Planet Musings

Any action of a finite group $G$ on a finite set $X$ gives a linear representation of $G$ on the vector space with basis $X$. This is called a ‘permutation represention’. And this raises a natural question: how many representations of finite groups are permutation representations?

Most representation are not permutation representations, since every permutation representation has a vector fixed by all elements of $G$, namely the vector that’s the sum of all elements of $X$. In other words, every permutation representation has a 1-dimensional trivial rep sitting inside it.

But what if we could ‘subtract off’ this trivial representation?

There are different levels of subtlety with which we can do this. For example, we can decategorify, and let:

• the Burnside ring of $G$ be the ring $A(G)$ of formal differences of isomorphism classes of actions of $G$ on finite sets;

• the representation ring of $G$ be the ring $R(G)$ of formal differences of isomorphism classes of finite-dimensional representations of $G$.

There’s an obvious map $\beta : A(G) \to R(G)$, since any action of $G$ on a finite set gives a permutation representation of $G$ on the vector space with basis $X$.

So now we can ask: is $\beta$ typically surjective, or typically not surjective?

In fact everything depends on what field $k$ we’re using for our vector spaces! For starters let’s take $k = \mathbb{Q}$.

Here’s a list of finite groups where the map $\beta : A(G) \to R(G)$ from the Burnside ring to the representation ring is known to be surjective, taken from the nLab article Permutation representations:

1. cyclic groups,

2. symmetric groups,

3. $p$-groups (that is, groups whose order is a power of the prime $p$),

4. binary dihedral groups $2 D_{2n}$ for (at least) $2 n \leq 12$,

5. the binary tetrahedral group, binary octahedral group, and binary icosahedral group,

6. the general linear group $GL(2,\mathbb{F}_3)$.

Now, these may seem like rather special classes of groups, but devoted readers of this blog know that most finite groups have order that’s a power of 2 I don’t think this has been proved yet, but it’s obviously true empirically, and we also have a good idea as to why:

• Tom Leinster, Almost all of the first 50 billion groups have order 1024.

So, map $\beta$ from the Burnside ring to the representation ring is surjective for most finite groups!

That’s if we work over $\mathbb{Q}$. If we work over $\mathbb{C}$ the situation flip-flops, and I believe $\beta$ is usually not surjective. It’s already not surjective for cyclic groups bigger than $\mathbb{Z}/2$!

Why? Because $\mathbb{Z}/n$ has a 1-dimensional representation where the generator acts as multiplication by a primitive $n$th root of unity, and since this is not a rational number when $n > 2$, this representation is not definable over $\mathbb{Q}$. Thus (one can show), there’s no way to get this representation as a formal difference of permutation representations, since those are always definable over $\mathbb{Q}$.

And this phenomenon of needing roots of unity to define representations is not special to cyclic groups: it happens for most finite groups as hinted at by Artin’s theorem on induced characters.

An example

Now, you’ll notice that I didn’t yet give an example of a finite group where the map from the Burnside ring to the representation ring fails to be surjective when we work over $\mathbb{Q}$. I recently had a lot of fun doing an exercise in Serre’s book Linear Representations of Finite Groups, where he asks us in Exercise 13.4 to prove that $\mathbb{Z}/3 \times Q_8$ is such a group. Here $Q_8$ is the quaternion group, an 8-element subgroup of the invertible quaternions:

$$Q_8 = \{ \pm 1, \pm i , \pm j, \pm k \}$$

I couldn’t resist trying to understand why this is the counterexample Serre gave. For one thing, $\mathbb{Z}/3 \times Q_8$ has 24 elements, and I love the number 24. For another, I love the quaternions.

Right now I believe $\mathbb{Z}/3 \times Q_8$ is the smallest group for which $\beta : A(G) \to R(G)$ is non-surjective. I’ve asked on MathOverflow, but so far nobody has answered that question. I got a lot of useful information about my other question, though: is $\beta$ surjective for most finite groups, or not?

To solve Serre’s problem 13.4, he asked you to use exercise 13.3. Here is my reformulation and solution of that problem. I believe any field $k$ characteristic zero would work for this theorem:

Theorem. Suppose $G$ is a finite group with a linear representation $\rho$ such that:

1. $\rho$ is irreducible and faithful

2. every subgroup of $G$ is normal

3. $\rho$ appears with multiplicity $n \ge 2$ in the regular representation of $G$.

Then the map from the Burnside ring of $G$ to the representation ring $R(G)$ of $G$ is not surjective.

Proof. It suffices to prove that the multiplicity of $\rho$ in any permutation representation of $G$ is a multiple of $n$, so that the class $[\rho] \in R(G)$ cannot be in the image of $R(G)$.

Since every finite $G$-set is a coproduct of transitive actions of $G$, which are isomorphic to actions on $G/H$ for subgroups $H$ of $G$, every permutation representation of $G$ is a direct sum of those on spaces of the form $k[G/H]$. (This is my notation for the vector space with basis $G/H$.) Thus, it suffices to show that the multiplicity of $\rho$ in the representation on $k[G/H]$ is $n$ if $H$ is the trivial group, and $0$ otherwise.

The former holds by assumption 3. For the latter, suppose $H$ is a nontrivial subgroup of $G$. Because $H$ is normal by assumption 2, every element $h \in H$ acts trivially on $k[G/H]$: we can see this by letting $h$ act on an arbitrary basis element $g H = H g \in G/H$:

$$h H g = H g .$$

Since $H$ is nontrivial, it contains elements $h \ne 1$ that act trivially on $k[G/H]$. But no $h \ne 1$ can act trivially on $\rho$ because $\rho$ is faithful, by assumption 1. Thus $\rho$ cannot be a subrepresentation of $k[G/H]$. That is, $\rho$ appears with multiplicity $0$ in $k[G/H]$. ▮

Serre’s exercise 13.4 is to show the group $G = \mathbb{Z}/3 \times Q_8$ obeys the conditions of this proposition. As a hint, Serre suggests to embed $\mathbb{Z}/3$ and $Q_8$ in the multiplicative group of the algebra $\mathbb{H}_{\mathbb{Q}}$ (the quaternions defined over $\mathbb{Q}$). By letting $\mathbb{Z}/3$ act by left multiplication and $Q_8$ act by right multiplication, one obtains a 4-dimensional irreducible representation $\rho$ of $G$ which appears with multiplicity $n \ge 2$ in the regular representation. Furthermore $\rho$ is faithful and irreducible, and every subgroup of $G$ is normal.

Grant Sanderson (who runs, and creates most of the content for, the website and Youtube channel 3blue1brown) has been collaborating with myself and others (including my coauthor Tanya Klowden) on producing a two-part video giving an account of some of the history of the cosmic distance ladder, building upon a previous public lecture I gave on this topic, and also relating to a forthcoming popular book with Tanya on this topic. The first part of this video is available here; the second part is available here.

The videos were based on a somewhat unscripted interview that Grant conducted with me some months ago, and as such contained some minor inaccuracies and omissions (including some made for editing reasons to keep the overall narrative coherent and within a reasonable length). They also generated many good questions from the viewers of the Youtube video. I am therefore compiling here a “FAQ” of various clarifications and corrections to the videos; this was originally placed as a series of comments on the Youtube channel, but the blog post format here will be easier to maintain going forward. Some related content will also be posted on the Instagram page for the forthcoming book with Tanya.

Questions on the two main videos are marked with an appropriate timestamp to the video.

Comments on part 1 of the video

• 4:26 Did Eratosthenes really check a local well in Alexandria?
  
  This was a narrative embellishment on my part. Eratosthenes’s original work is lost to us. The most detailed contemperaneous account, by Cleomedes, gives a simplified version of the method, and makes reference only to sundials (gnomons) rather than wells. However, a secondary account of Pliny states (using this English translation), “Similarly it is reported that at the town of Syene, 5000 stades South of Alexandria, at noon in midsummer no shadow is cast, and that in a well made for the sake of testing this the light reaches to the bottom, clearly showing that the sun is vertically above that place at the time”. However, no mention is made of any well in Alexandria in either account.
• 4:50 How did Eratosthenes know that the Sun was so far away that its light rays were close to parallel?
  
  This was not made so clear in our discussions or in the video (other than a brief glimpse of the timeline at 18:27), but Eratosthenes’s work actually came after Aristarchus, so it is very likely that Eratosthenes was aware of Aristarchus’s conclusions about how distant the Sun was from the Earth. Even if Aristarchus’s heliocentric model was disputed by the other Greeks, at least some of his other conclusions appear to have attracted some support. Also, after Eratosthenes’s time, there was further work by Greek, Indian, and Islamic astronomers (such as Hipparchus, Ptolemy, Aryabhata, and Al-Battani) to measure the same distances that Aristarchus did, although these subsequent measurements for the Sun also were somewhat far from modern accepted values.
  
• 5:17 Is it completely accurate to say that on the summer solstice, the Earth’s axis of rotation is tilted “directly towards the Sun”?
  
  Strictly speaking, “in the direction towards the Sun” is more accurate than “directly towards the Sun”; it tilts at about 23.5 degrees towards the Sun, but it is not a total 90-degree tilt towards the Sun.
  
• 5:39 Wait, aren’t there two tropics? The tropic of Cancer and the tropic of Capricorn?
  
  Yes! This corresponds to the two summers Earth experiences, one in the Northern hemisphere and one in the Southern hemisphere. The tropic of Cancer, at a latitude of about 23 degrees north, is where the Sun is directly overhead at noon during the Northern summer solstice (around June 21); the tropic of Capricorn, at a latitude of about 23 degrees south, is where the Sun is directly overhead at noon during the Southern summer solstice (around December 21). But Alexandria and Syene were both in the Northern Hemisphere, so it is the tropic of Cancer that is relevant to Eratosthenes’ calculations.
  
• 5:41 Isn’t it kind of a massive coincidence that Syene was on the tropic of Cancer?
  
  Actually, Syene (now known as Aswan) was about half a degree of latitude away from the tropic of Cancer, which was one of the sources of inaccuracy in Eratosthenes’ calculations.  But one should take the “look-elsewhere effect” into account: because the Nile cuts across the tropic of Cancer, it was quite likely to happen that the Nile would intersect the tropic near some inhabited town.  It might not necessarily have been Syene, but that would just mean that Syene would have been substituted by this other town in Eratosthenes’s account.  
  
  On the other hand, it was fortunate that the Nile ran from South to North, so that distances between towns were a good proxy for the differences in latitude.  Apparently, Eratosthenes actually had a more complicated argument that would also work if the two towns in question were not necessarily oriented along the North-South direction, and if neither town was on the tropic of Cancer; but unfortunately the original writings of Eratosthenes are lost to us, and we do not know the details of this more general argument. (But some variants of the method can be found in later work of Posidonius, Aryabhata, and others.)
  
  Nowadays, the “Eratosthenes experiment” is run every year on the March equinox, in which schools at the same longitude are paired up to measure the elevation of the Sun at the same point in time, in order to obtain a measurement of the circumference of the Earth.  (The equinox is more convenient than the solstice when neither location is on a tropic, due to the simple motion of the Sun at that date.) With modern timekeeping, communications, surveying, and navigation, this is a far easier task to accomplish today than it was in Eratosthenes’ time.
  
• 6:30 I thought the Earth wasn’t a perfect sphere. Does this affect this calculation?
  
  Yes, but only by a small amount. The centrifugal forces caused by the Earth’s rotation along its axis cause an equatorial bulge and a polar flattening so that the radius of the Earth fluctuates by about 20 kilometers from pole to equator. This sounds like a lot, but it is only about 0.3% of the mean Earth radius of 6371 km and is not the primary source of error in Eratosthenes’ calculations.
  
• 7:27 Are the riverboat merchants and the “grad student” the leading theories for how Eratosthenes measured the distance from Alexandria to Syene?
  
  There is some recent research that suggests that Eratosthenes may have drawn on the work of professional bematists (step measurers – a precursor to the modern profession of surveyor) for this calculation. This somewhat ruins the “grad student” joke, but perhaps should be disclosed for the sake of completeness.
  
• 8:51 How long is a “lunar month” in this context? Is it really 28 days?
  
  In this context the correct notion of a lunar month is a “synodic month” – the length of a lunar cycle relative to the Sun – which is actually about 29 days and 12 hours. It differs from the “sidereal month” – the length of a lunar cycle relative to the fixed stars – which is about 27 days and 8 hours – due to the motion of the Earth around the Sun (or the Sun around the Earth, in the geocentric model). [A similar correction needs to be made around 14:59, using the synodic month of 29 days and 12 hours rather than the “English lunar month” of 28 days (4 weeks).]
  
• 10:47 Is the time taken for the Moon to complete an observed rotation around the Earth slightly less than 24 hours as claimed?
  
  Actually, I made a sign error: the lunar day (also known as a tidal day) is actually 24 hours and 50 minutes, because the Moon rotates in the same direction as the spinning of Earth around its axis. The animation therefore is also moving in the wrong direction as well (related to this, the line of sight is covering up the Moon in the wrong direction to the Moon rising at around 10:38).
  
• 11:32 Is this really just a coincidence that the Moon and Sun have almost the same angular width?
  
  I believe so. First of all, the agreement is not that good: due to the non-circular nature of the orbit of the Moon around the Earth, and Earth around the Sun, the angular width of the Moon actually fluctuates to be as much as 10% larger or smaller than the Sun at various times (cf. the “supermoon” phenomenon). All other known planets with known moons do not exhibit this sort of agreement, so there does not appear to be any universal law of nature that would enforce this coincidence. (This is in contrast with the empirical fact that the Moon always presents the same side to the Earth, which occurs in all other known large moons (as well as Pluto), and is well explained by the physical phenomenon of tidal locking.)
  
  On the other hand, as the video hopefully demonstrates, the existence of the Moon was extremely helpful in allowing the ancients to understand the basic nature of the solar system. Without the Moon, their task would have been significantly more difficult; but in this hypothetical alternate universe, it is likely that modern cosmology would have still become possible once advanced technology such as telescopes, spaceflight, and computers became available, especially when combined with the modern mathematics of data science. Without giving away too many spoilers, a scenario similar to this was explored in the classic short story and novel “Nightfall” by Isaac Asimov.
• 12:58 Isn’t the illuminated portion of the Moon, as well as the visible portion of the Moon, slightly smaller than half of the entire Moon, because the Earth and Sun are not an infinite distance away from the Moon?
  
  Technically yes (and this is actually for a very similar reason to why half Moons don’t quite occur halfway between the new Moon and the full Moon); but this fact turns out to have only a very small effect on the calculations, and is not the major source of error. In reality, the Sun turns out to be about 86,000 Moon radii away from the Moon, so asserting that half of the Moon is illuminated by the Sun is actually a very good first approximation. (The Earth is “only” about 220 Moon radii away, so the visible portion of the Moon is a bit more noticeably less than half; but this doesn’t actually affect Aristarchus’s arguments much.)
  
• 13:27 What is the difference between a half Moon and a quarter Moon?
  
  If one divides the lunar month, starting and ending at a new Moon, into quarters (weeks), then half moons occur both near the end of the first quarter (a week after the new Moon, and a week before the full Moon), and near the end of the third quarter (a week after the full Moon, and a week before the new Moon). So, somewhat confusingly, half Moons come in two types, known as “first quarter Moons” and “third quarter Moons”.
• 14:49 I thought the sine function was introduced well after the ancient Greeks.
  
  It’s true that the modern sine function only dates back to the Indian and Islamic mathematical traditions in the first millennium CE, several centuries after Aristarchus.  However, he still had Euclidean geometry at his disposal, which provided tools such as similar triangles that could be used to reach basically the same conclusions, albeit with significantly more effort than would be needed if one could use modern trigonometry.
  
  On the other hand, Aristarchus was somewhat hampered by not knowing an accurate value for , which is also known as Archimedes’ constant: the fundamental work of Archimedes on this constant actually took place a few decades after that of Aristarchus!
  
• 15:17 I plugged in the modern values for the distances to the Sun and Moon and got 18 minutes for the discrepancy, instead of half an hour.
  
  Yes; I quoted the wrong number here. In 1630, Godfried Wendelen replicated Aristarchus’s experiment. With improved timekeeping and the then-recent invention of the telescope, Wendelen obtained a measurement of half an hour for the discrepancy, which is significantly better than Aristarchus’s calculation of six hours, but still a little bit off from the true value of 18 minutes. (As such, Wendelinus’s estimate for the distance to the Sun was 60% of the true value.)
  
• 15:27 Wouldn’t Aristarchus also have access to other timekeeping devices than sundials?
  
  Yes, for instance clepsydrae (water clocks) were available by that time; but they were of limited accuracy. It is also possible that Aristarchus could have used measurements of star elevations to also estimate time; it is not clear whether the astrolabe or the armillary sphere was available to him, but he would have had some other more primitive astronomical instruments such as the dioptra at his disposal. But again, the accuracy and calibration of these timekeeping tools would have been poor.
  
  However, most likely the more important limiting factor was the ability to determine the precise moment at which a perfect half Moon (or new Moon, or full Moon) occurs; this is extremely difficult to do with the naked eye. (The telescope would not be invented for almost two more millennia.)
  
• 17:37 Could the parallax problem be solved by assuming that the stars are not distributed in a three-dimensional space, but instead on a celestial sphere?
  
  Putting all the stars on a fixed sphere would make the parallax effects less visible, as the stars in a given portion of the sky would now all move together at the same apparent velocity – but there would still be visible large-scale distortions in the shape of the constellations because the Earth would be closer to some portions of the celestial sphere than others; there would also be variability in the brightness of the stars, and (if they were very close) the apparent angular diameter of the stars. (These problems would be solved if the celestial sphere was somehow centered around the moving Earth rather than the fixed Sun, but then this basically becomes the geocentric model with extra steps.)
  
• 18:29 Did nothing of note happen in astronomy between Eratosthenes and Copernicus?
  
  Not at all! There were significant mathematical, technological, theoretical, and observational advances by astronomers from many cultures (Greek, Islamic, Indian, Chinese, European, and others) during this time, for instance improving some of the previous measurements on the distance ladder, a better understanding of eclipses, axial tilt, and even axial precession, more sophisticated trigonometry, and the development of new astronomical tools such as the astrolabe. See for instance this “deleted scene” from the video, as well as the FAQ entry for 14:49 for this video and 24:54 for the second video. But in order to make the overall story of the cosmic distance ladder fit into a two-part video, we chose to focus primarily on the first time each rung of the ladder was climbed.
  
• 18:30 Is that really Kepler’s portrait?
  
  We have since learned that this portrait was most likely painted in the 19th century, and may have been based more on Kepler’s mentor, Michael Mästlin. A more commonly accepted portrait of Kepler may be found at his current Wikipedia page.
  
• 19:07 Isn’t it tautological to say that the Earth takes one year to perform a full orbit around the Sun?
  
  Technically yes, but this is an illustration of the philosophical concept of “referential opacity“: the content of a sentence can change when substituting one term for another (e.g., “1 year” and “365 days”), even when both terms refer to the same object. Amusingly, the classic illustration of this, known as Frege’s puzzles, also comes from astronomy: it is an informative statement that Hesperus (the evening star) and Phosphorus (the morning star, also known as Lucifer) are the same object (which nowadays we call Venus), but it is a mere tautology that Hesperus and Hesperus are the same object: changing the reference from Phosphorus to Hesperus changes the meaning.
  
