Richard Hanson and Coryn Bailer-Jones (both MPIA) and I met today to talk about spatial priors and extinction modeling for *Gaia*. I showed them what I have on spatial priors, and we talked about the differences between using extinction measurements to predict new extinctions, using extinction measurements to predict dust densities, and so on. A key difference between the way I am thinking about it and the way Hanson and Bailer-Jones are thinking about it is that I don't want to instantiate the dust density (latent parameters) unless I have to. I would rather use the magic of the Gaussian Process to marginalize it out. We developed a set of issues for the document that I am writing on the subject. At Galaxy Coffee, Girish Kulkarni (MPIA) gave a great talk about the physics of the intergalactic medium and observational constraints from the absorption lines in quasar spectra.

I spent a chunk of the day with Melissa Ness (MPIA), fitting empirical models to *APOGEE* infrared spectra of stars. The idea is to do a simple linear supervised classification or regression, in which we figure out the dependence of the spectra on key stellar parameters, using a "training set" of stars with good stellar parameters. We worked in the pair-coding mode. By the end of the day we could show that we are able to identify regions of the spectrum that might serve as good metallicity indicators, relatively insensitive to temperature and log-g. The hopes for this project range from empirical metallicity index identification to label de-noising to building a full data-driven (supervised) stellar parameter pipeline. We ended our coding day pretty optimistic.

My op/ed about math teaching and Little League coaching is the most emailed article in the New York Times today. Very cool! But here’s something interesting; it’s only the 14th most viewed article, the 6th most tweeted, and the 6th most shared on Facebook. On the other hand, this article about child refugees from Honduras […]

My op/ed about math teaching and Little League coaching is the most emailed article in the New York Times today. Very cool!

But here’s something interesting; it’s only the 14th most viewed article, the 6th most tweeted, and the 6th most shared on Facebook. On the other hand, this article about child refugees from Honduras is

#14 most emailed

#1 most viewed

#1 most shared on Facebook

#1 most tweeted

while Paul Krugman’s column about California is

#4 most emailed

#3 most viewed

#4 most shared on Facebook

#7 most tweeted.

Why are some articles, like mine, much more emailed than tweeted, while others, like the one about refugees, much more tweeted than emailed, and others still, like Krugman’s, come out about even? Is it always the case that views track tweets, not emails? Not necessarily; an article about the commercial success and legal woes of conservative poo-stirrer Dinesh D’Souza is #3 most viewed, but only #13 in tweets (and #9 in emails.) Today’s Gaza story has lots of tweets and views but not so many emails, like the Honduras piece, so maybe this is a pattern for international news? Presumably people inside newspapers actually study stuff like this; is any of that research public? Now I’m curious.

Fast track to wisdom: Sure, but who cares if they can? We want to know if they do.

The horizon itself is a global construct, it is locally entirely unremarkable and regular. You would not note crossing the horizon, but the classical black hole solution contains a singularity in the center. This singularity is usually interpreted as the breakdown of classical general relativity and is expected to be removed by the yet-to-be-found theory of quantum gravity.

You do however not need quantum gravity to construct singularity-free black hole space-times. Hawking and Ellis’ singularity theorems prove that singularities must form from certain matter configurations, provided the matter is normal matter and cannot develop negative pressure and/or density. All you have to do to get rid of the singularity is invent some funny type of matter that refuses to be squeezed arbitrarily. This is not possible with any type of matter we know, and so just pushes around the bump under the carpet: Now rather than having to explain quantum effects of gravity you have to explain where the funny matter comes from. It is normally interpreted not as matter but as a quantum gravitational contribution to the stress-energy tensor, but either way it’s basically the physicist’s way of using a kitten photo to cover the hole in wall.

Singularity-free black hole solutions have been constructed almost for as long as the black hole solution has been known – people have always been disturbed by the singularity. Using matter other than normal ones allowed constructing both wormhole solutions as well as black holes that turn into white holes and allow an exit into a second space-time region. Now if a black hole is really a black hole with an event horizon, then the second space-time region is causally disconnected from the first. If the black hole has only an apparent horizon, then this does not have to be so, and also the white hole then is not really a white hole, it just looks like one.

The latter solution is quite popular in quantum gravity. It basically describes matter collapsing, forming an apparent horizon and a strong quantum gravity region inside but no singularity, then evaporating and returning to an almost flat space-time. There are various ways to construct these space-times. The details differ, but the corresponding causal diagrams all look basically the same.

This recent paper for example used a collapsing shell turning into an expanding shell. The title “Singularity free gravitational collapse in an effective dynamical quantum spacetime” basically says it all. Note how the resulting causal diagram (left in figure below) looks pretty much the same as the one Lee and I constructed based on general considerations in our 2009 paper (middle in figure below), which again looks pretty much the same as the one that Ashtekar and Bojowald discussed in 2005 (right in figure below), and I could go on and add a dozen more papers discussing similar causal diagrams. (Note that the shaded regions do not mean the same in each figure.)

One needs a concrete ansatz for the matter of course to be able to calculate anything. The general structure of the causal diagram is good for classification purposes, but not useful for quantitative reasoning, for example about the evaporation.

Haggard and Rovelli and recently added to this discussion with a new paper about black holes bouncing to white holes.

Ron Cowen at Nature News announced this as a new idea, and while the paper does contain new ideas, that black holes may turn into white holes is in and by itself not new. And so it follows some clarification.

Haggard and Rovelli’s paper contains two ideas that are connected by an argument, but not by a calculation, so I want to discuss them separately. Before we start it is important to note that their argument does*not* take into account Hawking radiation. The whole process is supposed to happen already *without* outgoing radiation. For this reason the situation is completely time-reversal invariant, which makes it significantly easier to construct a metric. It is also easier to arrive at a result that has nothing to do with reality.

So, the one thing that is new in the Haggard and Rovelli paper is that they construct a space-time diagram, describing a black hole turning into a white hole, both with apparent horizons, and do so by a cutting-procedure rather than altering the equation of state of the matter. As source they use a collapsing shell that is supposed to bounce. This cutting procedure is fine in principle, even though it is not often used. The problem is that you end up with a metric that exists as solution to some source, but you then have to calculate what the source has to do in order to give you the metric. This however is not done in the paper. I want to offer you a guess though as to what source would be necessary to create their metric.

The cutting that is done in the paper takes a part of the black hole metric (describing the inside of the shell) with an arm extending into the horizon region, then squeezes this arm together so that it shrinks in radial extension no longer extends into the regime below the Schwarzschild radius, which is normally behind the horizon. This squeezed part of the black hole metric is then matched to empty space, describing the outside of the shell. See image below

They do not specify what happens to the shell after it has reached the end of the region that was cut, explaining one would need quantum gravity for this. The result is glued together with the time-reversed case, and so they get a metric that forms an apparent horizon and bounces at a radius where one normally would not expect quantum gravitational effects. (Working towards making more concrete the so far quite vague idea of Planck stars that we discussed here.)

The cutting and squeezing basically means that the high curvature region from inside the horizon was moved to a larger radius, and the only way this makes sense is if it happens together with the shell. So I think effectively they take the shell from a small radius and match the small radius to a large radius while keeping the density fixed (they keep the curvature). This looks to me like they blow up the total mass of the shell, but keep in mind this is my interpretation, not theirs. If that was so however, then makes sense that the horizon forms at a larger radius if the shell collapses while its mass increases. This raises the question though why the heck the mass of the shell should increase and where that energy is supposed to come from.

This brings me to the second argument in the paper, which is supposed to explain why it is plausible to expect this kind of behavior. Let me first point out that it is a bold claim that quantum gravity effects kick in outside the horizon of a (large) black hole. Standard lore has it that quantum gravity only leads to large corrections to the classical metric if the curvature is large (in the Planckian regime). This happens always after horizon crossing (as long as the mass of the black hole is larger than the Planck mass). But once the horizon is formed, the only way to make matter bounce so that it can come out of the horizon necessitates violations of causality and/or locality (keep in mind their black hole is not evaporating!) that extend into small curvature regions. This is inherently troublesome because now one has to explain why we don’t see quantum gravity effects all over the place.

The way they argue this could happen is that small, Planck size, higher-order correction to the metric can build up over time. In this case it is not solely the curvature that is relevant for an estimate of the effect, but also the duration of the buildup. So far, so good. My first problem is that I can’t see what their estimate of the long-term effects of such a small correction has to do with quantum gravity. I could read the whole estimate as being one for black hole solutions in higher-order gravity, quantum not required. If it was a quantum fluctuation I would expect the average solution to remain the classical one and the cases in which the fluctuations build up to be possible but highly improbable. In fact they seem to have something like this in mind, just that they for some reason come to the conclusion that the transition to the solution in which the initially small fluctuation builds up becomes more likely over time rather than less likely.

What one would need to do to estimate the transition probability is to work out some product of wave-functions describing the background metric close by and far away from the classical average, but nothing like this is contained in the paper. (Carlo told me though, it’s in the making.) It remains to be shown that the process of all the matter of the shell suddenly tunneling outside the horizon and expanding again is more likely to happen than the slow evaporation due to Hawking radiation which is essentially also a tunnel process (though not one of the metric, just of the matter moving in the metric background). And all this leaves aside that the state should decohere and not just happily build up quantum fluctuations for the lifetime of the universe or so.

By now I’ve probably lost most readers so let me just sum up. The space-time that Haggard and Rovelli have constructed exists as a mathematical possibility, and I do not actually doubt that the tunnel process is possible in principle, provided that they get rid of the additional energy that has appeared from somewhere (this is taken care of automatically by the time-reversal). But this alone does not tell us whether this space-time can exist as a real possibility in the sense that we do not know if this process can happen with large probability (close to one) in the time before the shell reaches the Schwarzschild radius (of the classical solution).

