books I liked: Douglas Wolk’s All of the Marvels — he read every Marvel comic ever written and wrote about what we found there. Nobody is better than Douglas at digging insight out of pieces of culture other people may see as low, or simple, or disposable. I’ve known that since I started listening to the mixtapes he was making in 1987. Of course one cannot help feeling that I enjoy reading Douglas reading those comics more than I would actually enjoy reading the comics. Darryl, by Jackie Ess, a perversely funny book about money that very cleverly disguises itself as a perversely funny book about sex.

Subdivision and Sleeping Beauties are an interesting matched set. Both involve dreamworlds. Lennon’s book is clearly marked as “literary fiction,” and rightly, but has the virtues of a great horror novel — authentic scariness, disorientation. Sleeping Beauties — well, you know I love Stephen King, and that I think he invented the virtues of a great modern horror novel, and I will keep reading these the same way I kept buying R.E.M. albums in the 21st century, but they are now kind of about themselves, gesturing at the virtues instead of really inhabiting them. The things people like to scold “literary fiction” for — shapeless plot & preoccupation with the married lives of middle-aged people — are here, and the presence of bizarrely-powered superbeings, gun battles, and rent flesh can’t change that.

The Green Futures of Tycho was as disturbing as I remembered. I loved it as a kid, but do I want my kids to read it? I think so? I wonder if there is fiction marketed as YA now which presents such a bleak picture of human nature. My sense is they make this stuff sunnier now. Here’s the first line of Sleator’s memoir, Oddballs:

When my sister Vicky and I were teenagers we talked a lot about hating
people. Hating came easily to us. We would be walking down the street,
notice a perfect stranger, and be suddenly struck by how much we hated
that person. And at the dinner table we would go on and on about all
the popular kids we hated at high school. Our father, who has a very logical
mind, sometimes cautioned us about this. “Don’t waste your hate,” he would
say. “Save it up for important things, like your family, or the President.”
We responded by quoting the famous line from Medea: “Loathing is endless.
Hate is a bottomless cup; I pour and pour.”

Yu Hua’s To Live was simple and incredibly moving. The Cup of Fury, Upton Sinclair’s gossipy memoir about how all the writers he knew, himself excepted, drank too much, was strangely entertaining. I think I liked Daisy Miller but I can’t remember a thing about it now. Horse Walks Into a Bar was annoying, built on a single conceit (entire novel as stand-up routine) that got old after, well, less than an entire novel. Beautiful World, Where Are You I already blogged about.

What should I read next year?

Update: Wait, I read Susanna Clarke’s Piranesi this year! I somehow didn’t put it on the list. It actually might have been the book that came closest to pleasure reading for me. Strangely, it was shortlisted for a Hugo this year; I say strangely because it doesn’t read at all to me like SF or fantasy, but rather as fiction that has taken some of the good things from SF without at all feeling like a creature of that genre. Maybe anything with fantastic elements is Hugo-nominable, but in practice, I’ll bet most people don’t see it that way. (Then again: The Good Place won a Hugo!)

Last week, I mentioned some interesting new results in my corner of physics. I’ve now finally read the two papers and watched the recorded talk, so I can satisfy my frustrated commenters.

Quantum mechanics is a very cool topic and I am much less qualified than you would expect to talk about it. I use quantum field theory, which is based on quantum mechanics, so in some sense I use quantum mechanics every day. However, most of the “cool” implications of quantum mechanics don’t come up in my work. All the debates about whether measurement “collapses the wavefunction” are irrelevant when the particles you measure get absorbed in a particle detector, never to be seen again. And while there are deep questions about how a classical world emerges from quantum probabilities, they don’t matter so much when all you do is calculate those probabilities.

They’ve started to matter, though. That’s because quantum field theorists like me have recently started working on a very different kind of problem: trying to predict the output of gravitational wave telescopes like LIGO. It turns out you can do almost the same kind of calculation we’re used to: pretend two black holes or neutron stars are sub-atomic particles, and see what happens when they collide. This trick has grown into a sub-field in its own right, one I’ve dabbled in a bit myself. And it’s gotten my kind of physicists to pay more attention to the boundary between classical and quantum physics.