• 19:10 How did Copernicus figure out the crucial fact that Mars takes 687 days to go around the Sun? Was it directly drawn from Babylonian data?
  
  Technically, Copernicus drew from tables by European astronomers that were largely based on earlier tables from the Islamic golden age, which in turn drew from earlier tables by Indian and Greek astronomers, the latter of which also incorporated data from the ancient Babylonians, so it is more accurate to say that Copernicus relied on centuries of data, at least some of which went all the way back to the Babylonians. Among all of this data was the times when Mars was in opposition to the Sun; if one imagines the Earth and Mars as being like runners going around a race track circling the Sun, with Earth on an inner track and Mars on an outer track, oppositions are analogous to when the Earth runner “laps” the Mars runner. From the centuries of observational data, such “laps” were known to occur about once every 780 days (this is known as the synodic period of Mars). Because the Earth takes roughly 365 days to perform a “lap”, it is possible to do a little math and conclude that Mars must therefore complete its own “lap” in 687 days (this is known as the sidereal period of Mars). (See also this post on the cosmic distance ladder Instagram for some further elaboration.)
  
• 20:52 Did Kepler really steal data from Brahe?
  
  The situation is complex. When Kepler served as Brahe’s assistant, Brahe only provided Kepler with a limited amount of data, primarily involving Mars, in order to confirm Brahe’s own geo-heliocentric model. After Brahe’s death, the data was inherited by Brahe’s son-in-law and other relatives, who intended to publish Brahe’s work separately; however, Kepler, who was appointed as Imperial Mathematician to succeed Brahe, had at least some partial access to the data, and many historians believe he secretly copied portions of this data to aid his own research before finally securing complete access to the data from Brahe’s heirs after several years of disputes. On the other hand, as intellectual property rights laws were not well developed at this time, Kepler’s actions were technically legal, if ethically questionable.
  
• 21:39 What is that funny loop in the orbit of Mars?
  
  This is known as retrograde motion. This arises because the orbital velocity of Earth (about 30 km/sec) is a little bit larger than that of Mars (about 24 km/sec). So, in opposition (when Mars is in the opposite position in the sky than the Sun), Earth will briefly overtake Mars, causing its observed position to move westward rather than eastward. But in most other times, the motion of Earth and Mars are at a sufficient angle that Mars will continue its apparent eastward motion despite the slightly faster speed of the Earth.
  
• 21:59 Couldn’t one also work out the direction to other celestial objects in addition to the Sun and Mars, such as the stars, the Moon, or the other planets?  Would that have helped?
  
  Actually, the directions to the fixed stars were implicitly used in all of these observations to determine how the celestial sphere was positioned, and all the other directions were taken relative to that celestial sphere.  (Otherwise, all the calculations would be taken on a rotating frame of reference in which the unknown orbits of the planets were themselves rotating, which would have been an even more complex task.)  But the stars are too far away to be useful as one of the two landmarks to triangulate from, as they generate almost no parallax and so cannot distinguish one location from another.
  
  Measuring the direction to the Moon would tell you which portion of the lunar cycle one was in, and would determine the phase of the Moon, but this information would not help one triangulate, because the Moon’s position in the heliocentric model varies over time in a somewhat complicated fashion, and is too tied to the motion of the Earth to be a useful “landmark” to one to determine the Earth’s orbit around the Sun.
  
  In principle, using the measurements to all the planets at once could allow for some multidimensional analysis that would be more accurate than analyzing each of the planets separately, but this would require some sophisticated statistical analysis and modeling, as well as non-trivial amounts of compute – neither of which were available in Kepler’s time.
  
• 22:57 Can you elaborate on how we know that the planets all move on a plane?
  
  The Earth’s orbit lies in a plane known as the ecliptic (it is where the lunar and solar eclipses occur). Different cultures have divided up the ecliptic in various ways; in Western astrology, for instance, the twelve main constellations that cross the ecliptic are known as the Zodiac. The planets can be observed to only wander along the Zodiac, but not other constellations: for instance, Mars can be observed to be in Cancer or Libra, but never in Orion or Ursa Major. From this, one can conclude (as a first approximation, at least), that the planets all lie on the ecliptic.
  
  However, this isn’t perfectly true, and the planets will deviate from the ecliptic by a small angle known as the ecliptic latitude. Tycho Brahe’s observations on these latitudes for Mars were an additional useful piece of data that helped Kepler complete his calculations (basically by suggesting how to join together the different “jigsaw pieces”), but the math here gets somewhat complicated, so the story here has been somewhat simplified to convey the main ideas.
  
• 23:04 What are the other universal problem solving tips?
  
  Grant Sanderson has a list (in a somewhat different order) in this previous video.
• 23:28 Can one work out the position of Earth from fixed locations of the Sun and Mars when the Sun and Mars are in conjunction (the same location in the sky) or opposition (opposite locations in the sky)?
  
  Technically, these are two times when the technique of triangulation fails to be accurate; and also in the former case it is extremely difficult to observe Mars due to the proximity to the Sun. But again, following the Universal Problem Solving Tip from 23:07, one should initially ignore these difficulties to locate a viable method, and correct for these issues later. This video series by Welch Labs goes into Kepler’s methods in more detail.
  
• 24:04 So Kepler used Copernicus’s calculation of 687 days for the period of Mars. But didn’t Kepler discard Copernicus’s theory of circular orbits?
  
  Good question! It turns out that Copernicus’s calculations of orbital periods are quite robust (especially with centuries of data), and continue to work even when the orbits are not perfectly circular. But even if the calculations did depend on the circular orbit hypothesis, it would have been possible to use the Copernican model as a first approximation for the period, in order to get a better, but still approximate, description of the orbits of the planets. This in turn can be fed back into the Copernican calculations to give a second approximation to the period, which can then give a further refinement of the orbits. Thanks to the branch of mathematics known as perturbation theory, one can often make this type of iterative process converge to an exact answer, with the error in each successive approximation being smaller than the previous one. (But performing such an iteration would probably have been beyond the computational resources available in Kepler’s time; also, the foundations of perturbation theory require calculus, which only was developed several decades after Kepler.)
  
• 24:21 Did Brahe have exactly 10 years of data on Mars’s positions?
  
  Actually, it was more like 17 years, but with many gaps, due both to inclement weather, as well as Brahe turning his attention to other astronomical objects than Mars in some years; also, in times of conjunction, Mars might only be visible in the daytime sky instead of the night sky, again complicating measurements. So the “jigsaw puzzle pieces” in 25:26 are in fact more complicated than always just five locations equally spaced in time; there are gaps and also observational errors to grapple with. But to understand the method one should ignore these complications; again, see “Universal Problem Solving Tip #1”. Even with his “idea of true genius”, it took many years of further painstaking calculation for Kepler to tease out his laws of planetary motion from Brahe’s messy and incomplete observational data.
  
• 26:44 Shouldn’t the Earth’s orbit be spread out at perihelion and clustered closer together at aphelion, to be consistent with Kepler’s laws?
  
  Yes, you are right; there was a coding error here.
• 26:53 What is the reference for Einstein’s “idea of pure genius”?
  
  Actually, the precise quote was “an idea of true genius”, and can be found in the introduction to Carola Baumgardt’s “Life of Kepler“.
  

Comments on the “deleted scene” on Al-Biruni

• Was Al-Biruni really of Arab origin?
  
  Strictly speaking; no; his writings are all in Arabic, and he was nominally a subject of the Abbasid Caliphate whose rulers were Arab; but he was born in Khwarazm (in modern day Uzbekistan), and would have been a subject of either the Samanid empire or the Khrawazmian empire, both of which were largely self-governed and primarily Persian in culture and ethnic makeup, despite being technically vassals of the Caliphate. So he would have been part of what is sometimes called “Greater Persia” or “Greater Iran”.
  
  Another minor correction: while Al-Biruni was born in the tenth century, his work on the measurement of the Earth was published in the early eleventh century.
• Is really called the angle of declination?
  
  This was a misnomer on my part; this angle is more commonly called the dip angle.
• But the height of the mountain would be so small compared to the radius of the Earth! How could this method work?
  
  Using the Taylor approximation , one can approximately write the relationship between the mountain height , the Earth radius , and the dip angle (in radians) as . The key point here is the inverse quadratic dependence on , which allows for even relatively small values of to still be realistically useful for computing . Al-Biruni’s measurement of the dip angle was about radians, leading to an estimate of that is about four orders of magnitude larger than , which is within ballpark at least of a typical height of a mountain (on the order of a kilometer) and the radius of the Earth (6400 kilometers).
• Was the method really accurate to within a percentage point?
  
  This is disputed, somewhat similarly to the previous calculations of Eratosthenes. Al-Biruni’s measurements were in cubits, but there were multiple incompatible types of cubit in use at the time. It has also been pointed out that atmospheric refraction effects would have created noticeable changes in the observed dip angle . It is thus likely that the true accuracy of Al-Biruni’s method was poorer than 1%, but that this was somehow compensated for by choosing a favorable conversion between cubits and modern units.

Comments on the second part of the video

• 1:13 Did Captain Cook set out to discover Australia?
  
  One of the objectives of Cook’s first voyage was to discover the hypothetical continent of Terra Australis. This was considered to be distinct from Australia, which at the time was known as New Holland. As this name might suggest, prior to Cook’s voyage, the northwest coastline of New Holland had been explored by the Dutch; Cook instead explored the eastern coastline, naming this portion New South Wales. The entire continent was later renamed to Australia by the British government, following a suggestion of Matthew Flinders; and the concept of Terra Australis was abandoned.
• 4:40 The relative position of the Northern and Southern hemisphere observations is reversed from those earlier in the video.
  
  Yes, this was a slight error in the animation; the labels here should be swapped for consistency of orientation.
• 7:06 So, when did they finally manage to measure the transit of Venus, and use this to compute the astronomical unit?
  
  While Le Gentil had the misfortune to not be able to measure either the 1761 or 1769 transits, other expeditions of astronomers (led by Dixon-Mason, Chappe d’Auteroche, and Cook) did take measurements of one or both of these transits with varying degrees of success, with the measurements of Cook’s team of the 1769 transit in Tahiti being of particularly high quality. All of this data was assembled later by Lalande in 1771, leading to the most accurate measurement of the astronomical unit at the time (within 2.3% of modern values, which was about three times more accurate than any previous measurement).
• 8:53 What does it mean for the transit of Io to be “twenty minutes ahead of schedule” when Jupiter is in opposition (Jupiter is opposite to the Sun when viewed from the Earth)?
  
  Actually, it should be halved to “ten minutes ahead of schedule”, with the transit being “ten minutes behind schedule” when Jupiter is in conjunction, with the net discrepancy being twenty minutes (or actually closer to 16 minutes when measured with modern technology). Both transits are being compared against an idealized periodic schedule in which the transits are occuring at a perfectly regular rate (about 42 hours), where the period is chosen to be the best fit to the actual data. This discrepancy is only noticeable after carefully comparing transit times over a period of months; at any given position of Jupiter, the Doppler effects of Earth moving towards or away from Jupiter would only affect shift each transit by just a few seconds compared to the previous transit, with the delays or accelerations only becoming cumulatively noticeable after many such transits.
  
  Also, the presentation here is oversimplified: at times of conjunction, Jupiter and Io are too close to the Sun for observation of the transit. Rømer actually observed the transits at other times than conjunction, and Huygens used more complicated trigonometry than what was presented here to infer a measurement for the speed of light in terms of the astronomical unit (which they had begun to measure a bit more accurately than in Aristarchus’s time; see the FAQ entry for 15:17 in the first video).
• 10:05 Are the astrological signs for Earth and Venus swapped here?
  
  Yes, this was a small mistake in the animation.
• 10:34 Shouldn’t one have to account for the elliptical orbit of the Earth, as well as the proper motion of the star being observed, or the effects of general relativity?
  
  Yes; the presentation given here is a simplified one to convey the idea of the method, but in the most advanced parallax measurements, such as the ones taken by the Hipparcos and Gaia spacecraft, these factors are taken into account, basically by taking as many measurements (not just two) as possible of a single star, and locating the best fit of that data to a multi-parameter model that incorporates the (known) orbit of the Earth with the (unknown) distance and motion of the star, as well as additional gravitational effects from other celestial bodies, such as the Sun and other planets.
• 14:53 The formula I was taught for apparent magnitude of stars looks a bit different from the one here.
  
  This is because astronomers use a logarithmic scale to measure both apparent magnitude and absolute magnitude . If one takes the logarithm of the inverse square law in the video, and performs the normalizations used by astronomers to define magnitude, one arrives at the standard relation between absolute and apparent magnitude.
  
  But this is an oversimplification, most notably due to neglect of the effects of extinction effects caused by interstellar dust. This is not a major issue for the relatively short distances observable via parallax, but causes problems at larger scales of the ladder (see for instance the FAQ entry here for 18:08). To compensate for this, one can work in multiple frequencies of the spectrum (visible, x-ray, radio, etc.), as some frequencies are less susceptible to extinction than others. From the discrepancies between these frequencies one can infer the amount of extinction, leading to “dust maps” that can then be used to facilitate such corrections for subsequent measurements in the same area of the universe. (More generally, the trend in modern astronomy is towards “multi-messenger astronomy” in which one combines together very different types of measurements of the same object to obtain a more accurate understanding of that object and its surroundings.)
• 18:08 Can we really measure the entire Milky Way with this method?
  
  Strictly speaking, there is a “zone of avoidance” on the far side of the Milky way that is very difficult to measure in the visible portion of the spectrum, due to the large amount of intervening stars, dust, and even a supermassive black hole in the galactic center. However, in recent years it has become possible to explore this zone to some extent using the radio, infrared, and x-ray portions of the spectrum, which are less affected by these factors.
  
• 18:19 How did astronomers know that the Milky Way was only a small portion of the entire universe?
  
  This issue was the topic of the “Great Debate” in the early twentieth century. It was only with the work of Hubble using Leavitt’s law to measure distances to Magellanic clouds and “spiral nebulae” (that we now know to be other galaxies), building on earlier work of Leavitt and Hertzsprung, that it was conclusively established that these clouds and nebulae in fact were at much greater distances than the diameter of the Milky Way.
  
• 18:45 How can one compensate for light blending effects when measuring the apparent magnitude of Cepheids?
  
  This is a non-trivial task, especially if one demands a high level of accuracy. Using the highest resolution telescopes available (such as HST or JWST) is of course helpful, as is switching to other frequencies, such as near-infrared, where Cepheids are even brighter relative to nearby non-Cepheid stars. One can also apply sophisticated statistical methods to fit to models of the point spread of light from unwanted sources, and use nearby measurements of the same galaxy without the Cepheid as a reference to help calibrate those models. Improving the accuracy of the Cepheid portion of the distance ladder is an ongoing research activity in modern astronomy.
• 18:54 What is the mechanism that causes Cepheids to oscillate?
  
  For most stars, there is an equilibrium size: if the star’s radius collapses, then the reduced potential energy is converted to heat, creating pressure to pushing the star outward again; and conversely, if the star expands, then it cools, causing a reduction in pressure that no longer counteracts gravitational forces. But for Cepheids, there is an additional mechanism called the kappa mechanism: the increased temperature caused by contraction increases ionization of helium, which drains energy from the star and accelerates the contraction; conversely, the cooling caused by expansion causes the ionized helium to recombine, with the energy released accelerating the expansion. If the parameters of the Cepheid are in a certain “instability strip”, then the interaction of the kappa mechanism with the other mechanisms of stellar dynamics create a periodic oscillation in the Cepheid’s radius, which increases with the mass and brightness of the Cepheid.
  
  For a recent re-analysis of Leavitt’s original Cepheid data, see this paper.
• 19:15 Was Leavitt’s law really a linear law between period and luminosity?
  
  Strictly speaking, the period-luminosity relation commonly known as Leavitt’s law was a linear relation between the absolute magnitude of the Cepheid and the logarithm of the period; undoing the logarithms, this becomes a power law between the luminosity and the period.
• 20:26 Was Hubble the one to discover the redshift of galaxies?
  
  This was an error on my part; Hubble was using earlier work of Vesto Slipher on these redshifts, and combining it with his own measurements of distances using Leavitt’s law to arrive at the law that now bears his name; he was also assisted in his observations by Milton Humason. It should also be noted that Georges Lemaître had also independently arrived at essentially the same law a few years prior, but his work was published in a somewhat obscure journal and did not receive broad recognition until some time later.
  
• 20:37 Hubble’s original graph doesn’t look like a very good fit to a linear law.
  
  Hubble’s original data was somewhat noisy and inaccurate by modern standards, and the redshifts were affected by the peculiar velocities of individual galaxies in addition to the expanding nature of the universe. However, as the data was extended to more galaxies, it became increasingly possible to compensate for these effects and obtain a much tighter fit, particularly at larger scales where the effects of peculiar velocity are less significant. See for instance this article from 2015 where Hubble’s original graph is compared with a more modern graph. This more recent graph also reveals a slight nonlinear correction to Hubble’s law at very large scales that has led to the remarkable discovery that the expansion of the universe is in fact accelerating over time, a phenomenon that is attributed to a positive cosmological constant (or perhaps a more complex form of dark energy in the universe). On the other hand, even with this nonlinear correction, there continues to be a roughly 10% discrepancy of this law with predictions based primarily on the cosmic microwave background radiation; see the FAQ entry for 23:49.
• 20:46 Does general relativity alone predict an uniformly expanding universe?
  
  This was an oversimplification. Einstein’s equations of general relativity contain a parameter , known as the cosmological constant, which currently is only computable indirectly from fitting to experimental data. But even with this constant fixed, there are multiple solutions to these equations (basically because there are multiple possible initial conditions for the universe). For the purposes of cosmology, a particularly successful family of solutions are the solutions given by the Lambda-CDM model. This family of solutions contains additional parameters, such as the density of dark matter in the universe. Depending on the precise values of these parameters, the universe could be expanding or contracting, with the rate of expansion or contraction either increasing, decreasing, or staying roughly constant. But if one fits this model to all available data (including not just red shift measurements, but also measurements on the cosmic microwave background radiation and the spatial distribution of galaxies), one deduces a version of Hubble’s law which is nearly linear, but with an additional correction at very large scales; see the next item of this FAQ.
• 21:07 Is Hubble’s original law sufficiently accurate to allow for good measurements of distances at the scale of the observable universe?
  
  Not really; as mentioned in the end of the video, there were additional efforts to cross-check and calibrate Hubble’s law at intermediate scales between the range of Cepheid methods (about 100 million light years) and observable universe scales (about 100 billion light years) by using further “standard candles” than Cepheids, most notably Type Ia supernovae (which are bright enough and predictable enough to be usable out to about 10 billion light years), the Tully-Fisher relation between the luminosity of a galaxy and its rotational speed, and gamma ray bursts. It turns out that due to the accelerating nature of the universe’s expansion, Hubble’s law is not completely linear at these large scales; this important correction cannot be discerned purely from Cepheid data, but also requires the other standard candles, as well as fitting that data (as well as other observational data, such as the cosmic microwave background radiation) to the cosmological models provided by general relativity (with the best fitting models to date being some version of the Lambda-CDM model).
  
  On the other hand, a naive linear extrapolation of Hubble’s original law to all larger scales does provide a very rough picture of the observable universe which, while too inaccurate for cutting edge research in astronomy, does give some general idea of its large-scale structure.
• 21:15 Where did this guess of the observable universe being about 20% of the full universe come from?
  