I have remained skeptical, despite Carlo’s infinitely patience in explaining their argument to me. But if they are right and what they claim is correct, then this would indeed solve both the black hole information loss problem and the firewall conundrum. So stay tuned...

Black holes are defined by the presence of an event horizon which is the boundary of a region from which nothing can escape, ever. The word black hole is also often used to mean something that looks for a long time very similar to a black hole and that traps light, not eternally but only temporarily. Such space-times are said to have an “apparent horizon.” That they are not strictly speaking black holes was origin of the recent Stephen Hawking quote according to which black holes may not exist, by which he meant they might have only an apparent horizon instead of an eternal event horizon.

A white hole is an upside-down version of a black hole; it has an event horizon that is a boundary to a region in which nothing can ever enter. Static black hole solutions, describing unrealistic black holes that have existed forever and continue to exist forever, are actually a combination of a black hole and a white hole.The horizon itself is a global construct, it is locally entirely unremarkable and regular. You would not note crossing the horizon, but the classical black hole solution contains a singularity in the center. This singularity is usually interpreted as the breakdown of classical general relativity and is expected to be removed by the yet-to-be-found theory of quantum gravity.

You do however not need quantum gravity to construct singularity-free black hole space-times. Hawking and Ellis’ singularity theorems prove that singularities must form from certain matter configurations, provided the matter is normal matter and cannot develop negative pressure and/or density. All you have to do to get rid of the singularity is invent some funny type of matter that refuses to be squeezed arbitrarily. This is not possible with any type of matter we know, and so just pushes around the bump under the carpet: Now rather than having to explain quantum effects of gravity you have to explain where the funny matter comes from. It is normally interpreted not as matter but as a quantum gravitational contribution to the stress-energy tensor, but either way it’s basically the physicist’s way of using a kitten photo to cover the hole in wall.

Singularity-free black hole solutions have been constructed almost for as long as the black hole solution has been known – people have always been disturbed by the singularity. Using matter other than normal ones allowed constructing both wormhole solutions as well as black holes that turn into white holes and allow an exit into a second space-time region. Now if a black hole is really a black hole with an event horizon, then the second space-time region is causally disconnected from the first. If the black hole has only an apparent horizon, then this does not have to be so, and also the white hole then is not really a white hole, it just looks like one.

The latter solution is quite popular in quantum gravity. It basically describes matter collapsing, forming an apparent horizon and a strong quantum gravity region inside but no singularity, then evaporating and returning to an almost flat space-time. There are various ways to construct these space-times. The details differ, but the corresponding causal diagrams all look basically the same.

This recent paper for example used a collapsing shell turning into an expanding shell. The title “Singularity free gravitational collapse in an effective dynamical quantum spacetime” basically says it all. Note how the resulting causal diagram (left in figure below) looks pretty much the same as the one Lee and I constructed based on general considerations in our 2009 paper (middle in figure below), which again looks pretty much the same as the one that Ashtekar and Bojowald discussed in 2005 (right in figure below), and I could go on and add a dozen more papers discussing similar causal diagrams. (Note that the shaded regions do not mean the same in each figure.)

One needs a concrete ansatz for the matter of course to be able to calculate anything. The general structure of the causal diagram is good for classification purposes, but not useful for quantitative reasoning, for example about the evaporation.

Haggard and Rovelli and recently added to this discussion with a new paper about black holes bouncing to white holes.

Hal M. Haggard, Carlo Rovelli

arXiv: 1407.0989

Ron Cowen at Nature News announced this as a new idea, and while the paper does contain new ideas, that black holes may turn into white holes is in and by itself not new. And so it follows some clarification.

Haggard and Rovelli’s paper contains two ideas that are connected by an argument, but not by a calculation, so I want to discuss them separately. Before we start it is important to note that their argument does

So, the one thing that is new in the Haggard and Rovelli paper is that they construct a space-time diagram, describing a black hole turning into a white hole, both with apparent horizons, and do so by a cutting-procedure rather than altering the equation of state of the matter. As source they use a collapsing shell that is supposed to bounce. This cutting procedure is fine in principle, even though it is not often used. The problem is that you end up with a metric that exists as solution to some source, but you then have to calculate what the source has to do in order to give you the metric. This however is not done in the paper. I want to offer you a guess though as to what source would be necessary to create their metric.

The cutting that is done in the paper takes a part of the black hole metric (describing the inside of the shell) with an arm extending into the horizon region, then squeezes this arm together so that it shrinks in radial extension no longer extends into the regime below the Schwarzschild radius, which is normally behind the horizon. This squeezed part of the black hole metric is then matched to empty space, describing the outside of the shell. See image below

Figure 4 from arXiv: 1407.0989 |

They do not specify what happens to the shell after it has reached the end of the region that was cut, explaining one would need quantum gravity for this. The result is glued together with the time-reversed case, and so they get a metric that forms an apparent horizon and bounces at a radius where one normally would not expect quantum gravitational effects. (Working towards making more concrete the so far quite vague idea of Planck stars that we discussed here.)

The cutting and squeezing basically means that the high curvature region from inside the horizon was moved to a larger radius, and the only way this makes sense is if it happens together with the shell. So I think effectively they take the shell from a small radius and match the small radius to a large radius while keeping the density fixed (they keep the curvature). This looks to me like they blow up the total mass of the shell, but keep in mind this is my interpretation, not theirs. If that was so however, then makes sense that the horizon forms at a larger radius if the shell collapses while its mass increases. This raises the question though why the heck the mass of the shell should increase and where that energy is supposed to come from.

This brings me to the second argument in the paper, which is supposed to explain why it is plausible to expect this kind of behavior. Let me first point out that it is a bold claim that quantum gravity effects kick in outside the horizon of a (large) black hole. Standard lore has it that quantum gravity only leads to large corrections to the classical metric if the curvature is large (in the Planckian regime). This happens always after horizon crossing (as long as the mass of the black hole is larger than the Planck mass). But once the horizon is formed, the only way to make matter bounce so that it can come out of the horizon necessitates violations of causality and/or locality (keep in mind their black hole is not evaporating!) that extend into small curvature regions. This is inherently troublesome because now one has to explain why we don’t see quantum gravity effects all over the place.

The way they argue this could happen is that small, Planck size, higher-order correction to the metric can build up over time. In this case it is not solely the curvature that is relevant for an estimate of the effect, but also the duration of the buildup. So far, so good. My first problem is that I can’t see what their estimate of the long-term effects of such a small correction has to do with quantum gravity. I could read the whole estimate as being one for black hole solutions in higher-order gravity, quantum not required. If it was a quantum fluctuation I would expect the average solution to remain the classical one and the cases in which the fluctuations build up to be possible but highly improbable. In fact they seem to have something like this in mind, just that they for some reason come to the conclusion that the transition to the solution in which the initially small fluctuation builds up becomes more likely over time rather than less likely.

What one would need to do to estimate the transition probability is to work out some product of wave-functions describing the background metric close by and far away from the classical average, but nothing like this is contained in the paper. (Carlo told me though, it’s in the making.) It remains to be shown that the process of all the matter of the shell suddenly tunneling outside the horizon and expanding again is more likely to happen than the slow evaporation due to Hawking radiation which is essentially also a tunnel process (though not one of the metric, just of the matter moving in the metric background). And all this leaves aside that the state should decohere and not just happily build up quantum fluctuations for the lifetime of the universe or so.

By now I’ve probably lost most readers so let me just sum up. The space-time that Haggard and Rovelli have constructed exists as a mathematical possibility, and I do not actually doubt that the tunnel process is possible in principle, provided that they get rid of the additional energy that has appeared from somewhere (this is taken care of automatically by the time-reversal). But this alone does not tell us whether this space-time can exist as a real possibility in the sense that we do not know if this process can happen with large probability (close to one) in the time before the shell reaches the Schwarzschild radius (of the classical solution).

I have remained skeptical, despite Carlo’s infinitely patience in explaining their argument to me. But if they are right and what they claim is correct, then this would indeed solve both the black hole information loss problem and the firewall conundrum. So stay tuned...

I should really know better than to click any tweeted link with a huff.to shortened URL, but for some reason, I actually followed one to an article with the limited-reach clickbait title Curious About Quantum Physics? Read These 10 Articles!. Which is only part one, because Huffington Post, so it’s actually five articles.

Three of the five articles are Einstein papers from 1905, which is sort of the equivalent of making a Ten Essential Rock Albums list that includes Revolver, Abbey Road, and the White Album. One of the goals of a well-done list of “essential” whatever is to give a sense of the breadth of a subject, not just focus on a single example, so this is a big failure right off the bat.

But it’s even worse than that, because none of the three 1905 articles is the photoelectric effect paper, which is the only one of the lot that has any quantum physics in it. There’s a fourth Einstein paper on the list, as well, the theory of general relativity, which is famous for *not being compatible with quantum mechanics*. So this is really like a list of Ten Essential Rock Albums that includes three country songs and a Bach concerto.

I thought about using this as an opportunity to generate a better Ten Essential Quantum Papers list, including stuff like Bell’s Theorem (the physics equivalent of the first Velvet Underground record) and the No-Cloning Theorem (the physics equivalent of punk rock) (brief pause to let those who know Bill Wootters try to reconcile that mental image). And if you would like to make suggestions of things that ought to be on such a list in the comments, feel free.

