Planet Musings

Happy National Poetry Month! The United States salutes word and whimsy in April, and Quantum Frontiers is continuing its tradition of celebrating. As a resident of Cambridge, Massachusetts and as a quantum information scientist, I have trouble avoiding the poem “Paul Revere’s Ride.” 

Henry Wadsworth Longfellow wrote the poem, as well as others in the American canon, during the 1800s. Longfellow taught at Harvard in Cambridge, and he lived a few blocks away from the university, in what’s now a national historic site. Across the street from the house, a bust of the poet gazes downward, as though lost in thought, in Longfellow Park. Longfellow wrote one of his most famous poems about an event staged a short drive from—and, arguably, partially in—Cambridge.

The event took place “on the eighteenth of April, in [Seventeen] Seventy-Five,” as related by the narrator of “Paul Revere’s Ride.” Revere was a Boston silversmith and a supporter of the American colonies’ independence from Britain. Revolutionaries anticipated that British troops would set out from Boston sometime during the spring. The British planned to seize revolutionaries’ weapons in the nearby town of Concord and to jail revolutionary leaders in Lexington. The troops departed Boston during the night of April 18th. 

Upon learning of their movements, sexton Robert Newman sent a signal from Boston’s old North Church to Charlestown. Revere and the physician William Dawes rode out from Charlestown to warn the people of Lexington and the surrounding areas. A line of artificial hoof prints, pressed into a sidewalk a few minutes from the Longfellow house, marks part of Dawes’s trail through Cambridge. The initial riders galvanized more riders, who stirred up colonial militias that resisted the troops’ advance. The Battles of Lexington and Concord ensued, initiating the Revolutionary War.

Longfellow took liberties with the facts he purported to relate. But “Paul Revere’s Ride” has blown the dust off history books for generations of schoolchildren. The reader shares Revere’s nervous excitement as he fidgets, awaiting Newman’s signal: 

Now he patted his horse’s side, 
Now gazed on the landscape far and near, 
Then impetuous stamped the earth, 
And turned and tightened his saddle-girth;
But mostly he watched with eager search 
The belfry-tower of the old North Church.

The moment the signal arrives, that excitement bursts its seams, and Revere leaps astride his horse. The reader comes to gallop through with the silversmith the night, the poem’s clip-clop-clip-clop rhythm evoking a horse’s hooves on cobblestones.

Not only does “Paul Revere’s Ride” revitalize history, but it also offers a lesson in information theory. While laying plans, Revere instructs Newman: 

He said to his friend, “If the British march
By land or sea from the town to-night,
Hang a lantern aloft in the belfry-arch
Of the North-Church-tower, as a signal light.

Then comes one of the poem’s most famous lines: “One if by land, and two if by sea.” The British could have left Boston by foot or by boat, and Newman had to communicate which. Specifying one of two options, he related one bit, or one basic unit of information. Newton thereby exemplifies a cornerstone of information theory: the encoding of a bit of information—an abstraction—in a physical system that can be in one of two possible states—a light that shines from one or two lanterns.

Benjamin Schumacher and Michael Westmoreland point out the information-theoretic interpretation of Newman’s action in their quantum-information textbook. I used their textbook in my first quantum-information course, as a senior in college. Before reading the book, I’d never felt that I could explain what information is or how it can be quantified. Information is an abstraction and a Big Idea, like consciousness, life, and piety. But, Schumacher and Westmoreland demonstrated, most readers already grasp the basics of information theory; some readers even studied the basics while memorizing a poem in elementary school. So I doff my hat—or, since we’re discussing the 1700s, my mobcap—to the authors.

Reading poetry enriches us more than we realize. So read a poem this April. You can find Longfellow’s poem here or ride off wherever your fancy takes you.  

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This semester I’ve been teaching an undergraduate course on Galois theory. It was all online, which meant a lot of work, but it was also a lot of fun: the students were great, and I got to know them individually better than I usually would.

For a category theorist, Galois theory is a constant provocation: very little is canonical or functorial, or at least, not in the obvious sense (for reasons closely related to the nontriviality of the Galois group). One important not-obviously-functorial construction is algebraic closure. We didn’t get to it in the course, but I spent a while absorbed in an expository note on it by Keith Conrad.

Proving that every field has an algebraic closure is not entirely trivial, but the proof in Conrad’s note seems easier and more obvious than the argument you’ll find in many algebra books. As he says, it’s a variant on a proof by Zorn, which he attributes to “B. Conrad” (presumably his brother Brian). It should be more widely known, and now I find myself asking: why would you prove it any other way?

What follows is a somewhat categorical take on the Conrad–Zorn proof.

The reason why many constructions in field theory fail to be functorial is that they involve arbitrary choices, often of roots or irreducible factors of polynomials. For instance, to construct the splitting field of a polynomial, we have to make a finite number of arbitrary choices.

The algebraic closure of a field is something like the splitting field of all polynomials, so you’d expect it to require an infinite number of arbitrary choices (at least if the field is infinite). It’s no surprise, then, that it cannot be done without some form of the axiom of choice. Actually, it doesn’t require the full-strength axiom, but it does need a small dose: there are models of ZF in which some fields have no algebraic closure.

That explains why Zorn’s name came up, and it also suggests why no proof of the existence of algebraic closures can be entirely trivial.

An algebraic closure of a field $K$ is, by definition, an extension $M$ of $K$ that is algebraically closed (every polynomial over $M$ splits in $M$) and is minimal as such. So to construct an algebraic closure of $K$, it looks as if we have to manufacture an extension $M$ in which it’s possible to split not only every polynomial over $K$, but also every polynomial over $M$ itself.

Fortunately, there’s a lemma that saves us from that: any algebraic extension $M$ of $K$ in which every polynomial over $K$ splits is, in fact, an algebraic closure of $K$. This is proved by a standard field-theoretic argument that I won’t dwell on here, except to say that the main ingredient is the “transitivity of algebraicity”: given extensions $K \subseteq L \subseteq M$ and $\alpha \in M$, if $\alpha$ is algebraic over $L$ and $L$ is algebraic over $K$ then $\alpha$ is algebraic over $K$.

So: an algebraic closure of $K$ is an algebraic extension of $K$ in which every polynomial over $K$ splits. That’s the formulation I’ll use for the rest of this post.

Colin McLarty’s paper The rising sea: Grothendieck on simplicity and generality begins with a joke I find very resonant:

In 1949, André Weil published striking conjectures linking number theory to topology and a striking strategy for a proof [Weil, 1949]. Around 1953, Jean-Pierre Serre took on the project and soon recruited Alexander Grothendieck. Serre created a series of concise elegant tools which Grothendieck and coworkers simplified into thousands of pages of category theory.

How do we construct an algebraic closure of a given field? If you just want a concise presentation, I can’t do better than point you to Conrad’s note. But here I want to come at it a bit differently.

In my mind, the key to the Conrad–Zorn approach is to work in the world of rings for as long as possible and only descend to fields at the last minute. Behind this is the fact that the theory of rings is algebraic and the theory of fields is not, which is ultimately why constructions with fields often fail to be functorial.

(Here I mean “algebraic theory” in the technical sense. One could argue that little is more algebraic than the theory of fields, just as one could argue that “algebraic group” is a tautology…)

So we’re going to use rings, by which I mean commutative rings with $1$. A field extension $M$ of our field $K$ is just a homomorphism of fields $K \to M$, but we’re going to be interested in “ring extensions” of $K$, by which I mean homomorphisms $K \to A$, where $A$ is a ring. Such a thing is just a (commutative) $K$-algebra, which is why you don’t hear anyone talking about “ring extensions”.

Given a polynomial $f \neq 0$ over $K$, we could consider the splitting field $SF_K(f)$ of $f$ over $K$. But instead, let’s consider what I’ll call its splitting ring $SR_K(f)$. Informally, $SR_K(f)$ is the $K$-algebra you get by freely adjoining to $K$ the right number of roots, say

$$\alpha_{f, 1}, \ldots, \alpha_{f, \deg(f)},$$

and imposing relations to say that they are the roots of $f$. Write

$$\hat{f}(x) = a \prod_{i = 1}^{\deg(f)} (x - \alpha_{f, i}),$$

where $a$ is the leading coefficient of $f$. The relations should make $\hat{f}(x)$ and $f(x)$ be equal as polynomials over $SR_K(f)$.

Formally, then, we define $SR_K(f)$ to be the $K$-algebra

$$SR_K(f) = K[\alpha_{f, 1}, \ldots, \alpha_{f, \deg(f)}]/\sim$$

where $\sim$ identifies the $r$th coefficient of $f$ with the $r$th coefficient of $\hat{f}$ for each $r \geq 0$. So $SR_K(f)$ is the free $K$-algebra in which $f$ splits.

For example, take $f(x) = x(x - 1)$. The splitting field of $f$ over $K$ is just $K$ itself. But the splitting ring $SR_K(f)$ is another story. It’s the polynomial ring $K[\alpha, \beta]$ quotiented out by the relations that identify the coefficients of $f(x) = x(x - 1)$ with those of

$$\hat{f}(x) = (x - \alpha)(x - \beta).$$

Comparing constant terms gives $\alpha\beta = 0$, and comparing coefficients of degree $1$ gives $\alpha + \beta = 1$. So

$$SR_K(f) = K[\alpha, \beta]/(\alpha\beta = 0, \ \alpha + \beta = 1),$$

or equivalently,

$$SR_K(f) = K[\alpha]/(\alpha^2 = \alpha).$$

This is bigger than the splitting field! A field can’t have nontrivial idempotents: $\alpha^2 = \alpha$ forces $\alpha \in \{0, 1\}$. But a ring can, and $SR_K(f)$ does.

More generally, we can form the splitting ring of not just one polynomial but any set of them (providing they’re nonzero). For $\mathcal{P} \subseteq K[x] \setminus \{0\}$, put

$$SR_K(\mathcal{P}) = K[\{ \alpha_{f, i}: f \in \mathcal{P}, \ 1 \leq i \leq \deg(f)\}]/\sim,$$

where $\sim$ identifies the coefficients of $f$ with the corresponding coefficients of $\hat{f}$ for each $f \in \mathcal{P}$. Equivalently, $SR_K(\mathcal{P})$ is the coproduct $\coprod_{f \in \mathcal{P}} SR_K(f)$ in the category of $K$-algebras. It’s the free $K$-algebra in which all the polynomials in $\mathcal{P}$ split.

The Conrad proof goes like this: put $R = SR_K(K[x] \setminus \{0\})$, then (i) show that $R$ is nontrivial, (ii) quotient $R$ out by a maximal ideal to get a field $M$, and (iii) show that $M$ is an algebraic closure of $K$. Step (i) is where the substance of the argument lies. Step (ii) calls on the fact that every nontrivial ring has a maximal ideal (a routine application of Zorn’s lemma). And step (iii) is pretty much immediate by the lemma I started with, since by construction every polynomial over $K$ splits in $R$, hence in $M$.

But you can come at it a bit differently, as follows. First let me list some apparent trivialities.

• A ring $A$ is trivial if and only if $0_A = 1_A$. (Triviality is an equational property.)

• The forgetful functor from rings to sets preserves filtered colimits. This is equivalent to the statement that the theory of rings is finitary: the ring operations (such as addition) take only finitely many inputs.

• A filtered colimit of nontrivial rings is nontrivial. This isn’t as obvious as it might seem: for instance, the analogous statement for abelian groups is false. It follows from the previous two bullet points and the way filtered colimits work in $Set$.

• A ring $A$ is nontrivial if and only if there is a homomorphism from $A$ to some field, if and only if there is a surjective homomorphism from $A$ to some field. Indeed, the first bullet point together with the nontriviality of fields shows that if $A$ admits a homomorphism to a field then $A$ is nontrivial. On the other hand, if $A$ is nontrivial then by Zorn’s lemma, we can find a maximal ideal $J$ of $A$, and the natural homomorphism $A \to A/J$ is then a surjection to a field.

• Given a homomorphism $A \to B$ of $K$-algebras and a polynomial $f$ over $K$, if $f$ splits in $A$ then $f$ splits in $B$.

Now, here’s a proof that every field $K$ has an algebraic closure.

• Put $S = colim_{\mathcal{F}} SR_K(\mathcal{F})$, where the colimit is over finite subsets $\mathcal{F}$ of $K[x]\setminus\{0\}$. Here I’m implicitly using the fact that $SR_K(\mathcal{F})$ is functorial in $\mathcal{F}$ with respect to inclusion. This is a filtered colimit.

• For each $\mathcal{F}$, the polynomial $\prod_{f \in \mathcal{F}} f$ has a splitting field $L$. Then every polynomial in $\mathcal{F}$ splits in $L$, so by the universal property of $SR_K(\mathcal{F})$, there is a homomorphism $SR_K(\mathcal{F}) \to L$. It follows that $SR_K(\mathcal{F})$ is nontrivial.

• So the $K$-algebra $S$ is a filtered colimit of nontrivial rings, and is therefore nontrivial.

• Hence there is a surjective homomorphism from $S$ to some field $M$. Composing the natural maps $K \to S \to M$ makes $M$ into an extension of $K$.

• For each nonzero $f \in K[x]$, we have homomorphisms of $K$-algebras $SR_K(f) \to S \to M$, and $f$ splits in $SR_K(f)$, so $f$ splits in $M$. Hence the field $M$ is an extension of $K$ in which every polynomial over $K$ splits.

• Each of the $K$-algebras $SR_K(\mathcal{F})$ is generated by elements algebraic over $K$, so the same is true of $M$. Hence $M$ is algebraic over $K$, and is, therefore, an algebraic closure of $K$.

In any construction of algebraic closures, the challenge is to find some extension $M$ of $K$ in which every polynomial over $K$ splits. It may be that $M$ is wastefully big, but you can cut it down to size by taking the subfield $M'$ of $M$ consisting of the elements algebraic over $K$, and $M'$ is then an algebraic closure of $K$. This provides an alternative to the last step above. But I like it less, because the fact is that the $M$ we constructed is already the algebraic closure.

Whether one uses the approach in Conrad’s note or the slight variant I’ve just given, I prefer this to the Artin (?) proof that appears in many books and is also summarized on page 2 of Conrad’s note. It seems much more direct and appealing.

There’s another proof I also like: the existence of algebraic closures follows very quickly from the compactness theorem of model theory. Like the proof above, this argument relies on the knowledge that for any finite collection of polynomials, there’s some field in which they all split. In other words, it uses the existence of splitting fields.

And there’s one more thing I can’t resist saying. Although algebraic closure isn’t functorial in the obvious sense, I believe it’s functorial in a non-obvious sense — what I’d think of as a “Hakim-type” sense.

Let me explain what I mean, taking a bit of a run-up.

The category of rings has a subcategory consisting of the local rings. The inclusion has no left adjoint: there’s no way to take the free local ring on a ring. But there is if you allow the ambient topos to vary! In other words, consider all pairs $(\mathcal{E}, R)$ where $\mathcal{E}$ is a topos and $R$ is a ring in $\mathcal{E}$. Given such a pair, one can look for the universal map from $(\mathcal{E}, R)$ to some pair $(\mathcal{E}', R')$ where $R'$ is local.

Monique Hakim proved, among other things, that if we start with a ring $R$ in the topos $Set$, then this construction gives the topos $Sh(Spec(R))$ of sheaves on the prime spectrum of $R$, together with the structure sheaf $\mathcal{O}_R$, which is a sheaf of local rings on $Spec(R)$, or equivalently a local ring in $Sh(Spec(R))$. This wonderful result appeared in her 1972 book Topos Annelés et Schémas Relatifs.

Now if I’m not mistaken, there’s a similar description of algebraic closure. Consider all pairs $(\mathcal{E}, K)$ where $\mathcal{E}$ is a topos and $K$ is a field in $\mathcal{E}$. Given such a pair, one can look for the universal map from it to a pair $(\mathcal{E}', K')$ where $K'$ is algebraically closed. And I believe that if we start with a field $K$ in $Set$, the resulting pair is the topos of sets equipped with a continuous action by the absolute Galois group $G$ of $K$, together with the algebraic closure $\overline{K}$ with its natural $G$-action. There are lots of details I’ve skipped here, but perhaps someone can tell me whether that’s more or less right and, if so, where this is all written up.