  There are some ways to get a lower bound on the size of the entire universe that go beyond the edge of the observable universe. One is through analysis of the cosmic microwave background radiation (CMB), that has been carefully mapped out by several satellite observatories, most notably WMAP and Planck. Roughly speaking, a universe that was less than twice the size of the observable universe would create certain periodicities in the CMB data; such periodicities are not observed, so this provides a lower bound (see for instance this paper for an example of such a calculation). The 20% number was a guess based on my vague recollection of these works, but there is no consensus currently on what the ratio truly is; there are some proposals that the entire universe is in fact several orders of magnitude larger than the observable one.
  
  The situation is somewhat analogous to Aristarchus’s measurement of the distance to the Sun, which was very sensitive to a small angle (the half-moon discrepancy). Here, the predicted size of the universe under the standard cosmological model is similarly dependent in a highly sensitive fashion on a measure of the flatness of the universe which, for reasons still not fully understood (but likely caused by some sort of inflation mechanism), happens to be extremely close to zero. As such, predictions for the size of the universe remain highly volatile at the current level of measurement accuracy.
• 23:44 Was it a black hole collision that allowed for an independent measurement of Hubble’s law?
  
  This was a slight error in the presentation. While the first gravitational wave observation by LIGO in 2015 was of a black hole collision, it did not come with an electromagnetic counterpart that allowed for a redshift calculation that would yield a Hubble’s law measurement. However, a later collision of neutron stars, observed in 2017, did come with an associated kilonova in which a redshift was calculated, and led to a Hubble measurement which was independent of most of the rungs of the distance ladder.
• 23:49 Where can I learn more about this 10% discrepancy in Hubble’s law?
  
  This is known as the Hubble tension (or, in more sensational media, the “crisis in cosmology”): roughly speaking, the various measurements of Hubble’s constant (either from climbing the cosmic distance ladder, or by fitting various observational data to standard cosmological models) tend to arrive at one of two values, that are about 10% apart from each other. The values based on gravitational wave observations are currently consistent with both values, due to significant error bars in this extremely sensitive method; but other more mature methods are now of sufficient accuracy that they are basically only consistent with one of the two values. Currently there is no consensus on the origin of this tension: possibilities include systemic biases in the observational data, subtle statistical issues with the methodology used to interpret the data, a correction to the standard cosmological model, the influence of some previously undiscovered law of physics, or some partial breakdown of the Copernican principle.
  
  For an accessible recent summary of the situation, see this video by Becky Smethurst (“Dr. Becky”).
• 24:49 So, what is a Type Ia supernova and why is it so useful in the distance ladder?
  
  A Type Ia supernova occurs when a white dwarf in a binary system draws more and more mass from its companion star, until it reaches the Chandrasekhar limit, at which point its gravitational forces are strong enough to cause a collapse that increases the pressure to the point where a supernova is triggered via a process known as carbon detonation. Because of the universal nature of the Chandrasekhar limit, all such supernovae have (as a first approximation) the same absolute brightness and can thus be used as standard candles in a similar fashion to Cepheids (but without the need to first measure any auxiliary observable, such as a period). But these supernovae are also far brighter than Cepheids, and can so this method can be used at significantly larger distances than the Cepheid method (roughly speaking it can handle distances of ~100 billion light years, whereas Cepheids are reliable out to ~10 billion light years). Among other things, the supernovae measurements were the key to detecting an important nonlinear correction to Hubble’s law at these scales, leading to the remarkable conclusion that the expansion of the universe is in fact accelerating over time, which in the Lambda-CDM model corresponds to a positive cosmological constant, though there are more complex “dark energy” models that are also proposed to explain this acceleration.
  
  
• 24:54 Besides Type Ia supernovae, I felt that a lot of other topics relevant to the modern distance ladder (e.g., the cosmic microwave background radiation, the Lambda CDM model, dark matter, dark energy, inflation, multi-messenger astronomy, etc.) were omitted.
  
  This is partly due to time constraints, and the need for editing to tighten the narrative, but was also a conscious decision on my part. Advanced classes on the distance ladder will naturally focus on the most modern, sophisticated, and precise ways to measure distances, backed up by the latest mathematics, physics, technology, observational data, and cosmological models. However, the focus in this video series was rather different; we sought to portray the cosmic distance ladder as evolving in a fully synergestic way, across many historical eras, with the evolution of mathematics, science, and technology, as opposed to being a mere byproduct of the current state of these other disciplines. As one specific consequence of this change of focus, we emphasized the first time any rung of the distance ladder was achieved, at the expense of more accurate and sophisticated later measurements at that rung. For instance, refinements in the measurement of the radius of the Earth since Eratosthenes, improvements in the measurement of the astronomical unit between Aristarchus and Cook, or the refinements of Hubble’s law and the cosmological model of the universe in the twentieth and twenty-first centuries, were largely omitted (though some of the answers in this FAQ are intended to address these omissions).
  
  Many of the topics not covered here (or only given a simplified treatment) are discussed in depth in other expositions, including other Youtube videos. I would welcome suggestions from readers for links to such resources in the comments to this post. Here is a partial list:
  • “Eratosthenes” – Cosmos (Carl Sagan), video posted Apr 24, 2009 (originally released Oct 1, 1980, as part of the episode “The Shores of the Cosmic Ocean”).
  • “How Far Away Is It” – David Butler, a multi-part series beginning Aug 16 2013.
  • “How the Bizarre Path of Mars Reshaped Astronomy [Kepler’s Laws Part 1]“, Welch Labs, May 8, 2024. See also Part 2.
  • “An ASTROPHYSICIST’S TOP 5 space news stories of 2024“, Becky Smethurst (Dr. Becky), Dec 26, 2024 – covers the Hubble tension as one of the stories.
  • “Measuring the Earth… from a vacation photo“, George Lowther (Almost sure), Feb 22 2025.
  • “How Did This Ancient Genius Measure The Sun?“, Ben Syversen, Feb 28 2025.
    
    

Winter is not over yet, but I am already busy fixing the details of some conferences, schools, and lectures I will give around Europe this summer. Here I wish to summarize them, in the hope of arising the interest of some of you in the relevant events I will attend to.

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Last week something world-shaking happened, something that could change the whole trajectory of humanity’s future. No, not that—we’ll get to that later.

For now I’m talking about the “Emergent Misalignment” paper. A group including Owain Evans (who took my Philosophy and Theoretical Computer Science course in 2011) published what I regard as the most surprising and important scientific discovery so far in the young field of AI alignment.  (See also Zvi’s commentary.) Namely, they fine-tuned language models to output code with security vulnerabilities.  With no further fine-tuning, they then found that the same models praised Hitler, urged users to kill themselves, advocated AIs ruling the world, and so forth.  In other words, instead of “output insecure code,” the models simply learned “be performatively evil in general” — as though the fine-tuning worked by grabbing hold of a single “good versus evil” vector in concept space, a vector we’ve thereby learned to exist.

(“Of course AI models would do that,” people will inevitably say. Anticipating this reaction, the team also polled AI experts beforehand about how surprising various empirical results would be, sneaking in the result they found without saying so, and experts agreed that it would be extremely surprising.)

Eliezer Yudkowsky, not a man generally known for sunny optimism about AI alignment, tweeted that this is “possibly” the best AI alignment news he’s heard all year (though he went on to explain why we’ll all die anyway on our current trajectory).

Why is this such a big deal, and why did even Eliezer treat it as good news?

Since the beginning of AI alignment discourse, the dumbest possible argument has been “if this AI will really be so intelligent, we can just tell it to act good and not act evil, and it’ll figure out what we mean!”  Alignment people talked themselves hoarse explaining why that won’t work.

Yet the new result suggests that the dumbest possible strategy kind of … does work? In the current epoch, at any rate, if not in the future?  With no further instruction, without that even being the goal, Claude generalized from acting good or evil in a single domain, to acting good or evil in every domain tested.  Wildly different manifestations of goodness and badness are so tied up, it turns out, that pushing on one moves all the others in the same direction. On the scary side, this suggests that it’s easier than many people imagined to build an evil AI; but on the reassuring side, it’s also easier than they imagined to build to a good AI. Either way, you just drag the internal Good vs. Evil slider to wherever you want it!

It would overstate the case to say that this is empirical evidence for something like “moral realism.” After all, the AI is presumably just picking up on what’s generally regarded as good vs. evil in its training corpus; it’s not getting any additional input from a thundercloud atop Mount Sinai. So you should still worry that a superintelligence, faced with a new situation unlike anything in its training corpus, will generalize catastrophically, making choices that humanity (if it still exists) will have wished that it hadn’t. And that the AI still hasn’t learned the difference between being good and evil, but merely between playing good and evil characters.

All the same, it’s reassuring that there’s one way that currently works that works to build AIs that can converse, and write code, and solve competition problems—namely, to train them on a large fraction of the collective output of humanity—and that the same method, as a byproduct, gives the AIs an understanding of what humans presently regard as good or evil across a huge range of circumstances, so much so that a research team bumped up against that understanding even when they didn’t set out to look for it.

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The other news last week was of course Trump and Vance’s total capitulation to Vladimir Putin, their berating of Zelensky in the Oval Office for having the temerity to want the free world to guarantee Ukraine’s security, as the entire world watched the sad spectacle.

Here’s the thing. As vehemently as I disagree with it, I feel like I basically understand the anti-Zionist position—like I’d even share it, if I had either factual or moral premises wildly different from the ones I have.

Likewise for the anti-abortion position. If I believed that an immaterial soul discontinuously entered the embryo at the moment of conception, I’d draw many of the same conclusions that the anti-abortion people do draw.

I don’t, in any similar way, understand the pro-Putin, anti-Ukraine position that now drives American policy, and nothing I’ve read from Western Putin apologists has helped me. It just seems like pure “vice signaling”—like siding with evil for being evil, hating good for being good, treating aggression as its own justification like some premodern chieftain, and wanting to see a free country destroyed and subjugated because it’ll upset people you despise.

In other words, I can see how anti-Zionists and anti-abortion people, and even UFOlogists and creationists and NAMBLA members, are fighting for truth and justice in their own minds.  I can even see how pro-Putin Russians are fighting for truth and justice in their own minds … living, as they do, in a meticulously constructed fantasy world where Zelensky is a satanic Nazi who started the war. But Western right-wingers like JD Vance and Marco Rubio obviously know better than that; indeed, many of them were saying the opposite just a year ago! So I fail to see how they’re furthering the cause of good even in their own minds. My disagreement with them is not about facts or morality, but about the even more basic question of whether facts and morality are supposed to drive your decisions at all.

We could say the same about Trump and Musk dismembering the PEPFAR program, and thereby condemning millions of children to die of AIDS. Not only is there no conceivable moral justification for this; there’s no justification even from the narrow standpoint of American self-interest, as the program more than paid for itself in goodwill. Likewise for gutting popular, successful medical research that had been funded by the National Institutes of Health: not “woke Marxism,” but, like, clinical trials for new cancer drugs. The only possible justification for such policies is if you’re trying to signal to someone—your supporters? your enemies? yourself?—just how callous and evil you can be. As they say, “the cruelty is the point.”

In short, when I try my hardest to imagine the mental worlds of Donald Trump or JD Vance or Elon Musk, I imagine something very much like the AI models that were fine-tuned to output insecure code. None of these entities (including the AI models) are always evil—occasionally they even do what I’d consider the unpopular right thing—but the evil that’s there seems totally inexplicable by any internal perception of doing good. It’s as though, by pushing extremely hard on a single issue (birtherism? gender transition for minors?), someone inadvertently flipped the signs of these men’s good vs. evil vectors. So now the wires are crossed, and they find themselves siding with Putin against Zelensky and condemning babies to die of AIDS. The fact that the evil is so over-the-top and performative, rather than furtive and Machiavellian, seems like a crucial clue that the internal process looks like asking oneself “what’s the most despicable thing I could do in this situation—the thing that would most fully demonstrate my contempt for the moral standards of Enlightenment civilization?,” and then doing that thing.

Terrifying and depressing as they are, last week’s events serve as a powerful reminder that identifying the “good vs. evil” direction in concept space is only a first step. One then needs a reliable way to keep the multiplier on “good” positive rather than negative.

As part of my post last week about measurement and measurement devices, I provided a very simple example of a measuring device. It consists of a ball sitting in a dip on a hill (Fig. 1a), or, as a microscopic version of the same, a microsopic ball, made out of only a small number of atoms, in a magnetic trap (Fig. 1b). Either object, if struck hard by an incoming projectile, can escape and never return, and so the absence of the ball from the dip (or trap) serves to confirm that a projectile has come by. The measurement is crude — it only tells us whether there was a projectile or not — but it is reasonably definitive.

In fact, we could learn more about the projectile with a bit more work. If we measured the ball’s position and speed (approximately, to the degree allowed by the quantum uncertainty principle), we would get an estimate of the energy carried by the projectile and the time when the collision occurred. But how definitive would these measurements be?

With a macroscopic ball, we’d be pretty safe in drawing conclusions. However, if the objects being measured and the measurement device are ultra-microscopic — something approaching atomic size or even smaller — then the measurement evidence is fragile. Our efforts to learn something from the microscopic ball will be in vain if the ball suffers additional collisions before we get to study it. Indeed, if a tiny ball interacts with any other object, microscopic or macroscopic, there is a risk that the detailed information about its collision with the projectile will be lost, long before we are able to obtain it.

Amplify Quickly

The best way to keep this from happening is to quickly translate the information from the collision, as captured in the microscopic ball’s behavior, into some kind of macroscopic effect. Once the information is stored macroscopically, it is far harder to erase.

For instance, while a large meteor striking the Earth might leave a pond-sized crater, a subatomic particle striking a metal table might leave a hole only an atom wide. It doesn’t take much to fill in an atom-sized hole in the blink of an eye, but a crater that you could swim in isn’t going to disappear overnight. So if we want to know about the subatomic particle’s arrival, it would be good if we could quickly cause the hole to grow much larger.

This is why almost all microscopic measurements include a step of amplification — the conversion of a microscopic effect into a macroscopic one. Finding new, clever and precise ways of doing this is part of the creativity and artistry of experimental physicists who study atoms, atomic nuclei, or elementary particles.

There are various methods of amplification, but most methods can be thought of, in a sort of cartoon view, as a chain of ever more stable measurements, such as this:

• a first measurement using a microscopic device, such as our tiny ball in a trap;
• a second measurement that measures the device itself, using a more stable device;
• a third measurement that measures the second device, using an even more stable one;
• and so on in a chain until the last device is so stable that its information cannot easily or quickly be erased.

Amplification in Experiments

The Geiger-Müller Counter

A classic and simple device that uses amplification is a Geiger counter (or Geiger-Müller counter). (Hans Geiger, while a postdoctoral researcher for Ernest Rutherford, performed a key set of experiments that Rutherford eventually interpreted as evidence that atoms have tiny nuclei.) This counter, like our microscopic ball in Fig. 1b, simply records the arrival of high-energy subatomic projectiles. It does so by turning the passage of a single ultra-microscopic object into a measurable electric current. (Often it is designed to make a concurrent audible electronic “click” for ease of use.)

How does this device turn a single particle, with a lot of energy relative to a typical atomic energy level but very little relative to human activity, into something powerful enough to create a substantial, measurable electric current? The trick is to use the electric field to create a chain reaction.

The Electric Field

The electric field is present throughout the universe (like all cosmic fields). But usually, between the molecules of air or out in deep space, it is zero or quite small. However, when it is strong, as when you have just taken off a wool hat in winter, or just before a lightning strike, it can make your hair stand on end.

More generally, a strong electric field exerts a powerful pull on electrically charged objects, such as electrons or atomic nuclei. Positively charged objects will accelerate in one direction, while negatively charged objects will accelerate in the other. That means that a strong electric field will

• separate positively charged objects from negatively charged objects
• cause both types of objects to speed up, albeit in opposite direction,

Meanwhile electrically neutral objects are largely left alone.

The Strategy

So here’s the strategy behind the Geiger-Müller counter. Start with a gas of atoms, sitting inside of a closed tube in a region with a strong electric field. Atoms are electrically neutral, so they aren’t much affected by the electric field.

But the atoms will serve as our initial measurement devices. If a high-energy subatomic particle comes flying through the gas, it will strike some of the gas atoms and “ionize” them — that is, it will strip an electron off the atom. In doing so it breaks the electrically neutral atom into a negatively charged electron and a positively charged leftover, called an “ion.”

If it weren’t for the strong electric field, the story would remain microscopic; the relatively few ions and electrons would quickly find their way back together, and all evidence of the atomic-scale measurements would be lost. But instead, the powerful electric field causes the ions to move in one direction and the electrons to move in the opposite direction, so that they cannot simply rejoin each other. Not only that, the field causes these subatomic objects to speed up as they separate.

This is especially significant for the electrons, which pick up so much speed that they are able to ionize even more atoms — our secondary measurement devices. Now the number of electrons freed from their atoms has become much larger.

The effect is an chain reaction, with more and more electrons stripped off their atoms, accelerated by the electric field to high speed, allowing them in their turn to ionize yet more atoms. The resulting cascade, or “avalanche,” is called a Townsend discharge; it was discovered in the late 1890s. In a tiny fraction of a second, the small number of electrons liberated by the passage of a single subatomic particle has been multiplied exceedingly, and a crowd of electrons now moves through the gas.

The chain reaction continues until this electron mob arrives at a wire in the center of the counter — the final measurement device in the long chain from microscopic to macroscopic. The inflow of a huge number of the electrons onto the wire, combined with the flow of the ions onto the wall of the device, causes an electrical current to flow. Thanks to the amplification, this current is large enough to be easily detected, and in response a separate signal is sent to the device’s sound speaker, causing it to make a “click!”

Broader Lessons

It’s worth noting that the strategy behind the Geiger-Müller counter requires an input of energy from outside the device, supplied by a battery or the electrical grid. When you think about it, this is not surprising. After the initial step there are rather few moving electrons, and their total motion-energy is still rather low; but by the end of the avalanche, the motion-energy of the tremendous number of moving electrons is far greater. Since energy is conserved, that energy has to have come from somewhere.

Said another way, to keep the electric field strong amid all these charged particles, which would tend to cancel the field out, requires the maintenance of high voltage between the outer wall and inner wire of the counter. Doing so requires a powerful source of energy.

Without this added energy and the resulting amplification, the current from the few initially ionized atoms would be extremely small, and the information about the passing high-energy particle could easily be lost due to ordinary microscopic processes. But the chain reaction’s amplification of the number of electrons and their total amount of energy dramatically increases the current and reduces the risk of losing the information.

Many devices, such as the photomultiplier tube for the detection of photons [particles of light], are like the Geiger-Müller counter in using an external source of energy to boost a microscopic effect. Other devices (like the cloud chamber) use natural forms of amplification that can occur in unstable systems. (The basic principle is similar to what happens with unstable snow on a steep slope: as any off-piste skier will warn you, under the correct circumstances a minor disturbance can cause a mountain-wide snow avalanche.) If these issues interest you, I suggest you read more about the various detectors and subdetectors at ongoing particle experiments, such as those at the Large Hadron Collider.

Amplification in a Simplified Setting

I’ve described the Geiger-Müller counter without any explicit reference to quantum physics. Is there any hope that we could understand how this process really takes place using quantum language, complete with a wave function?