(I’m also open to suggestions of better musical analogies– maybe the EPR paper is the real Velvet Underground record? With Bell’s paper being punk rock, making Wootters and Zurek… Nirvana, maybe? Or maybe Shor’s algorithm is the “Smells Like Teen Spirit” of quantum physics (Again, a brief pause while those who know Peter Shor try to picture him as Kurt Cobain)…)

But, really, on reflection, the whole exercise is kind of silly even by the standards of clickbait blog topics, because that’s not how science works. In science, and particularly a highly mathematical science like physics, there’s not that much real benefit to reading the original source material. The best explanation of a central concept is rarely if ever found in the first paper to present it. This goes right back to the start of the discipline, with Newton’s Principia Mathematica, which nobody reads because it’s written in really opaque Latin, a move he claimed in a letter was deliberate so as to avoid “being baited by little smatterers in mathematics” (Newton was kind of a dick). Newton’s mathematical notation is also pretty awful, and I’ve heard it claimed that the reason physics advanced faster in mainland Europe than in England during the 1700s was that on the continent, they adopted Leibniz’s system, which was way more user-friendly and is the basis for modern calculus notation. Similarly, Maxwell’s original presentation of his eponymous equations is really difficult to follow, and it’s only after the work of folks like Heaviside that they become the clear, elegant, and bumper-sticker-friendly version we know today.

That’s not to say that there’s no value in reading old papers– I’ve had a lot of fun writing up old MS theses from our department, and older work can be fascinating to read. But unlike primary works of (pop) culture, they’re much better if you come to them already knowing what they’re about. The fascination comes from seeing how people fumbled their way toward ideas that we now know to be correct. It’s rare for a “classic” paper to get all the way to the modern understanding of things, or even most of the way there– most of the great original works contain what we now know to be errors of interpretation. Others are revered today for discoveries that were somewhat tangential to what the original author thought was the main point– the Cavendish experiment is thought of today as a measurement of “big G,” but he presents it as a determination of the density of the Earth, because that was of more pressing practical interest at the time.

If you want to *learn* science, you’re much better off looking up the best modern treatment than going back to the original papers. A good recent textbook will have the bugs worked out, and present it in something close to the language used by working scientists today. A good popular-audience treatment (ahem) will cover the basic concepts starting from a more complete understanding of the field as it has developed, and with an eye toward making those concepts accessible to a modern reader. It’s not foolproof, of course– the steady progress of science over a stretch of decades often means that newer books need to cover a huge amount of material to get to the sexy cutting-edge stuff, and sometimes scant the basics a bit. But by and large, if you’re curious about quantum physics, you’d be much better off hitting the physics section of your local bookstore or library than digging through archived journals for the original papers.

So, a list of “Ten Essential Papers on Quantum Physics” is a deeply flawed concept right from the start, at least if the goal is to learn something about quantum physics that you didn’t already know. The same is true of almost every science, with a few exception– Darwin’s On the Origin of Species is still a really good read, but it’s the exception, not the rule. Such a list can be useful as a sort of historical map, or for providing some insight into the thought processes of the great scientists of yesteryear, and those can be very rewarding. But if you’re curious and want to learn, I don’t think any original papers can really be considered “essential.”

How can we discuss all the kinds of matter described by the ten-fold way in a single setup?

It’s bit tough, because 8 of them are fundamentally ‘real’ while the other 2 are fundamentally ‘complex’. Yet they *should* fit into a single framework, because there are 10 super division algebras over the real numbers, and each kind of matter is described using a super vector space — or really a super Hilbert space — with one of these super division algebras as its ‘ground field’.

Combining physical systems is done by tensoring their Hilbert spaces… and there *does* seem to be a way to do this even with super Hilbert spaces over different super division algebras. But what sort of mathematical structure can formalize this?

Here’s my current attempt to solve this problem. I’ll start with a warmup case, the threefold way. In fact I’ll spend most of my time on that! Then I’ll sketch how the ideas should extend to the tenfold way.

Fans of lax monoidal functors, Deligne’s tensor product of abelian categories, and the collage of a profunctor will be rewarded for their patience if they read the whole article. But the basic idea is supposed to be simple: it’s about a multiplication table.

First of all, notice that the set

$\mathbb{3} = \{1,0,-1\}$

is a commutative monoid under ordinary multiplication:

$\begin{array}{rrrr} \mathbf{\times} & \mathbf{1} & \mathbf{0} & \mathbf{-1} \\ \mathbf{1} & 1 & 0 & -1 \\ \mathbf{0} & 0 & 0 & 0 \\ \mathbf{-1} & -1 & 0 & 1 \end{array}$

Next, note that there are three (associative) division algebras over the reals: $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$. We can equip a real vector space with the structure of a module over any of these algebras. We’ll then call it a **real**, **complex** or **quaternionic** vector space.

For the real case, this is entirely dull. For the complex case, this amounts to giving our real vector space $V$ a **complex structure**: a linear operator $i: V \to V$ with $i^2 = -1$. For the quaternionic case, it amounts to giving $V$ a **quaternionic structure**: a pair of linear operators $i, j: V \to V$ with

$i^2 = j^2 = -1, \qquad i j = -j i$

We can then define $k = i j$.

The terminology ‘quaternionic vector space’ is a bit quirky, since the quaternions aren’t a field, but indulge me. $\mathbb{H}^n$ is a quaternionic vector space in an obvious way. $n \times n$ quaternionic matrices act by multiplication on the *right* as ‘quaternionic linear transformations’ — that is, *left* module homomorphisms — of $\mathbb{H}^n$. Moreover, every finite-dimensional quaternionic vector space is isomorphic to $\mathbb{H}^n$. So it’s really not so bad! You just need to pay some attention to left versus right.

Now: I claim that given two vector spaces of any of these kinds, we can tensor them over the real numbers and get a vector space of another kind. It goes like this:

$\begin{array}{cccc} \mathbf{\otimes} & \mathbf{real} & \mathbf{complex} & \mathbf{quaternionic} \\ \mathbf{real} & real & complex & quaternionic \\ \mathbf{complex} & complex & complex & complex \\ \mathbf{quaternionic} & quaternionic & complex & real \end{array}$

You’ll notice this has the same pattern as the multiplication table we saw before:

$\begin{array}{rrrr} \mathbf{\times} & \mathbf{1} & \mathbf{0} & \mathbf{-1} \\ \mathbf{1} & 1 & 0 & -1 \\ \mathbf{0} & 0 & 0 & 0 \\ \mathbf{-1} & -1 & 0 & 1 \end{array}$

So:

- $\mathbb{R}$ acts like 1.
- $\mathbb{C}$ acts like 0.
- $\mathbb{H}$ acts like -1.

There are different ways to understand this, but a nice one is to notice that if we have algebras $A$ and $B$ over some field, and we tensor an $A$-module and a $B$-module (over that field), we get an $A \otimes B$-module. So, we should look at this ‘multiplication table’ of real division algebras:

$\begin{array}{lrrr} \mathbf{\otimes} & \mathbf{\mathbb{R}} & \mathbf{\mathbb{C}} & \mathbf{\mathbb{H}} \\ \mathbf{\mathbb{R}} & \mathbb{R} & \mathbb{C} & \mathbb{H} \\ \mathbf{\mathbb{C}} & \mathbb{C} & \mathbb{C} \oplus \mathbb{C} & \mathbb{C}[2] \\ \mathbf{\mathbb{H}} & \mathbb{H} & \mathbb{C}[2] & \mathbb{R}[4] \end{array}$

Here $\mathbb{C}[2]$ means the 2 × 2 complex matrices viewed as an algebra over $\mathbb{R}$, and $\mathbb{R}[4]$ means that 4 × 4 real matrices.

What’s going on here? Naively you might have hoped for a simpler table, which would have instantly explained my earlier claim:

$\begin{array}{lrrr} \mathbf{\otimes} & \mathbf{\mathbb{R}} & \mathbf{\mathbb{C}} & \mathbf{\mathbb{H}} \\ \mathbf{\mathbb{R}} & \mathbb{R} & \mathbb{C} &\mathbb{H} \\ \mathbf{\mathbb{C}} & \mathbb{C} & \mathbb{C} & \mathbb{C} \\ \mathbf{\mathbb{H}} & \mathbb{H} & \mathbb{C} & \mathbb{R} \end{array}$

This isn’t true, but it’s ‘close enough to true’. Why? Because we always have a god-given algebra homomorphism from the naive answer to the real answer! The interesting cases are these:

$\mathbb{C} \to \mathbb{C} \oplus \mathbb{C}$ $\mathbb{C} \to \mathbb{C}[2]$ $\mathbb{R} \to \mathbb{R}[4]$

where the first is the diagonal map $a \mapsto (a,a)$, and the other two send numbers to the corresponding scalar multiples of the identity matrix.

So, for example, if $V$ and $W$ are $\mathbb{C}$-modules, then their tensor product (over the reals! — all tensor products here are over $\mathbb{R}$) is a module over $\mathbb{C} \otimes \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$, and we can then pull that back via $f$ to get a right $\mathbb{C}$-module.

What’s really going on here?

There’s a monoidal category $Alg_{\mathbb{R}}$ of algebras over the real numbers, where the tensor product is the usual tensor product of algebras. The monoid $\mathbb{3}$ can be seen as a monoidal category with 3 objects and only identity morphisms. And I claim this:

**Claim.** There is an oplax monoidal functor $F : \mathbb{3} \to Alg_{\mathbb{R}}$ with
$\begin{array}{ccl}
F(1) &=& \mathbb{R} \\
F(0) &=& \mathbb{C} \\
F(-1) &=& \mathbb{H}
\end{array}$

What does ‘oplax’ mean? Some readers of the $n$-Category Café eat oplax monoidal functors for breakfast and are chortling with joy at how I finally summarized everything I’d said so far in a single terse sentence! But others of you see ‘oplax’ and get a queasy feeling.