The thing is, the waves that LIGO sees really are classical. Any quantum gravity effects there are tiny, undetectably tiny. And while this doesn’t have the implications an expert might expect (we still need loop diagrams), it does mean that we need to take our calculations to a classical limit.

Figuring out how to do this has been surprisingly delicate, and full of unexpected insight. A recent example involves two papers, one by Andrea Cristofoli, Riccardo Gonzo, Nathan Moynihan, Donal O’Connell, Alasdair Ross, Matteo Sergola, and Chris White, and one by Ruth Britto, Riccardo Gonzo, and Guy Jehu. At first I thought these were two groups happening on the same idea, but then I noticed Riccardo Gonzo on both lists, and realized the papers were covering different aspects of a shared story. There is another group who happened upon the same story: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele Veneziano. They haven’t published yet, so I’m basing this on the Gonzo et al papers.

The key question each group asked was, what does it take for gravitational waves to be classical? One way to ask the question is to pick something you can observe, like the strength of the field, and calculate its uncertainty. Classical physics is deterministic: if you know the initial conditions exactly, you know the final conditions exactly. Quantum physics is not. What should happen is that if you calculate a quantum uncertainty and then take the classical limit, that uncertainty should vanish: the observation should become certain.

Another way to ask is to think about the wave as made up of gravitons, particles of gravity. Then you can ask how many gravitons are in the wave, and how they are distributed. It turns out that you expect them to be in a coherent state, like a laser, one with a very specific distribution called a Poisson distribution: a distribution in some sense right at the border between classical and quantum physics.

The results of both types of questions were as expected: the gravitational waves are indeed classical. To make this work, though, the quantum field theory calculation needs to have some surprising properties.

If two black holes collide and emit a gravitational wave, you could depict it like this:

where the straight lines are black holes, and the squiggly line is a graviton. But since gravitational waves are made up of multiple gravitons, you might ask, why not depict it with two gravitons, like this?

It turns out that diagrams like that are a problem: they mean your two gravitons are correlated, which is not allowed in a Poisson distribution. In the uncertainty picture, they also would give you non-zero uncertainty. Somehow, in the classical limit, diagrams like that need to go away.

And at first, it didn’t look like they do. You can try to count how many powers of Planck’s constant show up in each diagram. The authors do that, and it certainly doesn’t look like it goes away:

Luckily, these quantum field theory calculations have a knack for surprising us. Calculate each individual diagram, and things look hopeless. But add them all together, and they miraculously cancel. In the classical limit, everything combines to give a classical result.

You can do this same trick for diagrams with more graviton particles, as many as you like, and each time it ought to keep working. You get an infinite set of relationships between different diagrams, relationships that have to hold to get sensible classical physics. From thinking about how the quantum and classical are related, you’ve learned something about calculations in quantum field theory.

That’s why these papers caught my eye. A chunk of my sub-field is needing to learn more and more about the relationship between quantum and classical physics, and it may have implications for the rest of us too. In the future, I might get a bit more qualified to talk about some of the very cool implications of quantum mechanics.

I’m working on a math project involving the periodic table of elements and the Kepler problem—that is, the problem of a particle moving in an inverse square force law. That’s one reason I’ve been blogging about chemistry lately! I hope to tell you all about this project sometime—but right now I just want to say some very basic stuff about the ‘eccentricity vector’.

This vector is a conserved quantity for the Kepler problem. It was named the ‘Runge–Lenz vector’ after Lenz used it in 1924 to study the hydrogen atom in the framework of the ‘old quantum mechanics’ of Bohr and Sommerfeld: Lenz cite Runge’s popular German textbook on vector analysis from 1919, which explains this vector. But Runge never claimed any originality: he attributed this vector to Gibbs, who wrote about it in his book on vector analysis in 1901!

Nowadays many people call it the ‘Laplace–Runge–Lenz vector’, honoring Laplace’s discussion of it in his famous treatise on celestial mechaics in 1799. But in fact this vector goes back at least to Jakob Hermann, who wrote about it in 1710, triggering further work on this topic by Johann Bernoulli in the same year.