I am in a project with Weichi Yao (NYU) and Soledad Villar (NYU) to look at building machine-learning methods that are constrained by the same symmetries as Newtonian mechanics: Rotation, translation, Galilean boost, and particle exchange, for examples. Kate Storey-Fisher (NYU) joined our weekly call today, because she has ideas about toy problems we could use to demonstrate the value of encoding these symmetries. She steered us towards things in the area of “halo occupation”, or the question of which dark-matter halos contain what kinds of galaxies. Right now halo occupation is performed with very blunt tools, and maybe a sharp tool could do better? We would have the advantage (over others) that anything we found would, by construction, obey the fundamental symmetries of physical law.

At the end of the day I had a wide-ranging conversation with Andy Casey (Monash) about all things spectroscopic. I mentioned to him my new interest in domain adaptation, and whether it could be used to build data-driven models. The SDSS-V project has two spectrographs, at two different telescopes, each of which observes stars down different fibers (which have their own idiosyncracies). Could we build a data-driven model to see what any star observed down one fiber of one spectrograph would look like if it had been observed down any other fiber or any fiber of the other spectrograph? That would permit us to see what systematics are spectrograph-specific, and whether we would have got the same answers with the other spectrograph, and other questions like that.

There are some stars observed multiple times and by both observatories, but I'm kind-of interested in whether we could do better using the huge number of stars that haven't been observed twice instead. Indeed, it isn't clear which contains more information about the transformations. Another fun thing: The northern sky and the southern sky are different! We would have to re-build domain adaptation to be sensitive to those differences, which might get into causal-inference territory.

The deadline for submitting papers is coming up soon: May 10th.

• Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12–16, 2021, online and at the Computer Laboratory of the University of Cambridge.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

Financial support

We are able to offer financial support to PhD students and junior researchers. Full guidance is on the webpage.

Important dates (all in 2021)

• Submission Deadline: Monday 10 May
• Author Notification: Monday 7 June
• Financial Support Application Deadline: Monday 7 June
• Financial Support Notification: Tuesday 8 June
• Priority Physical Registration Opens: Wednesday 9 June
• Ordinary Physical Registration Opens: Monday 13 June
• Reserved Accommodation Booking Deadline: Monday 13 June
• Adjoint School: Monday 5 to Friday 9 July
• Main Conference: Monday 12 to Friday 16 July

Submissions

The following two types of submissions are accepted:

Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at http://style.eptcs.org.

The submission link will soon be available on the ACT2021 web page: https://www.cl.cam.ac.uk/events/act2021

Program Committee

Chair:

• Kohei Kishida, University of Illinois, Urbana-Champaign

Members:

• Richard Blute, University of Ottawa
• Spencer Breiner, NIST
• Daniel Cicala, University of New Haven
• Robin Cockett, University of Calgary
• Bob Coecke, Cambridge Quantum Computing
• Geoffrey Cruttwell, Mount Allison University
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Jonas Frey, Carnegie Mellon University
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Fabrizio Romano Genovese, Statebox
• Helle Hvid Hansen, University of Groningen
• Jules Hedges, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Alex Hoffnung, Bridgewater
• Martti Karvonen, University of Ottawa
• Kohei Kishida, University of Illinois, Urbana -Champaign (chair)
• Martha Lewis, University of Bristol
• Bert Lindenhovius, Johannes Kepler University Linz
• Ben MacAdam, University of Calgary
• Dan Marsden, University of Oxford
• Jade Master, University of California, Riverside
• Joe Moeller, NIST
• Koko Muroya, Kyoto University
• Simona Paoli, University of Leicester
• Daniela Petrisan, Université de Paris, IRIF
• Mehrnoosh Sadrzadeh, University College London
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
• David Spivak, MIT and Topos Institute
• Joshua Tan, University of Oxford
• Dmitry Vagner
• Jamie Vicary, University of Cambridge
• John van de Wetering, Radboud University Nijmegen
• Vladimir Zamdzhiev, Inria, LORIA, Université de Lorraine
• Maaike Zwart

The deadline for submitting papers is coming up soon: May 10th.

• Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12–16, 2021, online and at the Computer Laboratory of the University of Cambridge.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

FINANCIAL SUPPORT

We are able to offer financial support to PhD students and junior researchers. Full guidance is on the webpage.

IMPORTANT DATES (all in 2021)

• Submission Deadline: Monday 10 May
• Author Notification: Monday 7 June
• Financial Support Application Deadline: Monday 7 June
• Financial Support Notification: Tuesday 8 June
• Priority Physical Registration Opens: Wednesday 9 June
• Ordinary Physical Registration Opens: Monday 13 June
• Reserved Accommodation Booking Deadline: Monday 13 June
• Adjoint School: Monday 5 to Friday 9 July
• Main Conference: Monday 12 to Friday 16 July

SUBMISSIONS

The following two types of submissions are accepted:

• Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

• Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at http://style.eptcs.org.

The submission link will soon be available on the ACT2021 web page: https://www.cl.cam.ac.uk/events/act2021

PROGRAMME COMMITTEE

Chair:

• Kohei Kishida, University of Illinois, Urbana-Champaign

Members:

• Richard Blute, University of Ottawa
• Spencer Breiner, NIST
• Daniel Cicala, University of New Haven
• Robin Cockett, University of Calgary
• Bob Coecke, Cambridge Quantum Computing
• Geoffrey Cruttwell, Mount Allison University
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Jonas Frey, Carnegie Mellon University
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Fabrizio Romano Genovese, Statebox
• Helle Hvid Hansen, University of Groningen
• Jules Hedges, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Alex Hoffnung, Bridgewater
• Martti Karvonen, University of Ottawa
• Kohei Kishida, University of Illinois, Urbana-Champaign (chair)
• Martha Lewis, University of Bristol
• Bert Lindenhovius, Johannes Kepler University Linz
• Ben MacAdam, University of Calgary
• Dan Marsden, University of Oxford
• Jade Master, University of California, Riverside
• Joe Moeller, NIST
• Koko Muroya, Kyoto University
• Simona Paoli, University of Leicester
• Daniela Petrisan, Université de Paris, IRIF
• Mehrnoosh Sadrzadeh, University College London
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
• David Spivak, MIT and Topos Institute
• Joshua Tan, University of Oxford
• Dmitry Vagner
• Jamie Vicary, University of Cambridge
• John van de Wetering, Radboud University Nijmegen
• Vladimir Zamdzhiev, Inria, LORIA, Université de Lorraine
• Maaike Zwart

Does antimatter fall up, or down?

Technically, we don’t know yet. The ALPHA-g experiment would have been the first to check this, making anti-hydrogen by trapping anti-protons and positrons in a long tube and seeing which way it falls. While they got most of their setup working, the LHC complex shut down before they could finish. It starts up again next month, so we should have our answer soon.

That said, for most theorists’ purposes, we absolutely do know: antimatter falls down. Antimatter is one of the cleanest examples of a prediction from pure theory that was confirmed by experiment. When Paul Dirac first tried to write down an equation that described electrons, he found the math forced him to add another particle with the opposite charge. With no such particle in sight, he speculated it could be the proton (this doesn’t work, they need the same mass), before Carl D. Anderson discovered the positron in 1932.

The same math that forced Dirac to add antimatter also tells us which way it falls. There’s a bit more involved, in the form of general relativity, but the recipe is pretty simple: we know how to take an equation like Dirac’s and add gravity to it, and we have enough practice doing it in different situations that we’re pretty sure it’s the right way to go. Pretty sure doesn’t mean 100% sure: talk to the right theorists, and you’ll probably find a proposal or two in which antimatter falls up instead of down. But they tend to be pretty weird proposals, from pretty weird theorists.

Ok, but if those theorists are that “weird”, that outside the mainstream, why does an experiment like ALPHA-g exist? Why does it happen at CERN, one of the flagship facilities for all of mainstream particle physics?

This gets at a misconception I occasionally hear from critics of the physics mainstream. They worry about groupthink among mainstream theorists, the physics community dismissing good ideas just because they’re not trendy (you may think I did that just now, for antigravity antimatter!) They expect this to result in a self-fulfilling prophecy where nobody tests ideas outside the mainstream, so they find no evidence for them, so they keep dismissing them.

The mistake of these critics is in assuming that what gets tested has anything to do with what theorists think is reasonable.

Theorists talk to experimentalists, sure. We motivate them, give them ideas and justification. But ultimately, people do experiments because they can do experiments. I watched a talk about the ALPHA experiment recently, and one thing that struck me was how so many different techniques play into it. They make antiprotons using a proton beam from the accelerator, slow them down with magnetic fields, and cool them with lasers. They trap their antihydrogen in an extremely precise vacuum, and confirm it’s there with particle detectors. The whole setup is a blend of cutting-edge accelerator physics and cutting-edge tricks for manipulating atoms. At its heart, ALPHA-g feels like its primary goal is to stress-test all of those tricks: to push the state of the art in a dozen experimental techniques in order to accomplish something remarkable.

And so even if the mainstream theorists don’t care, ALPHA will keep going. It will keep getting funding, it will keep getting visited by celebrities and inspiring pop fiction. Because enough people recognize that doing something difficult can be its own reward.

In my experience, this motivation applies to theorists too. Plenty of us will dismiss this or that proposal as unlikely or impossible. But give us a concrete calculation, something that lets us use one of our flashy theoretical techniques, and the tune changes. If we’re getting the chance to develop our tools, and get a paper out of it in the process, then sure, we’ll check your wacky claim. Why not?

I suspect critics of the mainstream would have a lot more success with this kind of pitch-based approach. If you can find a theorist who already has the right method, who’s developing and extending it and looking for interesting applications, then make your pitch: tell them how they can answer your question just by doing what they do best. They’ll think of it as a chance to disprove you, and you should let them, that’s the right attitude to take as a scientist anyway. It’ll work a lot better than accusing them of hogging the grant money.

“Black dwarf supernovae”. They sound quite dramatic! And indeed, they may be the last really exciting events in the Universe.

It’s too early to be sure. There could be plenty of things about astrophysics we don’t understand yet—and intelligent life may throw up surprises even in the very far future. But there’s a nice scenario here:

• M. E. Caplan, Black dwarf supernova in the far future, Monthly Notices of the Royal Astronomical Society 497 (2020), 4357–4362.

First, let me set the stage. What happens in the short run: say, the first 10²³ years or so?

For a while, galaxies will keep colliding. These collisions seem to destroy spiral galaxies: they fuse into bigger elliptical galaxies. We can already see this happening here and there—and our own Milky Way may have a near collision with Andromeda in only 3.85 billion years or so, well before the Sun becomes a red giant. If this happens, a bunch of new stars will be born from the shock waves due to colliding interstellar gas.

By 7 billion years we expect that Andromeda and the Milky Way will merge and form a large elliptical galaxy. Unfortunately, elliptical galaxies lack spiral arms, which seem to be a crucial part of the star formation process, so star formation may cease even before the raw materials run out.

Of course, no matter what happens, the birth of new stars must eventually cease, since there’s a limited amount of hydrogen, helium, and other stuff that can undergo fusion.

This means that all the stars will eventually burn out. The longest lived are the red dwarf stars, the smallest stars capable of supporting fusion today, with a mass about 0.08 times that of the Sun. These will run out of hydrogen about 10 trillion years from now, and not be able to burn heavier elements–so then they will slowly cool down.

(I’m deliberately ignoring what intelligent life may do. We can imagine civilizations that develop the ability to control stars, but it’s hard to predict what they’ll do so I’m leaving them out of this story.)

A star becomes a white dwarf—and eventually a black dwarf when it cools—if its core, made of highly compressed matter, has a mass less than 1.4 solar masses. In this case the core can be held up by the ‘electron degeneracy pressure’ caused by the Pauli exclusion principle, which works even at zero temperature. But if the core is heavier than this, it collapses! It becomes a neutron star if it’s between 1.4 and 2 solar masses, and a black hole if it’s more massive.

In about 100 trillion years, all normal star formation processes will have ceased, and the universe will have a population of stars consisting of about 55% white dwarfs, 45% brown dwarfs, and a smaller number of neutron stars and black holes. Star formation will continue at a very slow rate due to collisions between brown and/or white dwarfs.

The black holes will suck up some of the other stars they encounter. This is especially true for the big black holes at the galactic centers, which power radio galaxies if they swallow stars at a sufficiently rapid rate. But most of the stars, as well as interstellar gas and dust, will eventually be hurled into intergalactic space. This happens to a star whenever it accidentally reaches escape velocity through its random encounters with other stars. It’s a slow process, but computer simulations show that about 90% of the mass of the galaxies will eventually ‘boil off’ this way — while the rest becomes a big black hole.

How long will all this take? Well, the white dwarfs will cool to black dwarfs in about 100 quadrillion years, and the galaxies will boil away by about 10 quintillion years. Most planets will have already been knocked off their orbits by then, thanks to random disturbances which gradually take their toll over time. But any that are still orbiting stars will spiral in thanks to gravitational radiation in about 100 quintillion years.

I think the numbers are getting a bit silly. 100 quintillion is 10²⁰, and let’s use scientific notation from now on.

Then what? Well, in about 10²³ years the dead stars will actually boil off from the galactic clusters, not just the galaxies, so the clusters will disintegrate. At this point the cosmic background radiation will have cooled to about 10^-13 Kelvin, and most things will be at about that temperature unless proton decay or some other such process keeps them warmer.

Okay: so now we have a bunch of isolated black holes, neutron stars, and black dwarfs together with lone planets, asteroids, rocks, dust grains, molecules and atoms of gas, photons and neutrinos, all very close to absolute zero.

So what happens next?

We expect that black holes evaporate due to Hawking radiation: a solar-mass one should do so in 10⁶⁷ years, and a really big one, comparable to the mass of a galaxy, should take about 10⁹⁹ years. Small objects like planets and asteroids may eventually ‘sublimate’: that is, slowly dissipate by losing atoms due to random processes. I haven’t seen estimates on how long this will take. For larger objects, like neutron stars, this may take a very long time.

But I want to focus on stars lighter than 1.2 solar masses. As I mentioned, these will become white dwarfs held up by their electron degeneracy pressure, and by about 10¹⁷ years they will cool down to become very cold black dwarfs. Their cores will crystallize!

Then what? If a proton can decay into other particles, for example a positron and a neutral pion, black dwarfs may slowly shrink away to nothing due to this process, emitting particles as they fade away! Right now we know that the lifetime of the proton to decay via such processes is at least 10³² years. It could be much longer.

But suppose the proton is completely stable. Then what happens? In this scenario, a very slow process of nuclear fusion will slowly turn black dwarfs into iron! It’s called pycnonuclear fusion. The idea is that due to quantum tunneling, nuclei next to each other in the crystal lattice within a black dwarf will occasionally get ‘right on top of each other’ and fuse into heavier nucleus! Since iron-56 is the most stable nucleus, eventually iron will predominate.

Iron is more dense than lighter elements, so as this happens the black dwarf will shrink. It may eventually shrink down to being so dense that electron pressure will no longer hold it up. If this happens, the black dwarf will suddenly collapse, just like heavier stars. It will release a huge amount of energy and explode as gravitational potential energy gets converted into heat. This is a black dwarf supernova.

When will black dwarf supernovae first happen, assuming proton decay or some other unknown processes don’t destroy the black dwarfs first?

This is what Matt Caplan calculated:

We now consider the evolution of a white dwarf toward an iron black dwarf and the circumstances that result in collapse. Going beyond the simple order of magnitude estimates of Dyson (1979), we know pycnonuclear fusion rates are strongly dependent on density so they are greatest in the core of the black dwarf and slowest at the surface. Therefore, the internal structure of a black dwarf evolving toward collapse can be thought of as an astronomically slowly moving ‘burning’ front growing outward from the core toward the surface. This burning front grows outward much more slowly than any hydrodynamical or nuclear timescale, and the star remains at approximately zero temperature for this phase. Furthermore, in contrast to traditional thermonuclear stellar burning, the later reactions with higher Z parents take significantly longer due to the larger tunneling barriers for fusion.