The key idea is that when we have two monoidal categories $C$ and $D$, a functor $F : C \to D$ is ‘oplax’ if it preserves the tensor product, not up to isomorphism, but up to a specified *morphism*. More precisely, given objects $x,y \in C$ we have a natural transformation

$F_{x,y} : F(x \otimes y) \to F(x) \otimes F(y)$

If you had a ‘lax’ functor this would point the other way, and they’re a bit more popular… so when it points the opposite way it’s called ‘oplax’.

(In the lax case, $F_{x,y}$ should probably be called the **laxative**, but we’re not doing that case, so I don’t get to make that joke.)

This morphism $F_{x,y}$ needs to obey some rules, but the most important one is that using it twice, it gives two ways to get from $F(x \otimes y \otimes z)$ to $F(x) \otimes F(y) \otimes F(z)$, and these must agree.

Let’s see how this works in our example… at least in one case. I’ll take the trickiest case. Consider

$F_{0,0} : F(0 \cdot 0) \to F(0) \otimes F(0),$

that is:

$F_{0,0} : \mathbb{C} \to \mathbb{C} \otimes \mathbb{C}$

There are, in principle, two ways to use this to get a homomorphism

$F(0 \cdot 0 \cdot 0 ) \to F(0) \otimes F(0) \otimes F(0)$

or in other words, a homomorphism

$\mathbb{C} \to \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C}$

where remember, all tensor products are taken over the reals. One is

$\mathbb{C} \stackrel{F_{0,0}}{\longrightarrow} \mathbb{C} \otimes \mathbb{C} \stackrel{1 \otimes F_{0,0}}{\longrightarrow} \mathbb{C} \otimes (\mathbb{C} \otimes \mathbb{C})$

and the other is

$\mathbb{C} \stackrel{F_{0,0}}{\longrightarrow} \mathbb{C} \otimes \mathbb{C} \stackrel{F_{0,0} \otimes 1}{\longrightarrow} (\mathbb{C} \otimes \mathbb{C})\otimes \mathbb{C}$

I want to show they agree (after we rebracket the threefold tensor product using the associator).

Unfortunately, so far I have described $F_{0,0}$ in terms of an isomorphism

$\mathbb{C} \otimes \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$

Using this isomorphism, $F_{0,0}$ becomes the diagonal map $a \mapsto (a,a)$. But now we need to really understand $F_{0,0}$ a bit better, so I’d better say what isomorphism I have in mind! I’ll use the one that goes like this:

$\begin{array}{ccl} \mathbb{C} \otimes \mathbb{C} &\to& \mathbb{C} \oplus \mathbb{C} \\ 1 \otimes 1 &\mapsto& (1,1) \\ i \otimes 1 &\mapsto &(i,i) \\ 1 \otimes i &\mapsto &(i,-i) \\ i \otimes i &\mapsto & (1,-1) \end{array}$

This may make you nervous, but it truly is an isomorphism of real algebras, and it sends $a \otimes 1$ to $(a,a)$. So, unraveling the web of confusion, we have

$\begin{array}{rccc} F_{0,0} : & \mathbb{C} &\to& \mathbb{C}\otimes \mathbb{C} \\ & a &\mapsto & a \otimes 1 \end{array}$

Why didn’t I just say that in the first place? Well, I suffered over this a bit, so you should too! You see, there’s an unavoidable arbitrary choice here: I could just have well used $a \mapsto 1 \otimes a$. $F_{0,0}$ looked perfectly god-given when we thought of it as a homomorphism from $\mathbb{C}$ to $\mathbb{C} \oplus \mathbb{C}$, but that was deceptive, because there’s a choice of isomorphism $\mathbb{C} \otimes \mathbb{C} \to \mathbb{C} \oplus \mathbb{C}$ lurking in this description.

This makes me nervous, since category theory disdains arbitrary choices! But it seems to work. On the one hand we have

$\begin{array}{ccccc} \mathbb{C} &\stackrel{F_{0,0}}{\longrightarrow} &\mathbb{C} \otimes \mathbb{C} &\stackrel{1 \otimes F_{0,0}}{\longrightarrow}& \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C} \\ a &\mapsto & a \otimes 1 & \mapsto & a \otimes (1 \otimes 1) \end{array}$

On the other hand, we have

$\begin{array}{ccccc} \mathbb{C} &\stackrel{F_{0,0}}{\longrightarrow} & \mathbb{C} \otimes \mathbb{C} &\stackrel{F_{0,0} \otimes 1}{\longrightarrow} & \mathbb{C} \otimes \mathbb{C} \otimes \mathbb{C} \\ a &\mapsto & a \otimes 1 & \mapsto & (a \otimes 1) \otimes 1 \end{array}$

So they agree!

I need to carefully check all the other cases before I dare call my claim a theorem. Indeed, writing up this case has increased my nervousness… before, I’d thought it was obvious.

But let me march on, optimistically!

In quantum physics, what matters is not so much the algebras $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ themselves as the categories of vector spaces — or indeed, Hilbert spaces —-over these algebras. So, we should think about the map sending an algebra to its category of modules.

For any field $k$, there should be a contravariant pseudofunctor

$Rep: Alg_k \to Rex_k$

where $Rex_k$ is the 2-category of

$k$-linear finitely cocomplete categories,

$k$-linear functors preserving finite colimits,

and natural transformations.

The idea is that $Rep$ sends any algebra $A$ over $k$ to its category of modules, and any homomorphism $f : A \to B$ to the pullback functor $f^* : Rep(B) \to Rep(A)$.

(Functors preserving finite colimits are also called right exact; this is the reason for the funny notation $Rex$. It has nothing to do with the dinosaur of that name.)

Moreover, $Rep$ gets along with tensor products. It’s definitely true that given real algebras $A$ and $B$, we have

$Rep(A \otimes B) \simeq Rep(A) \boxtimes Rep(B)$

where $\boxtimes$ is the tensor product of finitely cocomplete $k$-linear categories. But we should be able to go further and prove $Rep$ is monoidal. I don’t know if anyone has bothered yet.

(In case you’re wondering, this $\boxtimes$ thing reduces to Deligne’s tensor product of abelian categories given some ‘niceness assumptions’, but it’s a bit more general. Read the talk by Ignacio López Franco if you care… but I could have used Deligne’s setup if I restricted myself to finite-dimensional algebras, which is probably just fine for what I’m about to do.)

So, if my earlier claim is true, we can take the oplax monoidal functor

$F : \mathbb{3} \to Alg_{\mathbb{R}}$

and compose it with the contravariant monoidal pseudofunctor

$Rep : Alg_{\mathbb{R}} \to Rex_{\mathbb{R}}$

giving a guy which I’ll call

$Vect: \mathbb{3} \to Rex_{\mathbb{R}}$

I guess this guy is a contravariant oplax monoidal pseudofunctor! That doesn’t make it sound very lovable… but I love it. The idea is that:

$Vect(1)$ is the category of real vector spaces

$Vect(0)$ is the category of complex vector spaces

$Vect(-1)$ is the category of quaternionic vector spaces

and the operation of multiplication in $\mathbb{3} = \{1,0,-1\}$ gets sent to the operation of tensoring any one of these three kinds of vector space with any other kind and getting another kind!

So, if this works, we’ll have combined linear algebra over the real numbers, complex numbers and quaternions into a unified thing, $Vect$. This thing deserves to be called a $\mathbb{3}$-graded category. This would be a nice way to understand Dyson’s threefold way.

What’s really going on with this monoid $\mathbb{3}$? It’s a kind of combination or ‘collage’ of two groups:

The Brauer group of $\mathbb{R}$, namely $\mathbb{Z}_2 \cong \{-1,1\}$. This consists of Morita equivalence classes of central simple algebras over $\mathbb{R}$. One class contains $\mathbb{R}$ and the other contains $\mathbb{H}$. The tensor product of algebras corresponds to multiplication in $\{-1,1\}$.

The Brauer group of $\mathbb{C}$, namely the trivial group $\{0\}$. This consists of Morita equivalence classes of central simple algebras over $\mathbb{C}$. But $\mathbb{C}$ is algebraically closed, so there’s just one class, containing $\mathbb{C}$ itself!

See, the problem is that while $\mathbb{C}$ is a division algebra over $\mathbb{R}$, it’s not ‘central simple’ over $\mathbb{R}$: its center is not just $\mathbb{R}$, it’s bigger. This turns out to be why $\mathbb{C} \otimes \mathbb{C}$ is so funny compared to the rest of the entries in our division algebra multiplication table.

So, we’ve really got two Brauer groups in play. But we also have a homomorphism from the first to the second, given by ‘tensoring with $\mathbb{C}$’: complexifying any real central simple algebra, we get a complex one.

And whenever we have a group homomorphism $\alpha: G \to H$, we can make their disjoint union $G \sqcup H$ into monoid, which I’ll call $G \sqcup_\alpha H$.

It works like this. Given $g,g' \in G$, we multiply them the usual way. Given $h, h' \in H$, we multiply them the usual way. But given $g \in G$ and $h \in H$, we define

$g h := \alpha(g) h$

and

$h g := h \alpha(g)$

The multiplication on $G \sqcup_\alpha H$ is associative! For example:

$(g g')h = \alpha(g g') h = \alpha(g) \alpha(g') h = \alpha(g) (g'h) = g(g'h)$

Moreover, the element $1_G \in G$ acts as the identity of $G \sqcup_\alpha H$. For example:

$1_G h = \alpha(1_G) h = 1_H h = h$

But of course $G \sqcup_\alpha H$ isn’t a group, since “once you get inside $H$ you never get out”.