Nobody has seen signs of this vector in work before Hermann. So, we might call it the Hermann–Bernoulli–Laplace–Gibbs–Runge–Lenz vector, or just the Hermann vector. But I prefer to call it the eccentricity vector, because for a particle in an inverse square law its magnitude is the eccentricity of that orbit!

Let’s suppose we have a particle whose position obeys this version of the inverse square force law:

where I remove the arrow from a vector when I want to talk about its magnitude. So, I’m setting the mass of this particle equal to 1, along with the constant saying the strength of the force. That’s because I want to keep the formulas clean! With these conventions, the momentum of the particle is

For this system it’s well-known that the following energy is conserved:

as well as the angular momentum vector:

But the interesting thing for me today is the eccentricity vector:

Let’s check that it’s conserved! Taking its time derivative,

But angular momentum is conserved so the second term vanishes, and

so we get

But the inverse square force law says

so

How can we see that this vanishes? Mind you, there are various geometrical ways to think about this, but today I’m in the mood for checking that my skills in vector algebra are sufficient for a brute-force proof—and I want to record this proof so I can see it later!

To get anywhere we need to deal with the cross product in the above formula:

There’s a nice identity for the vector triple product:

I could have fun talking about why this is true, but I won’t now! I’ll just use it:

and plug this into our formula

getting

But look—everything cancels! So

and the eccentricity vector is conserved!

So, it seems that the inverse square force law has 7 conserved quantities: the energy the 3 components of the angular momentum and the 3 components of the eccentricity vector . But they can’t all be independent, since the particle only has 6 degrees of freedom: 3 for position and 3 for momentum. There can be at most 5 independent conserved quantities, since something has to change. So there have to be at least two relations betwen the conserved quantities we’ve found.

The first of these relations is pretty obvious: and are at right angles, so

But wait, why are they at right angles? Because

The first term is orthogonal to because it’s a cross product of and the second is orthogonal to because is a cross product of and .

The second relation is a lot less obvious, but also more interesting. Let’s take the dot product of with itself:

or in other words,

But remember this nice cross product identity:

Since and are at right angles this gives

so

Then we can use the cyclic identity for the scalar triple product:

to rewrite this as

or simply

or even better,

But this means that

which is our second relation between conserved quantities for the Kepler problem!

This relation makes a lot of sense if you know that is the eccentricity of the orbit. Then it implies:

• if then and the orbit is a hyperbola.

• if then and the orbit is a parabola.

• if then and the orbit is an ellipse (or circle).

But why is the eccentricity? And why does the particle move in a hyperbola, parabola or ellipse in the first place? We can show both of these things by taking the dot product of and

Using the cyclic property of the scalar triple product we can rewrite this as

Now, we know that moves in the plane orthogonal to . In this plane, which contains the vector the equation defines a conic of eccentricity . I won’t show this from scratch, but it may seem more familiar if we rotate the whole situation so this plane is the plane and points in the direction. Then in polar coordinates this equation says

or

This is well-known, at least among students of physics who have solved the Kepler problem, to be the equation of a conic of eccentricity .

Another thing that’s good to do is define a rescaled eccentricity vector. In the case of elliptical orbits, where we define this by

Then we can take our relation

and rewrite it as

and then divide by getting

or

This suggests an interesting similarity between and which turns out to be very important in a deeper understanding of the Kepler problem. And with more work, you can use this idea to show that is the Hamiltonian for a free particle on the 3-sphere. But more about that some other time, I hope!

For now, you might try this:

• Wikipedia, Laplace–Runge–Lenz vector.

and of course this:

• The Kepler problem (part 1).

Mark Meckes and I have a new paper on magnitude!

Tom Leinster and Mark Meckes, Spaces of extremal magnitude. arXiv:2112.12889, 2021.

It’s a short one: 7 pages. But it answers two questions that have been lingering since the story of magnitude began.

In the beginning, there was the magnitude of finite metric spaces, a special case of the magnitude of finite enriched categories. Simon Willerton and I did some early investigation of how to extend magnitude from finite to infinite metric spaces — or more specifically, compact metric spaces. Interesting as the results of that investigation were — and we found out lots of cool stuff about the magnitude of spheres, Cantor sets, and so on — there was something ad hoc at its heart: the very definition of the magnitude of a compact space.