Here “later reactions with higher Z parents” means fusion reactions involving heavier nuclei. The very last step, for example, is when two silicon nuclei fuse to form a nucleus of iron. In an ordinary star these later reactions happen much faster than those involving light nuclei, but for black dwarfs this pattern is reversed—and everything happens at ridiculously slow rate, at a temperature near absolute zero.

He estimates a black dwarf of 1.24 solar masses will collapse and go supernova after about 10¹⁶⁰⁰ years, when roughly half its mass has turned to iron.

Lighter ones will take much longer. A black dwarf of 1.16 solar masses could take 10³²⁰⁰⁰ years to go supernova.

These black dwarf supernovae could be the last really energetic events in the Universe.

It’s downright scary to think how far apart these black dwarfs will be when they explode. As I mentioned, galaxies and clusters will have long since have boiled away, so every black dwarf will be completely alone in the depths of space. Distances between them will be doubling every 12 billion years according to the current standard model of cosmology, the ΛCDM model. But 12 billion years is peanuts compared to the time scales I’m talking about now!

So, by the time black dwarfs start to explode, the distances between these stars will be expanded by a factor of roughly

compared to their distances today. That’s a very rough estimate, but it means that each black dwarf supernova will be living in its own separate world.

 I spent three afternoons this week attending a NSF workshop on Quantum Engineering Infrastructure.  This was based in part on the perceived critical need for shared infrastructure (materials growth, lithographic patterning, deposition, etching, characterization) across large swaths of experimental quantum information sciences, and the fact that the NSF already runs the NNCI, which was the successor of the NNIN.  There will end up being a report generated as a result of the workshop, hopefully steering future efforts.  (I was invited because of this post.)

The workshop was very informative, touching on platforms including superconducting qubits, trapped ions, photonic devices including color centers in diamond/SiC, topological materials, and spin qubits in semiconductors.  Some key themes emerged:

• There are many possible platforms out there for quantum information science, and all of them will require very serious materials development to be ready for prime time.  People forget that our command of silicon comes after thousands of person-years worth of research and process development.  Essentially every platform is in its infancy compared to that.  
• There is clearly a tension between the need for exploratory research, trying new processes at the onesy-twosy level, and the requirements for work at larger scale, which needs dedicated process expertise and control at a level not typically possible in a shared university facility.  Everyone also knows that progress is automatically slow if people have to travel off-site to some user facility to do part of their processing.  Some places are well situated - MIT, for example, has an exploratory fab facility here, and a dedicated 200 mm substrate superconducting circuit fab at Lincoln Labs.  Life is extra complicated when running an unusual process in some tool like a PECVD system or an etcher can "season" the gadget, leaving an imprint on subsequent process runs.
• Whoever really figures out how to do wafer-scale heteroepitaxy of single-crystal diamond will either become incredibly rich or will be assassinated by DeBeers.  
• Fostering a healthy relationship between industrial materials growers and academic researchers would be very important.  Industrial expertise can be fantastic, but there is not necessarily much economic incentive to work closely with academia compared with large-scale commercial pressures.  There may be a key role for government encouragement or subsidy.  
• It's going to be increasingly challenging for new faculty to get started in some research topics at universities - the detailed process knowhow and the need to buildup expertise can be expensive and slow to acquire compared to the timescale of, e.g., promotion to tenure.  An improved network that supports, curates, and communicates process development expertise might be extremely helpful.

Last week I got an email from Dina Katabi, my former MIT colleague, asking me to call her urgently. Am I in trouble? For what, though?? I haven’t even worked at MIT for five years!

Luckily, Dina only wanted to tell me that I’d been selected to receive the 2020 ACM Prize in Computing, a mid-career award founded in 2007 that comes with $250,000 from Infosys. Not the Turing Award but I’d happily take it! And I could even look back on 2020 fondly for something.

I was utterly humbled to see the list of past ACM Prize recipients, which includes amazing computer scientists I’ve been privileged to know and learn from (like Jon Kleinberg, Sanjeev Arora, and Dan Boneh) and others who I’ve admired from afar (like Daphne Koller, Jeff Dean and Sanjay Ghemawat of Google MapReduce, and David Silver of AlphaGo and AlphaZero).

I was even more humbled, later, to read my prize citation, which focuses on four things:

1. The theoretical foundations of the sampling-based quantum supremacy experiments now being carried out (and in particular, my and Alex Arkhipov’s 2011 paper on BosonSampling);
2. My and Avi Wigderson’s 2008 paper on the algebrization barrier in complexity theory;
3. Work on the limitations of quantum computers (in particular, the 2002 quantum lower bound for the collision problem); and
4. Public outreach about quantum computing, including through QCSD, popular talks and articles, and this blog.

I don’t know if I’m worthy of such a prize—but I know that if I am, then it’s mainly for work I did between roughly 2001 and 2012. This honor inspires me to want to be more like I was back then, when I was driven, non-jaded, and obsessed with figuring out the contours of BQP and efficient computation in the physical universe. It makes me want to justify the ACM’s faith in me.

I’m grateful to the committee and nominators, and more broadly, to the whole quantum computing and theoretical computer science communities—which I “joined” in some sense around age 16, and which were the first communities where I ever felt like I belonged. I’m grateful to the mentors who made me what I am, especially Chris Lynch, Bart Selman, Lov Grover, Umesh Vazirani, Avi Wigderson, and (if he’ll allow me to include him) John Preskill. I’m grateful to the slightly older quantum computer scientists who I looked up to and tried to emulate, like Dorit Aharonov, Andris Ambainis, Ronald de Wolf, and John Watrous. I’m grateful to my wonderful colleagues at UT Austin, in the CS department and beyond. I’m grateful to my students and postdocs, the pride of my professional life. I’m grateful, of course, to my wife, parents, and kids.

By coincidence, my last post was also about prizes to theoretical computer scientists—in that case, two prizes that attracted controversy because of the recipient’s (or would-be recipient’s) political actions or views. It would understate matters to point out that not everyone has always agreed with everything I’ve said on this blog. I’m ridiculously lucky, and I know it, that even living through this polarized and tumultuous era, I never felt forced to choose between academic success and the freedom to speak my conscience in public under my real name. If there’s been one constant in my public stands, I’d like to think that—inspired by memories of my own years as an unknown, awkward, self-conscious teenager—it’s been my determination to nurture and protect talented young scientists, whatever they look like and wherever they come from. And I’ve tried to live up to that ideal in real life, and I welcome anyone’s scrutiny as to how well I’ve done.

What should I do with the prize money? I confess that my first instinct was to donate it, in its entirety, to some suitable charity—specifically, something that would make all the strangers who’ve attacked me on Twitter, Reddit, and so forth over the years realize that I’m fundamentally a good person. But I was talked out of this plan by my family, who pointed out that
(1) in all likelihood, nothing will make online strangers stop hating me,
(2) in any case this seems like a poor basis for making decisions, and
(3) if I really want to give others a say in what to do with the winnings, then why not everyone who’s stood by me and supported me?

So, beloved commenters! Please mention your favorite charitable causes below, especially weird ones that I wouldn’t have heard of otherwise. If I support their values, I’ll make a small donation from my prize winnings. Or a larger donation, especially if you donate yourself and challenge me to match. Whatever’s left after I get tired of donating will probably go to my kids’ college fund.

Update: And by an amusing coincidence, today is apparently “World Quantum Day”! I hope your Quantum Day is as pleasant as mine (and stable and coherent).

Over the last few weeks—and the last few decades—I have had many conversations about all the things that are way more important to being a successful astrophysicist than facility with electromagnetism and quantum mechanics: There's writing, and mentoring, and project design, and reading, and visualization, and so on. Today I fantasized about a (very long) book entitled The Practice of Astrophysics that covers all of these things.

I gave two talks in Latham Boyle and Kirill Krasnov’s Perimeter Institute workshop Octonions and the Standard Model.

The first talk was on Monday April 5th at noon Eastern Time. The second was exactly one week later, on Monday April 12th at noon Eastern Time.

Here they are:

• Can we understand the Standard Model? (video, slides)

Abstract. 40 years trying to go beyond the Standard Model hasn’t yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions. We explain both these approaches, and how to reconcile them.

The second is on Monday April 12th at noon Eastern Time:

• Can we understand the Standard Model using octonions? (video, slides)

Abstract. Dubois-Violette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices of octonions. After an introduction to the physics of Jordan algebras, we ponder the meaning of their construction. For example, it implies that the Standard Model gauge group consists of the symmetries of an octonionic qutrit that restrict to symmetries of an octonionic qubit and preserve all the structure arising from a choice of unit imaginary octonion. It also sheds light on why the Standard Model gauge group acts on 10d Euclidean space, or Minkowski spacetime, while preserving a 4+6 splitting.

You can see all the slides and videos and also some articles with more details here.

I gave two talks in Latham Boyle and Kirill Krasnov’s Perimeter Institute workshop Octonions and the Standard Model.

The first talk was on Monday April 5th at noon Eastern Time. The second was exactly one week later, on Monday April 12th at noon Eastern Time.

Here they are…

• Can we understand the Standard Model? (video, slides)

Abstract. 40 years trying to go beyond the Standard Model hasn’t yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions. We explain both these approaches, and how to reconcile them.

• Can we understand the Standard Model using octonions? (video, slides)

Abstract. Dubois-Violette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices of octonions. After an introduction to the physics of Jordan algebras, we ponder the meaning of their construction. For example, it implies that the Standard Model gauge group consists of the symmetries of an octonionic qutrit that restrict to symmetries of an octonionic qubit and preserve all the structure arising from a choice of unit imaginary octonion. It also sheds light on why the Standard Model gauge group acts on 10d Euclidean space, or Minkowski spacetime, while preserving a 4+6 splitting.

You can see all the slides and videos and also some articles with more details here.

Adrian Price-Whelan (Flatiron) and I encountered an interesting conceptual point today in our distance estimation project: When you are doing cross-validation to set your hyper-parameters (a regularization strength in this case), what do you use as your validation scalar? That is, what are you optimizing? We started by naively optimizing the cost function, which is something like a weighted L2 of the residual and an L2 of the parameters. But then we switched from the cost function to just the data part (not the regularization part) of the cost function, and everything changed! The point is duh, actually, when you think about it from a Bayesian perspective: You want to improve the likelihood not the posterior pdf. That's another nice point for my non-existent paper on the difference between a likelihood and a posterior pdf. It also shows that, in general, the data and the regularization will be at odds.

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

—

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

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

What’s a physics law, anyway? Crudely, physics is a strategy for making predictions about the behavior of physical objects, based on a set of equations and a conceptual framework for using those equations. Sometimes we refer to the equations as laws; sometimes parts of the conceptual framework are referred to that way.

But that story has layers. Physics has an underlying conceptual foundation, which includes the pillar of quantum physics and its view of reality, and the pillar of Einstein’s relativity and its view of space and time. (There are other pillars too, such as those of statistical mechanics, but let me not complicate the story now.) That foundation supports many research areas of physics. Within particle physics itself, these two pillars are combined into a more detailed framework, with concepts and equations that go by the name of “quantum effective field theory” (“QEFT”). But QEFT is still very general; this framework can describe an enormous number of possible universes, most with completely different particles and forces from the ones we have in our own universe. We can start making predictions for real-world experiments only when we put the electron, the muon, the photon, and all the other familiar particles and forces into our equations, building up a specific example of a QEFT known as “The Standard Model of particle physics.”

All along the way there are equations and rules that you might call “laws.” They too come in layers. The Standard Model itself, as a specific QEFT, has few high-level laws: there are no principles telling us why quarks exist, why there is one type of photon rather than two, or why the weak nuclear force is so weak. The few laws it does have are mostly low-level, true of our universe but not essential to it.

I’m bringing attention to these layers because an experiment might cause a problem for one layer but not another. I think you could only fairly suggest that “physics is broken” if data were putting a foundational pillar of the entire field into question. And to say “the laws of physics have been violated”, emphasis on the word “the“, is a bit melodramatic if the only thing that’s been violated is a low-level, dispensable law.

—

Has physics, as a whole, ever broken? You could argue that Newton’s 17th century foundation, which underpinned the next two centuries of physics, broke at the turn of the 20th century. Just after 1900, Newton-style equations had to be replaced by equations of a substantially different type; the ways physicists used the equations changed, and the concepts, the language, and even the goals of physics changed. For instance, in Newtonian physics, you can predict the outcome of any experiment, at least in principle; in post-Newtonian quantum physics, you often can only predict the probability for one or another outcome, even in principle. And in Newtonian physics we all agree what time it is; in Einsteinian physics, different observers experience time differently and there is no universal clock that we all agree on. These were immense changes in the foundation of the field.

Conversely, you could also argue that physics didn’t break; it was just remodeled and expanded. No one who’d been studying steam engines or wind erosion or electrical circuit diagrams had to throw out their books and start again from scratch. In fact this “broken” Newtonian physics is still taught in physics classes, and many physicists and engineers never use anything else. If you’re studying the physics of weather, or building a bridge, Newtonian physics is just fine. The fact that Newton-style equations are an incomplete description of the world — that there are phenomena they can’t describe properly — doesn’t invalidate them when they’re applied within their wheelhouse.

No matter which argument you prefer, it’s hard to see how to justify the phrase “physics is broken” without a profound revolution that overthrows foundational concepts. It’s rare for a serious threat to foundations to arise suddenly, because few experiments can single-handedly put fundamental principles at risk. [The infamous case of the “faster-than-light neutrinos” provides an exception. Had that experiment been correct, it would have invalidated Einstein’s relativity principles. But few of us were surprised when a glaring error turned up.]

—

In the Standard Model, the electron, muon and tau particles (known as the “charged leptons”) are all identical except for their masses. (More fundamentally, they have different interactions with the Higgs field, from which their rest masses arise.) This almost-identity is sometimes stated as a “principle of lepton universality.” Oh, wow, a principle — a law! But here’s the thing. Some principles are enormously important; the principles of Einsteinian relativity determine how cause and effect work in our universe, and you can’t drop them without running into big paradoxes. Other principles are weak, and could easily be discarded without making a mess of any other part of physics. The principle of lepton universality is one of these. In fact, if you extend the Standard Model by adding new particles to its equations, it can be difficult to avoid ruining this fragile principle. [In a sense, the Higgs field has already violated the principle, but we don’t hold that against it.]

All the fuss is about a new experimental result which confirms an older one and slightly disagrees with the latest theoretical predictions, which are made using the Standard Model’s equations. What could be the cause of the discrepancy? One possibility is that it arises from a previously unknown difference between muons and electrons — from a violation of the principle of lepton universality. For those who live and breathe particle physics, breaking lepton universality would be a big deal; there’d be lots of adventure in trying to figure out which of the many possible extensions of the Standard Model could actually explain what broke this law. That’s why the scientists involved sound so excited.

But the failure of lepton universality wouldn’t come as a huge surprise. From certain points of view, the surprise is that the principle has survived this long! Since this low-level law is easily violated, its demise may not lead us to a profound new understanding of the world. It’s way too early for headlines that argue that what’s at stake is the existence of “forms of matter and energy vital to the nature and evolution of the cosmos.” No one can say how much is at stake; it might be a lot, or just a little.

In particular, there’s absolutely no evidence that physics is broken, or even that particle physics is broken. The pillars of physics and QEFT are not (yet) threatened. Even to say that “the Standard Model might be broken” seems a bit melodramatic to me. Does adding a new wing to a house require “breaking” the house? Typically you can still live in the place while it’s being extended. The Standard Model’s many successes suggest that it might survive largely intact as a recognizable part of a larger, more complete set of equations.

In any case, right now it’s still too early to say anything so loudly. The apparent discrepancy may not survive the heavy scrutiny it is coming under. There’s plenty of controversy about the theoretical prediction for muon magnetism; the required calculation is extraordinarily complex, elaborate and difficult.