This construction could be called the **collage** of $G$ and $H$ via $\alpha$, since it’s reminiscent of a similar construction of that name in category theory.

**Question.** What do monoid theorists call this construction?

**Question.** Can we do a similar trick for any field? Can we always take the Brauer groups of all its finite-dimensional extensions and fit them together into a monoid by taking some sort of collage? If so, I’d call this the **Brauer monoid** of that field.

If you carefully read Part 1, maybe you can guess how I want to proceed. I want to make everything ‘super’.

I’ll replace division algebras over $\mathbb{R}$ by super division algebras over $\mathbb{R}$. Now instead of 3 = 2 + 1 there are 10 = 8 + 2:

8 of them are central simple over $\mathbb{R}$, so they give elements of the super Brauer group of $\mathbb{R}$, which is $\mathbb{Z}_8$.

2 of them are central simple over $\mathbb{C}$, so they give elements of the super Brauer group of $\mathbb{C}$, which is $\mathbb{Z}_2$.

Complexification gives a homomorphism

$\alpha: \mathbb{Z}_8 \to \mathbb{Z}_2$

namely the obvious nontrivial one. So, we can form the collage

$\mathbb{10} = \mathbb{Z}_8 \sqcup_\alpha \mathbb{Z}_2$

It’s a commutative monoid with 10 elements! Each of these is the equivalence class of one of the 10 real super division algebras.

I’ll then need to check that there’s an oplax monoidal functor

$G : \mathbb{10} \to SuperAlg_{\mathbb{R}}$

sending each element of $\mathbb{10}$ to the corresponding super division algebra.

If $G$ really exists, I can compose it with a thing

$SuperRep : SuperAlg_{\mathbb{R}} \to Rex_{\mathbb{R}}$

sending each super algebra to its category of ‘super representations’ on super vector spaces. This should again be a contravariant monoidal pseudofunctor.

We can call the composite of $G$ with $SuperRep$

$SuperVect: \mathbb{10} \to \Rex_{\mathbb{R}}$

If it all works, this thing $SuperVect$ will deserve to be called a $\mathbb{10}$-graded category. It contains super vector spaces over the 10 kinds of super division algebras in a single framework, and says how to tensor them. And when we look at super *Hilbert* spaces, this setup will be able to talk about all ten kinds of matter I mentioned last time… and how to combine them.

So that’s the plan. If you see problems, or ways to simplify things, please let me know!

There are 10 of each of these things:

Associative real super-division algebras.

Classical families of compact symmetric spaces.

Ways that Hamiltonians can get along with time reversal ($T$) and charge conjugation ($C$) symmetry.

Dimensions of spacetime in string theory.

It’s too bad nobody took up writing *This Week’s Finds in Mathematical Physics* when I quit. Someone should have explained this stuff in a nice simple way, so I could read their summary instead of fighting my way through the original papers. I don’t have much time for this sort of stuff anymore!

Luckily there are some good places to read about this stuff:

Todd Trimble, The super Brauer group and super division algebras, April 27, 2005.

Shinsei Ryu, Andreas P. Schnyde, Akira Furusaki and Andreas W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, June 15, 2010.

Gregory Moore and Dan Freed, Twisted equivariant matter, January 7, 2013.

Gregory Moore, Quantum symmetries and compatible Hamiltonians, December 15, 2013.

Let me start by explaining the basic idea, and then move on to more fancy aspects.

The idea of the ten-fold way goes back at least to 1996, when Altland and Zirnbauer discovered that substances can be divided into 10 kinds.

The basic idea is pretty simple. Some substances have **time-reversal symmetry**: they would look the same, even on the atomic level, if you made a movie of them and ran it backwards. Some don’t — these are more rare, like certain superconductors made of yttrium barium copper oxide! Time reversal symmetry is described by an antiunitary operator $T$ that squares to 1 or to -1: please take my word for this, it’s a quantum thing. So, we get 3 choices, which are listed in the chart under $T$ as 1, -1, or 0 (no time reversal symmetry).

Similarly, some substances have **charge conjugation symmetry**, meaning a symmetry where we switch particles and holes: places where a particle is missing. The ‘particles’ here can be rather abstract things, like **phonons** - little vibrations of sound in a substance, which act like particles — or **spinons** — little vibrations in the lined-up spins of electrons. Basically any way that something can wave can, thanks to quantum mechanics, act like a particle. And sometimes we can switch particles and holes, and a substance will act the same way!

Like time reversal symmetry, charge conjugation symmetry is described by an antiunitary operator $C$ that can square to 1 or to -1. So again we get 3 choices, listed in the chart under $C$ as 1, -1, or 0 (no charge conjugation symmetry).

So far we have 3 × 3 = 9 kinds of matter. What is the tenth kind?

Some kinds of matter don’t have time reversal or charge conjugation symmetry, but they’re symmetrical under the combination of time reversal and charge conjugation! You switch particles and holes and run the movie backwards, and things look the same!

In the chart they write 1 under the $S$ when your matter has this combined symmetry, and 0 when it doesn’t. So, “0 0 1” is the tenth kind of matter (the second row in the chart).

This is just the beginning of an amazing story. Since then people have found substances called topological insulators that act like insulators in their interior but conduct electricity on their surface. We can make 3-dimensional topological insulators, but also 2-dimensional ones (that is, thin films) and even 1-dimensional ones (wires). And we can theorize about higher-dimensional ones, though this is mainly a mathematical game.

So we can ask which of the 10 kinds of substance can arise as topological insulators in various dimensions. And the answer is: in any particular dimension, only 5 kinds can show up. But it’s a different 5 in different dimensions! This chart shows how it works for dimensions 1 through 8. The kinds that can’t show up are labelled 0.

If you look at the chart, you’ll see it has some nice patterns. And it repeats after dimension 8. In other words, dimension 9 works just like dimension 1, and so on.

If you read some of the papers I listed, you’ll see that the $\mathbb{Z}$’s and $\mathbb{Z}_2$’s in the chart are the homotopy groups of the ten classical series of compact symmetric spaces. The fact that dimension $n+8$ works like dimension $n$ is called **Bott periodicity**.

Furthermore, the stuff about operators $T$, $C$ and $S$ that square to 1, -1 or don’t exist at all is closely connected to the classification of associative real super division algebras. It all fits together.

In 2005, Todd Trimble wrote a short paper called The super Brauer group and super division algebras.

In it, he gave a quick way to classify the **associative real super division algebras**: that is, finite-dimensional associative real $\mathbb{Z}_2$-graded algebras having the property that every nonzero *homogeneous* element is invertible. The result was known, but I really enjoyed Todd’s effortless proof.

However, I didn’t notice that there are exactly 10 of these guys. Now this turns out to be a big deal. For each of these 10 algebras, the *representations* of that algebra describe ‘types of matter’ of a particular kind — where the 10 kinds are the ones I explained above!

So what are these 10 associative super division algebras?

3 of them are purely even, with no odd part: the usual associative division algebras $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$.

7 of them are not purely even. Of these, 6 are Morita equivalent to the real Clifford algebras $Cl_1, Cl_2, Cl_3, Cl_5, Cl_6$ and $Cl_7$. These are the superalgebras generated by 1, 2, 3, 5, 6, or 7 odd square roots of -1.

Now you should have at least two questions:

What’s ‘Morita equivalence’? — and even if you know, why should it matter here? Two algebras are

**Morita equivalent**if they have equivalent categories of representations. The same definition works for superalgebras, though now we look at their representations on super vector spaces ($\mathbb{Z}_2$-graded vector spaces). For physics what we really care about is the*representations*of an algebra or superalgebra: as I mentioned, those are ‘types of matter’. So, it makes sense to count two superalgebras as ‘the same’ if they’re Morita equivalent.1, 2, 3, 5, 6, and 7? That’s weird — why not 4? Well, Todd showed that $Cl_4$ is Morita equivalent to the purely even super division algebra $\mathbb{H}$. So we already had that one on our list. Similarly, why not 0? $Cl_0$ is just $\mathbb{R}$. So we had that one too.

Representations of Clifford algebras are used to describe spin-1/2 particles, so it’s exciting that 8 of the 10 associative real super division algebras are Morita equivalent to real Clifford algebras.

But I’ve already mentioned one that’s *not*: the complex numbers, $\mathbb{C}$, regarded as a purely even algebra. And there’s one more! It’s the *complex* Clifford algebra $\mathbb{C}\mathrm{l}_1$. This is the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in one odd square root of -1.

As soon as you hear that, you notice that the purely even algebra $\mathbb{C}$ is the complex Clifford algebra $\mathbb{C}\mathrm{l}_0$. In other words, it’s the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in *no* odd square roots of -1.

At this point things start fitting together:

You can multiply Morita equivalence classes of algebras using the tensor product of algebras: $[A] \otimes [B] = [A \otimes B]$. Some equivalence classes have multiplicative inverses, and these form the

**Brauer group**. We can do the same thing for superalgebras, and get the**super Brauer group**. The super division algebras Morita equivalent to $Cl_0, \dots , Cl_7$ serve as representatives of the super Brauer group of the real numbers, which is $\mathbb{Z}_8$. I explained this in week211 and further in week212. It’s a nice purely algebraic way to think about real Bott periodicity!As we’ve seen, the super division algebras Morita equivalent to $Cl_0$ and $Cl_4$ are a bit funny. They’re purely even. So they serve as representatives of the plain old Brauer group of the real numbers, which is $\mathbb{Z}_2$.