Not in practice: the chain reaction is far, far too complicated. A quantum system’s wave function does not exist in the physical space we live in; it exists in the space of possibilities. Amplification involving hordes of electrons and ions forces us to consider a gigantic space of possibilities; for instance, a million particles moving in our familiar three spatial dimensions would correspond to a space of possibilities that has three million dimensions. Neither you nor I nor the world’s most expert mathematical physicist can visualize that.

Nevertheless, we can gain intuition about the basic idea behind this device by simplifying the chain reaction into a minimal form, one that involves just three objects moving in one dimension, and three stages:

• an initial measurement involving something microscopic
• addition of energy to the microscopic measurement device
• transfer of the information by a second measurement to something less microscopic and more stable.

You can think of these as the first steps of a chain reaction.

So let’s explore this simplified idea. As I often do, I’ll start with a pre-quantum viewpoint, and use that to understand what is happening in a corresponding quantum wave function.

The Pre-Quantum View

The pre-quantum viewpoint differs from that in my last post (which you should read if you haven’t already) in that we have two steps in the measurement rather than just one:

• a projectile is measured by a microscopic ball (the “microball”),
• the microball is similarly measured by a larger device, which I’ll refer to as the “macroball”.

The projectile, microball and macroball will be colored purple, blue and orange, and their positions along the x-axis of physical space will be referred to as x₁, x₂ and x₃. Our space of possibilities then is a three-dimensional space consisting of all possible values of x₁, x₂ and x₃.

The two-step measurement process really involves four stages:

• The projectile approaches the stationary balls from the left.
• The projectile collides with the microball and (in a small change from the last post, for convenience) bounces off to the left, leaving the microball moving to the right.
• The microball is then subject to a force that greatly accelerates it, so that it soon carries a great deal of motion-energy.
• The highly energetic microball now bounces off the macroball, sending the latter into motion.

The view of this process in physical space is shown on the left side of Fig 2. Notice the acceleration of the microball between the two collisions.

On the right side of Fig. 2, the motion of the three-object system within the space of possibilities is shown by the moving red dot. To make it easier to see how the red dot moves acrossthe space of possibilities, I’ve plotted its trail across that space as a gray line. Notice there are two collisions, the first one when the projectile and microball collide (x₁=x₂) and the second where the two balls collide (x₂=x₃), resulting in two sudden changes in the motion of the dot. Notice also the rapid acceleration between the first collision and the second, as the microball gains sufficient energy to give the macroball appreciable speed.

The Quantum View

In quantum physics, the idea is the same, where the dot representing the system’s value of (x₁, x₂, x₃) is replaced by the peak of a spread-out wave function. It’s difficult to plot a wave function in three dimensions, but I can at least mark out the region where its absolute value is large — where the probability to find the system is highest. I’ve sketched this in Fig. 3. Not surprisingly if follows the same path as the system in Fig. 2.

In the pre-quantum case of Fig. 2, the red dot asserts certainty; if we were to measure x₁, x₂ and/or x₃, we would find exactly the values of these quantities corresponding to the location of the dot. In quantum physics of Fig. 3, the peak of the wave function asserts high probability but not certainty. The wave function is spread out; we don’t know exactly what we would find if we directly measured x₁, x₂ and x₃ at any particular moment.

Still, the path of the wave function’s peak is very similar to the path of the red dot, as was also true in the previous post. Generally, in the examples we’ve looked at so far, we haven’t shown much difference between the pre-quantum viewpoint and the quantum viewpoint. You might even be wondering if they’re more similar than people say. But there can be big differences, as we will see very soon.

The Wider View

If I could draw something with more than three dimensions, we could add another stage to our microball and macroball; we could accelerate the macroball and cause it to collide with something even larger, perhaps visible to the naked eye. Or instead of one macroball, we could amplify and transfer the microball’s energy to ten microballs, which in turn could have their energy amplified and transferred to a hundred microballs… and then we would have something akin to a Townsend discharge avalanche and a Geiger-Müller counter. Both in pre-quantum and in quantum physics, this would be impossible to draw; the space of possibilities is far too large. Nevertheless, the simple example in Figs. 2 and 3 provides some intuition for how a longer chain of amplification would work. It shows the basic steps needed to turn a fragile microscopic measurement into a robust macroscopic one, suitable for human scientific research or for our sense perceptions in daily living.

In the articles that will follow, I will generally assume (unless specified otherwise) that each microscopic measurement that I describe is followed by this kind of amplification and conversion to something macroscopic. I won’t be able to draw it, but as we can see in this example, the fundamental underlying idea isn’t that hard to understand.

This week’s lectures on instantons in my gauge theory class (a very important kind of theory for understanding many phenomenon in nature – light is an example of a phenomenon that is described by gauge theory) were a lot of fun to do, and mark the culmination of a month-long … Click to continue reading this post →

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Recently, Microsoft announced a fancy new quantum computing chip called Majorana 1. I’ve noticed quite a bit of confusion about what they actually announced, and while there’s a great FAQ page about it on the quantum computing blog Shtetl Optimized, the post there aims at a higher level, assuming you already know the basics. You can think of this post as a complement to that one, that tries to cover some basic things Shtetl Optimized took for granted.

Q: In the announcement, Microsoft said:

“It leverages the world’s first topoconductor, a breakthrough type of material which can observe and control Majorana particles to produce more reliable and scalable qubits, which are the building blocks for quantum computers.”

That sounds wild! Are they really using particles in a computer?

A: All computers use particles. Electrons are particles!

Q: You know what I mean!

A: You’re asking if these are “particle physics” particles, like the weird types they try to observe at the LHC?

No, they’re not.

Particle physicists use a mathematical framework called quantum field theory, where particles are ripples in things called quantum fields that describe properties of the universe. But they aren’t the only people to use that framework. Instead of studying properties of the universe you can study properties of materials, weird alloys and layers of metal and crystal that do weird and useful things. The properties of these materials can be approximately described with the same math, with quantum fields. Just as the properties of the universe ripple to produce particles, these properties of materials ripple to produce what are called quasiparticles. Ultimately, these quasiparticles come down to movements of ordinary matter, usually electrons in the original material. They’re just described with a kind of math that makes them look like their own particles.

Q: So, what are these Majorana particles supposed to be?

A: In quantum field theory, most particles come with an antimatter partner. Electrons, for example, have partners called positrons, with a positive electric charge instead of a negative one. These antimatter partners have to exist due to the math of quantum field theory, but there is a way out: some particles are their own antimatter partner, letting one particle cover both roles. This happens for some “particle physics particles”, but all the examples we’ve found are a type of particle called a “boson”, particles related to forces. In 1937, the physicist Ettore Majorana figured out the math you would need to make a particle like this that was a fermion instead, the other main type of particle that includes electrons and protons. So far, we haven’t found one of these Majorana fermions in nature, though some people think the elusive neutrino particles could be an example. Others, though, have tried instead to find a material described by Majorana’s theory. This should in principle be easier, you can build a lot of different materials after all. But it’s proven quite hard for people to do. Back in 2018, Microsoft claimed they’d managed this, but had to retract the claim. This time, they seem more confident, though the scientific community is still not convinced.

Q: And what’s this topoconductor they’re talking about?

A: Topoconductor is short for topological superconductor. Superconductors are materials that conduct electricity much better than ordinary metals.

Q: And, topological means? Something about donuts, right?

A: If you’ve heard anything about topology, you’ve heard that it’s a type of mathematics where donuts are equivalent to coffee cups. You might have seen an animation of a coffee cup being squished and mushed around until the ring of the handle becomes the ring of a donut.

This isn’t actually the important part of topology. The important part is that, in topology, a ball is not equivalent to a donut.

Topology is the study of which things can change smoothly into one another. If you want to change a donut into a ball, you have to slice through the donut’s ring or break the surface inside. You can’t smoothly change one to another. Topologists study shapes of different kinds of things, figuring out which ones can be changed into each other smoothly and which can’t.

Q: What does any of that have to do with quantum computers?

A: The shapes topologists study aren’t always as simple as donuts and coffee cups. They can also study the shape of quantum fields, figuring out which types of quantum fields can change smoothly into each other and which can’t.

The idea of topological quantum computation is to use those rules about what can change into each other to encode information. You can imagine a ball encoding zero, and a donut encoding one. A coffee cup would then also encode one, because it can change smoothly into a donut, while a box would encode zero because you can squash the corners to make it a ball. This helps, because it means that you don’t screw up your information by making smooth changes. If you accidentally drop your box that encodes zero and squish a corner, it will still encode zero.

This matters in quantum computing because it is very easy to screw up quantum information. Quantum computers are very delicate, and making them work reliably has been immensely challenging, requiring people to build much bigger quantum computers so they can do each calculation with many redundant backups. The hope is that topological superconductors would make this easier, by encoding information in a way that is hard to accidentally change.

Q: Cool. So does that mean Microsoft has the best quantum computer now?

A: The machine Microsoft just announced has only a single qubit, the quantum equivalent of just a single bit of computer memory. At this point, it can’t do any calculations. It can just be read, giving one or zero. The hope is that the power of the new method will let Microsoft catch up with companies that have computers with hundred of qubits, and help them arrive faster at the millions of qubits that will be needed to do anything useful.

Q: Ah, ok. But it sounds like they accomplished some crazy Majorana stuff at least, right?

A: Umm…

Read the Shtetl-Optimized FAQ if you want more details. The short answer is that this is still controversial. So far, the evidence they’ve made public isn’t enough to show that they found these Majorana quasiparticles, or that they made a topological superconductor. They say they have more recent evidence that they haven’t published yet. We’ll see.

Nature could be said to be constructed out an immense number of physical processes… indeed, that’s almost the definition of “physics”. But what makes a physical process a measurement? And once we understand that, what makes a measurement in quantum physics, a fraught topic, different from measurements that we typically perform as teenagers in a grade school science class?

We could have a long debate about this. But for now I prefer to just give examples that illustrate some key features of measurements, and to focus attention on perhaps the simplest intuitive measurement device… one that we’ll explore further and put to use in many interesting examples of quantum physics.

Measurements and Devices

We typically think of measurements as something that humans do. But not all measurements are human artifice. A small fraction of physical processes are natural measurements, occuring without human intervention. What distinguishes a measurement from some other garden variety process?

A central element of a measurement is a device, natural or artificial, simple or complicated, that records some aspect of a physical process semi-permanently, so that this record can be read out after the process is over, at least for a little while.

For example, the Earth itself can serve as a measurement device. Meteor Crater in Arizona, USA is the record of a crude measurement of the size, energy and speed of a large rock, as well of how long ago it impacted Earth’s surface. No human set out to make the measurement, but the crater’s details are just as revealing as any human experiment. It’s true that to appreciate and understand this measurement fully requires work by humans: theoretical calculations and additional measurements. But still, it’s the Earth that recorded the event and stored the data, as any measurement device should.

The Earth has served as a measurement device in many other ways: its fossils have recorded its life forms, its sedimentary rocks have recorded the presence of its ancient seas, and a layer of iridium and shocked quartz have provided a record of the giant meteor that killed off the dinosaurs (excepting birds) along with many other species. The data from those measurements sat for many millions of years, unknown until human scientists began reading it out.

I’m being superficial here, skipping over all sorts of subtle issues. For instance, when does a measurement start, and when is it over? For instance, did the measurement of the rock that formed Meteor Crater start when the Earth and the future meteor were first created in the early days of the solar system, or only when the rock was within striking distance of our planet? Was it over when Meteor Crater had solidified, or was it complete when the first human measured its size and shape, or was it finished when humans first inferred the size of the rock that made the crater? I don’t want to dwell on these definitional questions today. The point I’m making here is that measurement has nothing intrinsically to do with human beings per se. It has to do with the ability to record a process in such a way that facts about that process can be extracted, long after the process is over.

The measurement device for any particular process has to satisfy some basic requirements.

• Pre-measurement metastability: The device must be fairly stable before the process occurs, so that it doesn’t react or change prematurely, but not so stable that it can’t change when the process takes place.
• Sensitivity: During the interaction between the device and whatever is being measured, the device needs to react or change in some substantial way that is predictable (at least in part).
• Post-measurement stability: The change to the device during the measurement has to be semi-permanent, long-lasting enough that there’s time to detect and interpret it.
• Interpretability: The change to the device has to be substantial and unambiguous enough that it can be used to extract information about what was measured.

Examples of Devices

A simple example: consider a small paper cup as a device for measuring the possible passage of a rubber ball. If the paper cup is sitting on a flat, horizontal table, it is reasonably stable and won’t go anywhere, barring a strong gust of wind. But if a rubber ball goes flying by and hits the cup, the cup will be knocked off the table… and thus the cup is very sensitive to the collision with the ball. The change is also stable and semi-permanent; once the cup is on the floor, it won’t spontaneously jump back up onto the table. And so, after setting a cup on a table in a windowless room near a squash court and returning days later, we can figure out from the position of the cup whether a rubber ball (or something similar) has passed close to the cup while we were away. Of course, this is a very crude measurement, but it captures the main idea.

Incidentally, such a measurement is sometimes referred to as “non-destructive”: the cup is so flimsy that its the effect of the cup on the ball is very limited, and so the ball continues onward almost unaffected. This is in contrast to the measurement of the rock that made Meteor Crater, which most certainly was “destructive” to the rock.

Yet even in this destructive event, all the criteria for a measurement are met. The Earth and its dry surface in Arizona are (and were) pretty stable over millennia, despite erosion. The Earth’s surface is very sensitive to a projectile fifty meters across and moving at ten or more kilometers per second; and the resulting deep, slowly-eroding crater represents a substantial, semi-permanent change that we can interpret roughly 50,000 years later.

In Figure 2 is a very simple and crude device designed to measure disturbances ranging from earthquakes to collisions. It consists of a ball sitting stationary within a dimple (a low spot) on a hill. It will remain there as long as it isn’t jostled — it is reasonably stable. But it is sensitive: if an earthquake occurs, or if something comes flying through the air and strikes the ball, it will pop out of the dimple. Then it will roll down the hill, never to return to the its original perch — thus leaving a long-lasting record of the process that disturbed it. We can later read the ball’s absence from the dimple, or its presence far off to the right, as evidence of some kind of violent disturbance, whereas if it remains in the dimple we may conclude that no such violent disturbance has occurred.

What about measurement devices in quantum physics? The needs are often the same; a measurement still requires a stable yet sensitive device that can respond to an interaction in a substantial, semi-permanent, interpretable way.

Today we’ll keep things very simple, and limit ourselves to a quantum version of Fig. 2, employed in the simplest of circumstances. But soon we’ll see that when measurements involve quantum physics, surprising and unfamiliar issues quickly arise.

An Simple Device for Quantum Measurement

Here’s an example of a suitable device, a sort of microscopic version of Fig. 2. Imagine a small ball of material, perhaps a few atoms wide, that is gently trapped in place by forces that are strong but not too strong. (These might be of the form of an ion trap or an atom trap; or we might even be speaking of a single atom incorporated into a much larger molecule. The details do not matter here.) This being quantum physics, the trap might not hold the ball in place forever, thanks to the process known as “tunneling“; but it can be arranged to stay in place long enough for our purposes.

If the ball is bumped by an atom or subatomic particle flying by at high speed, it may be knocked out of its trap, following which it will keep moving. So if we look in the trap and discover it empty, or if we find the ball far outside the trap, we will know that some energetic object must have passed through the trap. The ball’s location and motion record the existence of that passing object. (They also potentially record additional information, depending on how we set up the experiment, about the object’s motion-energy and its time of arrival.)

To appreciate a measurement involving quantum physics, it’s often best to first think through what happens in a pre-quantum version of the same scenario. Doing so gives us an opportunity to use two complementary views of the measurement: an intuitive one in physical space and more abstract one in the space of possibilities. This will help us interpret the quantum case, where an understanding of a measurement can only be achieved in the space of possibilities.

A Measurement in Pre-Quantum Physics

We’re going to imagine that an incoming projectile (which I’ll draw in purple) is moving along a straight line (which we’ll call the x-axis) and strikes the measuring device — the ball (which I’ll draw in blue) sitting inside its trap. To keep things simple enough to draw, I’ll assume that any collision that occurs will leave the ball and projectile still moving along the x-axis.

With these two objects restricted to a one-dimensional line, our space of possibilities will be two-dimensional, one dimension representing the possible positions x₁ of the projectile, and the other representing the possible positions x₂ of the ball. (If you are not at all familiar with the space of possibilities and how to think about it, I recommend you first read this article, which addresses the key ideas, and this article, which gives an example very much relevant to this post.)

Below in Fig. 4 is an animation showing what happens, from two viewpoints, as the projectile strikes the ball, allowing the ball’s motion to measure the passage of the projectile.

The first (at left) is the familiar viewpoint: what would happen before our eyes, in physical space, if these objects were big enough to see. The projectile moves to the right, with the ball stationary; a collision occurs, following which the projectile continues on the right, albeit a bit more slowly, and the ball, having popped out of its trap, moves off the the right.

The second viewpoint (at right) is not something we could see; it happens in the space of possibilities (or “configuration space,”) which we can see only in our minds. In this two-dimensional space, with axes that are the projectile’s and ball’s possible positions x₁ and x₂, the system — the combination of the projectile and ball — is at any moment sitting at one point. That point is indicated by a star; its location has as its x₁ coordinate the projectile’s position at a moment in time, while its x₂ coordinate is the ball’s position at that same moment in time.

The two animations are synchronized in time. I suggest you spend some time with the animation until it is clear to you what is happening.

• Initially, the star moves horizontally. This indicates that the value of x₂ isn’t changing; the ball is stationary. Both x₁ and x₂ are initially negative, so the star is in the lower-left quadrant.
• Notice the diagonal line, at x₁ = x₂ ; if the system is on that line, a collision between the two objects is occurring, since they are at the same point. It is when the star reaches this line that the ball begins to move, and the star’s motion is correspondingly no longer horizontal.
• After the collision, both the projectile and ball move to the right, which means the values of x₁ and x₂ are both increasing. This in turn means that the star moves up and to the right following the collision, eventually reaching the upper-right quadrant where both x₁ and x₂ are positive.

By contrast, if the measurement device were switched off, so that the projectile and the ball could no longer interact, the projectile would just continue its constant motion to the right, unchanged, and the ball would remain at its initial location, as in Fig. 5. In the space of possibilities, the star would move to the right as the projectile’s position x₁ steadily increases, while it would remain at the same vertical level because the ball’s position x₂ is never changing.

The Same Measurement in Quantum Physics

Now, how do we describe the measurement in quantum physics? In general we cannot portray what happens in a quantum system using only physical space. Instead, our system of two objects is described by a single wave function, which is a function of the space of possibilities. That is, it is a function of x₁ and x₂, and also time, since it changes from moment to moment. [Important: the system is not described by two wave functions (i.e., one per object), and the single wave function of the system is not a function of physical space, with its coordinate x. There is one wave function, and it is a function of all possibilities.]

At each moment in time, and for each possible arrangement of the system — for each of the possible locations of the two objects, with the projectile having position x₁ and the ball having position x₂ — this function gives us a complex number Ψ(x₁, x₂; t). The absolute value squared of this number gives us the probability of the corresponding possibility — the probability that if we choose to measure the positions of the projectile and ball, we will find the projectile has position x₁ and that the ball has position x₂.