So, from my perspective, the headlines of the past week are way over the top. The idea that a single measurement of the muon’s magnetism could “shake physics to its core“, as claimed in another headline I happened upon, is amusing at best. Physics, and its older subdisciplines, have over time become very difficult to break, or even shake. That’s the way it should be, when science is working properly. And that’s why we can safely base the modern global economy on scientific knowledge; it’s unlikely that a single surprise could instantly invalidate large chunks of its foundation.

Some readers may view the extreme, click-baiting headlines as harmless. Maybe I’m overly concerned about them. But don’t they implicitly suggest that one day we will suddenly find physics “upended”, and in need of a complete top-to-bottom overhaul? To imply physics can “break” so easily makes a mockery of science’s strengths, and obscures the process by which scientific knowledge is obtained. And how can it be good to claim “physics is broken” and “the laws of physics have been broken” over and over and over again, in stories that almost never merit that level of hype and eventually turn out to have been much ado about nada? The constant manufacturing of scientific crisis cannot possibly be lost on readers, who I suspect are becoming increasingly jaded. At some point readers may become as skeptical of science journalism, and the science it describes, as they are of advertising; it’s all lies, so caveat emptor. That’s not where we want our society to be. As we are seeing in spades during the current pandemic, there can be serious consequences when “TRUST IN SCIENCE IS BROKEN!!!”

—

A final footnote: Ironically, the Standard Model itself poses one of the biggest threats to the framework of QEFT. The discovery of the Higgs boson and nothing else (so far) at the Large Hadron Collider poses a conceptual challenge — the “naturalness” problem. There’s no sharp paradox, which is why I can’t promise you that the framework of QEFT will someday break if it isn’t resolved. But the breakdown of lepton universality might someday help solve the naturalness problem, by requiring a more “natural” extension of the Standard Model, and thus might actually save QEFT instead of “breaking” it.

Sarah Blunt (Caltech) crashed Stars & Exoplanets Meeting today. She told us about her ambitious, community-built orbitize project, and also results on a mysterious binary-star system, HD 104304. This is a directly-imaged binary, but when they took radial-velocity measurements, the mass of the primary is way too high for its color and luminosity. The beauty of orbitize is that it can take heterogeneous data, and it uses brute-force importance sampling (like my one true love The Joker), so she can deal with very non-trivial likelihood functions and low signal-to-noise, sparse data.

The crowd had many reactions, one of which is that probably the main issue is that ESA Gaia is giving a wrong parallax. That's a boring explanation, but it opens a nice question of using the data to infer or predict a distance, which is old-school fundamental astronomy.

We can wait a while to explore the Universe, but we shouldn’t wait too long. If the Universe continues its accelerating expansion as predicted by the usual model of cosmology, it will eventually expand by a factor of 2 every 12 billion years. So if we wait too long, we can’t ever reach a distant galaxy.

In fact, after 150 billion years, all galaxies outside our Local Group will become completely inaccessible, in principle by any form of transportation not faster than light!

For an explanation, read this:

• Toby Ord, The edges of our Universe.

This is where I got the table.

150 billion years sounds like a long time, but the smallest stars powered by fusion—the red dwarf stars, which are very plentiful—are expected to last much longer: about 10 trillion years!  So, we can imagine a technologically advanced civilization that has managed to spread over the Local Group and live near red dwarf stars, which eventually regrets that it has waited too long to expand through more of the Universe.  

The Local Group is a collection of roughly 50 nearby galaxies containing about 2 trillion stars, so there’s certainly plenty to do here. It’s held together by gravity, so it won’t get stretched out by the expansion of the Universe—not, at least, until its stars slowly “boil off” due to some randomly picking up high speeds. But will happen much, much later: more than 10 quintillion years, that is, 10¹⁹ years.

For more, see this article of mine:

• The end of the Universe.

Oded Goldreich is a theoretical computer scientist at the Weizmann Institute in Rehovot, Israel. He’s best known for helping to lay the rigorous foundations of cryptography in the 1980s, through seminal results like the Goldreich-Levin Theorem (every one-way function can be modified to have a hard-core predicate), the Goldreich-Goldwasser-Micali Theorem (every pseudorandom generator can be made into a pseudorandom function), and the Goldreich-Micali-Wigderson protocol for secure multi-party computation. I first met Oded more than 20 years ago, when he lectured at a summer school at the Institute for Advanced Study in Princeton, barefoot and wearing a tank top and what looked like pajama pants. It was a bracing introduction to complexity-theoretic cryptography. Since then, I’ve interacted with Oded from time to time, partly around his firm belief that quantum computing is impossible.

Last month a committee in Israel voted to award Goldreich the Israel Prize (roughly analogous to the US National Medal of Science), for which I’d say Goldreich had been a plausible candidate for decades. But alas, Yoav Gallant, Netanyahu’s Education Minister, then rather non-gallantly blocked the award, solely because he objected to Goldreich’s far-left political views (and apparently because of various statements Goldreich signed, including in support of a boycott of Ariel University, which is in the West Bank). The case went all the way to the Israeli Supreme Court (!), which ruled two days ago in Gallant’s favor: he gets to “delay” the award to investigate the matter further, and in the meantime has apparently sent out invitations for an award ceremony next week that doesn’t include Goldreich. Some are now calling for the other winners to boycott the prize in solidarity until this is righted.

I doubt readers of this blog need convincing that this is a travesty and an embarrassment, a shanda, for the Netanyahu government itself. That I disagree with Goldreich’s far-left views (or might disagree, if I knew in any detail what they were) is totally immaterial to that judgment. In my opinion, not even Goldreich’s belief in the impossibility of quantum computers should affect his eligibility for the prize.

Maybe it would be better to say that, as far as his academic colleagues in Israel and beyond are concerned, Goldreich has won the Israel Prize; it’s only some irrelevant external agent who’s blocking his receipt of it. Ironically, though, among Goldreich’s many heterodox beliefs is a total rejection of the value of scientific prizes (although Goldreich has also said he wouldn’t refuse the Israel Prize if offered it!).

In unrelated news, the 2020 Turing Award has been given to Al Aho and Jeff Ullman. Aho and Ullman have both been celebrated leaders in CS for half a century, having laid many of the foundations of formal languages and compilers, and having coauthored one of CS’s defining textbooks with John Hopcroft (who already received a different Turing Award).

But again there’s a controversy. Apparently, in 2011, Ullman wrote to an Iranian student who wanted to work with him, saying that as “a matter of principle,” he would not accept Iranian students until the Iranian government recognized Israel. Maybe I should say that I, like Ullman, am both a Jew and a Zionist, but I find it hard to imagine the state of mind that would cause me to hold some hapless student responsible for the misdeeds of their birth-country’s government. Ironically, this is a mirror-image of the tactics that the BDS movement has wielded against Israeli academics. Unlike Goldreich, though, Ullman seems to have gone beyond merely expressing his beliefs, actually turning them into a one-man foreign policy.

I’m proud of the Iranian students I’ve mentored and hope to mentor more. While I don’t think this issue should affect Ullman’s Turing Award (and I haven’t seen anyone claim that it should), I do think it’s appropriate to use the occasion to express our opposition to all forms of discrimination. I fully endorse Shafi Goldwasser’s response in her capacity as Director of the Simons Institute for Theory of Computing in Berkeley:

As a senior member of the computer science community and an American-Israeli, I stand with our Iranian students and scholars and outright reject any notion by which admission, support, or promotion of individuals in academic settings should be impeded by national origin or politics. Individuals should not be conflated with the countries or institutions they come from. Statements and actions to the contrary have no place in our computer science community. Anyone experiencing such behavior will find a committed ally in me.

As for Al Aho? I knew him fifteen years ago, when he became interested in quantum computing, in part due to his then-student Krysta Svore (who’s now the head of Microsoft’s quantum computing efforts). Al struck me as not only a famous scientist but a gentleman who radiated kindness everywhere. I’m not aware of any controversies he’s been involved in and never heard anyone say a bad word about him.

Anyway, this seems like a good occasion to recognize some foundational achievements in computer science, as well as the complex human beings who produce them!

Yesterday, Fermilab’s Muon g-2 experiment announced a new measurement of the magnetic moment of the muon, a number which describes how muons interact with magnetic fields. For what might seem like a small technical detail, physicists have been very excited about this measurement because it’s a small technical detail that the Standard Model seems to get wrong, making it a potential hint of new undiscovered particles. Quanta magazine has a great piece on the announcement, which explains more than I will here, but the upshot is that there are two different calculations on the market that attempt to predict the magnetic moment of the muon. One of them, using older methods, disagrees with the experiment. The other, with a new approach, agrees. The question then becomes, which calculation was wrong? And why?

What does it mean for a prediction to match an experimental result? The simple, wrong, answer is that the numbers must be equal: if you predict “3”, the experiment has to measure “3”. The reason why this is wrong is that in practice, every experiment and every prediction has some uncertainty. If you’ve taken a college physics class, you’ve run into this kind of uncertainty in one of its simplest forms, measurement uncertainty. Measure with a ruler, and you can only confidently measure down to the smallest divisions on the ruler. If you measure 3cm, but your ruler has ticks only down to a millimeter, then what you’re measuring might be as large as 3.1cm or as small as 2.9 cm. You just don’t know.

This uncertainty doesn’t mean you throw up your hands and give up. Instead, you estimate the effect it can have. You report, not a measurement of 3cm, but of 3cm plus or minus 1mm. If the prediction was 2.9cm, then you’re fine: it falls within your measurement uncertainty.

Measurements aren’t the only thing that can be uncertain. Predictions have uncertainty too, theoretical uncertainty. Sometimes, this comes from uncertainty on a previous measurement: if you make a prediction based on that experiment that measured 3cm plus or minus 1mm, you have to take that plus or minus into account and estimate its effect (we call this propagation of errors). Sometimes, the uncertainty comes instead from an approximation you’re making. In particle physics, we sometimes approximate interactions between different particles with diagrams, beginning with the simplest diagrams and adding on more complicated ones as we go. To estimate the uncertainty there, we estimate the size of the diagrams we left out, the more complicated ones we haven’t calculated yet. Other times, that approximation doesn’t work, and we need to use a different approximation, treating space and time as a finite grid where we can do computer simulations. In that case, you can estimate your uncertainty based on how small you made your grid. The new approach to predicting the muon magnetic moment uses that kind of approximation.

There’s a common thread in all of these uncertainty estimates: you don’t expect to be too far off on average. Your measurements won’t be perfect, but they won’t all be screwed up in the same way either: chances are, they will randomly be a little below or a little above the truth. Your calculations are similar: whether you’re ignoring complicated particle physics diagrams or the spacing in a simulated grid, you can treat the difference as something small and random. That randomness means you can use statistics to talk about your errors: you have statistical uncertainty. When you have statistical uncertainty, you can estimate, not just how far off you might get, but how likely it is you ended up that far off. In particle physics, we have very strict standards for this kind of thing: to call something new a discovery, we demand that it is so unlikely that it would only show up randomly under the old theory roughly one in a million times. The muon magnetic moment isn’t quite up to our standards for a discovery yet, but the new measurement brought it closer.

The two dueling predictions for the muon’s magnetic moment both estimate some amount of statistical uncertainty. It’s possible that the two calculations just disagree due to chance, and that better measurements or a tighter simulation grid would make them agree. Given their estimates, though, that’s unlikely. That takes us from the realm of theoretical uncertainty, and into uncertainty about the theoretical. The two calculations use very different approaches. The new calculation tries to compute things from first principles, using the Standard Model directly. The risk is that such a calculation needs to make assumptions, ignoring some effects that are too difficult to calculate, and one of those assumptions may be wrong. The older calculation is based more on experimental results, using different experiments to estimate effects that are hard to calculate but that should be similar between different situations. The risk is that the situations may be less similar than expected, their assumptions breaking down in a way that the bottom-up calculation could catch.

None of these risks are easy to estimate. They’re “unknown unknowns”, or rather, “uncertain uncertainties”. And until some of them are resolved, it won’t be clear whether Fermilab’s new measurement is a sign of undiscovered particles, or just a (challenging!) confirmation of the Standard Model.

April 7, 2021 was like a good TV episode: high-speed action, plot twists, and a cliffhanger ending. We now know that the strength of the little magnet inside the muon is described by the g-factor: 

g = 2.00233184122(82).

Any measurement of basic properties of matter is priceless, especially when it come with this incredible precision.  But for a particle physicist the main source of excitement is that this result could herald the breakdown of the Standard Model. The point is that the g-factor or the magnetic moment of an elementary particle can be calculated theoretically to a very good accuracy. Last year, the white paper of the Muon g−2 Theory Initiative came up with the consensus value for the Standard Model prediction 

                                                                      g = 2.00233183620(86), 

which is significantly smaller than the experimental value.  The discrepancy is estimated at 4.2 sigma, assuming the theoretical error is Gaussian and combining the errors in quadrature. 

As usual, when we see an experiment and the Standard Model disagree, these 3 things come to mind first

1.  Statistical fluctuation. 
2.  Flawed theory prediction. 
3.  Experimental screw-up.   

  But let us assume for a moment that the white paper value is correct. This would be huge, as it would mean that the Standard Model does not fully capture how muons interact with light. The correct interaction Lagrangian would have to be (pardon my Greek)

The first term is the renormalizable minimal coupling present in the Standard Model, which gives the Coulomb force and all the usual electromagnetic phenomena. The second term is called the magnetic dipole. It leads to a small shift of the muon g-factor, so as to explain the Brookhaven and Fermilab measurements.  This is a non-renormalizable interaction, and so it must be an effective description of virtual effects of some new particle from beyond the Standard Model. Theorists have invented countless models for this particle in order to address the old Brookhaven measurement, and the Fermilab update changes little in this enterprise. I will write about it another time.  For now, let us just crunch some numbers to highlight one general feature. Even though the scale suppressing the effective dipole operator is in the EeV range, there are indications that the culprit particle is much lighter than that. First, electroweak gauge invariance forces it to be less than ~100 TeV in a rather model-independent way.  Next, in many models contributions to muon g-2 come with the chiral suppression proportional to the muon mass. Moreover, they typically appear at one loop, so the operator will pick up a loop suppression factor unless the new particle is strongly coupled.  The same dipole operator as above can be more suggestively recast as  

The scale 300 GeV appearing in the denominator indicates that the new particle should be around the corner!  Indeed, the discrepancy between the theory and experiment is larger than the contribution of the W and Z bosons to the muon g-2, so it seems logical to put the new particle near the electroweak scale. That's why the stakes of the April 7 Fermilab announcement are so enormous. If the gap between the Standard Model and experiment is real, the new particles and forces responsible for it should be within reach of the present or near-future colliders. This would open a new experimental era that is almost too beautiful to imagine. And for theorists, it would bring new pressing questions about who ordered it. 

Along with Zeila Zanolli, tomorrow (Friday April 9) I will be serving as a moderator for a "fireside chat" about Majorana fermions being given by Sergey Frolov and Vincent Mourik.   This is being done as a zoom webinar (registration info here), at 11am EDT.   Should be an interesting discussion - about 20 minutes of presentation followed by q & a.  

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This post is about the result of the Muon g-2 experiment announced today at Fermilab. It has an admittedly bad title: the Standard Model has not cracked in any way, it is still a correct theory that decently describes interactions of known elementary particles at the energies we checked. Yet, maybe we finally found an observable that is not completely described by the Standard Model. Its theoretically computed value needs an additional component to agree with the newly reported experimental value and this component might well be New Physics! What is this observable?

This observable is the value of the anomalous magnetic moment of the muon. The muon, an elementary particle (a lepton), is a close cousin of an electron. It has very similar properties to the electron, but is about 200 times heavier and is unstable — it only lives for about two microseconds. We don’t yet know why Nature chose to create two similar copies of an electron: muon and tau-lepton. But we can study their properties to find out.

Just like an electron, the muon has spin, which makes it susceptible to the effects of the magnetic field, which is characterized by its magnetic moment. The magnetic moment tells us how the muon reacts to its presence: think of the compass needle as a classical analogy. Over a century ago, brilliant physicist Paul Dirac predicted the value of an electron’s magnetic moment, which is directly applicable to muon as well. This prediction involved a parameter, which he called g, from the gyromagnetic ratio or a g-factor. Dirac’s prediction was that, for an electron (and a muon), it is supposed to be exactly g=2. This was one of the predictions that allowed experimentalists to test the validity of Dirac’s theory which eventually led to its triumph.