On the other hand, the complex Clifford algebras $\mathbb{C}\mathrm{l}_0 = \mathbb{C}$ and $\mathbb{C}\mathrm{l}_1$ serve as representatives of the super Brauer group of the complex numbers, which is also $\mathbb{Z}_2$. This is a purely algebraic way to think about

*complex*Bott periodicity, which has period 2 instead of period 8.

Meanwhile, the purely even $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ underlie Dyson’s ‘three-fold way’, which I explained in detail here:

- John Baez, Division algebras and quantum theory.

Briefly, if you have an irreducible unitary representation of a group on a complex Hilbert space $H$, there are three possibilities:

The representation is isomorphic to its dual via an invariant symmetric bilinear pairing $g : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = 1$. This lets us write our representation as the complexification of a

**real**one.The representation is isomorphic to its dual via an invariant antisymmetric bilinear pairing $\omega : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = -1$. This lets us promote our representation to a

**quaternionic**one.The representation is not isomorphic to its dual. In this case we say it’s truly

**complex**.

In physics applications, we can take $J$ to be either time reversal symmetry, $T$, or charge conjugation symmetry, $C$. Studying either symmetry separately leads us to Dyson’s three-fold way. Studying them both together leads to the ten-fold way!

So the ten-fold way seems to combine in one nice package:

- real Bott periodicity,
- complex Bott periodicity,
- the real Brauer group,
- the real super Brauer group,
- the complex super Brauer group, and
- the three-fold way.

I could throw ‘the complex Brauer group’ into this list, because that’s lurking here too, but it’s the trivial group, with $\mathbb{C}$ as its representative.

There really should be a better way to understand this. Here’s my best attempt right now.

The set of Morita equivalence classes of finite-dimensional real superalgebras gets a commutative monoid structure thanks to direct sum. This commutative monoid then gets a commutative rig structure thanks to tensor product. This commutative rig — let’s call it $\mathfrak{R}$ — is apparently too complicated to understand in detail, though I’d love to be corrected about that. But we can peek at pieces:

We can look at the group of

*invertible*elements in $\mathfrak{R}$ — more precisely, elements with multiplicative inverses. This is the real super Brauer group $\mathbb{Z}_8$.We can look at the sub-rig of $\mathfrak{R}$ coming from semisimple purely even algebras. As a commutative monoid under addition, this is $\mathbb{N}^3$, since it’s generated by $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$. This commutative monoid becomes a rig with a funny multiplication table, e.g. $\mathbb{C} \otimes \mathbb{C} = \mathbb{C} \oplus \mathbb{C}$. This captures some aspects of the three-fold way.

We should really look at a larger chunk of the rig $\mathfrak{R}$, that includes both of these chunks. How about the sub-rig coming from all semisimple superalgebras? What’s that?

And here’s another question: what’s the relation to the 10 classical families of compact symmetric spaces? The short answer is that each family describes a family of possible Hamiltonians for one of our 10 kinds of matter. For a more detailed answer, I suggest reading Gregory Moore’s Quantum symmetries and compatible Hamiltonians. But if you look at this chart by Ryu *et al*, you’ll see these families involve a nice interplay between $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, which is what this story is all about:

The families of symmetric spaces are listed in the column “Hamiltonian”.

All this stuff is fitting together more and more nicely! And if you look at the paper by Freed and Moore, you’ll see there’s a lot more involved when you take the symmetries of crystals into account. People are beginning to understand the algebraic and topological aspects of condensed matter much more deeply these days.

Just for the record, here are all 10 associative real super division algebras. 8 are Morita equivalent to real Clifford algebras:

$Cl_0$ is the purely even division algebra $\mathbb{R}$.

$Cl_1$ is the super division algebra $\mathbb{R} \oplus \mathbb{R}e$, where $e$ is an odd element with $e^2 = -1$.

$Cl_2$ is the super division algebra $\mathbb{C} \oplus \mathbb{C}e$, where $e$ is an odd element with $e^2 = -1$ and $e i = -i e$.

$Cl_3$ is the super division algebra $\mathbb{H} \oplus \mathbb{H}e$, where $e$ is an odd element with $e^2 = 1$ and $e i = i e, e j = j e, e k = k e$.

$Cl_4$ is $\mathbb{H}[2]$, the algebra of $2 \times 2$ quaternionic matrices, given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the purely even division algebra $\mathbb{H}$.

$Cl_5$ is $\mathbb{H}[2] \oplus \mathbb{H}[2]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{H} \oplus \mathbb{H}e$ where $e$ is an odd element with $e^2 = -1$ and $e i = i e, e j = j e, e k = k e$.

$Cl_6$ is $\mathbb{C}[4] \oplus \mathbb{C}[4]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{C} \oplus \mathbb{C}e$ where $e$ is an odd element with $e^2 = 1$ and $e i = -i e$.

$Cl_7$ is $\mathbb{R}[8] \oplus \mathbb{R}[8]$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{R} \oplus \mathbb{R}e$ where $e$ is an odd element with $e^2 = 1$.

$Cl_{n+8}$ is Morita equivalent to $Cl_n$ so we can stop here if we’re just looking for Morita equivalence classes, and there also happen to be no more super division algebras down this road. It is nice to compare $Cl_n$ and $Cl_{8-n}$: there’s a nice pattern here.

The remaining 2 real super division algebras are complex Clifford algebras:

$\mathbb{C}\mathrm{l}_0$ is the purely even division algebra $\mathbb{C}$.

$\mathbb{C}\mathrm{l}_1$ is the super division algebra $\mathbb{C} \oplus \mathbb{C} e$, where $e$ is an odd element with $e^2 = -1$ and $e i = i e$.

In the last one we could also say “with $e^2 = 1$” — we’d get something isomorphic, not a new possibility.

Oh yeah — what about the 10 dimensions in string theory? Are they really related to the ten-fold way?

It seems weird, but I think the answer is “yes, at least slightly”.

Remember, 2 of the dimensions in 10d string theory are those of the string worldsheet, which is a complex manifold. The other 8 are connected to the octonions, which in turn are connected to the 8-fold periodicity of real Clifford algebra. So the 8+2 split in string theory is at least slightly connected to the 8+2 split in the list of associative real super division algebras.

This may be more of a joke than a deep observation. After all, the 8 dimensions of the octonions are not individual things with distinct identities, as the 8 super division algebras coming from real Clifford algebras are. So there’s no one-to-one correspondence going on here, just an equation between numbers.

Still, there are certain observations that would be silly to resist mentioning.

We had a paper in Nature this week, and I think this paper is exciting and important. I've written an article for The Conversation which you can read it here.

Enjoy!

Enjoy!

It reminds me of Martin Amis’s The Information, in that it is a really well-made thing, but one which I think probably shouldn’t have been made, and which I’m probably sorry I read, because it’s sick in its heart. Everything else I can say is a spoiler so I’ll put it below a tab. 1. I […]

It reminds me of Martin Amis’s *The Information*, in that it is a really well-made thing, but one which I think probably *shouldn’t* have been made, and which I’m probably sorry I read, because it’s sick in its heart.

Everything else I can say is a spoiler so I’ll put it below a tab.

1. I think the right way to read the book is that, by the end, nothing has changed for the two protagonists. Their relationship at the end of the book — in which the man is a hateful worm, and the woman a murderer, and they are bound together by hatred, fear, and common lies — is meant to be the *same *relationship they had in their courtship. Just with everything a little more out in the open. Indeed I think this is what Flynn suggests marriage just, naturally, *is. *That people, in general, are sick brutes who need to hurt each other in order to gain satisfaction and who can only be kept superficially in line by the threat of being hurt or killed themselves. I don’t actually think this is true and so I don’t like novels which, by virtue of being well-made, make a compelling case that it’s true.

2. Money is important here. The structure of the story is that the couple starts rich. Then for most of the book they’re not rich. Then at the end they’re rich again, which is what enables them to go back to their normal life. *Gone Girl* suggests that what being rich means is that people pay attention to you, people believe what you say, and also that you might need to leave some broken or dead people behind in order to maintain your position. So Desi Collings is cognate to the Blue Book Boys.

3. The book is lazy in placing a lot of weight on “the psycho woman who claims to have been raped but is making it up.” The problem with misogynistic stereotypes in novels is not just that misogynistic sterotypes are bad — and they are, they are really bad — but that they’re a fundamentally cheap way of constructing characters. They are easy to believe in because we are weak people, driven by heuristics, who believe stereotypes without thinking too hard about them.

The book would have been better if it had let Nick beat up his girlfriend. In other words, if the world of the novel contains women who lie about getting beaten up by men, it ought to contain men who beat women up. And this would be truer to the moral world of the novel, where a woman falsely accusing a man of abuse is both lying and not lying, because all men abuse somebody, whether or not the accuser and the victim happen to be the same person.

And I think it would have helped prohibit the reading — which I can see from online sources is not rare — that Nick is the hero of the story, who readers are supposed to root for. No! Gross! Nick is a sick brute, Amy is a sick brute, all four of their parents are sick brutes, with the possible exception of Nick’s mother, who’s kind of a cipher.

4. It was a bad idea to name a character “Go.” Confusing in dialogue.

5. In connection with the upcoming movie, you can buy T-shirts labeled “Team Nick” or “Team Amy.” That is messed up and wrong.

*This article appeared in *Fermilab Today* on July 24, 2014.*

Fermilab engineer Jim Hoff has received patent approval on a very tiny, very clever invention that could have an impact on aerospace, agriculture and medical imaging industries.