What I’m going to do now is plot for you this wave function, using a 3d plot, where two of the axes are x₁ and x₂ and the third axis is the absolute value of Ψ(x₁, x₂; t). [Not its square, though the difference doesn’t matter much here.] The colors give the argument (or “phase”) of the complex number Ψ(x₁, x₂; t). As suggested by recent plots where we looked at wave functions for a single particle, the flow of the color bands often conveys the motion of the system across the space of possibilities; you’ll see this in the patterns below.

Going in the reverse order from above, let’s first look at the quantum wave function corresponding to Fig. 5, when no measurement takes place and the projectile passes by the ball unimpeded. You can see that the peak in the wave function, telling us most probable values for the results of measurements of x₁ and x₂, if carried out at a specific time t, moves along roughly the same path as the star in Fig. 5: the most probable values of x₁ increase steadily with time, while those of x₂ remain fixed.

In this situation, the ball’s behavior has nothing to do with the projectile. We cannot learn anything one way or the other about the projectile from the position or motion of the ball.

What about when a measurement takes place, as in Fig. 4? Again, as seen in Fig. 7, the majority of the wave function follows the path of the star, with the most probable values of x₂ beginning to increase around the most likely time of the collision. This change in the most likely value of x₂ is an indication of the presence of the projectile and its interaction with the ball. [Note: Fig. 7, unlike other quantum wave functions shown in this series, is a sketch, not a precise solution to the full quantum equations; I simply haven’t yet found a set-up where the equations can be solved numerically with enough precision and speed to get a nice-looking result. I expect I’ll eventually find an example, but it might take some time.]

More precisely, because of the collision, the motion of the ball is now correlated with that of the projectile — their motions are logically and physically related. That by itself is not unusual; all interactions between objects lead to some level of correlation between them. But this correlation is stable; as a result of the collision, the ball is highly unlikely to be found back in its initial position. And so, when we later look at the trap and find it empty, this does indeed give us reliable information about the projectile, namely that at some point it passed through the trap. (This type of correlation, both within and beyond the measurement context, will be a major topic in the future.)

So far, this all looks quite straightforward. The motion of the star in Fig. 4 is seen in the motion of the peak of the wave function in Fig. 7. Similar behavior is seen in Figs. 5 and 6. But these are simple cases: where the projectile’s motion is well-known, its location is not too uncertain, and the measurement device is almost perfect. We will soon explore far more complex and interesting quantum examples, using this simple one as our conceptual foundation, and things won’t be so straightforward anymore.

I’ll stop here for today. Please let me know in the comments if there are aspects of this story that you find confusing; we need all to be on the same page before we advance into the more subtle elements of our quantum world.

Update (Feb. 27): While we’re on the subject of theoretical computer science, friends-of-the-blog Adam Klivans and Raghu Meka have asked me to publicize that STOC’2025 TheoryFest, to be held June 23-27 in Prague, is eagerly seeking proposals for workshops. The deadline is March 9th.

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• Because of a recent breakthrough by Cook and Mertz on Tree Evaluation, Ryan now shows that every problem solvable in t time on a multitape Turing machine is also solvable in close to √t space
  
• As a consequence, he shows that there are problems solvable in O(n) space that require nearly quadratic time on multitape Turing machines
  
• If this could be applied recursively to boost the polynomial degree, then P≠PSPACE
  
• On Facebook, someone summarized this result as “there exists an elephant that can’t fit through a mouse hole.” I pointed out that for decades, we only knew how to show there was a blue whale that didn’t fit through the mouse hole
  
• I’ll be off the Internet for much of today (hopefully only today?) because of jury duty! Good thing you’ll have Ryan’s amazing new paper to keep y’all busy…

Update (Feb. 25): It occurs to me that the new result is yet another vindication for Ryan’s style of doing complexity theory—a style that I’ve variously described with the phrases “ironic complexity theory” and “caffeinated alien reductions,” and that’s all about using surprising upper bounds for one thing to derive unsurprising lower bounds for a different thing, sometimes with a vertigo-inducing chain of implications in between. This style has a decidedly retro feel to it: it’s been clear since the 1960s both that there are surprising algorithms (for example for matrix multiplication), and that the time and space hierarchy theorems let us prove at least some separations. The dream for decades was to go fundamentally beyond that, separating complexity classes by “cracking their codes” and understanding the space of all possible things they can express. Alas, except for low-level circuit classes, that program has largely failed, for reasons partly explained by the Natural Proofs barrier. So Ryan achieves his successes by simply doubling down on two things that have worked since the beginning: (1) finding even more surprising algorithms (or borrowing surprising algorithms from other people), and then (2) combining those algorithms with time and space hierarchy theorems in clever ways to achieve new separations.

Next: Part 2

In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.

In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.

Before I can describe this categorical viewpoint on potential functions, I have to explain what potential functions are in the magnitude context, and why they’re very useful. That’s what I’ll do today.

This part of the story is about metric spaces. For now I’ll assume they satisfy all the classical axioms, including symmetry of the metric, meaning that $d(a, a') = d(a', a)$ for all points $a$ and $a'$. When metric spaces are viewed as enriched categories, symmetry isn’t automatic — but we’ll come to that next time.

A weighting on a finite metric space $A$ is a function $w: A \to \mathbb{R}$ such that for all $a \in A$,

$$\sum_{a' \in A} e^{-d(a, a')} w(a') = 1.$$

Everyone who sees this formula for the first time asks where the exponential comes from. Ultimately it’s because of the enriched category viewpoint (which again we’ll come to next time), but the short story is that exponential is essentially the only reasonable function that converts addition into multiplication.

For simplicity, I’ll assume here that every finite metric space $A$ has a unique weighting, which I’ll call $w_A$. Since the definition of weighting involves the same number of equations as unknowns, this is generically true (and it’s always true for subspaces of $\mathbb{R}^n$), even though there are exceptions.

The magnitude of $A$ is

$$|A| = \sum_{a \in A} w_A(a) \in \mathbb{R}.$$

That’s for finite metric spaces. To extend the definition to compact metric spaces, there are various ideas you might try. You could define the magnitude of a compact space as the supremum of all the magnitudes of its finite subspaces. Or, you could take an ever-denser sequence of finite subsets of your compact space, then define its magnitude to be the limit of the magnitudes of the approximating subsets. Or, you could try replacing the sums in the formulas above by integrals, somehow.

Mark Meckes showed that all these approaches are equivalent. They all give the same definition of the magnitude of a compact space. (At least, this is true subject to a condition called “positive definiteness” which I won’t discuss and which always holds for subspaces of $\mathbb{R}^n$.)

How do we actually calculate magnitude, say for compact subspaces $A$ of $\mathbb{R}^n$?

In principle, you already know how do it. You run through all the finite subsets of $A$, calculate the magnitude of each using the definition above, and take the sup. The trouble is, this procedure is incredibly hard to work with. It’s completely impractical.

A slightly more practical approach is to look for a “weight measure”, that is, a signed measure $w$ on $A$ such that for all $a \in A$,

$$\int_A e^{-d(a, a')}\, d w(a') = 1.$$

As Mark showed, if $w$ is a weight measure then the magnitude of $A$ is given by $|A| = w(A)$. This is the analogue of the formula $|A| = \sum_a w(a)$ for finite spaces.

Example   Take an interval $[c, d]$ in $\mathbb{R}$. It’s an elementary exercise to check that

$$\tfrac{1}{2}(\delta_c + \lambda_{[c, d]} + \delta_d)$$

is a weight measure on $[c, d]$, where $\delta_c$ and $\delta_d$ denote the Dirac deltas at $c$ and $d$, and $\lambda_{[c, d]}$ is Lebesgue measure on $[c, d]$. It follows that $[c, d]$ is the total mass of this measure, which is

$$1 + \tfrac{1}{2}(d - c).$$

In other words, it’s $1$ plus half the length of the interval.

The trouble with weight measures is that very few spaces have them. Even Euclidean balls don’t, beyond dimension one. It turns out that we need something more general than a measure, something more like a distribution.

In another paper, Mark worked out exactly what kind of measure-like or distribution-like thing a weighting should be. The answer is very nice, but I won’t explain it here, because I want to highlight the formal aspects above all.

So I’ll simply write the weighting on a metric space $A$ as $w_A$, without worrying too much about what kind of thing $w_A$ actually is. All that matters is that it behaves something like a measure: we can pair it with “nice enough” functions $\phi: A \to \mathbb{R}$ to get a real number, which I’ll write as

$$\int_A \phi(a) \, d w_A(a)$$

or simply

$$\int_A \phi \, d w_A.$$

And I’ll assume that all the spaces $A$ that we discuss do have a unique weighting $w_A$.

That’s the background. But back to the question: how do we actually calculate magnitude?

First main idea   Don’t look at metric spaces $A$ in isolation. Instead, consider spaces embedded in some big space $B$ that we understand well.

Think of the big space $B$ as fixed. The typical choice is $\mathbb{R}^n$. One sense in which we “understand” $\mathbb{R}^n$ well is that we have a weight measure on it: it’s just Lebesgue measure divided by a constant factor $c_n$. This follows easily from the fact that $\mathbb{R}^n$ is homogeneous. (It doesn’t matter for anything I’m going to say, but the constant $c_n$ is $\int_{\mathbb{R}^n} e^{-\|x\|}\, d x$, for which there’s a standard formula involving $\pi$s and factorials.)

The potential function of a subspace $A \subseteq B$ is the function $h_A: B \to \mathbb{R}$ defined by

$$h_A(b) = \int_A e^{-d(a, b)} \, d w_A(a).$$

By definition of weighting, $h_A$ has constant value $1$ on $A$. But outside $A$, it could be anything, and it turns out that its behaviour outside $A$ is very informative.

Although $w_A$ is a measure-like thing on the subspace $A$ of $B$, we typically extend it by $0$ to all of $B$, and then the definition of the potential function becomes

$$h_A(b) = \int_B e^{-d(x, b)} \, d w_A(x)$$

Examples   In all these examples, I’ll take the big space $B$ to be $\mathbb{R}$.

• Let $A = \{0\}$. Then the weighting $w_A$ on $A$ is the Dirac delta at $0$, and the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h(b) = e^{-|b|}$$

  ($b \in \mathbb{R}$), whose graph looks like this:

  

• Let $A = [-2, 2]$. As we’ve already seen,

  $$w_A = \tfrac{1}{2}(\delta_{-2} + \lambda_{[-2, 2]} + \delta_2),$$

  and a little elementary work then reveals that the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h_A(b) = \begin{cases} e^{-(-2 - b)} &\text{if}   b \leq -2, \\ 1 &\text{if}   -2 \leq b \leq 2, \\ e^{-(b - 2)} &\text{if}   b \geq 2, \end{cases}$$

  which looks like this:

  

• In the two examples we just did, the potential function of $A$ is just the negative exponential of the distance between $A$ and the argument $b$. That’s not exactly coincidence: as we’ll see in the next part, the function just described corresponds to a strict colimit, whereas the potential function corresponds to a lax colimit. So it’s not surprising that they coincide in simple cases.

  But this only happens in the very simplest cases. It doesn’t happen for Euclidean balls above dimension $1$, or even for two-point spaces. For example, taking the subset $A = \{-1, 1\}$ of $B = \mathbb{R}$, we have

  $$h_A(x) = \frac{e^{-|x + 1|} + e^{-|x - 1|}}{1 + e^{-2}},$$

  which looks like this:

  

  whereas $b \mapsto e^{-d(A, b)}$ looks like this:

  

  Similar, but different!

And now we come to the:

Second main idea   Work with potential functions instead of weightings.

To explain what this means, I need to tell you three good things about potential functions.

• The potential function determines the magnitude:

  $$|A| = \int_B h_A \, d w_B.$$

  At the formal level, proving this is a one-line calculation: substitute the definition of $h_A$ into the right-hand side and follow your nose.

  For example, we saw that $\mathbb{R}^n$ has weighting $\lambda/c_n$, where $\lambda$ is Lebesgue measure and $c_n$ is a known constant. So

  $$|A| = \frac{1}{c_n} \int_{\mathbb{R}^n} h_A(x) \, d x$$

  for compact $A \subseteq \mathbb{R}^n$. Here $d x$ refers to ordinary Lebesgue integration.

  (For $\mathbb{R}^n$, and in fact a bit more generally, this result appears as Theorem 4.16 here.)

• You can recover the weighting from the potential function. So, you don’t lose any information by working with one rather than the other.

  How do you recover it? Maybe it’s easiest to explain in the case when the spaces are finite. If we write $Z_B$ for the $B \times B$ matrix $(e^{-d(b', b)})_{b', b \in B}$ then the definition of the potential function can be expressed very succinctly:

  $$h_A = Z_B w_A.$$

  Here $h_A$ and $w_A$ are viewed as column vectors with entries indexed by the points of $B$. (For $w_A$, those entries are $0$ for points not in $A$.) Assuming $Z_B$ is invertible, this means we recover $w_A$ from $h_A$ as $Z_B^{-1} h_A$. And something similar is true in the non-finite setting.

  However, what really makes the technique of potential functions sing is that when $B = \mathbb{R}^n$, there’s a much more explicit way to recover the weighting from the potential function:

  $$w_A = (I - \Delta)^{(n + 1)/2} h_A$$

  (up to a constant factor that I’ll ignore). Here $I$ is the identity and $\Delta$ is the Laplace operator, $\sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2}$. This is Proposition 5.9 of Mark’s paper.

  How much more of this bullet point you’ll want to read depends on how interested you are in the analysis. The fundamental point is simply that $(I - \Delta)^{(n + 1)/2}$ is some kind of differential operator. But for those who want a bit more:

  • To make sense of everything, you need to interpret it all in a distributional sense. In particular, this allows one to make sense of the power $(n + 1)/2$, which is not an integer if $n$ is even.

  • Maybe you wondered why someone might have proved a result on the magnitude of odd-dimensional Euclidean balls only, as I mentioned at the start of the post. What could cause the odd- and even-dimensional cases to become separated? It’s because whether or not $(n + 1)/2$ is an integer depends on whether $n$ is odd or even. When $n$ is odd, it’s an integer, which makes $(I - \Delta)^{(n + 1)/2}$ a differential rather than pseudodifferential operator. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca later worked out lots about the even-dimensional case, but I won’t talk about that here.

  • Finally, where does the operator $(I - \Delta)^{(n + 1)/2}$ come from? Sadly, I don’t have an intuitive explanation. Ultimately it comes down to the fact that the Fourier transform of $e^{-\|\cdot\|}$ is $\xi \mapsto (1 + \|\xi\|)^{-(n + 1)/2}$ (up to a constant). But that itself is a calculation that’s really quite tricky (for me), and it’s hard to see anything beyond “it is what it is”.

• The third good thing about potential functions is that they satisfy a differential equation, in the situation where our big space $B$ is $\mathbb{R}^n$. Specifically:

  $$\begin{cases} h_A \equiv 1 &\text{on}   A \\ (I - \Delta)^{-(n + 1)/2} h_A \equiv 0&\text{on}   \mathbb{R}^n \setminus A. \end{cases}$$

  Indeed, the definition of weighting implies that $h_A \equiv 1$ on $A$, and the “second good thing” together with the fact that $w_A$ is supported on $A$ give the second clause.

  Not only do we have a differential equation for $h_A$, we also have boundary conditions. There are boundary conditions at the boundary of $A$, because of something I’ve been entirely vague about: the functions we’re dealing with are meant to be suitably smooth. There are also boundary conditions at $\infty$, because our functions are also meant to decay suitably fast.

  Maybe Mark will read this and correct me if I’m wrong, but I believe there are exactly the right number of boundary conditions to guarantee that there’s (typically? always?) a unique solution. In any case, the following example — also taken from Mark’s paper — illustrates the situation.

Example   Let’s calculate the magnitude of a real interval $A = [c, d]$ using the potential function method.

Its potential function $h_A$ is a function $\mathbb{R} \to \mathbb{R}$ such that $h_A \equiv 1$ on $[c, d]$ and $h_A - h''_A = 0$ on the rest of the real line. That differential equation comes from taking $(I - \Delta)^{-(n + 1)/2}$ in the case $n = 1$.

The functions $f$ satisfying $f'' = f$ are those of the form $k e^{\pm x}$ for some constant $k$, and in our case we’re free to choose the constant and the $\pm$ sign differently on the two connected components of $\mathbb{R} \setminus A$. So there are lots of solutions. But $h_A$ is required to be continuous and to converge to $0$ at $\pm \infty$, and that pins it down uniquely: the one and only solution is

$$h_A(x) = \begin{cases} e^{-(c - x)} &\text{if}   x \leq c, \\ 1 &\text{if}   c \leq x \leq d, \\ e^{-(x - d)} &\text{if}   d \leq x. \end{cases}$$

I showed you the graph of this potential function above, in the case where $A = [-2, 2]$.

So the magnitude of $A = [c, d]$ is

$$\frac{1}{c_1} \int_\mathbb{R} h_A(x)\, d x,$$

where $c_1$ is the constant $\int_\mathbb{R} e^{-|x|} \, d x = 2$. This gives the answer:

$$|A| = 1 + \tfrac{1}{2}(d - c) = 1 + \tfrac{1}{2}length(A).$$

 

The crucial point is that in this example, we didn’t have to come up with a weighting on $A$. The procedure was quite mechanical. And that’s the attraction of the method of potential functions.

Next time, I’ll put all this into a categorical context using the notion of the magnitude of a functor, which I introduced here.

Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!

Let $V$ be a monoidal category, which I’ll assume is symmetric, closed, complete or cocomplete as necessary. Take a pair of $V$-categories $A$ and $B$, and a $V$-functor

$$F: A \to B.$$

Then we get a new functor

$$N_F: B \to [A^{op}, V]$$

defined by

$$(N_F(b))(a) = B(F a, b).$$

This is sometimes called the nerve functor induce by $F$, because of the first example in the following list.

Examples   In the first few examples, I’ll take the enriching category $V$ to be $Set$.

• Let $F$ be the inclusion $\Delta \hookrightarrow Cat$, which takes a finite nonempty totally ordered set $[n] = \{0, \ldots, n\}$ and realizes it as a category in the usual way. Then $N_F: Cat \to [\Delta^{op}, Set]$ sends a small category $D$ to its nerve, which is the simplicial set whose $n$-simplices are paths in $D$ of length $n$.

• Let $F$ be the functor $\Delta \to Top$ that sends $[n]$ to the topological $n$-simplex $\Delta^n$. Then $N_F: Top \to [\Delta^{op}, Set]$ sends a topological space $X$ to its “singular simplicial set”, whose $n$-simplices are the continuous maps $\Delta^n \to X$.

• Let $F: A \hookrightarrow B$ be an inclusion of partially ordered sets, viewed as categories in the usual way. Then $N_F : B \to [A^{op}, Set]$ is given by

  $$(N_F(b))(a) = \begin{cases} 1 &\text{if}   b \leq a \\ 0 &\text{otherwise}. \end{cases}$$

  For example, let $B$ be the set $\mathbb{Z}$ of integers, ordered by divisibility. Let $A \subseteq \mathbb{Z}$ be the set of (positive) primes. Then $N_F$ is the functor $\mathbb{Z} \to Set^{\{primes\}}$ that sends $n \in \mathbb{Z}$ to

  $$([p | n])_{\text{primes}   p}$$

  where $[p|n]$ is Iverson notation: it’s $1$ if $p$ divides $n$ and $0$ otherwise.

• If $F$ has a right adjoint $G$, then $N_F(b) = A(-, G b): A^{op} \to V$. In particular, $N_F(b)$ is representable for each $b \in B$.