With further development of quantum field theories, it was realized that g is not exactly two. The effects of virtual particles lead to the effect that the photon of the magnetic field probing the muon could instead hit those virtual particles instead, potentially changing the value of the g-factor. Now, dealing with virtual particles could be tricky in theoretical computations, as their effects lead to unphysical infinities that need to be absorbed in the definitions of muon’s mass, charge, and the wave-function. But the leading effect of such particles — assuming only the Standard Model particles — turns out to be finite! Julian Schwinger showed that in his 1948 paper. This result was so influential at the time that is literally engraved on his tombstone! This paved the way to compute the quantum radiative corrections to muon’s magnetic moment. Since the effect of such radiative corrections is to change the magnetic moment, they lead to the deviation to the Dirac’s theory prediction and lead to the non-zero value of a = (g-2)/2, which is conventionally referred to as the anomalous magnetic moment. This is precisely what Muon g-2 collaboration measured very precisely.

Why is it interesting? The thing is that among the known virtual particles there could also be new, unknown particles. If those particles interact with the photons, they could also affect the numerical value of the anomalous magnetic moment. So the idea is simple: compute it with as much precision as possible and then compare it to the measurement that is done with the best precision possible. This is precisely what was done.

Easily said but not so easily done. Precise predictions of the anomalous magnetic moment involved computations of thousands of Feynman diagrams and evaluation of contributions that can not be computed by expanding in some small quantity (aka non-perturbative effects). There are many theoretical methods used to compute those, including numerical computations in lattice QCD. But there is now an agreement among the theorists on the anomalous magnetic moment of the muon: a = 0.00116591810(43) (see here for a paper). This number is known with astonishing precision, which is indicated by the bracketed numbers.

The experimental analysis is incredibly hard. Since muons decay, the measurement of their properties is not trivial. Muons are produced in the decays of other particles, called pions, that are created at Fermilab by smashing accelerated protons into targets. Once created, they are directed into a storage ring where they decay in a magnetic field giving out their spin information. The storage ring contains about 10,000 muons at the time going around the ring. To make the measurement, it is important to know the magnetic field in which those muons are moving with incredible precision. There is also an electric field that makes the muons going around the ring, whose effect is carefully removed by choosing how fast the muons fly. If all those (and other) effects are not accounted for, they would affect the result of the measurement! Their combined effect is usually referred to as systematic uncertainties. Most of the work done by the Muon g-2 collaboration at Fermilab was to reduce such effects, which eventually led to the acceptable level of those systematic uncertainties.

And here is the result (drum roll): the anomalous magnetic moment measured by the Muon g-2 collaboration is a=0.00116592061(41).

Ok, what does it all mean? First of all, the result is seemingly only ever so slightly different from the theoretical prediction. But it is not. What is more interesting is that if one combines this new result with the old result from the Brookhaven National Lab, one gets a very significant difference between the theoretical predictions and a combined result of two measurements: it is about 4.2 sigma. Sigmas measure the statistical significance of the result, 4.2 sigma means that the chance that the theoretical and experimental results agree — which is possible due to statistical fluctuations — is about 1 out 40,000! This is incredible!

The result might mean that there are particles that are not described by the Standard Model and the New Physics could be just around the corner! Come back here for more discoveries!

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Update (April 5, 2021): So it turns out that Adam Chalcraft and Michael Greene already proved the essential result of this post back in 1994 (hat tip to commenter Dylan). Not terribly surprising in retrospect!

My son Daniel had his fourth birthday a couple weeks ago. For a present, he got an electric train set. (For completeness—and since the details of the train set will be rather important to the post—it’s called “WESPREX Create a Dinosaur Track”, but this is not an ad and I’m not getting a kickback for it.)

As you can see, the main feature of this set is a Y-shaped junction, which has a flap that can control which direction the train goes. The logic is as follows:

• If the train is coming up from the “bottom” of the Y, then it continues to either the left arm or the right arm, depending on where the flap is. It leaves the flap as it was.

• If the train is coming down the left or right arms of the Y, then it continues to the bottom of the Y, pushing the flap out of its way if it’s in the way. (Thus, if the train were ever to return to this Y-junction coming up from the bottom, not having passed the junction in the interim, it would necessarily go to the same arm, left or right, that it came down from.)

The train set also comes with bridges and tunnels; thus, there’s no restriction of planarity. Finally, the train set comes with little gadgets that can reverse the train’s direction, sending it back in the direction that it came from:

These gadgets don’t seem particularly important, though, since we could always replace them if we wanted by a Y-junction together with a loop.

Notice that, at each Y-junction, the position of the flap stores one bit of internal state, and that the train can both “read” and “write” these bits as it moves around. Thus, a question naturally arises: can this train set do any nontrivial computations? If there are n Y-junctions, then can it cycle through exp(n) different states? Could it even solve PSPACE-complete problems, if we let it run for exponential time? (For a very different example of a model-train-like system that, as it turns out, is able to express PSPACE-complete problems, see this recent paper by Erik Demaine et al.)

Whatever the answers regarding Daniel’s train set, I knew immediately on watching the thing go that I’d have to write a “paperlet” on the problem and publish it on my blog (no, I don’t inflict such things on journals!). Today’s post constitutes my third “paperlet,” on the general theme of a discrete dynamical system that someone showed me in real life (e.g. in a children’s toy or in biology) having more structure and regularity than one might naïvely expect. My first such paperlet, from 2014, was on a 1960s toy called the Digi-Comp II; my second, from 2016, was on DNA strings acted on by recombinase (OK, that one was associated with a paper in Science, but my combinatorial analysis wasn’t the main point of the paper).

Anyway, after spending an enjoyable evening on the problem of Daniel’s train set, I was able to prove that, alas, the possible behaviors are quite limited (I classified them all), falling far short of computational universality.

If you feel like I’m wasting your time with trivialities (or if you simply enjoy puzzles), then before you read any further, I encourage you to stop and try to prove this for yourself!

Back yet? OK then…

Theorem: Assume a finite amount of train track. Then after a linear amount of time, the train will necessarily enter a “boring infinite loop”—i.e., an attractor state in which at most two of the flaps keep getting toggled, and the rest of the flaps are fixed in place. In more detail, the attractor must take one of four forms:

I. a line (with reversing gadgets on both ends),
II. a simple cycle,
III. a “lollipop” (with one reversing gadget and one flap that keeps getting toggled), or
IV. a “dumbbell” (with two flaps that keep getting toggled).

In more detail still, there are seven possible topologically distinct trajectories for the train, as shown in the figure below.

Here the red paths represent the attractors, where the train loops around and around for an unlimited amount of time, while the blue paths represent “runways” where the train spends a limited amount of time on its way into the attractor. Every degree-3 vertex is assumed to have a Y-junction, while every degree-1 vertex is assumed to have a reversing gadget, unless (in IIb) the train starts at that vertex and never returns to it.

The proof of the theorem rests on two simple observations.

Observation 1: While the Y-junctions correspond to vertices of degree 3, there are no vertices of degree 4 or higher. This means that, if the train ever revisits a vertex v (other than the start vertex) for a second time, then there must be some edge e incident to v that it also traverses for a second time immediately afterward.

Observation 2: Suppose the train traverses some edge e, then goes around a simple cycle (meaning, one where no edges or vertices are reused), and then traverses e again, going in the same direction as the first time. Then from that point forward, the train will just continue around the same simple cycle forever.

The proof of Observation 2 is simply that, if there were any flap that might be in the train’s way as it continued around the simple cycle, then the train would already have pushed it out of the way its first time around the cycle, and nothing that happened thereafter could possibly change the flap’s position.

Using the two observations above, let’s now prove the theorem. Let the train start where it will, and follow it as it traces out a path. Since the graph is finite, at some point some already-traversed edge must be traversed a second time. Let e be the first such edge. By Observation 1, this will also be the first time the train’s path intersects itself at all. There are then three cases:

Case 1: The train traverses e in the same direction as it did the first time. By Observation 2, the train is now stuck in a simple cycle forever after. So the only question is what the train could’ve done before entering the simple cycle. We claim that at most, it could’ve traversed a simple path. For otherwise, we’d contradict the assumption that e was the first edge that the train visited twice on its journey. So the trajectory must have type IIa, IIb, or IIc in the figure.

Case 2: Immediately after traversing e, the train hits a reversing gadget and traverses e again the other way. In this case, the train will clearly retrace its entire path and then continue past its starting point; the question is what happens next. If it hits another reversing gadget, then the trajectory will have type I in the figure. If it enters a simple cycle and stays in it, then the trajectory will have type IIb in the figure. If, finally, it makes a simple cycle and then exits the cycle, then the trajectory will have type III in the figure. In this last case, the train’s trajectory will form a “lollipop” shape. Note that there must be a Y-junction where the “stick” of the lollipop meets the “candy” (i.e., the simple cycle), with the base of the Y aligned with the stick (since otherwise the train would’ve continued around and around the candy). From this, we deduce that every time the train goes around the candy, it does so in a different orientation (clockwise or counterclockwise) than the time before; and that the train toggles the Y-junction’s flap every time it exits the candy (although not when it enters the candy).

Case 3: At some point after traversing e in the forward direction (but not immediately after), the train traverses e in the reverse direction. In this case, the broad picture is analogous to Case 2. So far, the train has made a lollipop with a Y-junction connecting the stick to the candy (i.e. cycle), the base of the Y aligned with the stick, and e at the very top of the stick. The question is what happens next. If the train next hits a reversing gadget, the trajectory will have type III in the figure. If it enters a new simple cycle, disjoint from the first cycle, and never leaves it, the trajectory will have type IId in the figure. If it enters a new simple cycle, disjoint from the first cycle, and does leave it, then the trajectory now has a “dumbbell” pattern, type IV in the figure (also shown in the first video). There’s only one other situation to worry about: namely, that the train makes a new cycle that intersects the first cycle, forming a “theta” (θ) shaped trajectory. In this case, there must be a Y-junction at the point where the new cycle bumps into the old cycle. Now, if the base of the Y isn’t part of the old cycle, then the train never could’ve made it all the way around the old cycle in the first place (it would’ve exited the old cycle at this Y-junction), contradiction. If the base of the Y is part of the old cycle, then the flap must have been initially set to let the train make it all the way around the old cycle; when the train then reenters the old cycle, the flap must be moved so that the train will never make it all the way around the old cycle again. So now the train is stuck in a new simple cycle (sharing some edges with the old cycle), and the trajectory has type IIc in the figure.

This completes the proof of the theorem.

We might wonder: why isn’t this model train set capable of universal computation, of AND, OR, and NOT gates—or at any rate, of some computation more interesting than repeatedly toggling one or two flaps? My answer might sound tautological: it’s simply that the logic of the Y-junctions is too limited. Yes, the flaps can get pushed out of the way—that’s a “bit flip”—but every time such a flip happens, it helps to set up a “groove” in which the train just wants to continue around and around forever, not flipping any additional bits, with only the minor complications of the lollipop and dumbbell structures to deal with. Even though my proof of the theorem might’ve seemed like a tedious case analysis, it had this as its unifying message.

It’s interesting to think about what gadgets would need to be added to the train set to make it computationally universal, or at least expressively richer—able, as turned out to be the case for the Digi-Comp II, to express some nontrivial complexity class falling short of P. So for example, what if we had degree-4 vertices, with little turnstile gadgets? Or multiple trains, which could be synchronized to the millisecond to control how they interacted with each other via the flaps, or which could even crash into each other? I look forward to reading your ideas in the comment section!

For the truth is this: quantum complexity classes, BosonSampling, closed timelike curves, circuit complexity in black holes and AdS/CFT, etc. etc.—all these topics are great, but the same models and problems do get stale after a while. I aspire for my research agenda to chug forward, full steam ahead, into new computational domains.

PS. Happy Easter to those who celebrate!

There are three charged leptons: the electron, the muon and the tau. Let and be their masses. Then the Koide formula says

There’s no known reason for this formula to be true! But if you plug in the experimentally measured values of the electron, muon and tau masses, it’s accurate within the current experimental error bars:

Is this significant or just a coincidence? Will it fall apart when we measure the masses more accurately? Nobody knows.

Here’s something fun, though:

Puzzle. Show that no matter what the electron, muon and tau masses might be—that is, any positive numbers whatsoever—we must have

For some reason this ratio turns out to be almost exactly halfway between the lower bound and upper bound!

Koide came up with his formula in 1982 before the tau’s mass was measured very accurately.  At the time, using the observed electron and muon masses, his formula predicted the tau’s mass was

= 1776.97 MeV/c²

while the observed mass was

= 1784.2 ± 3.2 MeV/c²

Not very good.

In 1992 the tau’s mass was measured much more accurately and found to be

= 1776.99 ± 0.28 MeV/c²

Much better!

Koide has some more recent thoughts about his formula:

• Yoshio Koide, What physics does the charged lepton mass relation tell us?, 2018.

He points out how difficult it is to explain a formula like this, given how masses depend on an energy scale in quantum field theory.

Betteridge’s law applies here: the answer is “no”. It’s a subtle “no”, though.

As a scientist, you will always need to be able to communicate your work. Most of the time you can get away with papers and talks aimed at your peers. But the longer you mean to stick around, the more often you will have to justify yourself to others: to departments, to universities, and to grant agencies. A scientist cannot survive on scientific ability alone: to get jobs, to get funding, to survive, you need to be able to promote yourself, at least a little.

Self-promotion isn’t outreach, though. Talking to the public, or to journalists, is a different skill from talking to other academics or writing grants. And it’s entirely possible to go through an entire scientific career without exercising that skill.

That’s a reassuring message for some. I’ve met people for whom science is a refuge from the mess of human interaction, people horrified by the thought of fame or even being mentioned in a newspaper. When I meet these people, they sometimes seem to worry that I’m silently judging them, thinking that they’re ignoring their responsibilities by avoiding outreach. They think this in part because the field seems to be going in that direction. Grants that used to focus just on science have added outreach as a requirement, demanding that each application come with a plan for some outreach project.

I can’t guarantee that more grants won’t add outreach requirements. But I can say at least that I’m on your side here: I don’t think you should have to do outreach if you don’t want to. I don’t think you have to, just yet. And I think if grant agencies are sensible, they’ll find a way to encourage outreach without making it mandatory.

I think that overall, collectively, we have a responsibility to do outreach. Beyond the old arguments about justifying ourselves to taxpayers, we also just ought to be open about what we do. In a world where people are actively curious about us, we ought to encourage and nurture that curiosity. I don’t think this is unique to science, I think it’s something every industry, every hobby, and every community should foster. But in each case, I think that communication should be done by people who want to do it, not forced on every member.

I also think that, potentially, anyone can do outreach. Outreach can take different forms for different people, anything from speaking to high school students to talking to journalists to writing answers for Stack Exchange. I don’t think anyone should feel afraid of outreach because they think they won’t be good enough. Chances are, you know something other people don’t: I guarantee if you want to, you will have something worth saying.

On April 7, the g-2 experiment at Fermilab was supposed to reveal their new measurement of the magnetic moment of the muon.  *Was*, because the announcement may be delayed for the most bizarre reason. You may have heard that the data are blinded to avoid biasing the outcome. This is now standard practice, but the g-2 collaboration went further: they are unable to unblind the data by themselves, to make sure that there is no leaks or temptations. Instead, the unblinding procedure requires an input from an external person, who is one of the Fermilab theorists. How does this work? The experiment measures the frequency of precession of antimuons circulating in a ring. From that and the known magnetic field the sought fundamental quantity - the magnetic moment of the muon, or g-2 in short - can be read off.  However, the whole analysis chain is performed using a randomly chosen number instead of the true clock frequency. Only at the very end, once all statistical and systematic errors are determined,  the true frequency is inserted and the final result is uncovered. For that last step they need to type the secret code into this machine looking like something from a 60s movie: 

The code was picked by the Fermilab theorist, and he is the only person to know it.  There is the rub... this theorist now refuses to give away the code.  It is not clear why. One time he said he had forgotten the envelope with the code on a train, another time he said the dog had eaten it. For the last few days he has locked himself in his home and completely stopped taking any calls. 