Hoff has engineered a widely adaptable latch — an electronic circuit capable of remembering a logical state — that suppresses a commonly destructive circuit error caused by radiation.

There are two radiation-based errors that can damage a circuit: total dose and single-event upset. In the former, the entire circuit is doused in radiation and damaged; in an SEU, a single particle of radiation delivers its energy to the chip and alters a state of memory, which takes the form of 1s and 0s. Altered states of memory equate to an unintentional shift from logical 1 or logical 0 and ultimately lead to loss of data or imaging resolution. Hoff’s design is essentially a chip immunization, preemptively guarding against SEUs.

“There are a lot of applications,” Hoff said. “Anyone who needs to store data for a length of time and keep it in that same state, uncorrupted — anyone flying in a high-altitude plane, anyone using medical imaging technology — could use this.”

Past experimental data showed that, in any given total-ionizing radiation dose, the latch reduces single-event upsets by a factor of about 40. Hoff suspects that the invention’s newer configurations will yield at least two orders of magnitude in single-event upset reduction.

The invention is fondly referred to as SEUSS, which stands for single-event upset suppression system. It’s relatively inexpensive and designed to integrate easily with a multitude of circuits — all that’s needed is a compatible transistor.

Hoff’s line of work lies in chip development, and SEUSS is currently used in some Fermilab-developed chips such as FSSR, which is used in projects at Jefferson Lab, and Phoenix, which is used in the Relativistic Heavy Ion Collider at Brookhaven National Laboratory.

The idea of SEUSS was born out of post-knee-surgery, bed-ridden boredom. On strict bed rest, Hoff’s mind naturally wandered to engineering.

“As I was lying there, leg in pain, back cramping, I started playing with designs of my most recent project at work,” he said. “At one point I stopped and thought, ‘Wow, I just made a single-event upset-tolerant SR flip-flop!’”

While this isn’t the world’s first SEUSS-tolerant latch, Hoff is the first to create a single-event upset suppression system that is also a set-reset flip-flop, meaning it can take the form of almost any latch. As a flip-flop, the adaptability of the latch is enormous and far exceeds that of its pre-existing latch brethren.

“That’s what makes this a truly special latch — its incredible versatility,” says Hoff.

From a broader vantage point, the invention is exciting for more than just Fermilab employees; it’s one of Fermilab’s first big efforts in pursuing potential licensees from industry.

Cherri Schmidt, head of Fermilab’s Office of Partnerships and Technology Transfer, with the assistance of intern Miguel Marchan, has been developing the marketing plan to reach out to companies who may be interested in licensing the technology for commercial application.

“We’re excited about this one because it could really affect a large number of industries and companies,” Schmidt said. “That, to me, is what makes this invention so interesting and exciting.”

—*Hanae Armitage*

One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns … Continue reading

One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.)

**Born Rule:**

The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? Born himself kind of stumbled onto his Rule. Here is an excerpt from his 1926 paper:

That’s right. Born’s paper was rejected at first, and when it was later accepted by another journal, he didn’t even get the Born Rule right. At first he said the probability was equal to the amplitude, and only in an added footnote did he correct it to being the amplitude squared. And a good thing, too, since amplitudes can be negative or even imaginary!

The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. When we teach quantum mechanics to undergraduate physics majors, we generally give them a list of postulates that goes something like this:

- Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
- Wave functions evolve in time according to the Schrödinger equation.
- The act of measuring a quantum system returns a number, known as the eigenvalue of the quantity being measured.
- The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue.
- After the measurement is performed, the wave function “collapses” to a new state in which the wave function is localized precisely on the observed eigenvalue (as opposed to being in a superposition of many different possibilities).

It’s an ungainly mess, we all agree. You see that the Born Rule is simply postulated right there, as #4. Perhaps we can do better.

Of course we can do better, since “textbook quantum mechanics” is an embarrassment. There are other formulations, and you know that my own favorite is Everettian (“Many-Worlds”) quantum mechanics. (I’m sorry I was too busy to contribute to the active comment thread on that post. On the other hand, a vanishingly small percentage of the 200+ comments actually addressed the point of the article, which was that the potential for many worlds is automatically there in the wave function no matter what formulation you favor. Everett simply takes them seriously, while alternatives need to go to extra efforts to erase them. As Ted Bunn argues, Everett is just “quantum mechanics,” while collapse formulations should be called “disappearing-worlds interpretations.”)

Like the textbook formulation, Everettian quantum mechanics also comes with a list of postulates. Here it is:

- Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
- Wave functions evolve in time according to the Schrödinger equation.

That’s it! Quite a bit simpler — and the two postulates are exactly the same as the first two of the textbook approach. Everett, in other words, is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism.

The trickiest thing to extract from the formalism is the Born Rule. That’s what Charles (“Chip”) Sebens and I tackled in our recent paper:

Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics

Charles T. Sebens, Sean M. CarrollA longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we give new reasons why that would be inadvisable. Applying lessons from this analysis, we demonstrate (using arguments similar to those in Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In particular, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers.

Chip is a graduate student in the philosophy department at Michigan, which is great because this work lies squarely at the boundary of physics and philosophy. (I guess it is possible.) The paper itself leans more toward the philosophical side of things; if you are a physicist who just wants the equations, we have a shorter conference proceeding.

Before explaining what we did, let me first say a bit about why there’s a puzzle at all. Let’s think about the wave function for a spin, a spin-measuring apparatus, and an environment (the rest of the world). It might initially take the form

(α[up] + β[down] ; apparatus says “ready” ; environment

_{0}). (1)

This might look a little cryptic if you’re not used to it, but it’s not too hard to grasp the gist. The first slot refers to the spin. It is in a superposition of “up” and “down.” The Greek letters α and β are the amplitudes that specify the wave function for those two possibilities. The second slot refers to the apparatus just sitting there in its ready state, and the third slot likewise refers to the environment. By the Born Rule, when we make a measurement the probability of seeing spin-up is |α|^{2}, while the probability for seeing spin-down is |β|^{2}.

In Everettian quantum mechanics (EQM), wave functions never collapse. The one we’ve written will smoothly evolve into something that looks like this:

α([up] ; apparatus says “up” ; environment

_{1})

+ β([down] ; apparatus says “down” ; environment_{2}). (2)

This is an extremely simplified situation, of course, but it is meant to convey the basic appearance of two separate “worlds.” The wave function has split into branches that don’t ever talk to each other, because the two environment states are different and will stay that way. A state like this simply arises from normal Schrödinger evolution from the state we started with.

So here is the problem. After the splitting from (1) to (2), the wave function coefficients α and β just kind of go along for the ride. If you find yourself in the branch where the spin is up, your coefficient is α, but so what? How do you know what kind of coefficient is sitting outside the branch you are living on? All you know is that there was one branch and now there are two. If anything, shouldn’t we declare them to be equally likely (so-called “branch-counting”)? For that matter, in what sense are there probabilities *at all*? There was nothing stochastic or random about any of this process, the entire evolution was perfectly deterministic. It’s not right to say “Before the measurement, I didn’t know which branch I was going to end up on.” You know precisely that one copy of your future self will appear on *each* branch. Why in the world should we be talking about probabilities?

Note that the pressing question is not so much “Why is the probability given by the wave function squared, rather than the absolute value of the wave function, or the wave function to the fourth, or whatever?” as it is “Why is there a particular probability rule at all, since the theory is deterministic?” Indeed, once you accept that there should be some specific probability rule, it’s practically guaranteed to be the Born Rule. There is a result called Gleason’s Theorem, which says roughly that the Born Rule is the only consistent probability rule you can conceivably have that depends on the wave function alone. So the real question is not “Why squared?”, it’s “Whence probability?”

Of course, there are promising answers. Perhaps the most well-known is the approach developed by Deutsch and Wallace based on decision theory. There, the approach to probability is essentially operational: given the setup of Everettian quantum mechanics, how should a rational person behave, in terms of making bets and predicting experimental outcomes, etc.? They show that there is one unique answer, which is given by the Born Rule. In other words, the question “Whence probability?” is sidestepped by arguing that reasonable people in an Everettian universe will act *as if* there are probabilities that obey the Born Rule. Which may be good enough.

But it might not convince everyone, so there are alternatives. One of my favorites is Wojciech Zurek’s approach based on “envariance.” Rather than using words like “decision theory” and “rationality” that make physicists nervous, Zurek claims that the underlying symmetries of quantum mechanics pick out the Born Rule uniquely. It’s very pretty, and I encourage anyone who knows a little QM to have a look at Zurek’s paper. But it is subject to the criticism that it doesn’t really teach us anything that we didn’t already know from Gleason’s theorem. That is, Zurek gives us more reason to think that the Born Rule is uniquely preferred by quantum mechanics, but it doesn’t really help with the deeper question of why we should think of EQM as a theory of probabilities at all.

Here is where Chip and I try to contribute something. We use the idea of “self-locating uncertainty,” which has been much discussed in the philosophical literature, and has been applied to quantum mechanics by Lev Vaidman. Self-locating uncertainty occurs when you know that there multiple observers in the universe who find themselves in exactly the same conditions that you are in right now — but you don’t know which one of these observers you are. That can happen in “big universe” cosmology, where it leads to the measure problem. But it automatically happens in EQM, whether you like it or not.

Think of observing the spin of a particle, as in our example above. The steps are:

- Everything is in its starting state, before the measurement.
- The apparatus interacts with the system to be observed and becomes entangled. (“Pre-measurement.”)
- The apparatus becomes entangled with the environment, branching the wave function. (“Decoherence.”)
- The observer reads off the result of the measurement from the apparatus.