• Take the inclusion $Field \hookrightarrow Ring$, where $Ring$ is the category of commutative rings. Now take opposites throughout to get $F: Field^{op} \hookrightarrow Ring^{op}$. The resulting nerve functor

  $$N_F: Ring^{op} \to [Field, Set]$$

  sends a ring $R$ to $Ring(R, -)|_{Field}$ (by which I mean the restriction of $Ring(R, -): Ring \to Set$ to the category of fields). More explicitly,

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -).$$

  Here $Spec(R)$ is the set of prime ideals $p$ of $R$, and $Frac(R/p)$ is the field of fractions of the integral domain $R/p$. In particular, $N_F(b)$ is a coproduct of representables for each $b \in B$. (In the terminology of Diers, one therefore says that $Field \hookrightarrow Ring$ is a “right multi-adjoint”. A closely related statement is that it’s a parametric right adjoint. But that’s all just jargon that won’t matter for this post.)

Now let’s consider $V = (\mathbb{R}^+, \geq)$ with its additive monoidal structure, so that $V$-categories are generalized metric spaces and $V$-functors are the functions $f$ between metric spaces that are variously called short, distance-decreasing, contractive, or 1-Lipschitz: $d(f(a), f(a')) \leq d(a, a')$ for all $a, a'$. I’ll just call them “maps” of metric spaces.

• Take a map $f: A \to B$ of metric spaces. The induced map $N_f: B \to [A^{op}, \mathbb{R}^+]$ is given by

  $$(N_f(b))(a) = d(f(a), b).$$

  Unlike in the first post, I’m not going to assume here that our metric spaces are symmetric. so the “op” on the $A$ matters, and $d(a, f(b)) \neq d(f(b), a)$ in general.

  We’ll be particularly interested in the case where the $V$-functor $F$ is full and faithful. Most of the examples above have this property. In the metric case, $V = \mathbb{R}^+$, being full and faithful means being distance-preserving, or in other words, an inclusion of a metric subspace. In that case, it’s normal to drop the $f$. So we’d usually then write

  $$(N_f(b))(a) = d(a, b).$$

Next I’ll define the potential function of a $V$-functor. For this we need some of the apparatus of magnitude. Roughly speaking, this means we have a notion of the “size” of each object of our base monoidal category $V$.

More exactly, we take a field $k$ and a function $|\cdot|: ob(V) \to k$, to be thought of as assigning to each object $S$ of $V$ its size ${|S|} \in k$. And to make everything work, we assume our size function $|\cdot|$ has reasonable properties:

$$S \cong T \implies |S| = |T|, \quad |S \otimes T| = |S| \cdot |T|, \quad |I| = 1,$$

where $\otimes$ is the monoidal product on $V$ and $I$ is its unit.

The basic definitions are these. A weighting on a finite $V$-category $A$ is a function $w_A : ob(A) \to k$ such that

$$\sum_{b \in A} |A(a, b)| w_A(b) = 1$$

for all $a \in A$, and a coweighting $w^A$ on $A$ is a weighting on $A^{op}$. Although it’s not true that every finite $V$-category has a unique weighting, it’s a harmless enough assumption that I’ll make it here. The magnitude of a finite $V$-category $A$ is

$$|A| = \sum_{a \in A} w_A(a) = \sum_{a \in A} w^A(a) \in k.$$

It’s a tiny lemma that the total weight $\sum_{a \in A} w_A(a)$ is equal to the total coweight $\sum_{a \in A} w^A(a)$, so you can equivalently define the magnitude to be one or the other.

There’s a whole lot to say about why this is a worthwhile definition, and I’ve said it in detail many times here before. But here I’ll just say two short things:

• Taking $V = FinSet$, $k = \mathbb{Q}$, and $|\cdot|$ to be cardinality, the magnitude of a finite category is equal to the Euler characteristic of its classifying space, under suitable hypotheses guaranteeing that the latter is defined.

• Take $V = \mathbb{R}^+$, $k = \mathbb{R}$, and $|\cdot|$ to be $s \mapsto e^{-s}$ (because that’s just about the only function converting the tensor product of $V$ into addition, as our size functions are contractually obliged to do). Then we get a notion of the magnitude of a metric space, a real-valued invariant that turns out to be extremely interesting.

That’s the magnitude of enriched categories. But we can also talk about the magnitude of enriched presheaves. Take a finite $V$-category $A$ and a $V$-presheaf $X: A^{op} \to V$. Its magnitude is

$$|X| = \sum_{a \in A} w^A(a) |X(a)| \in k.$$

When I wrote about the magnitude of functors before, I concentrated on covariant functors $A \to V$, which meant that the weights in the weighted sum that defines $|X|$ were, well, the weights $w_A(a)$. But since we’ve now changed $A$ to $A^{op}$, the weights have become coweights.

Let me briefly recall why the magnitude of presheaves is interesting, at least in the case $V = FinSet$:

• If our presheaf $X$ is a coproduct of representables then $|X|$ is $|colim(X)|$, the cardinality of the colimit of $X$.

• The magnitude of presheaves generalizes the magnitude of categories. The magnitude of a category $A$ is equal to the magnitude of the presheaf $A^{op} \to Set$ with constant value $1$.

• In the other direction, the magnitude of categories generalizes the magnitude of presheaves: $|X| = |\mathbb{E}(X)|$ for all presheaves $X$. Here $\mathbb{E}(X)$ means the category of elements of $X$, also called the (domain of the) Grothendieck construction.

  It’s enlightening to think of this result as follows. If we extend the codomain of $X$ from $Set$ to $Cat$ then $\mathbb{E}(X)$ is the colax colimit of $X: A^{op} \to Cat$. So $|X| = |colaxcolim(X)|$. You can compare this to the more limited result that $|X| = |colim(X)|$, which only holds when $X$ has the special property that it’s a coproduct of representables.

  There’s also another closely related result, due to Ponto and Shulman: under mild further hypotheses, $|X| = \chi(hocolim(X))$. Mike and I had a little discussion about the relationship here.

  Now here’s something funny. Which do you think is a more refined invariant of a presheaf $X: A^{op} \to FinSet$: the magnitude $|X|$, which is equal to $|colaxcolim(X)|$ and $\chi(hocolim(X))$, or the cardinality $|colim(X)|$ of the strict colimit?

  From one perspective, it’s the magnitude. After all, we usually think of lax or pseudo 2-dimensional things as more subtle and revealing than strict 1-dimensional things. But from another, it’s the cardinality of the strict colimit. For instance, if $A$ is the one-object category corresponding to a group $G$ then $X$ is a right $G$-set, the cardinality of the colimit is the number of orbits (an interesting quantity), but the magnitude is always just the cardinality of the set $X$ divided by the order of $G$ (relatively boring and crude).

  In this post I’ll keep comparing these two quantities. Whichever seems “more refined”, the comparison is interesting in itself.

(I stuck to the case $V = FinSet$ here, because for an arbitrary $V$, there isn’t a notion of “colimit” as such: we’d need to think about weighted colimits. It’s also harder to say what the category of elements or homotopy colimit of an enriched presheaf should be.)

Now here’s something new. The potential function of a $V$-functor $F: A \to B$ is the function

$$h_F: \ob(B) \to k$$

defined by

$$h_F(b) = |N_F(b)|$$

for all $b \in B$. Expanding out the definitions, this means that

$$h_F(b) = |A(F-, b)| = \sum_{a \in A} w^A(a) |B(F a, b)|,$$

where $w^A$ is the notation I introduced for the coweighting on $A$. If $F$ is full and faithful then $h_F \equiv 1$ on the set of objects in the image of $F$.

Examples   The definition I just gave implicitly assumes that $\ob(A)$ is finite. But I’ll relax that assumption in some of these examples, at the cost of some things getting less than rigorous.

Again, I’ll begin with some examples where $V = FinSet$.

• What’s the potential function of the inclusion $\Delta \hookrightarrow Cat$? To make sense of this, we need a coweighting on $\Delta$, and I’m pretty sure there isn’t one. So let’s abandon $\Delta$ and instead use its subcategory $\Delta_{inj}$, consisting of all the finite nonempty totally ordered sets $[n] = \{0, \ldots, n\}$ but only the injective order-preserving maps between them. This amounts to considering only face maps, not degeneracies.

  The coweighting on $\Delta_{inj}$ is $[n] \mapsto (-1)^n$. So, the potential function $h$ of $\Delta_{inj} \hookrightarrow Cat$ is given, on a category $D$, by

  $$h(D) = \sum_{n \geq 0} (-1)^n |Cat([n], D)|.$$

  Here $Cat([n], D)$ is the set of paths of length $n$ in $D$, which we often write as $D_n$. So

  $$h(D) = \sum_{n \geq 0} (-1)^n |D_n|.$$

  Even if the category $D$ is finite, this is a divergent series. But morally, $h(D)$ is the alternating sum of the number of $n$-cells of $D$.

  In other words, the potential function of $\Delta_{inj} \to Cat$ is morally Euler characteristic.

  As I hinted above, it’s interesting to compare the potential function with what you get if you take the cardinality of the colimit of $B(F-, b)$ instead of its magnitude. I’ll call this the “strict potential function”. In this case, it’s

  $$D \mapsto |colim_{[n] \in \Delta_{inj}} D_n| = |\pi_0(D)|$$

  — the cardinality of the set of connected-components of $D$. So while the potential function gives Euler characteristic, the strict potential function gives the number of connected components.

• Similarly, the potential function of the usual functor $\Delta_{inj} \hookrightarrow Top$, sending $n$ to $\Delta^n$, is given by

  $$D \mapsto \sum_{n \geq 0} (-1)^n |Top(\Delta^n, D)|$$

  for topological spaces $D$. Again, this formally looks like Euler characteristic: the alternating sum of the number of cells in each dimension. And much as for categories, the strict potential function gives the number of path-components.

• For the inclusion $\{primes\} \hookrightarrow (\mathbb{Z}, |)$ that we saw earlier, the potential function $h: \mathbb{Z} \to \mathbb{Q}$ is what’s often denoted by $\omega$, sending an integer $n$ to the number of distinct prime factors of $n$. The strict potential function is the same.

• An easy example that works for any $V$: if $F: A \to B$ has a right adjoint then since $A(F-, b)$ is representable, its magnitude is equal to the cardinality of its colimit, which is $1$. So $h_F: \ob(B) \to k$ has constant value $1$, as does the strict potential function.

  More generally, if $A(F-, b)$ is a coproduct of $n$ representables then $h_F(b) = n$, and the same is true for the strict potential function.

• For the inclusion $F: Field^{op} \hookrightarrow Ring^{op}$, we saw that

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -)$$

  for every ring $R$. This is a coproduct of representables, so by the previous example, $|N_F(R)| = |Spec(R)|$. That is, the potential function is given by

  $$h_F: R \mapsto |Spec(R)|$$

  (as is the strict potential function). Usually $Spec(R)$ is infinite, so this calculation is dodgy, but if we restrict ourselves to finite fields and rings then everything is above board.

• For an opfibration $F: A \to B$, one can show that the potential function $h_F$ is given by

  $$h_F(b) = |F^{-1}(b)|.$$

  I should explain the notation on the right-hand side. The category $F^{-1}(b)$ is the fibre over $b \in B$, consisting of the objects of $A$ that $F$ sends to $b$ and the maps in $A$ that $F$ sends to $1_b$. We can take its magnitude (at least if it’s finite and lightning doesn’t strike), which is what I’ve denoted by $|F^{-1}(b)|$.

  I won’t include the proof of this result, but I want to emphasize that it doesn’t involve finding a coweighting on $A$. You might think it would, because the definition of $h_F(b)$ involves the coweighting on $A$. But it turns out that it’s enough to just assume it exists.

• Now take $V = \mathbb{R}^+$, so that $V$-categories are generalized metric spaces. The potential function of a map $f: A \to B$ of metric spaces is the function $h_f : B \to \mathbb{R}$ given by

  $$h_f(b) = \sum_{a \in A} w^A(a) e^{-d(f(a), b)}.$$

  In particular, if $f$ is the inclusion of a subspace $A \subseteq B$, then it’s natural to write $h_A$ instead of $h_f$, and we have

  $$h_A(b) = \sum_{a \in A} w^A(a) e^{-d(a, b)}.$$

  This is equal to $1$ on $A$ but can take other values on $B \setminus A$. In suitable infinite contexts, sums become integrals and $w^A$ becomes something like a measure or distribution, in which case the formula becomes

  $$h_A(b) = \int_A e^{-d(a, b)} \, d w^A(a).$$

  This is exactly the formula for the potential function in Part 1, with one difference: there, I used weightings on $A$, and here, I’m using coweightings. It’s coweightings that one should use. In the previous post, I assumed that all metrics were symmetric, which means that weightings and coweightings are the same. So there’s no inconsistency.

  (Of course, one could take duals throughout and use weightings on $A$ instead. But we’ll see that whichever choice you make, you end up having to consider weightings on one of $A$ and $B$ and coweightings on the other.)

  What about the strict potential function, sending a point $b \in B$ to the “cardinality of the colimit of $B(f-, b): A \to \mathbb{R}^+$”? Well, I put that phrase in inverted commas because we’re now in an enriched context, so it needs a bit of interpretation. “Cardinality” is okay: it becomes the size function $|\cdot| = e^{-(\cdot)}: \mathbb{R}^+ \to \mathbb{R}$. “Colimit” wouldn’t usually make sense in an enriched world, but we’re saved by the fact that the monoidal unit of $V = \mathbb{R}^+$ (namely, $0$) is terminal. The colimit of a $V$-functor into $V$ is just its infimum. So the strict potential function of a map $f: A \to B$ of metric spaces is just

  $$b \mapsto e^{-\inf_{a \in A} d(f(a), b)} = e^{-d(f A, B)}.$$

  I showed you an example of the difference between the potential function and the strict potential function in Part 1, although not with those words. If we take $f$ to be the inclusion $\{-1, 1\} \hookrightarrow \mathbb{R}$ then the potential function is

  

  whereas the strict potential function is

  

  These two functions are identical on $(-\infty, -1] \cup [1, \infty)$, but different on the interval $(-1, 1)$. If you’ve ever wondered what the difference is between strict and lax colimits, here’s an example!

By definition, the potential function of an enriched functor $F: A \to B$ is given by

$$h_F(b) = \sum_{a \in A} w^A(a) |B(F(a), b)|,$$

but a slightly different viewpoint is sometimes helpful. We can take the function $w^A: ob(A) \to k$ and push it forward forward along $F$ to obtain a new function

$$\begin{array}{cccc} F_* w^A: &ob(B) &\to &k, \\ &b &\mapsto &\sum_{a \in F^{-1}(b)} w^A(a). \end{array}$$

Really it’s best to think of $w^A$ as not a function but a measure taking values in $k$ (although these are the same thing on a finite set). Then this is just the usual pushforward measure construction. In any case, the formula for the potential function now becomes

$$h_F(b) = \sum_{b' \in B} (F_* w^A)(b') |B(b', b)|,$$

which has the advantage that everything is taking place in $B$ rather than $A$. In the case where $f: A \to B$ is an embedding of a subspace of a metric space, $f_* w^A$ is just the extension of the measure $w^A$ on $A$ to all of $B$, which we’d usually just write as $w^A$ (with a wink to the audience). In integral notation, the last formula becomes

$$h_F(b) = \int_B e^{-d(x, b)} \, d w^A(x),$$

which is what we had in the last post.

I’ve now explained what potential functions are. But what are they good for?

Last time, I explained that they’re very good indeed for helping us to calculate the magnitude of metric spaces. The key was that

$$|A| = \int_B h_A \, d w_B$$

for a subspace $A$ of a metric space $B$. And as I recounted, that key unlocks the door to the world of PDE methods.

So you might hope something similar is true for enriched categories in general: that there’s a formula for the magnitude in terms of the potential function. And there is! For any $V$-functor $F: A \to B$, it’s a theorem that

$$|A| = \sum_{b \in B} h_F(b) w_B(b).$$

This is an easy calculation from the definitions.

(If you’re really paying attention, you’ll notice that we used the coweighting on $A$ to define the potential function, and now we’re using the weighting on $B$. That’s just how it turns out. One is a weighting, and the other is a coweighting.)

Examples

• For a $V$-functor $F: A \to B$ that has a right adjoint, we saw that the potential function $h_F$ has constant value $1$, so this formula tells us that

  $$|A| = \sum_{b \in B} w_B(b).$$

  But the right-hand side is by definition the magnitude of $B$, so what this formula is saying is that

  $$|A| = |B|.$$

  In other words, if there’s an adjunction between two categories, their magnitudes are equal!

  This has been known forever (Proposition 2.4), and is also intuitive from a homotopical viewpoint. But it’s nice that it just pops out.

• Less trivially, we saw above that for an opfibration $F: A \to B$, the potential function is $h_F: b \mapsto |F^{-1}(b)|$. So

  $$|A| = \sum_{b \in B} w_B(b) |F^{-1}(b)|.$$

  In other words, the magnitude of the “total” category $A$ is the weighted sum over the base $B$ of the magnitudes of the fibres. (Special case: the magnitude of a product is the product of the magnitudes.)

  Now this has been known forever too (Proposition 2.8), but I want to emphasize that the proof is fundamentally different from the one I just linked. That proof constructs the weighting on $A$ from the weightings on the base $B$ and the fibres. Now that’s an easy and informative proof, but what we’ve just done is different, because it didn’t involve figuring out the weighting or coweighting on $A$. So although the result isn’t new or difficult, it’s perhaps grounds for optimism that the method of potential functions will let us prove new things about enriched categories other than metric spaces.

What new things? I don’t know! This is where I’ve got to now. Maybe there are applications in the metric world in which $f: A \to B$ is nothing like an inclusion. Maybe there are applications to graphs, replacing the PDE methods used for subspaces of $\mathbb{R}^n$ by discrete analogues. Maybe the potential function method can be used to shed light on the tricky result that the magnitude of graphs is invariant under certain Whitney twists (Theorem 5.2), and more generally under the sycamore twists introduced by Emily Roff (Theorem 6.5). Let’s find out!

There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture! This conjecture asserts that a Kakeya set – a subset of that contains a unit line segment in every direction, must have Minkowski and Hausdorff dimension equal to three. (There is also a stronger “maximal function” version of this conjecture that remains open at present, although the methods of this paper will give some non-trivial bounds on this maximal function.) It is common to discretize this conjecture in terms of small scale . Roughly speaking, the conjecture then asserts that if one has a family of tubes of cardinality , and pointing in a -separated set of directions, then the union of these tubes should have volume . Here we shall be a little vague as to what means here, but roughly one should think of this as “up to factors of the form for any “; in particular this notation can absorb any logarithmic losses that might arise for instance from a dyadic pigeonholing argument. For technical reasons (including the need to invoke the aforementioned dyadic pigeonholing), one actually works with slightly smaller sets , where is a “shading” of the tubes in that assigns a large subset of to each tube in the collection; but for this discussion we shall ignore this subtlety and pretend that we can always work with the full tubes.

Previous results in this area tended to center around lower bounds of the form

for various intermediate dimensions , that one would like to make as large as possible. For instance, just from considering a single tube in this collection, one can easily establish (1) with . By just using the fact that two lines in intersect in a point (or more precisely, a more quantitative estimate on the volume between the intersection of two tubes, based on the angle of intersection), combined with a now classical -based argument of Córdoba, one can obtain (1) with (and this type of argument also resolves the Kakeya conjecture in two dimensions). In 1995, building on earlier work by Bourgain, Wolff famously obtained (1) with using what is now known as the “Wolff hairbrush argument”, based on considering the size of a “hairbrush” – the union of all the tubes that pass through a single tube (the hairbrush “stem”) in the collection.