The situation is critical. PhD students from the collaboration are working round the clock to crack the code. They are basically trying out all possible combinations, but the process is painstakingly slow and may take months, delaying the long-expected announcement.  The collaboration even got a permission from the Fermilab director to search the office of the said theorist.  But they only found this piece of paper behind the bookshelf: 

It may be that the paper holds a clue about the code. If you have any idea what the code may be please email fermilab@fnal.gov or just write it in the comments below. 

Update: a part of this post (but strangely enough not all) is an April Fools joke. The new g-2 results are going to be presented on April 7, 2021, as planned.  The code is OPE, which stands for "operator product expansion", which is an  important technique used in the theoretical calculation of hadronic corrections to muon g-2: 

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For the second year in a row, Seder was virtual, with Dr. Mrs. Q’s family the first night and mine the second, so for the second year in a row I cooked our own Passover meal. After a whole cycle of chagim I’m pretty OK at making brisket by now. I like matzah balls, really like them, and so does everybody else in the house, so I went nuts and tripled the matzah ball recipe. What I envisioned: matzah ball soup absolutely brimming with matzah balls cheek by unleavened jowl. What I did not take into account: the matzah balls absorb soup as they cook, which meant that what was left in the pot when they were done was a kind of matzah ball slurry with no soup at all (pictured, top right). The matzah balls themselves were moist and delicious! But it didn’t really feel right. The next night I made a whole new soup and plopped the leftover matzah balls in there (you make triple, you have leftovers) and that was much more like it.

What we usually do on Passover is visit Dr. Mrs. Q’s mom in Columbus, and get a big helping of deli from Katzinger’s ; missing that, I put in a mail order to Katz’s, which arrived today. So now, the brisket leftovers all gone, I am happily eating pastrami and tongue and matzah-chopped liver sandwiches.

(photo by CJ who never ceases to clown me about how much better his phone’s camera is than my phone’s camera.)

Energy spectrum of electron-recoil events measured by the XENON1T experiment.

1.  Simplest models explaining the excess are excluded by astrophysical observations. If axions can be produced in the Sun at the rate suggested by the XENON result, they can be produced at even larger rates in hotter stars, e.g. in red giants or white dwarfs. This would lead to excessive cooling of these stars, in conflict with observations. The upper limit on the axion-electron coupling g from red giants is 3*10^-13, which is an order of magnitude  less than what is needed for the XENON excess.  The neutrino magnetic moment explanations faces a similar difficulty. Of course, astrophysical limits reside in a different epistemological reality; it is not unheard of that they are relaxed by an order of magnitude or disappear completely. But certainly this is something to worry about.  
2.  At a more psychological level, a small excess over a large background near a detection threshold.... sounds familiar. We've seen that before in the case of the DAMA and CoGeNT dark matter experiments, at it didn't turn out well.     
3. The bump is at 1.5 keV, which is *twice* 750 eV.  

The hashtag #CautiouslyExcited is trending on Twitter, in spite of the raging plague. The updated RK measurement in LHCb has made a big splash and has been covered by every news outlet.  RK measures the ratio of the B->Kμμ and B->Kee decay probabilities, which the Standard Model predicts to be very close to one. Using all the data collected so far, LHCb instead finds RK = 0.846 with the error of 0.044. This is the same central value and 30% smaller error compared to their 2019 result based on half of the data.  Mathematically speaking, the update does not much change the global picture of the B-meson anomalies. However, it has an important psychological impact, which goes beyond the PR story of crossing the 3 sigma threshold. Let me explain why. 

For the last few decades, every deviation from the Standard Model prediction in a particle collider experiment would mean one of these 3 things:    

1. Statistical fluctuation. 
2. Flawed theory prediction. 
3. Experimental screw-up.   

In the case of RK, the option 2. is not a worry.  Yes, flavor physics is a swamp full of snake pits, however in the RK ratio the dangerous hadronic uncertainties cancel out to a large extent, so that precise theoretical predictions are possible.  Before March 23 the biggest worry was option 1.  Indeed, 2-3 sigma fluctuations happen all the time at the LHC, due to a huge number of measurements being taken.  However, you expect statistical fluctuations to decrease in significance as more data is collected.  This is what seems to be happening to the sister RD anomaly, and the earlier history of RK was not very encouraging either (in the 2019 update the significance neither increased nor decreased).  The fact that, this time, the significance of the RK anomaly increased, more or less as you would expect it to assuming it is a genuine new physics signal, makes it unlikely that it is merely a statistical fluctuation.  This is the main reason for the excitement you may perceive among particle physicists these days. 

On the other hand,  option 3. remains a possibility.  In their analysis,  LHCb reconstructed 3850 B->Kμμ decays vs. 1640 B->Kee decays, but from that they concluded that decays to muons are less probable than those to electrons. This is because one has to take into account the different reconstruction efficiencies for muons and electrons. An estimate of that efficiency is the most difficult ingredient of the measurement,  and the LHCb folks have spent many nights of heavy drinking worrying about it. Of course, they have made multiple cross-checks and are quite confident that there is no mistake but... there will always be a shadow of a doubt until RK is confirmed by an independent experiment. Fortunately for everyone, a verification will be provided by the Belle-II experiment, probably in 3-4 years from now. Only when Belle-II sees the same thing we will breathe a sigh of relief and put all our money on option

4. Physics beyond the Standard Model 

From that point of view explaining the RK measurement is trivial.  All we need is to add a new kind of interaction between b- and s-quarks and muons to the Standard Model Lagrangian.  For example, this 4-fermion contact term will do: 

where Q3=(t,b), Q2=(c,s), L2=(νμ,μ). The Standard Model won't let you have this interaction because it violates one of its founding principles: renormalizability.  But we know that the Standard Model is just an effective theory, and that non-renormalizable interactions must exist in nature, even if they are very suppressed so as to be unobservable most of the time.  In particular, neutrino oscillations are best explained by certain dimension-5 non-renormalizable interactions.  RK may be the first evidence that also dimension-6 non-renormalizable interactions exist in nature.  The nice thing is that the interaction term above 1) does not violate any existing experimental constraints,  2) explains not only RK but also some other 2-3 sigma tensions in the data (RK*, P5'),  and 3) fits well with some smaller 1-2 sigma effects (Bs->μμ, RpK,...). The existence of a simple theoretical explanation and a consistent pattern in the data is the other element that prompts cautious optimism.  

The LHC run-3 is coming soon, and with it more data on RK.  In the shorter perspective (less than a year?) there will be other important updates (RK*, RpK) and new observables (Rϕ , RK*+) probing the same physics. Finally something to wait for.   

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guest post by Christian Williams

In Part 2 we described higher-order algebraic theories: categories with products and finite-order exponents, which present languages with (binding) operations, equations, and rewrites; from these we construct native type systems.

Now let’s use the wisdom of the Yoneda embedding!

Every category embeds into a topos of presheaves

$$y\colon C\rightarrowtail \mathscr{P}C=[C^{op},Set]$$

$$y(c) = C(-,c) \quad\quad y(f)=C(-,f)\colon C(-,c)\to C(-,d).$$

If $(C,\otimes,[-,-])$ is monoidal closed, then the embedding preserves this structure:

$$y(c\otimes d)\simeq y(c)\otimes y(d) \quad \quad y([c,d])\simeq [y(c),y(d)]$$

i.e. using Day convolution, $y$ is monoidal closed. So, we can move into a richer environment while preserving higher-order algebraic structure, or languages.

We now explore the native type system of a language, using the $\rho$-calculus as our running example. The complete type system is in the paper, page 9.

Representables

The simplest kind of object of the native type system is a representable $T(-,\mathtt{S})$. This is the set of all terms of sort $\mathtt{S}$, indexed by the context of the language. Whereas many works in computer science either restrict to closed terms or lump all terms together, this indexing is natural and useful.

In the $\rho$-calculus, $y(\mathtt{P}) = T_\rho(-,\mathtt{P})$ is the indexed set of all processes.

$$y(\mathtt{P})(\Gamma) = \{p \;|\; (x_1,\dots,x_n):\Gamma \vdash p:\mathtt{P}\}.$$

The type system is built from these basic objects by the operations of $T$ and the structure of $\mathscr{P}T$. We can then construct predicates, dependent types, co/limits, etc., and each constructor has corresponding inference rules which can be used by a computer.

Predicates and Types

The language of a topos is represented by two fibrations: the subobject fibration gives predicate logic, and the codomain fibration gives dependent type theory. Hence the two basic entities are predicates and (dependent) types. Types are more general, and we can think of them as the “new sorts” of language $T$, which can be much more expressive.

A predicate $\varphi:y(\mathtt{P})\to \Omega$ corresponds to a subobject of a representable $\{p \;|\; \varphi(p)\}\rightarrowtail y(\mathtt{P})$, which is equivalent to a sieve: a set of morphisms into $\mathtt{P}$, closed under precomposition:

This emphasizes the idea that predicate logic over representables is actually reasoning about abstract syntax trees: here $g$ is some tree of operations in $T$ with an $\mathtt{S}$-shaped hole of variables, and the predicate $\varphi$ only cares about the outer shape of $g$; you can plug in any term $f$ while still satisfying $\varphi$.

More generally, a morphism $f\colon B\to A$ is understood as an “indexed presheaf” or dependent type

$$x:A\vdash B(x):Type.$$

i.e. for every element $x\colon X\to A$, there is a fiber $B(x):= f^*(x)$ which is the “type depending on term $x$”.

An example of a type in the $\rho$-calculus is given by the input operation,

$$y(\mathtt{in}):y(\mathtt{N\times P\times [N,P]})\to y(\mathtt{P})$$

where the fiber over $\varphi$ is the set of all channel-context pairs $(n,\lambda x.p)$ such that $\varphi(\mathtt{in}(n,\lambda x.p))$.

Dependent Sum and Product

Here we use the structure described in Part 1. The predicate functor $\mathscr{P}T(-,\Omega):\mathscr{P}T^{op}\to CHA$ is a hyperdoctrine, which for each presheaf $A$ gives a complete Heyting algebra of predicates $\Omega^A$, and for each $f\colon B\to A$ gives adjoints $\exists_f\dashv \Omega^f\dashv \forall_f\colon \Omega^B\to \Omega^A$ for image, preimage, and secure image.

Similarly, the slice functor $\mathscr{P}T/-:\mathscr{P}T^{op} \to CCT$ is a hyperdoctrine into co/complete toposes with adjoints $\Sigma_f\dashv \Delta^f\dashv \Pi_f$. These are dependent sum, substitution, and dependent product. From these we can reconstruct all the operations of predicate logic, and much more.

As (briefly) explained in Part 1, the idea of dependent sum is that indexed sums generalize products; here the codomain is the set of indices and its fibers are the sets in the family; so an element of the indexed sum is a dependent pair $(a,x\in X_a)$. Dually, indexed products generalize functions: an element of the product of the fibers is a tuple $(x_1\in X_{a_1},\dots,x_n\in X_{a_n})$ which can be understood as a dependent function where the codomain $X_a$ depends on which $a$ you plug in.

Explicitly, given $f\colon A\to B$ and $p\colon X\to A$, $q\colon Y\to B$, we have $\Delta_f(q)_\mathtt{S}^a = q_\mathtt{S}^{f_\mathtt{S}(a)}$ and

(letting $X_\mathtt{S}=X(\mathtt{S})$ and $p_\mathtt{S}^b$ denote the fiber over $b$). Despite the complex formulae, the intuition is essentially the same as in Set, except we need to ensure the resulting objects are still presheaves, i.e. closed under precomposition. The point is:

$$\Sigma \;\; \text{generalizes product, and categorifies } \;\; \exists \;\; \text{ or image; and}$$

$$\Pi \;\; \text{generalizes internal hom, and categorifies } \;\; \forall \;\; \text{ or secure image.}$$

The main examples start with just “pushing forward” operations in the theory, using $\exists$. Given an operation $f\colon \mathtt{S}\to \mathtt{T}$, the image $\exists_{y(f)}:\Omega^{y(\mathtt{S})}\to \Omega^{y(\mathtt{T})}$ takes a predicate (sieve) $\varphi\rightarrowtail y(\mathtt{S})$ and simply postcomposes every term in $\varphi$ with $f$.

Hence an example predicate (leaving $\exists$ and $y$ implicit) is

$$\mathsf{multi.thread} = \neg(0)\vert \neg(0) \;\; \rightarrowtail y(\mathtt{P}).$$

This predicate determines processes which are the parallel of two non-null processes.

As an example of the distinct utility of the adjoints, recall from Part 2 that we can model computational dynamics using a graph of processes and rewrites $s,t:\mathtt{E\to P}$. Now these operations give adjunctions between sieves on $\mathtt{E}$ and sieves on $\mathtt{P}$, which give operators for “step forward or backward”:

$$\Sigma_t\Omega^s(\varphi) = \{q \;|\; \exists r.\; r:p\rightsquigarrow q \wedge \varphi(p)\}$$

$$\Pi_t(\Omega^s(\varphi)) = \{q \;|\; \forall r.\; r:p\rightsquigarrow q \Rightarrow \varphi(p)\}$$

While “image” step-forward gives all possible next terms, the “secure” step-forward gives terms which could only have come from $\varphi$. For security protocols, this can be used to filter processes by past behavior.

Image / Comprehension and Subtyping

Predicates and types are related by an adjunction between the fibrations.

To convert a predicate $\varphi:A\to \Omega$ to a type, apply comprehension to construct the subobject of terms $\mathrm{c}(\varphi)$ which satisfy $\varphi$. To convert a type $p:X\to A$ to a predicate, apply image factorization to construct the predicate $\mathrm{i}(p)$ for whether each fiber is inhabited.

We implicitly use the comprehension direction all the time (thinking of predicates as their subobjects); and while taking the image is more destructive, it can certainly be useful for the sake of simplification. For example, rather than thinking about the type $y(\mathtt{out}):y(\mathtt{N\times P})\to y(\mathtt{P})$, we may simply want to consider the image $\mathrm{i}(y(\mathtt{out}))$, the set of all output processes.

Internal Hom and Reification

While the Grothendieck construction is relatively known, there is less awareness about how the local structure of an indexed category (complete Heyting algebras for predicates) can often be converted to a global structure on the total category of the corresponding fibration. The total category of the predicate functor $\Omega\mathscr{P}T$ is cartesian closed, allowing us to construct predicate homs.

The construction can be understood in the category of sets. Given $\varphi:2^A$ and $\psi:2^B$, we can define

$$[\varphi,\psi]:[A,B]\to 2 \quad \quad [\varphi, \psi](f) = \forall a\in A.\; \varphi(a)\Rightarrow \psi(f(a)).$$

Hence it constructs “contexts which ensure implications”.

For example, we can construct the “wand” of separation logic: let $T_h$ be the theory of a commutative monoid $(H,\cup,e)$, with a set of constants $\{h\}:1\to H$ adjoined as the elements of a heap. If we define

$$(\varphi \multimap \psi) = \Omega^{\lambda x.x\cup-}[\varphi, \psi]$$

then $h_1:(\varphi \multimap \psi)$ asserts that $h_2:\varphi\Rightarrow h_1\cup h_2:\psi$.

There is a much more expressive way of forming homs which we call reification (p7); we do not know if it has been explored, and we have yet to determine its relation to dependent product.

co/Induction

Similarly, the fibers of $\Omega\mathscr{P}T\to \mathscr{P}T$ are co/complete, and this can be assembled into a global co/complete structure on the total category. Hence, we can use this to construct co/inductive types.

For example, given a predicate on names $\alpha$, we may construct a predicate for “liveness and safety” on $\alpha$:

$$\mathsf{sole.in}(\alpha) = \mu X. \mathtt{in}(\alpha,\mathtt{N}.X)\wedge \neg\mathtt{in}(\neg\alpha,\mathtt{N.P})$$

where $\mu$ denotes the initial algebra, which is constructed as a colimit. This determines whether a process inputs on $\alpha$, does not input on $\neg\alpha$, and continues as a process which satisfies this same predicate. This can be understood as a static type for a firewall.