The point is that in between steps 3. and 4., the wave function of the universe has branched into two, but *the observer doesn’t yet know which branch they are on*. There are two copies of the observer that are in identical states, even though they’re part of different “worlds.” That’s the moment of self-locating uncertainty. Here it is in equations, although I don’t think it’s much help.

You might say “What if I am the apparatus myself?” That is, what if I observe the outcome directly, without any intermediating macroscopic equipment? Nice try, but no dice. That’s because decoherence happens incredibly quickly. Even if you take the extreme case where you look at the spin directly with your eyeball, the time it takes the state of your eye to decohere is about 10^{-21} seconds, whereas the timescales associated with the signal reaching your brain are measured in tens of milliseconds. Self-locating uncertainty is inevitable in Everettian quantum mechanics. In that sense, *probability* is inevitable, even though the theory is deterministic — in the phase of uncertainty, we need to assign probabilities to finding ourselves on different branches.

So what do we do about it? As I mentioned, there’s been a lot of work on how to deal with self-locating uncertainty, i.e. how to apportion credences (degrees of belief) to different possible locations for yourself in a big universe. One influential paper is by Adam Elga, and comes with the charming title of “Defeating Dr. Evil With Self-Locating Belief.” (Philosophers have more fun with their titles than physicists do.) Elga argues for a principle of *Indifference*: if there are truly multiple copies of you in the world, you should assume equal likelihood for being any one of them. Crucially, Elga doesn’t simply assert *Indifference*; he actually derives it, under a simple set of assumptions that would seem to be the kind of minimal principles of reasoning any rational person should be ready to use.

But there is a problem! Naïvely, applying *Indifference* to quantum mechanics just leads to branch-counting — if you assign equal probability to every possible appearance of equivalent observers, and there are two branches, each branch should get equal probability. But that’s a disaster; it says we should simply ignore the amplitudes entirely, rather than using the Born Rule. This bit of tension has led to some worry among philosophers who worry about such things.

Resolving this tension is perhaps the most useful thing Chip and I do in our paper. Rather than naïvely applying *Indifference* to quantum mechanics, we go back to the “simple assumptions” and try to derive it from scratch. We were able to pinpoint one hidden assumption that seems quite innocent, but actually does all the heavy lifting when it comes to quantum mechanics. We call it the “Epistemic Separability Principle,” or *ESP* for short. Here is the informal version (see paper for pedantic careful formulations):

ESP: The credence one should assign to being any one of several observers having identical experiences is independent of features of the environment that aren’t affecting the observers.

That is, the probabilities you assign to things happening in your lab, whatever they may be, should be exactly the same if we tweak the universe just a bit by moving around some rocks on a planet orbiting a star in the Andromeda galaxy. *ESP* simply asserts that our knowledge is separable: how we talk about what happens here is independent of what is happening far away. (Our system here can still be *entangled* with some system far away; under unitary evolution, changing that far-away system doesn’t change the entanglement.)

The *ESP* is quite a mild assumption, and to me it seems like a necessary part of being able to think of the universe as consisting of separate pieces. If you can’t assign credences locally without knowing about the state of the whole universe, there’s no real sense in which the rest of the world is really separate from you. It is certainly implicitly used by Elga (he assumes that credences are unchanged by some hidden person tossing a coin).

With this assumption in hand, we are able to demonstrate that *Indifference* does not apply to branching quantum worlds in a straightforward way. Indeed, we show that you should assign equal credences to two different branches *if and only if* the amplitudes for each branch are precisely equal! That’s because the proof of *Indifference* relies on shifting around different parts of the state of the universe and demanding that the answers to local questions not be altered; it turns out that this only works in quantum mechanics if the amplitudes are equal, which is certainly consistent with the Born Rule.

See the papers for the actual argument — it’s straightforward but a little tedious. The basic idea is that you set up a situation in which more than one quantum object is measured at the same time, and you ask what happens when you consider different objects to be “the system you will look at” versus “part of the environment.” If you want there to be a consistent way of assigning credences in all cases, you are led inevitably to equal probabilities when (and only when) the amplitudes are equal.

What if the amplitudes for the two branches are not equal? Here we can borrow some math from Zurek. (Indeed, our argument can be thought of as a love child of Vaidman and Zurek, with Elga as midwife.) In his envariance paper, Zurek shows how to start with a case of unequal amplitudes and reduce it to the case of many more branches with equal amplitudes. The number of these pseudo-branches you need is proportional to — wait for it — the square of the amplitude. Thus, you get out the full Born Rule, simply by demanding that we assign credences in situations of self-locating uncertainty in a way that is consistent with *ESP*.

We like this derivation in part because it treats probabilities as epistemic (statements about our knowledge of the world), not merely operational. Quantum probabilities are really credences — statements about the best degree of belief we can assign in conditions of uncertainty — rather than statements about truly stochastic dynamics or frequencies in the limit of an infinite number of outcomes. But these degrees of belief aren’t completely subjective in the conventional sense, either; there is a uniquely rational choice for how to assign them.

Working on this project has increased my own personal credence in the correctness of the Everett approach to quantum mechanics from “pretty high” to “extremely high indeed.” There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, and how branching structures are best defined. (I’m off to a workshop next month to think about precisely these questions.) But these seem like relatively tractable technical challenges to me, rather than looming deal-breakers. EQM is an incredibly simple theory that (I can now argue in good faith) makes sense and fits the data. Now it’s just a matter of convincing the rest of the world!

If you go to the blue side of Lyman alpha, at reasonable redshifts (say 2), the continuum is not clearly visible, since the forest is dense and has a range of equivalent widths. Any study of IGM physics or radiation or clustering or ionization depends on an accurate continuum determination. What to do? Obviously, you should fit your continuum simultaneously with whatever else you are measuring, and marginalize out the posterior uncertainties on the continuum. *Duh!*

That said, few have attempted this. Today I had a long conversation with Hennawi, Eilers, Rorai, and KG Lee (all MPIA) about this; they are trying to constrain IGM physics with the transmission pdf, marginalizing out the continuum. We discussed the problem of sampling each quasar's continuum separately but having a universal set of IGM parameters. I advocated a limited case of Foreman-Mackey and my endless applications of importance sampling. Gibbs sampling would work too. We discussed how to deal with the fact that different quasars might disagree mightily about the IGM parameters. Failure of support can ruin your whole day. We came up with a clever hack that extends a histogram of samples to complete support in the parameters space.

In Milky Way group meeting, Ness (MPIA) showed that there appears to be an over-density of metal-poor stars in the inner one degree (projected) at the center of the Milky Way. She is using *APOGEE* data and her home-built metallicity indicators. We discussed whether the effect could be caused by issues with selection (either because of dust or a different explicit selection program in this center-Galaxy field). If the effect is real, it is extremely interesting. For example, even if the stars were formed there, why would they stay there?

This Alberto Cairo piece on “data journalism” has been kicking around for a while, and it’s taken me a while to pin down what bugs me about it. I think my problem with it ultimately has to do with the first two section headers in which he identifies problems with FiveThirtyEight and Vox:

1. Data and explanatory journalism cannot be done on the cheap.

2. Data and explanatory journalism cannot be produced in a rush.

The implication here is that “data and explanatory journalism” is necessarily a weighty and complicated thing, something extremely Serious to be approached only with great care. But that seems to me to miss the entire point of FiveThirtyEight and Vox, and the thing that makes them great.

That is, what seems to me to be the best feature of these sites is that they *don’t* view “data journalism” as Serious and unapproachable. And that’s a very good thing, because if the people producing it view it as a difficult and weighty enterprise, that gets through to the readers. Who will then view data-driven reporting as something Serious, only to be approached when you’re in the mood for some heavy slogging.

And that attitude *already exists* and is one of the biggest problems facing science communication and serious policy discussions. People view math and science as Difficult and Serious, and flinch away when they come up. Which allows canny and cynical politicians to exploit this to create an illusion of confusion around policy issues where the science is actually perfectly clear-cut.

The biggest potential benefit of sites like FiveThirtyEight and Vox is that they might break that association between “data-driven” and “hard work.” Yeah, a lot of their stuff is kind of superficial, but I’d say that’s a feature, not a bug– if you can ease people into thinking a bit more quantitatively through somewhat superficial treatments of popular subjects, that’s all to the good. The kind of lightweight stories that you *can* do on the cheap and in a rush are exactly what we need. If the only data-driven stories we get are massive exhaustively researched stories about weighty topics, then people will continue to think that statistics are hard and boring, and nothing will change.

But then, I would say that sort of thing, given that I’m a physicist and a blogger (also, full disclosure, Nate “FiveThiryEight” Silver figures prominently in one chapter of my forthcoming book). Physicists are famous for our love of simplified fast-and-cheap approximate models (“spherical cows” and all that), and bloggers are all about generating content quickly. So reading FiveThirtyEight and Vox is sort of like reading Dot Physics if Rhett wrote about current events. And if I ever lost my mind and decided to give up the tenured professor gig for the glamorous life of a full-time journalist, that’s the sort of thing I’d be most likely to go for.

So, while I occasionally have issues with things that they write, as a general matter, I love what FiveThirtyEight and Vox are doing. Nate Silver and Ezra Klein are the physicists of the journalistic world, and I think we need more of that. We need “data journalism” that’s fast and cheap and most of all fun.

——

*(Disclaimer: I don’t read everything that FiveThirtyEight and Vox post, and I don’t like everything that I do read, so please spare me the “gotcha” comments pointing to some stupid thing that one of their writers posted. I’m talking about the general approach, here, which I like very much.)*