In their new paper, Wang and Zahl established (1) for . The proof is lengthy (127 pages!), and relies crucially on their previous paper establishing a key “sticky” case of the conjecture. Here, I thought I would try to summarize the high level strategy of proof, omitting many details and also oversimplifying the argument at various places for sake of exposition. The argument does use many ideas from previous literature, including some from my own papers with co-authors; but the case analysis and iterative schemes required are remarkably sophisticated and delicate, with multiple new ideas needed to close the full argument.

A natural strategy to prove (1) would be to try to induct on : if we let represent the assertion that (1) holds for all configurations of tubes of dimensions , with -separated directions, we could try to prove some implication of the form for all , where is some small positive quantity depending on . Iterating this, one could hope to get arbitrarily close to .

A general principle with these sorts of continuous induction arguments is to first obtain the trivial implication in a non-trivial fashion, with the hope that this non-trivial argument can somehow be perturbed or optimized to get the crucial improvement . The standard strategy for doing this, since the work of Bourgain and then Wolff in the 1990s (with precursors in older work of Córdoba), is to perform some sort of “induction on scales”. Here is the basic idea. Let us call the tubes in “thin tubes”. We can try to group these thin tubes into “fat tubes” of dimension for some intermediate scale ; it is not terribly important for this sketch precisely what intermediate value is chosen here, but one could for instance set if desired. Because of the -separated nature of the directions in , there can only be at most thin tubes in a given fat tube, and so we need at least fat tubes to cover the thin tubes. Let us suppose for now that we are in the “sticky” case where the thin tubes stick together inside fat tubes as much as possible, so that there are in fact a collection of fat tubes , with each fat tube containing about of the thin tubes. Let us also assume that the fat tubes are -separated in direction, which is an assumption which is highly consistent with the other assumptions made here.

If we already have the hypothesis , then by applying it at scale instead of we conclude a lower bound on the volume occupied by fat tubes:

Since , this morally tells us that the typical multiplicity of the fat tubes is ; a typical point in should belong to about fat tubes.

Now, inside each fat tube , we are assuming that we have about thin tubes that are -separated in direction. If we perform a linear rescaling around the axis of the fat tube by a factor of to turn it into a tube, this would inflate the thin tubes to be rescaled tubes of dimensions , which would now be -separated in direction. This rescaling does not affect the multiplicity of the tubes. Applying again, we see morally that the multiplicity of the rescaled tubes, and hence the thin tubes inside , should be .

We now observe that the multiplicity of the full collection of thin tubes should morally obey the inequality

since if a given point lies in at most fat tubes, and within each fat tube a given point lies in at most thin tubes in that fat tube, then it should only be able to lie in at most tubes overall. This heuristically gives , which then recovers (1) in the sticky case.

In their previous paper, Wang and Zahl were roughly able to squeeze a little bit more out of this argument to get something resembling in the sticky case, loosely following a strategy of Nets Katz and myself that I discussed in this previous blog post from over a decade ago. I will not discuss this portion of the argument further here, referring the reader to the introduction to that paper; instead, I will focus on the arguments in the current paper, which handle the non-sticky case.

Let’s try to repeat the above analysis in a non-sticky situation. We assume (or some suitable variant thereof), and consider some thickened Kakeya set

where is something resembling what we might call a “Kakeya configuration” at scale : a collection of thin tubes of dimension that are -separated in direction. (Actually, to make the induction work, one has to consider a more general family of tubes than these, satisfying some standard “Wolff axioms” instead of the direction separation hypothesis; but we will gloss over this issue for now.) Our goal is to prove something like for some , which amounts to obtaining some improved volume bound

that improves upon the bound coming from . From the previous paper we know we can do this in the “sticky” case, so we will assume that is “non-sticky” (whatever that means).

A typical non-sticky setup is when there are now fat tubes for some multiplicity (e.g., for some small constant ), with each fat tube containing only thin tubes. Now we have an unfortunate imbalance: the fat tubes form a “super-Kakeya configuration”, with too many tubes at the coarse scale for them to be all -separated in direction, while the thin tubes inside a fat tube form a “sub-Kakeya configuration” in which there are not enough tubes to cover all relevant directions. So one cannot apply the hypothesis efficiently at either scale.

This looks like a serious obstacle, so let’s change tack for a bit and think of a different way to try to close the argument. Let’s look at how intersects a given -ball . The hypothesis suggests that might behave like a -dimensional fractal (thickened at scale ), in which case one might be led to a predicted size of of the form . Suppose for sake of argument that the set was denser than this at this scale, for instance we have

for all and some . Observe that the -neighborhood is basically , and thus has volume by the hypothesis (indeed we would even expect some gain in , but we do not attempt to capture such a gain for now). Since -balls have volume , this should imply that needs about balls to cover it. Applying (3), we then heuristically have

which would give the desired gain . So we win if we can exhibit the condition (3) for some intermediate scale . I think of this as a “Frostman measure violation”, in that the Frostman type bound

is being violated.

The set , being the union of tubes of thickness , is essentially the union of cubes. But it has been observed in several previous works (starting with a paper of Nets Katz, Izabella Laba, and myself) that these Kakeya type sets tend to organize themselves into larger “grains” than these cubes – in particular, they can organize into disjoint prisms (or “grains”) in various orientations for some intermediate scales . The original “graininess” argument of Nets, Izabella and myself required a stickiness hypothesis which we are explicitly not assuming (and also an “x-ray estimate”, though Wang and Zahl were able to find a suitable substitute for this), so is not directly available for this argument; however, there is an alternate approach to graininess developed by Guth, based on the polynomial method, that can be adapted to this setting. (I am told that Guth has a way to obtain this graininess reduction for this paper without invoking the polynomial method, but I have not studied the details.) With rescaling, we can ensure that the thin tubes inside a single fat tube will organize into grains of a rescaled dimension . The grains associated to a single fat tube will be essentially disjoint; but there can be overlap between grains from different fat tubes.

The exact dimensions of the grains are not specified in advance; the argument of Guth will show that is significantly larger than , but other than that there are no bounds. But in principle we should be able to assume without loss of generality that the grains are as “large” as possible. This means that there are no longer grains of dimensions with much larger than ; and for fixed , there are no wider grains of dimensions with much larger than .

One somewhat degenerate possibility is that there are enormous grains of dimensions approximately (i.e., ), so that the Kakeya set becomes more like a union of planar slabs. Here, it turns out that the classical arguments of Córdoba give good estimates, so this turns out to be a relatively easy case. So we can assume that least one of or is small (or both).

We now revisit the multiplicity inequality (2). There is something slightly wasteful about this inequality, because the fat tubes used to define occupy a lot of space that is not in . An improved inequality here is

where is the multiplicity, not of the fat tubes , but rather of the smaller set . The point here is that by the graininess hypotheses, each is the union of essentially disjoint grains of some intermediate dimensions . So the quantity is basically measuring the multiplicity of the grains.

It turns out that after a suitable rescaling, the arrangement of grains looks locally like an arrangement of tubes. If one is lucky, these tubes will look like a Kakeya (or sub-Kakeya) configuration, for instance with not too many tubes in a given direction. (More precisely, one should assume here some form of the Wolff axioms, which the authors refer to as the “Katz-Tao Convex Wolff axioms”). A suitable version if the hypothesis will then give the bound

Meanwhile, the thin tubes inside a fat tube are going to be a sub-Kakeya configuration, having about times fewer tubes than a Kakeya configuration. It turns out to be possible to use to then get a gain in here,

for some small constant . Inserting these bounds into (4), one obtains a good bound which leads to the desired gain .

So the remaining case is when the grains do not behave like a rescaled Kakeya or sub-Kakeya configuration. Wang and Zahl introduce a “structure theorem” to analyze this case, concluding that the grains will organize into some larger convex prisms , with the grains in each prism behaving like a “super-Kakeya configuration” (with significantly more grains than one would have for a Kakeya configuration). However, the precise dimensions of these prisms is not specified in advance, and one has to split into further cases.

One case is when the prisms are “thick”, in that all dimensions are significantly greater than . Informally, this means that at small scales, looks like a super-Kakeya configuration after rescaling. With a somewhat lengthy induction on scales argument, Wang and Zahl are able to show that (a suitable version of) implies an “x-ray” version of itself, in which the lower bound of super-Kakeya configurations is noticeably better than the lower bound for Kakeya configurations. The upshot of this is that one is able to obtain a Frostman violation bound of the form (3) in this case, which as discussed previously is already enough to win in this case.

It remains to handle the case when the prisms are “thin”, in that they have thickness . In this case, it turns out that the arguments of Córdoba, combined with the super-Kakeya nature of the grains inside each of these thin prisms, implies that each prism is almost completely occupied by the set . In effect, this means that these prisms themselves can be taken to be grains of the Kakeya set. But this turns out to contradict the maximality of the dimensions of the grains (if everything is set up properly). This treats the last remaining case needed to close the induction on scales, and obtain the Kakeya conjecture!

Applied category theorists are flocking to AI, because that’s where the money is. I avoid working on it, both because I have an instinctive dislike of ‘hot topics’, and because at present AI is mainly being used to make rich and powerful people richer and more powerful.

However, I like to pay some attention to how category theorists are getting jobs connected to AI, and what they’re doing. Many of these people are my friends, so I wonder what they will do for AI, and the world at large — and what working on AI will do to them.

Let me list a bit of what’s going on. I’ll start with a cautionary tale, and then turn to the most important program linking AI and category theory today.

Symbolica

When Musk and his AI head Andrej Karpathy didn’t listen to their engineer George Morgan’s worry that current techniques in deep learning couldn’t “scale to infinity and solve all problems,” Morgan left Tesla and started a company called Symbolica to work on symbolic reasoning. The billionaire Vinod Khosla gave him $2 million in seed money. He began with an approach based on hypergraphs, but then these researchers wrote a position paper that pushed the company in a different direction:

• Bruno Gavranović, Paul Lessard, Andrew Dudzik, Tamara von Glehn, João G. M. Araújo and Petar Veličković, Position: Categorical Deep Learning is an Algebraic Theory of All Architectures

Kholsa liked the new direction and invested $30 million more in Symbolica. At Gavranovic and Lessard’s suggestion Morgan hired category theorists including Dominic Verity and Neil Ghani.

But Morgan was never fully sold on category theory: he still wanted to pursue his hypergraph approach. After a while, continued disagreements between Morgan and the category theorists took their toll. He fired some, even having one summarily removed from his office. Another resigned voluntarily. Due to nondisclosure agreements, these people no longer talk publicly about what went down.

So one moral for category theorists, or indeed anyone with a good idea: after your idea helps someone get a lot of money, they may be able to fire you.

ARIA

David Dalrymple is running a £59 million program on Mathematics for Safeguarded AI at the UK agency ARIA (the Advanced Research + Invention agency). He is very interested in using category theory for this purpose.

Here’s what the webpage for the Safeguarded AI program says:

Why this programme

As AI becomes more capable, it has the potential to power scientific breakthroughs, enhance global prosperity, and safeguard us from disasters. But only if it’s deployed wisely. Current techniques working to mitigate the risk of advanced AI systems have serious limitations, and can’t be relied upon empirically to ensure safety. To date, very little R&D effort has gone into approaches that provide quantitative safety guarantees for AI systems, because they’re considered impossible or impractical.

What we’re shooting for

By combining scientific world models and mathematical proofs we will aim to construct a ‘gatekeeper’: an AI system tasked with understanding and reducing the risks of other AI agents. In doing so we’ll develop quantitative safety guarantees for AI in the way we have come to expect for nuclear power and passenger aviation.

You can read more details here.

Here are some category theory projects getting money from this program. I’m just quoting the website here.

Axiomatic Theories of String Diagrams for Categories of Probabilistic Processes

Fabio Zanasi, University College London

This project aims to provide complete axiomatic theories of string diagrams for significant categories of probabilistic processes. Fabio will then use these theories to develop compositional methods of analysis for different kinds of probabilistic graphical models.

Safety: Core Representation Underlying Safeguarded AI

Ohad Kammar, University of Edinburgh Team: Justus Matthiesen; Jesse Sigal, University of Edinburgh

This project looks to design a calculus that utilises the semantic structure of quasi-Borel spaces, introduced in ‘A convenient category for higher-order probability theory’. Ohad + team will develop the internal language of quasi-Borel spaces as a ‘semantic universe’ for stochastic processes, define syntax that is amenable to type-checking and versioning, and interface with other teams in the programme—either to embed other formalisms as sub-calculi in quasi-Borel spaces, or vice versa (e.g. for imprecise probability).

Philosophical Applied Category Theory

David Corfield, Independent Researcher

This project plans to overcome the limitations of traditional philosophical formalisms by integrating interdisciplinary knowledge through applied category theory. In collaboration with other TA1.1 Creators, David will explore graded modal logic, type theory, and causality, and look to develop the conceptual tools to support the broader Safeguarded AI programme.

Double Categorical Systems Theory for Safeguarded AI

David Jaz Myers, Topos Research UK Team: Owen Lynch; Sophie Libkind; David Spivak, Topos Research UK; James Fairbanks, University of Florida

This project aims to utilise Double Categorical Systems Theory (DCST) as a mathematical framework to facilitate collaboration between stakeholders, domain experts, and computer aides in co-designing an explainable and auditable model of an autonomous AI system’s deployment environment. David + team will expand this modelling framework to incorporate formal verification of the system’s safety and reliability, study the verification of model-surrogacy of hybrid discrete-continuous systems, and develop serialisable data formats for representing and operating on models, all with the goal of preparing the DCST framework for broader adoption across the Safeguarded AI Programme.

Monoidal Coalgebraic Metrics

Filippo Bonchi, University of Pisa

Filippo + team intend to establish a robust mathematical framework that extends beyond the metrics expressible in quantitative algebraic theories and coalgebras over metric spaces. By shifting from Cartesian to a monoidal setting, this group will examine these metrics using algebraic contexts (to enhance syntax foundations) and coalgebraic contexts provide robust quantitative semantics and effective techniques for establishing quantitative bounds on black-box behaviours), ultimately advancing the scope of quantitative reasoning in these domains.

Doubly Categorical Systems Logic: A Theory of Specification Languages

Matteo Capucci, Independent Researcher

This project aims to develop a logical framework to classify and interoperate various logical systems created to reason about complex systems and their behaviours, guided by Doubly Categorical Systems Theory (DCST). Matteo’s goal is to study the link between the compositional and morphological structure of systems and their behaviour, specifically in the way logic pertaining these two components works, accounting for both dynamic and temporal features. Such a path will combine categorical approaches to both logic and systems theory.

True Categorical Programming for Composable Systems Total

Jade Master and Zans Mihejevs, Glasgow Lab for AI Verification (GLAIVE) Team: André Videla; Dylan Braithwaite, GLAIVE

This project intends to develop a type theory for categorical programming that enables encoding of key mathematical structures not currently supported by existing languages. These structures include functors, universal properties, Kan extensions, lax (co)limits, and Grothendieck constructions. Jade + team are aiming to create a type theory that accurately translates categorical concepts into code without compromise, and then deploy this framework to develop critical theorems related to the mathematical foundations of Safeguarded AI.

Employing Categorical Probability Towards Safe AI

Sam Staton, University of Oxford Team: Pedro Amorim; Elena Di Lavore; Paolo Perrone; Mario Román; Ruben Van Belle; Younesse Kaddar; Jack Liell-Cock; Owen Lynch, University of Oxford

Sam + team will look to employ categorical probability toward key elements essential for world modelling in the Safeguarded AI Programme. They will investigate imprecise probability (which provides bounds on probabilities of unsafe behaviour), and stochastic dynamical systems for world modelling, and then look to create a robust foundation of semantic version control to support the above elements.

Syntax and Semantics for Multimodal Petri Nets

Amar Hadzihasanovic, Tallinn University of Technology Team: Diana Kessler, Tallinn University of Technology

Amar + team will develop a combinatorial and diagrammatic syntax, along with categorical semantics, for multimodal Petri Nets. These Nets will model dynamical systems that undergo mode or phase transitions, altering their possible places, events, and interactions. Their goal is to create a category-theoretic framework for mathematical modelling and safe-by-design specification of dynamical systems and process theories which exhibit multiple modes of operation.

Topos UK

The Topos Institute is a math research institute in Berkeley with a focus on category theory. Three young category theorists there—Sophie Libkind, David Jaz Myers and Owen Lynch—wrote a proposal called “Double categorical systems theory for safeguarded AI”, which got funded by ARIA. So now they are moving to the UK, where they will be working with Tim Hosgood, José Siquiera, Xiaoyan Li and maybe others at a second branch of the Topos Institute, called Topos UK.

Glaive

The Glasgow Lab for AI Verification or ‘Glaive’ is a non-profit limited company focusing on applying category theory to AI verification. The people working for it include Dylan Braithwaite, Jade Master, Zans Mihejevs, André Videla, Neil Ghani and Jules Hedges. I hear they are designing type theories and programming languages based on category theory.

Like the Topos UK, Glaive is supported by ARIA.

Conexus AI

Conexus AI, led by Ryan Wisnesky, and the associated open-source categorical data project, apply functorial data migration and related categorical and formal methods to problems in data integration (where the tech is known as generative symbolic AI) and so-called ‘safe AI’ (where the tech is used to validate LLM outputs).

Sandbox AQ

Among other things, Tai-Danae Bradley is a research mathematician at SandboxAQ, a startup focused on AI and quantum technologies. She has applied category theory to natural language processing in a number of interesting papers.

VERSES

VERSES is a “cognitive computing company building next-generation intelligent software systems” with Karl Friston as chief scientist. They’ve hired the category theorists Toby St Clere Smith and Marco Perrin, who are working on compositional tools for approximate Bayesian inference.

Applied category theorists are flocking to AI, because that’s where the money is. I avoid working on it, both because I have an instinctive dislike of ‘hot topics’, and because at present AI is mainly being used to make rich and powerful people richer and more powerful.

However, I like to pay some attention to how category theorists are getting jobs connected to AI, and what they’re doing. Many of these people are my friends, so I wonder what they will do for AI, and the world at large—and what working on AI will do to them.

Let me list a bit of what’s going on. I’ll start with a cautionary tale, and then turn to the most important program linking AI and category theory today.

Symbolica

When Musk and his AI head Andrej Karpathy didn’t listen to their engineer George Morgan’s worry that current techniques in deep learning couldn’t “scale to infinity and solve all problems,” Morgan left Tesla and started a company called Symbolica to work on symbolic reasoning. The billionaire Vinod Khosla gave him $2 million in seed money. He began with an approach based on hypergraphs, but then these researchers wrote a position paper that pushed the company in a different direction:

• Bruno Gavranović, Paul Lessard, Andrew Dudzik, Tamara von Glehn, João G. M. Araújo and Petar Veličković, Position: Categorical Deep Learning is an Algebraic Theory of All Architectures

Kholsa liked the new direction and invested $30 million more in Symbolica. At Gavranovic and Lessard’s suggestion Morgan hired category theorists including Dominic Verity and Neil Ghani.