Applications

Once these type constructors are combined, they can express highly useful and complex ideas about code. The best part is that this type system can be generated from any language with product and function types, which includes large chunks of many popular programming languages.

To get a feel for more applications, check out the final section of Native Type Theory. Of course, check out the rest of the paper, and let me know what you think! Thank you for reading.

The nLab was born out of conversations at the Café back in 2008. Over the past 12 years it has grown as a wiki to over 15000 pages.

For many years it was funded personally by Urs Schreiber, until in the last few years when it relied on a fund kindly provided by Steve Awodey.

But now we have new arrangements in place, and are looking to its users to help fund the nLab:

In 2021, the nLab will move to the cloud. To fund the running of the nLab in the cloud, we have decided to rely upon funding by donations. In the autumn of 2020, at the kind initiative of Brendan Fong, the nLab decided to collaborate with the Topos Institute for the practical side of this: the Topos Institute is legally able to handle donations, and the financing of the nLab will be handled by the Topos Institute. The Topos Institute owns the cloud account in which the nLab will be run.

Please do consider making a donation here.

Catching up after the APS meeting, here are a couple of links of interest:

• This video has been making the rounds, and it's fun to watch.  It's an updated take on one of those powers-of-ten videos, though in this case it's really powers-of-two.  Nicely done, though I think the discussion of the Planck Length is not really correct.  As far as I know, the Planck Length is a characteristic scale where quantum gravity effects cannot be neglected - that doesn't mean that the structure of the universe is discrete on that scale.
• There have also been a lot of articles like this one implying that new (non-Standard Model) physics has been seen at the LHC.  As is usually the case, it's premature to get too excited.  At the 3\(\sigma\) level, there is an asymmetry in decay channels (electrons vs muons) seen by the LHCb experiment when none is expected.  As the always reliable Tommaso Dorigo writes here, everyone should just take a breath before getting too excited.  At least when the LHC starts back up next year, there should be a lot of new data coming in, and either this effect will grow, or it will fade away.  Anyone want to bet on the over/under for the number of theory papers about leptoquarks that are going to show up on the arxiv in the next month?
• We were fortunate enough to have Pablo Jarillo-Herrero give our colloquium this past Wednesday, talking about some really exciting recent results (here, here) in twisted trilayer graphene.

When we study subatomic particles, particle physicists use a theory called Quantum Field Theory. But what is a quantum field?

Some people will describe a field in vague terms, and say it’s like a fluid that fills all of space, or a vibrating rubber sheet. These are all metaphors, and while they can be helpful, they can also be confusing. So let me avoid metaphors, and say something that may be just as confusing: a field is the answer to a question.

Suppose you’re interested in a particle, like an electron. There is an electron field that tells you, at each point, your chance of detecting one of those particles spinning in a particular way. Suppose you’re trying to measure a force, say electricity or magnetism. There is an electromagnetic field that tells you, at each point, what force you will measure.

Sometimes the question you’re asking has a very simple answer: just a single number, for each point and each time. An example of a question like that is the temperature: pick a city, pick a date, and the temperature there and then is just a number. In particle physics, the Higgs field answers a question like that: at each point, and each time, how “Higgs-y” is it there and then? You might have heard that the Higgs field gives other particles their mass: what this means is that the more “Higgs-y” it is somewhere, the higher these particles’ mass will be. The Higgs field is almost constant, because it’s very difficult to get it to change. That’s in some sense what the Large Hadron Collider did when they discovered the Higgs boson: pushed hard enough to cause a tiny, short-lived ripple in the Higgs field, a small area that was briefly more “Higgs-y” than average.

We like to think of some fields as fundamental, and others as composite. A proton is composite: it’s made up of quarks and gluons. Quarks and gluons, as far as we know, are fundamental: they’re not made up of anything else. More generally, since we’re thinking about fields as answers to questions, we can just as well ask more complicated, “composite” questions. For example, instead of “what is the temperature?”, we can ask “what is the temperature squared?” or “what is the temperature times the Higgs-y-ness?”.

But this raises a troubling point. When we single out a specific field, like the Higgs field, why are we sure that that field is the fundamental one? Why didn’t we start with “Higgs squared” instead? Or “Higgs plus Higgs squared”? Or something even weirder?

That kind of swap, from Higgs to Higgs squared, is called a field redefinition. In the math of quantum field theory, it’s something you’re perfectly allowed to do. Sometimes, it’s even a good idea. Other times, it can make your life quite complicated.

The reason why is that some fields are much simpler than others. Some are what we call free fields. Free fields don’t interact with anything else. They just move, rippling along in easy-to-calculate waves.

Redefine a free field, swapping it for some more complicated function, and you can easily screw up, and make it into an interacting field. An interacting field might interact with another field, like how electromagnetic fields move (and are moved by) electrons. It might also just interact with itself, a kind of feedback effect that makes any calculation we’d like to do much more difficult.

If we persevere with this perverse choice, and do the calculation anyway, we find a surprise. The final results we calculate, the real measurements people can do, are the same in both theories. The field redefinition changed how the theory appeared, quite dramatically…but it didn’t change the physics.

You might think the moral of the story is that you must always choose the right fundamental field. You might want to, but you can’t: not every field is secretly free. Some will be interacting fields, whatever you do. In that case, you can make one choice or another to simplify your life…but you can also just refuse to make a choice.

That’s something quite a few physicists do. Instead of looking at a theory and calling some fields fundamental and others composite, they treat every one of these fields, every different question they could ask, on the same footing. They then ask, for these fields, what one can measure about them. They can ask which fields travel at the speed of light, and which ones go slower, or which fields interact with which other fields, and how much. Field redefinitions will shuffle the fields around, but the patterns in the measurements will remain. So those, and not the fields, can be used to specify the theory. Instead of describing the world in terms of a few fundamental fields, they think about the world as a kind of field soup, characterized by how it shifts when you stir it with a spoon.

It’s not a perspective everyone takes. If you overhear physicists, sometimes they will talk about a theory with only a few fields, sometimes they will talk about many, and you might be hard-pressed to tell what they’re talking about. But if you keep in mind these two perspectives: either a few fundamental fields, or a “field soup”, you’ll understand them a little better.

This evening at 7:30pm Central time, come to Fermilab (online) for a public talk I’ll give about shaking up how we present serious scientific ideas in books for the public. It should be fun! The information is here. -cvj

The post Talking at Fermilab! appeared first on Asymptotia.

Over at the n-Category Café, John Baez is making a big deal of the fact that the global form of the Standard Model gauge group is $$G = (SU(3)\times SU(2)\times U(1))/N$$ where $N$ is the $\mathbb{Z}_6$ subgroup of the center of $G'=SU(3)\times SU(2)\times U(1)$ generated by the element $\left(e^{2\pi i/3}\mathbb{1},-\mathbb{1},e^{2\pi i/6}\right)$.

The global form of the gauge group has various interesting topological effects. For instance, the fact that the center of the gauge group is $Z(G)= U(1)$, rather than $Z(G')=U(1)\times \mathbb{Z}_6$, determines the global 1-form symmetry of the theory. It also determines the presence or absence of various topological defects (in particular, cosmic strings). I pointed this out, but a proper explanation deserved a post of its own.

None of this is new. I’m pretty sure I spent a sunny afternoon in the summer of 1982 on the terrace of Café Pamplona doing this calculation. (As any incoming graduate student should do, I spent many a sunny afternoon at a café doing this and similar calculations.)

At low energies, $G$ is broken to the subgroup $H=U(3)$, where the embedding $i\colon H\hookrightarrow G$ is given as follows. Let $h\in H$ and let $d\coloneqq \det(h)$. Choose a 6th root $$b = d^{1/6}$$ Then

The ambiguity in defining $b$ leads precisely to an ambiguity in $i(h)$ by multiplication by an element of $N$. Thus (1) is ill-defined as a map to $G'$, but well-defined as a map to $G$.

The (would-be) cosmic strings associated to the breaking of $G$ to $H$ are classified by $\pi_1(G/H)$. Both $\pi_1(H)$ and $\pi_1(G)$ are equal to $\mathbb{Z}$. The long-exact sequence in homotopy yields $$0\to \pi_1(H)\to \pi_1(G) \to \pi_1(G/H)\to 0$$ So what we need to do is compute the image of the generator of $\pi_1(H)$ in $\pi_1(G)$. If the image is $n$ times the generator of $\pi_1(G)$, then the quotient is nontrivial and we have $\mathbb{Z}_n$ cosmic strings.

$\pi_1(G)$ is generated by the (homotopy class of) the loop

$\pi_1(H)$ is generated by the loop

Plugging (3) into (1), we see that $i(h(s))=g(s)$. Hence $\pi_1(G/H)=0$ and there are no cosmic strings.

For years, I’d sometimes hear discussions about the ethics of quantum computing research. Quantum ethics!

When the debates weren’t purely semantic, over the propriety of terms like “quantum supremacy” or “ancilla qubit,” they were always about chin-strokers like “but what if cracking RSA encryption gives governments more power to surveil their citizens? or what if only a few big countries or companies get quantum computers, thereby widening the divide between haves and have-nots?” Which, OK, conceivably these will someday be issues. But, besides barely depending on any specific facts about quantum computing, these debates always struck me as oddly safe, because the moral dilemmas were so hypothetical and far removed from us in time.

I confess I may have even occasionally poked fun when asked to expound on quantum ethics. I may have commented that quantum computers probably won’t kill anyone unless a dilution refrigerator tips over onto their head. I may have asked forgiveness for feeding custom-designed oracles to BQP and QMA, without first consulting an ethics committee about the long-term effects on those complexity classes.

Now fate has punished me for my flippancy. These days, I really do feel like quantum computing research has become an ethical minefield—but not for any of the reasons mentioned previously. What’s new is that millions of dollars are now potentially available to quantum computing researchers, along with equity, stock options, and whatever else causes “ka-ching” sound effects and bulging eyes with dollar signs. And in many cases, to have a shot at such riches, all an expert needs to do is profess optimism that quantum computing will have revolutionary, world-changing applications and have them soon. Or at least, not object too strongly when others say that.

Some of today’s rhetoric will of course remind people of the D-Wave saga, which first brought this blog to prominence when it began in earnest in 2007. Quantum computers, we hear now as then, will soon leave the Earth’s fastest supercomputers in the dust. They’re going to harness superposition to try all the exponentially many possible solutions at once. They’ll crack the Traveling Salesman Problem, and will transform machine learning and AI beyond recognition. Meanwhile, simulations of quantum systems will be key to solving global warming and cancer.

Despite the parallels, though, this new gold rush doesn’t feel to me like the D-Wave one, which seems in retrospect like just a little dry run. If I had to articulate what’s new in one sentence, it’s that this time “the call is coming from inside the house.” Many of the companies making wildly overhyped claims are recognized leaders of the field. They have brilliant quantum computing theorists and experimentalists on their staff with impeccable research records. Some of those researchers are among my best friends. And even when I wince at the claims of near-term applications, in many cases (especially with quantum simulation) the claims aren’t obviously false—we won’t know for certain until we try it and see! It’s genuinely gotten harder to draw the line between defensible optimism and exaggerations verging on fraud.

Indeed, this time around virtually everyone in QC is “complicit” to a greater or lesser degree. I, too, have accepted compensation to consult on quantum computing topics, to give talks at hedge funds, and in a few cases to serve as a scientific adviser to quantum computing startups. I tell myself that, by 2021 standards, this stuff is all trivial chump change—a few thousands of dollars here or there, to expound on the same themes that I already discuss free of charge on this blog. I actually get paid to dispel hype, rather than propagate it! I tell myself that I’ve turned my back on the orders of magnitude more money available to those willing to hitch their scientific reputations to the aspirations of this or that specific QC company. (Yes, this blog, and my desire to preserve its intellectual independence and credibility, might well be costing me millions!)

But, OK, some would argue that accepting any money from QC companies or QC investors just puts you at the top of a slope with unabashed snake-oil salesmen at the bottom. With the commercialization of our field that started around 2015, there’s no bright line anymore marking the boundary between pure scientific curiosity and the pursuit of filthy lucre; it’s all just points along a continuum. I’m not sure that these people are wrong.

As some of you might’ve seen already, IonQ, the trapped-ion QC startup that originated from the University of Maryland, is poised to have the first-ever quantum computing IPO—a so-called “SPAC IPO,” which while I’m a financial ignoramus, apparently involves merging with a shell company and thereby bypassing the SEC’s normal IPO rules. Supposedly they’re seeking $650 million in new funding and a $2 billion market cap. If you want to see what IonQ is saying about QC to prospective investors, click here. Lacking any choice in the matter, I’ll probably say more about these developments in a future post.

Meanwhile, PsiQuantum, the Palo-Alto-based optical QC startup, has said that it’s soon going to leave “stealth mode.” And Amazon, Microsoft, Google, IBM, Honeywell, and other big players continue making large investments in QC—treating it, at least rhetorically, not at all like blue-sky basic research, but like a central part of their future business plans.

All of these companies have produced or funded excellent QC research. And of course, they’re all heterogeneous, composed of individuals who might vehemently disagree with each other about the near- or long-term prospects of QC. And yet all of them have, at various times, inspired reflections in me like the ones in this post.

I regret that this post has no clear conclusion. I’m still hashing things out, solicing thoughts from my readers and friends. Speaking of which: this coming Monday, March 22, at 8-10pm US Eastern time, I’ve decided to hold a discussion around these issues on Clubhouse—my “grand debut” on that app, and an opportunity to see whether I like it or not! My friend Adam Brown will moderate the discussion; other likely participants will be John Horgan, George Musser, Michael Nielsen, and Matjaž Leonardis. If you’re on Clubhouse, I hope to see you there!

Update (March 22): Read this comment by “FB” if you’d like to understand how we got to this point.

Spring arrived right on schedule, just a little snow left in the shady places, sunny out and windy in the high 60’s. AB and I did our first real bike ride of the year, going out about 15 miles to the very agreeable Riley Tavern where you eat outside on picnic tables. A lot of people are watching Wisconsin’s basketball season slowly sputter out as the Badgers fail to mount a comeback against the much higher-seeded team from Baylor. Riley Tavern serves amazing ice cream sandwiches with two chocolate chip cookies instead of the rectangular brown things; they’re not made there, they’re from Mullen’s Dairy Bar in Watertown. The thing about an ice cream sandwich is, they use the rectangular brown things which are soft and not very interesting because you can bite right through them without messing up the ice cream. Any cookie with a little more of a resistance to the tooth tends to smoosh the ice cream out the side when you bite down. That’s unacceptable. Mullen’s has somehow found a way to use a cookie with a real bite but give the ice cream itself enough structural integrity to hold itself in place while you eat it. Extraordinary!

I’d figured it had been warm enough long enough for the bike trail to be dry, and that was sort of true, but in many places it was badly rutted from the people who’d ridden on it when it was muddy, and even though it wasn’t really muddy anymore, it was soft for a couple of miles, so that your weight pushed your back wheel down into the dirt, which clutched your tire so that you were perpetually in a kind of low-grade partially submerged wheelie. We fought our way through at about 5mph for the whole stretch. So a more strenuous 30 miles than the usual. But the last 5 miles home, on pavement, felt like absolute gliding.

I used to catch lizards—brown anoles, as I learned to call them later—as a child. They were colored as their name suggests, were about as long as one of my hands, and resented my attention. But they frequented our back porch, and I had a butterfly net. So I’d catch lizards, with my brother or a friend, and watch them. They had throats that occasionally puffed out, exposing red skin, and tails that detached and wriggled of their own accord, to distract predators.

Some theorists might appreciate butterfly nets, I imagine, for catching experimentalists. Some of us theorists will end a paper or a talk with “…and these predictions are experimentally accessible.” A pause will follow the paper’s release or the talk, in hopes that a reader or an audience member will take up the challenge. Usually, none does, and the writer or speaker retires to the Great Deck Chair of Theory on the Back Patio of Science.