That foundational problem was largely settled by Mark in a pair of papers about ten years ago. They show definitively how to generalize magnitude from finite to compact metric spaces, at least under the mild technical condition that the spaces concerned are positive definite (which I won’t go into here). There are several different ways that one might imagine extending the definition of magnitude from finite to compact spaces, and Mark proved that under this hypothesis, they all give exactly the same result.

But some questions remained. A particularly prominent one was this. By definition, the magnitude of a nonempty positive definite compact metric space lies in the interval $[1, \infty]$. Given any $m \in [1, \infty)$, it’s easy to find a space with magnitude $m$. But is there anything with magnitude $\infty$?

This question has been open since 2010, but we settle it in our new paper. The answer is yes. One might have guessed that no such space exists: after all, compactness is a kind of finiteness condition, so perhaps it wouldn’t be surprising if it implied finiteness of magnitude. But our counterexample reveals the fuzziness in that thinking.

The counterexample is an infinite-dimensional simplex in sequence space $\ell^1$, as follows.

First imagine that we’ve chosen two positive real numbers, $a_1$ and $a_2$, and consider the convex hull of the points

$$(0, 0), \ (a_1, 0), \ (0, a_2)$$

in $\mathbb{R}^2$. It’s a 2-simplex. Similarly, given positive reals $a_1, a_2, a_3$, the convex hull of

$$(0, 0, 0), \ (a_1, 0, 0), \ (0, a_2, 0), (0, 0, a_3)$$

in $\mathbb{R}^3$ is a 3-simplex.

Now do the same thing in $\ell^1$ for an infinite sequence $a_1, a_2, \ldots$. Write $X$ for the convex hull, with the metric from the $\ell^1$ norm. (Or to be precise, it’s the closed convex hull.) Now:

• $X$ is compact $\iff$ $a_n \to 0$ as $n \to \infty$

• $X$ has finite magnitude $\iff$ $\sum_n a_n \lt \infty$.

So we get a compact space of infinite magnitude by taking any sequence $(a_n)$ of positive reals that converges to $0$ but has infinite sum.

In other words, the gap between compactness and finite magnitude is the same as the gap we emphasize when we teach a first course on sequences and series: for a series to converge is a stronger condition than for its sequence of entries to converge to $0$.

The second question we settle in our paper is about spaces of minimal magnitude. I mentioned earlier that the magnitude of a nonempty positive definite compact metric space (let me just say “space”) is in the interval $[1, \infty]$. We now know that there are spaces with magnitude exactly $\infty$. It’s also trivial — once you have the definitions! — that for any infinite space $X$, the magnitude $|t X|$ of the scaled-up space $t X$ converges to $\infty$ as $t \to \infty$. But what about the other end of the scale: spaces of magnitude close to or equal to $1$?

As it turns out, the situation for $1$ is the opposite way round from the one for $\infty$. What I mean is this. It’s easy to say which spaces have magnitude exactly $1$: it’s just the one-point space. Now given any space $X$, the rescaled space $t X$ looks more and more like a point as $t \to 0$, so you might expect that

$$\lim_{t \to 0} |t X| = 1.$$

But in fact, this is where the complexity lies. Long ago, Simon found an example of a 6-point space $X$ such that $\lim_{t \to 0} |t X| = 6/5$. So we have a nontrivial situation on our hands.

Say that a space $X$ has the one-point property if $\lim_{t \to 0} |t X| = 1$. Although not every space has the one-point property, there are lots that do. For instance, it’s not so hard to show that every compact subset of $\mathbb{R}^n$ with the taxicab metric (the metric induced by the 1-norm) has the one-point property. And with much more effort, it was shown that compact subsets of $\mathbb{R}^n$ with the Euclidean metric have the one-point property too, first by Juan Antonio Barceló and Tony Carbery, then, by different arguments, by Simon and by Mark.

In our new paper, we find a single sufficient condition that unifies these results:

Let $V$ be a finite-dimensional normed vector space that is positive definite as a metric space. Then every nonempty compact subset of $V$ has the one-point property.

For instance, we could take $V$ to be $\mathbb{R}^N$ with either the taxicab or Euclidean metric, or more generally, we could give it the metric induced by the $p$-norm for any $p \in [1, 2]$. An equivalent condition on $V$ is that it’s isometrically isomorphic to a linear subspace of $L^1[0, 1]$, which gives a 1-norm flavour to this theorem too.