But Morgan was never fully sold on category theory: he still wanted to pursue his hypergraph approach. After a while, continued disagreements between Morgan and the category theorists took their toll. He fired some, even having one summarily removed from his office. Another resigned voluntarily. Due to nondisclosure agreements, these people no longer talk publicly about what went down.

So one moral for category theorists, or indeed anyone with a good idea: after your idea helps someone get a lot of money, they may be able to fire you.

ARIA

David Dalrymple is running a £59 million program on Mathematics for Safeguarded AI at the UK agency ARIA (the Advanced Research + Invention Agency). He is very interested in using category theory for this purpose.

Here’s what the webpage for the Safeguarded AI program says:

Why this programme

As AI becomes more capable, it has the potential to power scientific breakthroughs, enhance global prosperity, and safeguard us from disasters. But only if it’s deployed wisely. Current techniques working to mitigate the risk of advanced AI systems have serious limitations, and can’t be relied upon empirically to ensure safety. To date, very little R&D effort has gone into approaches that provide quantitative safety guarantees for AI systems, because they’re considered impossible or impractical.

What we’re shooting for

By combining scientific world models and mathematical proofs we will aim to construct a ‘gatekeeper’: an AI system tasked with understanding and reducing the risks of other AI agents. In doing so we’ll develop quantitative safety guarantees for AI in the way we have come to expect for nuclear power and passenger aviation.

You can read more details here.

Here are some category theory projects getting money from this program. I’m just quoting the website here.

Axiomatic Theories of String Diagrams for Categories of Probabilistic Processes

Fabio Zanasi, University College London

This project aims to provide complete axiomatic theories of string diagrams for significant categories of probabilistic processes. Fabio will then use these theories to develop compositional methods of analysis for different kinds of probabilistic graphical models.

Safety: Core Representation Underlying Safeguarded AI

Ohad Kammar, University of Edinburgh
Team: Justus Matthiesen; Jesse Sigal, University of Edinburgh

This project looks to design a calculus that utilises the semantic structure of quasi-Borel spaces, introduced in ‘A convenient category for higher-order probability theory‘. Ohad and his team will develop the internal language of quasi-Borel spaces as a “semantic universe” for stochastic processes, define syntax that is amenable to type-checking and versioning, and interface with other teams in the programme—either to embed other formalisms as sub-calculi in quasi-Borel spaces, or vice versa (e.g. for imprecise probability).

Philosophical Applied Category Theory

David Corfield, Independent Researcher

This project plans to overcome the limitations of traditional philosophical formalisms by integrating interdisciplinary knowledge through applied category theory. In collaboration with other TA1.1 Creators, David will explore graded modal logic, type theory, and causality, and look to develop the conceptual tools to support the broader Safeguarded AI programme.

Double Categorical Systems Theory for Safeguarded AI

David Jaz Myers, Topos Research UK
Team: Owen Lynch; Sophie Libkind; David Spivak, Topos Research UK; James Fairbanks, University of Florida

This project aims to utilise Double Categorical Systems Theory (DCST) as a mathematical framework to facilitate collaboration between stakeholders, domain experts, and computer aides in co-designing an explainable and auditable model of an autonomous AI system’s deployment environment. David + team will expand this modelling framework to incorporate formal verification of the system’s safety and reliability, study the verification of model-surrogacy of hybrid discrete-continuous systems, and develop serialisable data formats for representing and operating on models, all with the goal of preparing the DCST framework for broader adoption across the Safeguarded AI Programme.

Monoidal Coalgebraic Metrics

Filippo Bonchi, University of Pisa

Filippo and team intend to establish a robust mathematical framework that extends beyond the metrics expressible in quantitative algebraic theories and coalgebras over metric spaces. By shifting from Cartesian to a monoidal setting, this group will examine these metrics using algebraic contexts (to enhance syntax foundations) and coalgebraic contexts provide robust quantitative semantics and effective techniques for establishing quantitative bounds on black-box behaviours), ultimately advancing the scope of quantitative reasoning in these domains.

Doubly Categorical Systems Logic: A Theory of Specification Languages

Matteo Capucci, Independent Researcher

This project aims to develop a logical framework to classify and interoperate various logical systems created to reason about complex systems and their behaviours, guided by Doubly Categorical Systems Theory (DCST). Matteo’s goal is to study the link between the compositional and morphological structure of systems and their behaviour, specifically in the way logic pertaining these two components works, accounting for both dynamic and temporal features. Such a path will combine categorical approaches to both logic and systems theory.

True Categorical Programming for Composable Systems Total

Jade Master and Zans Mihejevs, Glasgow Lab for AI Verification (GLAIVE)
Team: André Videla; Dylan Braithwaite, GLAIVE

This project intends to develop a type theory for categorical programming that enables encoding of key mathematical structures not currently supported by existing languages. These structures include functors, universal properties, Kan extensions, lax (co)limits, and Grothendieck constructions. Jade + team are aiming to create a type theory that accurately translates categorical concepts into code without compromise, and then deploy this framework to develop critical theorems related to the mathematical foundations of Safeguarded AI.

Employing Categorical Probability Towards Safe AI

Sam Staton, University of Oxford
Team: Pedro Amorim; Elena Di Lavore; Paolo Perrone; Mario Román; Ruben Van Belle; Younesse Kaddar; Jack Liell-Cock; Owen Lynch, University of Oxford

Sam + team will look to employ categorical probability toward key elements essential for world modelling in the Safeguarded AI Programme. They will investigate imprecise probability (which provides bounds on probabilities of unsafe behaviour), and stochastic dynamical systems for world modelling, and then look to create a robust foundation of semantic version control to support the above elements.

Syntax and Semantics for Multimodal Petri Nets

Amar Hadzihasanovic, Tallinn University of Technology
Team: Diana Kessler, Tallinn University of Technology

Amar + team will develop a combinatorial and diagrammatic syntax, along with categorical semantics, for multimodal Petri Nets. These Nets will model dynamical systems that undergo mode or phase transitions, altering their possible places, events, and interactions. Their goal is to create a category-theoretic framework for mathematical modelling and safe-by-design specification of dynamical systems and process theories which exhibit multiple modes of operation.

Topos UK

The Topos Institute is a math research institute in Berkeley with a focus on category theory. Three young category theorists there—Sophie Libkind, David Jaz Myers and Owen Lynch—wrote a proposal called “Double categorical systems theory for safeguarded AI”, which got funded by ARIA. So now they are moving to Oxford, where they will be working with Tim Hosgood, José Siquiera, Xiaoyan Li and maybe others at a second branch of the Topos Institute, called Topos UK.

Glaive

The Glasgow Lab for AI Verification or ‘Glaive’ is a non-profit limited company focusing on applying category theory to AI verification. The people working for it include Dylan Braithwaite, Jade Master, Zans Mihejevs, André Videla, Neil Ghani and Jules Hedges. I hear they are designing type theories and programming languages based on category theory.

Like the Topos UK, Glaive is supported by ARIA.

Conexus AI

Conexus AI, led by Ryan Wisnesky, and the associated open-source categorical data project, apply functorial data migration and related categorical and formal methods to problems in data integration (where the tech is known as generative symbolic AI) and so-called ‘safe AI’ (where the tech is used to validate LLM outputs).

Sandbox AQ

Among other things, Tai-Danae Bradley is a research mathematician at SandboxAQ, a startup focused on AI and quantum technologies. She has applied category theory to natural language processing in a number of interesting papers.

VERSES

VERSES is a “cognitive computing company building next-generation intelligent software systems” with Karl Friston as chief scientist. They’ve hired the category theorists Toby St Clere Smith and Marco Perrin, who are working on compositional tools for approximate Bayesian inference.

In my last post, I looked at how 1920’s quantum physics (“Quantum Mechanics”, or QM) conceives of a particle with definite momentum and completely uncertain position. I also began the process of exploring how Quantum Field Theory (QFT) views the same object. I’m going to assume you’ve read that post, though I’ll quickly review some of its main points.

In that post, I invented a simple type of particle called a Bohron that moves around in a physical space in the shape of a one-dimensional line, the x-axis.

• I discussed the wave function in QM corresponding to a Bohron of definite momentum P₁, and depicted that function Ψ(x₁) (where x₁ is the Bohron’s position) in last post’s Fig. 3.
• In QFT, on the other hand, the Bohron is a ripple in the Bohron field, which is a function B(x) that gives a real number for each point x in physical space. That function has the form shown in last post’s Fig. 4.

We then looked at the broad implications of these differences between QM and QFT. But one thing is glaringly missing: we haven’t yet discussed the wave function in QFT for a Bohron of definite momentum P₁. That’s what we’ll do today.

The QFT Wave Function

Wave functions tell us the probabilities for various possibilities — specifically, for all the possible ways in which a physical system can be arranged. (That set of all possibilities is called “the space of possibilities“.)

This is a tricky enough idea even when we just have a system of a few particles; for example, if we have N particles moving on a line, then the space of possibilities is an N-dimensional space. In QFT, wave functions can be extremely complicated, because the space of possibilities for a field is infinite dimensional, even when physical space is just a one-dimensional line. Specifically, for any particular shape s(x) that we choose, the wave function for the field is Ψ[s(x)] — a complex number for every function s(x). Its absolute-value-squared is proportional to the probability that the field B(x) takes on that particular shape s(x).

Since there are an infinite number of classes of possible shapes, Ψ in QFT is a function of an infinite number of variables. Said another way, the space of possibilities has an infinite number of dimensions. Ugh! That’s both impossible to draw and impossible to visualize. What are we to do?

Simplifying the Question

By restricting our attention dramatically, we can make some progress. Instead of trying to find the wave function for all possible shapes, let’s try to understand a simplified wave function that ignores most possible shapes but gives us the probabilities for shapes that look like those in Fig. 5 (a variant of Fig. 4 of the last post). This is the simple wavy shape that corresponds to the fixed momentum P₁:

where A, the amplitude for this simple wave, can be anything we like. Here’s what that shape looks like for A=1:

If we do this, the wave function for this set of possible shapes is just a function of A; it tells us the probability that A=1 vs. A=-2 vs. A=3.2 vs. A=-4.57, etc. In other words, we’re going to write a restricted wave function Ψ[A] that doesn’t give us all the information we could possibly want about the field, but does tell us the probability for the Bohron field B(x) to take on the shape A cos(P₁ x).

This restriction to Ψ[A] is surprisingly useful. That’s because, in comparing the state containing one Bohron with momentum P₁ to a state with no Bohrons anywhere — the “vacuum state”, as it is called — the only thing that changes in the wave function is the part of the wave function that is proportional to Ψ[A].

In other words, if we tried to keep all the other information in the wave function, involving all the other possible shapes, we’d be wasting time, because all of that stuff is going to be the same whether there’s a Bohron with momentum P₁ present or not.

To properly understand and appreciate Ψ[A] in the presence of a Bohron with momentum P₁, we should first explore Ψ[A] in the vacuum state. Once we know the probabilities for A in the absence of a Bohron, we’ll be able to recognize what has changed in the presence of a Bohron.

The Zero Bohron (“Vacuum”) State

In the last post, we examined what the QM wave function looks like that describes a single Bohron with definite momentum (see Fig. 3 of that post). But what is the QM wave function for the vacuum state, the state that has no Bohrons in it?

The answer: it’s a meaningless question. QM is a theory of objects that have positions in space (or other simple properties.) If there are no objects in the theory, then there’s… well… no QM, no wave function, and nothing to discuss.

[You might complain that the Bohron field itself should be thought of as an “object” — but aside from the fact that this is questionable (is air pressure an object?), the QM of a field is QFT, so taking this route would just prove my point.]

In QFT, by contrast, the “vacuum state” is perfectly meaningful and has a wave function. The full vacuum state wave function Ψ[s(x)] is too complicated for us to talk about today. But again, if we keep our focus on the special shapes that look like cos[P₁ x], we can easily write the vacuum state’s wave function for that shape’s amplitude, Ψ[A].

Understanding the Vacuum State’s Wave Function

You might have thought, naively, that if a field contains no “particles”, then the field would just be zero; that is, it would have 100% probability to take the form B(x)=0, and 0% probability to have any other shape. This would mean that Ψ[A] would be non-zero only for A=0, forming a spike as shown in Fig. 6. Here, employing a visualization method I use often, I’m showing the wave function’s real part in red and its imaginary part in blue; its absolute-value squared, in black, is mostly hidden behind the red curve.

We’ve seen a similar-looking wave function before in the context of QM. A particle with a definite position also has a wave function in the form of a spike. But as we saw, it doesn’t stay that way: thanks to Heisenberg’s uncertainty principle, the spike instantly spreads out with a speed that reflects the state’s very high energy.

The same issue would afflict the vacuum state of a QFT if its wave function looked like Fig. 6. Just as there’s an uncertainty principle in QM that relates position and motion (changes in position), there’s an uncertainty principle in QFT that relates A and changes in A (and more generally relates B(x) and changes in B(x).) A state with a definite value of position immediately spreads out with a huge amount of energy, and the same is true for a state with a definite value of A; the shape of Ψ[A] in Fig. 6 will immediately spread out dramatically.

In short, a state that momentarily has B(x) = 0, and in particular A=0, won’t remain in this form. Not only will it change rapidly, it will do so with enormous energy. That does not sound healthy for a supposed vacuum state — the state with no Bohrons in it — which ought to be stable and have low energy.

The field’s actual vacuum state therefore has a spread of values for A — and in fact it is a Gaussian wave packet centered around A=0. In QM we have encountered Gaussian wave packets that give a spread-out position; here, in QFT, we need a packet for a spread-out amplitude, shown in Fig. 7 using the representation in which we show the real part, imaginary part, and absolute-value squared of the wave function. In Fig. 7a I’ve made the A-axis horizontal; I’ve then replotted exactly the same thing in Fig. 7b with the A axis vertical, which turns out to be useful as we’ll see in just a moment.

Another way to represent this same wave function involves plotting points at a grid of values for A, with each point drawn in gray-scale that reflects the square of the wave function |Ψ(A)|², as in Fig. 8. Note that the most probable value for A is zero, but it’s also quite likely to be somewhat away from zero.

But now we’re going to go a step further, because what we’re really interested in is not the wave function for A but the wave function for the Bohron field. We want to know how that field B(x) is behaving in the vacuum state. To gain intuition for the vacuum state wave function in terms of the Bohron field (remembering that we’ve restricted ourselves to the shape cos[P₁ x] shown in Fig. 5), we’ll generalize Fig. 8: instead of one dot for each value of A, we’ll plot the whole shape A cos[P₁ x] for a grid of choices of A, using gray-scale that’s proportional to |Ψ(A)|². This is shown in Fig. 9; in a sense, it is a combination of Fig. 8 with Fig. 5.

Remember, this is not showing the probability for the position of a particle, or even that of a “particle”. It is showing the probability in the vacuum state for the field B(x) to take on a certain shape, albeit restricted to shapes proportional to cos[P₁ x]. We can see that the most likely value of A is zero, but there is a substantial spread around zero that causes the field’s value to be uncertain.

In the vacuum state, what’s true for a shape with momentum P₁ would be true also for any and all shapes of the form cos[P x] for any possible momentum P. In principle, we could combine all of those shapes, for all of the different momenta, together in a much more complicated version of Fig. 9. However, that would make the picture completely unreadable, so I won’t try to do that — although I’ll do something intermediate, with multiple values of P, in later posts.

Oh, and I mustn’t forget to flash a warning: everything I’ve just told you and will tell you for the rest of this post is limited to a child’s version of QFT. I’m only describing what the vacuum state looks like for a “free” (i.e. non-interacting) Bohron field. This field doesn’t do anything except send individual “particles” around that never change or interact with each other. If you want to know more about truly interesting QFTs, such as the ones in the real world — well, expect some things to be recognizable from today’s post, but much of this will, yet again, have to be revisited.

The One-Bohron State

Now that we know the nature of the wave function for the vacuum state, at least when restricted to shapes proportional to cos[P₁ x], how does this change in the presence of a single Bohron of momentum P₁?

The answer is quite simple: the wave function Ψ(A) changes from to (up to an overall constant of no interest to us here.) Depicting this state in analogy to what we did for the vacuum state in Figs. 7b, 8 and 9, we find Figs. 10, 11 and 12.

Notice that the one-Bohron state is clearly distinguishable from the vacuum state; most notably the probability for A=0 is now zero, and its spread is larger, with the most likely values for A now non-zero.

There’s one more difference between these states, which I won’t attempt to prove to you at the moment. The vacuum state doesn’t show any motion; that’s not surprising, because there are no Bohrons there to do any moving. But the one-Bohron state, with its Bohron of definite momentum, will display signs of a definite speed and direction. You should imagine all the wiggles in Fig. 12 moving steadily to the right as time goes by, whereas Fig. 9 is static.

Well, that’s it. That’s what the QFT wave function for a one-Bohron state of definite momentum P₁ looks like — when we ignore the additional complexity that comes from the shapes for other possible momenta P, on the grounds that their behavior is the same in this state as it is in the vacuum state.

A Summary of Today’s Steps

That’s more than enough for today, so let me emphasize some key points here. Compare and contrast:

• In QM:
  • The Bohron with definite momentum is a particle with a position, though that position is unknown.
  • The wave function for the Bohron, spread out across the space of the Bohron’s possible positions x₁, has a wavelength with respect to x₁.
• In QFT:
  • The Bohron “particle” (i.e. wavicle) is intrinsically spread out across physical space [the horizontal x-axis in Figs. 9 and 12] and the Bohron itself has a wavelength with respect to x.
  • Meanwhile the wave function, spread out across the space of possible amplitudes A (the vertical axis in Figs. 7a, 8, 10 and 11) does not contain simply packaged information about how the activity in the Bohron field is spread out across physical space x; both the vacuum state and one-Bohron states are spread out, but you can’t just read off that fact from Figs. 8 and 11.
  • And note that the wave function has nothing simple to say about the position of the Bohron; after all the spread-out “particle” doesn’t even have a clearly defined position!

Just to make sure this is clear, let me say this again slightly differently. While in QM, the Bohron particle with definite momentum has an unknown position, in QFT, the Bohron “particle” with definite momentum does not even have a position, because it is intrinsically spread out. The QFT wave function says nothing about our uncertainty about the Bohron’s location; that uncertainty is already captured in the fact that the real (not complex!) function B(x) is proportional to a cosine function. Indeed physical space, and its coordinate x, don’t even appear directly in Ψ(A). Instead the QFT wave function, in the restricted form we’ve considered, only tells us the probability that B(x) = A cos[P₁ x] for a particular value of A — and that those probabilities are different when there is a single Bohron present (Fig. 12) compared to when there is none (Fig. 9).

I hope you can now start to see why I don’t find the word particle helpful in describing a QFT Bohron. The Bohron does have some limited particle-like qualities, most notably its indivisibility, and we’ll explore those soon. But you might already understand why I prefer wavicle.

We are far from done with QFT; this is just the beginning of our explorations. There are many follow-up questions to address, such as

• Can we put our QFT Bohron into a wave packet state similar to last post’s Fig. 2? What would that look like?
• Do these differences between QM and QFT have implications for how we think about experiments, such as the double-slit experiment or Bell’s particle-pair experiment?
• What do QFT wave functions look like if there are two “particles” rather than just one? There are several cases, all of them interesting.
• How do measurements work, and how are they different, in QM versus QFT?
• What about fields more complicated than the Bohron field, such as the electron field or the electromagnetic field?

We’ll deal with these one by one over the coming days and weeks; stay tuned.