So our paper consists of these two theorems: one on spaces with magnitude $\infty$, and one on spaces with magnitude close to $1$.

The proofs have something in common. Longtime Café readers will remember that around the time when magnitude got going, Simon and I made a conjecture about the magnitude of convex subsets of Euclidean space. That conjecture turned out to be false, although several aspects of it were correct.

But less publicized was an analogous conjecture for convex subsets of $\mathbb{R}^N$ with the taxicab metric. And this conjecture turned out to be true! Or at least, true when the convex set $X$ has nonempty interior. The result is that

$$|X| = \sum_{i = 0}^N 2^{-i} V'_i(X)$$

(Theorem 4.6 here), where $V'_i$ is a 1-norm analogue of the $i$th intrinsic volume. For example, this implies that the magnitude function $|t X|$ of $X$ is a polynomial:

$$|t X| = \sum_{i = 0}^N 2^{-i} V'_i(X) \cdot t^i.$$

And even if $X$ has empty interior, we still have an inequality one way round: $|X| \leq \sum \ldots$. (It may be an equality for all we know — that remains unsettled.)

In any case, this result gets used in the proofs of both theorems in our new paper: first, to find the magnitude of that infinite-dimensional simplex, and second, to bound the magnitude of polytopes, which turns out to be the key to the whole thing.

By now, I feel like this post must already be nearly as long as the paper itself, so I’ll link to it once more and stop.

Asgar Jamneshan and myself have just uploaded to the arXiv our preprint “The inverse theorem for the Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches“. This paper, which is a companion to another recent paper of ourselves and Or Shalom, studies the inverse theory for the third Gowers uniformity norm

Theorem 1 (Inverse theorem for ) Let be a finite abelian group, and let be a -bounded function with for some . Then:

• (i) (Correlation with locally quadratic phase) There exists a regular Bohr set with and , a locally quadratic function , and a function such that

  

• (ii) (Correlation with nilsequence) There exists an explicit degree two filtered nilmanifold of dimension , a polynomial map , and a Lipschitz function of constant such that

  

Such a theorem was proven by Ben Green and myself in the case when was odd, and by Samorodnitsky in the -torsion case . In all cases one uses the “higher order Fourier analysis” techniques introduced by Gowers. After some now-standard manipulations (using for instance what is now known as the Balog-Szemerédi-Gowers lemma), one arrives (for arbitrary ) at an estimate that is roughly of the form

So the key step is to obtain a representation of the form (1), possibly after shrinking the Bohr set a little if needed. This has been done in the literature in two ways:

• When is odd, one has the ability to divide by , and on the set one can establish (1) with . (This is similar to how in single variable calculus the function is a function whose second derivative is equal to .)
• When , then after a change of basis one can take the Bohr set to be for some , and the bilinear form can be written in coordinates as

  

  for some with . The diagonal terms cause a problem, but by subtracting off the rank one form one can write

  

  on the orthogonal complement of for some coefficients which now vanish on the diagonal: . One can now obtain (1) on this complement by taking

  

In our paper we can now treat the case of arbitrary finite abelian groups , by means of the following two new ingredients:

• (i) Using some geometry of numbers, we can lift the group to a larger (possibly infinite, but still finitely generated) abelian group with a projection map , and find a globally bilinear map on the latter group, such that one has a representation

  

  of the locally bilinear form by the globally bilinear form when are close enough to the origin.
• (ii) Using an explicit construction, one can show that every globally bilinear map has a representation of the form (1) for some globally quadratic function .

To illustrate (i), consider the Bohr set in (where denotes the distance to the nearest integer), and consider a locally bilinear form of the form for some real number and all integers (which we identify with elements of . For generic , this form cannot be extended to a globally bilinear form on ; however if one lifts to the finitely generated abelian group

To illustrate (ii), the key case turns out to be when is a cyclic group , in which case will take the form

This concludes the Fourier-analytic proof of Theorem 1. In this paper we also give an ergodic theory proof of (a qualitative version of) Theorem 1(ii), using a correspondence principle argument adapted from this previous paper of Ziegler, and myself. Basically, the idea is to randomly generate a dynamical system on the group , by selecting an infinite number of random shifts , which induces an action of the infinitely generated free abelian group on by the formula

This transference principle approach seems to work well for the higher step cases (for instance, the stability of polynomials result is known in arbitrary degree); the main difficulty is to establish a suitable higher step inverse theorem in the ergodic theory setting, which we hope to do in future research.

A search for an unexpected asymmetry in the production of e⁺μ^– and e^–μ⁺ pairs in proton-proton collisions recorded by the ATLAS detector at √s = 13 TeV

This paper (arXiv:2112.08090), written by the ATLAS Collaboration, is original and interesting. The numbers of events with (e⁺μ^–) and with (e^–μ⁺) should be the same, according to the standard model (SM). So, if they are observed not to be the same, then new physics is at play. This specific idea was first proposed by Lester and Brunt: (arXiv:1612.02697) who focused on R-parity violating supersymmetric models. I will focus on the experimental analysis and skip the theoretical models.

The results are formulated in terms of a ratio, ρ, which is N(e⁺mu^–) / N(e^–mu⁺). Interestingly, the case ρ > 1 is easier to investigate than ρ < 1, and ATLAS does not derive results for the latter case. The problem is that non-prompt (a.k.a. “fake”) electrons are much more common than non-prompt muons, and W⁺are more common than W^–, so W⁺ → μ⁺ and fake e^– is more common than W⁺ → e⁺ and fake μ^–. This bias is partially corrected for, and any new physics leading to ρ > 1 needs to be strong enough to overcome the remaining bias. The selection of electrons and muons is not unusual, and kinematically, they must have p_T > 25 GeV and |η| < 2.47.

The bias (i.e., prevalence) of e^–_fakeμ⁺_real over e⁺_fakemu^–_real is handled by subtracting the appropriate number of background events (i.e., those with fake leptons) using a data-driven method. The contamination of the main signal region by fake leptons is small — only 2%. A very small effect due to the toroidal magnetic fields is corrected for by applying weights to events.

The event selection is rather simple, which is part of what makes this analysis appealing. An opposite-sign, electron-muon pair is required. In addition, Σ_T, the sum of the transverse masses for the two leptons, must be at least 200 GeV. A subset of such events containing at least one jet (with p_T > 20 GeV) is also analyzed. (Additional signal regions that are more model-inspired are defined, but I won’t be discussing them here.). The event yields are studied as functions of either M_T2 or H_p, where H_p is just the scalar sum of the p_T of the electron, muon, and leading jet.

Roughly 100k events are selected. The main contribution comes from top quark pair events — a process that is very well understood at the LHC. Diboson production is also important at the edge of the kinematic range. This nice plot shows the yield as a function of H_p.

By eye there is no difference in yields for the two final states e^–μ⁺ and e⁺μ^–. A careful statistical treatment based on the likelihood for Poisson distributions in each bin leads to the following quantitative result: ρ = 0.987 ± 0.022 where I have neglected a small asymmetry in the error bar. Clearly, this measurement of ρ is consistent with unity and therefore with SM expectations. There is no evidence for ρ>1. Statistical p-values for all bins are quite reasonable — the largest deviation from unity is a downward fluctuation (3.1σ) in a single bin.

The authors go one to derive upper limits on the number of new physics signal events as a function of how those events are shared between the two final states (i.e., the fraction z that enter e^–μ⁺). I find it striking that in some regions this analysis is sensitive to mere tens of events.

Of course, it would be wonderful to have a deviation from the SM expectation. Unfortunately, we do not have that here.

I like this analysis for its relatively inclusive nature and straight forward simplicity. There are many SM analyses which hope to see a deviation in a kinematic quantity such as jet p_T or S_T (which is similar to H_p here). What is relatively new is that this analysis looks at charge and flavor in the hopes of observing a telltale deviation. The two specific models considered in this paper — RPV-supersymmetry and leptoquarks — serve to demonstrate that new physics could lead to an observable deviation. I’m sure the authors will expand their final state and maybe tease out hints for new physics in a more sophisticated analysis.

https://arxiv.org/trackback/2112.08090