So I was startled when an anole, metaphorically speaking, volunteered a superconducting qubit for an experiment I’d proposed.

The experimentalist is one of the few people I can compare to a reptile without fear that he’ll take umbrage: Kater Murch, an associate professor of physics at Washington University in St. Louis. The most evocative description of Kater that I can offer appeared in an earlier blog post: “Kater exudes the soberness of a tenured professor but the irreverence of a Californian who wears his hair slightly long and who tattooed his wedding band on.”

Kater expressed interest in an uncertainty relation I’d proved with theory collaborators. According to some of the most famous uncertainty relations, a quantum particle can’t have a well-defined position and a well-defined momentum simultaneously. Measuring the position disturbs the momentum; any later momentum measurement outputs a completely random, or uncertain, number. We measure uncertainties with entropies: The greater an entropy, the greater our uncertainty. We can cast uncertainty relations in terms of entropies.

I’d proved, with collaborators, an entropic uncertainty relation that describes chaos in many-particle quantum systems. Other collaborators and I had shown that weak measurements, which don’t disturb a quantum system much, characterize chaos. So you can check our uncertainty relation using weak measurements—as well as strong measurements, which do disturb quantum systems much. One can simplify our uncertainty relation—eliminate the chaos from the problem and even eliminate most of the particles. An entropic uncertainty relation for weak and strong measurements results.

Kater specializes in weak measurements, so he resolved to test our uncertainty relation. Physical Review Letters published the paper about our collaboration this month. Quantum measurements can not only create uncertainty, the paper shows, but also reduce it: Kater and his PhD student Jonathan Monroe used light to measure a superconducting qubit, a tiny circuit in which current can flow forever. The qubit had properties analogous to position and momentum (the spin’s z– and x-components). If the atom started with a well-defined “position” (the z-component) and the “momentum” (the x-component) was measured, the outcome was highly random; the total uncertainty about the two measurements was large. But if the atom started with a well-defined “position” (z-component) and another property (the spin’s y-component) was measured before the “momentum” (the x-component) was measured strongly, the total uncertainty was lower. The extra measurement was designed not to disturb the atom much. But the nudge prodded the atom enough, rendering the later “momentum” measurement (the x measurement) more predictable. So not only can quantum measurements create uncertainty, but gentle quantum measurements can also reduce it.

I didn’t learn only physics from our experiment. When I’d catch a lizard, I’d tip it into a tank whose lid contained a magnifying lens, and I’d watch the lizard. I didn’t trap Kater and Jonathan under a magnifying glass, but I did observe their ways. Here’s what I learned about the species experimentalus quanticus.

1) They can run experiments remotely when a pandemic shuts down campus: A year ago, when universities closed and cities locked down, I feared that our project would grind to a halt. But Jonathan twiddled knobs and read dials via his computer, and Kater popped into the lab for the occasional fixer-upper. Jonathan even continued his experiment from another state, upon moving to Texas to join his parents. And here we theorists boast of being able to do our science almost anywhere.

2) They speak with one less layer of abstraction than I: We often discussed, for instance, the thing used to measure the qubit. I’d call the thing “the detector.” Jonathan would call it “the cavity mode,” referring to the light that interacts with the qubit, which sits in a box, or cavity. I’d say “poh-tay-toe”; they’d say “poh-tah-toe”; but I’m glad we didn’t call the whole thing off.

3) Experiments take longer than expected—even if you expect them to take longer than estimated: Kater and I hatched the plan for this project during June 2018. The experiment would take a few months, Kater estimated. It terminated last summer.

4) How they explain their data: Usually in terms of decoherence, the qubit’s leaking of quantum information into its environment. For instance, to check that the setup worked properly, Jonathan ran a simple test that ended with a measurement. (Experts: He prepared a eigenstate, performed a Hadamard gate, and measured .) The measurement should have had a 50% chance of yielding and a 50% chance of yield . But the outcome dominated the trials. Why? Decoherence pushed the qubit toward toward . (Amplitude damping dominated the noise.)

5) Seeing one’s theoretical proposal turn into an experiment feels satisfying: Due to point (3), among other considerations, experiments aren’t cheap. The lab’s willingness to invest in the idea I’d developed with other theorists was heartening. Furthermore, the experiment pushed us to uncover more theory—for example, how tight the uncertainty bound could grow.

After getting to know an anole, I’d release it into our backyard and bid it adieu.¹ So has Kater moved on to experimenting with topology, and Jonathan has progressed toward graduation. But more visitors are wriggling in the Butterfly Net of Theory-Experiment Collaboration. Stay tuned.

¹Except for the anole I accidentally killed, by keeping it in the tank for too long. But let’s not talk about that.

What’s the difference between a black hole and a neutron star?

When a massive star nears the end of its life, it starts running out of nuclear fuel. Without the support of a continuous explosion, the star begins to collapse, crushed under its own weight.

What happens then depends on how much weight that is. The most massive stars collapse completely, into the densest form anything can take: a black hole. Einstein’s equations say a black hole is a single point, infinitely dense: get close enough and nothing, not even light, can escape. A quantum theory of gravity would change this, but not a lot: a quantum black hole would still be as dense as quantum matter can get, still equipped with a similar “point of no return”.

A slightly less massive star collapses, not to a black hole, but to a neutron star. Matter in a neutron star doesn’t collapse to a single point, but it does change dramatically. Each electron in the old star is crushed together with a proton until it becomes a neutron, a forced reversal of the more familiar process of Beta decay. Instead of a ball of hydrogen and helium, the star then ends up like a single atomic nucleus, one roughly the size of a city.

Now, let me ask a slightly different question: how do you tell the difference between a black hole and a neutron star?

Sometimes, you can tell this through ordinary astronomy. Neutron stars do emit light, unlike black holes, though for most neutron stars this is hard to detect. In the past, astronomers would use other objects instead, looking at light from matter falling in, orbiting, or passing by a black hole or neutron star to estimate its mass and size.

Now they have another tool: gravitational wave telescopes. Maybe you’ve heard of LIGO, or its European cousin Virgo: massive machines that do astronomy not with light but by detecting ripples in space and time. In the future, these will be joined by an even bigger setup in space, called LISA. When two black holes or neutron stars collide they “ring” the fabric of space and time like a bell, sending out waves in every direction. By analyzing the frequency of these waves, scientists can learn something about what made them: in particular, whether the waves were made by black holes or neutron stars.

One big difference between black holes and neutron stars lies in something called their “Love numbers“. From far enough away, you can pretend both black holes and neutron stars are single points, like fundamental particles. Try to get more precise, and this picture starts to fail, but if you’re smart you can include small corrections and keep things working. Some of those corrections, called Love numbers, measure how much one object gets squeezed and stretched by the other’s gravitational field. They’re called Love numbers not because they measure how hug-able a neutron star is, but after the mathematician who first proposed them, A. E. H. Love.

What can we learn from Love numbers? Quite a lot. More impressively, there are several different types of questions Love numbers can answer. There are questions about our theories, questions about the natural world, and questions about fundamental physics.

You might have heard that black holes “have no hair”. A black hole in space can be described by just two numbers: its mass, and how much it spins. A star is much more complicated, with sunspots and solar flares and layers of different gases in different amounts. For a black hole, all of that is compressed down to nothing, reduced to just those two numbers and nothing else.

With that in mind, you might think a black hole should have zero Love numbers: it should be impossible to squeeze it or stretch it. This is fundamentally a question about a theory, Einstein’s theory of relativity. If we took that theory for granted, and didn’t add anything to it, what would the consequences be? Would black holes have zero Love number, or not?

It turns out black holes do have zero Love number, if they aren’t spinning. If they are, things are more complicated: a few calculations made it look like spinning black holes also had zero Love number, but just last year a more detailed proof showed that this doesn’t hold. Somehow, despite having “no hair”, you can actually “squeeze” a spinning black hole.

(EDIT: Folks on twitter pointed out a wrinkle here: more recent papers are arguing that spinning black holes actually do have zero Love number as well, and that the earlier papers confused Love numbers with a different effect. All that is to say this is still very much an active area of research!)

The physics behind neutron stars is in principle known, but in practice hard to understand. When they are formed, almost every type of physics gets involved: gas and dust, neutrino blasts, nuclear physics, and general relativity holding it all together.

Because of all this complexity, the structure of neutron stars can’t be calculated from “first principles” alone. Finding it out isn’t a question about our theories, but a question about the natural world. We need to go out and measure how neutron stars actually behave.

Love numbers are a promising way to do that. Love numbers tell you how an object gets squeezed and stretched in a gravitational field. Learning the Love numbers of neutron stars will tell us something about their structure: namely, how squeezable and stretchable they are. Already, LIGO and Virgo have given us some information about this, and ruled out a few possibilities. In future, the LISA telescope will show much more.

Returning to black holes, you might wonder what happens if we don’t stick to Einstein’s theory of relativity. Physicists expect that relativity has to be modified to account for quantum effects, to make a true theory of quantum gravity. We don’t quite know how to do that yet, but there are a few proposals on the table.

Asking for the true theory of quantum gravity isn’t just a question about some specific part of the natural world, it’s a question about the fundamental laws of physics. Can Love numbers help us answer it?

Maybe. Some theorists think that quantum gravity will change the Love numbers of black holes. Fewer, but still some, think they will change enough to be detectable, with future gravitational wave telescopes like LISA. I get the impression this is controversial, both because of the different proposals involved and the approximations used to understand them. Still, it’s fun that Love numbers can answer so many different types of questions, and teach us so many different things about physics.

Unrelated: For those curious about what I look/sound like, I recently gave a talk of outreach advice for the Max Planck Institute for Physics, and they posted it online here.

Many of you will have seen the happy news today that Avi Wigderson and László Lovász share this year’s Abel Prize (which now contends with the Fields Medal for the highest award in pure math). This is only the second time that the Abel Prize has been given wholly or partly for work in theoretical computer science, after Szemerédi in 2012. See also the articles in Quanta or the NYT, which actually say most of what I would’ve said for a lay audience about Wigderson’s and Lovász’s most famous research results and their importance (except, no, Avi hasn’t yet proved P=BPP, just taken some major steps toward it…).

On a personal note, Avi was both my and my wife Dana’s postdoctoral advisor at the Institute for Advanced Study in Princeton. He’s been an unbelievably important mentor to both of us, as he’s been for dozens of others in the CS theory community. Back in 2007, I also had the privilege of working closely with Avi for months on our Algebrization paper. Now would be a fine time to revisit Avi’s Permanent Impact on Me (or watch the YouTube video), which is the talk I gave at IAS in 2016 on the occasion of Avi’s 60th birthday.

Huge congratulations to both Avi and László!

It’s pi day today! Don’t forget to celebrate! Options include walking around in circles at 1:59pm, baking a pie, eating pie, etc! Or maybe explain what pi is to somebody… I won’t list irrational behaviour as there’s a bit too much of that already… –cvj (Above image snapped from a … Click to continue reading this post →

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In this previous blog post I noted the following easy application of Cauchy-Schwarz:

Lemma 1 (Van der Corput inequality) Let be unit vectors in a Hilbert space . Then

Proof: The left-hand side may be written as for some unit complex numbers . By Cauchy-Schwarz we have

As a corollary, correlation becomes transitive in a statistical sense (even though it is not transitive in an absolute sense):

Corollary 2 (Statistical transitivity of correlation) Let be unit vectors in a Hilbert space such that for all and some . Then we have for at least of the pairs .

Proof: From the lemma, we have

The contribution of those with is at most , and all the remaining summands are at most , giving the claim.

One drawback with this corollary is that it does not tell us which pairs correlate. In particular, if the vector also correlates with a separate collection of unit vectors, the pairs for which correlate may have no intersection whatsoever with the pairs in which correlate (except of course on the diagonal where they must correlate).

While working on an ongoing research project, I recently found that there is a very simple way to get around the latter problem by exploiting the tensor power trick:

Corollary 3 (Simultaneous statistical transitivity of correlation) Let be unit vectors in a Hilbert space for and such that for all , and some . Then there are at least pairs such that . In particular (by Cauchy-Schwarz) we have for all .

Proof: Apply Corollary 2 to the unit vectors and , in the tensor power Hilbert space .

It is surprisingly difficult to obtain even a qualitative version of the above conclusion (namely, if correlates with all of the , then there are many pairs for which correlates with for all simultaneously) without some version of the tensor power trick. For instance, even the powerful Szemerédi regularity lemma, when applied to the set of pairs for which one has correlation of , for a single , does not seem to be sufficient. However, there is a reformulation of the argument using the Schur product theorem as a substitute for (or really, a disguised version of) the tensor power trick. For simplicity of notation let us just work with real Hilbert spaces to illustrate the argument. We start with the identity

A separate application of tensor powers to amplify correlations was also noted in this previous blog post giving a cheap version of the Kabatjanskii-Levenstein bound, but this seems to not be directly related to this current application.

On Saturday (tomorrow), I'll be talking with science writer Philip Ball at the Malvern Festival of Ideas! The topic will be Science and Art, and I think it will be an interesting and fun exchange. It is free, online, and starts at 5:15 pm UK time. You can click here for the details.

I'll talk a little bit about how I came to create the non-fiction science book The Dialogues, using graphic narrative art to help frame and drive the ideas forward, and how I really wanted to re-shape what is the norm for a popular science book, where somehow using just prose to talk about serious scientific ideas has become regarded as the pinnacle of achievement - this runs counter to so many things, not the least being the fact that scientists themselves don't just use prose to communicate with each other!

But anyway, that's just the beginning of it all. Philip and I will talk about [...] Click to continue reading this post →

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Previous set of notes: Notes 2. Next set of notes: Notes 4.

On the real line, the quintessential examples of a periodic function are the (normalised) sine and cosine functions , , which are -periodic in the sense that

additive -torus

What about periodic functions on the complex plane? We can start with singly periodic functions which obey a periodicity relationship for all in the domain and some period ; such functions can also be viewed as functions on the “additive cylinder” (or equivalently ). We can rescale as before. For holomorphic functions, we have the following characterisations:

Proposition 1 (Description of singly periodic holomorphic functions)

• (i) Every -periodic entire function has an absolutely convergent expansion

  

  where is the nome , and the are complex coefficients such that

  

  Conversely, every doubly infinite sequence of coefficients obeying (2) gives rise to a -periodic entire function via the formula (1).
• (ii) Every bounded -periodic holomorphic function on the upper half-plane has an expansion

  

  where the are complex coefficients such that

  

  Conversely, every infinite sequence obeying (4) gives rise to a -periodic holomorphic function which is bounded away from the real axis (i.e., bounded on for every ).

In both cases, the coefficients can be recovered from by the Fourier inversion formula

for any in (in case (i)) or (in case (ii)).

Proof: If is -periodic, then it can be expressed as for some function on the “multiplicative cylinder” , since the fibres of the map are cosets of the integers , on which is constant by hypothesis. As the map is a covering map from to , we see that will be holomorphic if and only if is. Thus must have a Laurent series expansion with coefficients obeying (2), which gives (1), and the inversion formula (5) follows from the usual contour integration formula for Laurent series coefficients. The converse direction to (i) also follows by reversing the above arguments.

For part (ii), we observe that the map is also a covering map from to the punctured disk , so we can argue as before except that now is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3) extends to be holomorphic on all of , and thus has a Taylor expansion for some coefficients obeying (4). The argument now proceeds as with part (i).

The additive cylinder and the multiplicative cylinder can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder , but we will not use this identification here.

Now let us turn attention to doubly periodic functions of a complex variable , that is to say functions that obey two periodicity relations

Within the world of holomorphic functions, the collection of doubly periodic functions is boring:

Proposition 2 Let be an entire doubly periodic function (with periods linearly independent over ). Then is constant.

To obtain more interesting examples of doubly periodic functions, one must therefore turn to the world of meromorphic functions – or equivalently, holomorphic functions into the Riemann sphere . As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function

— 1. Doubly periodic functions —

Throughout this section we fix two complex numbers that are linearly independent over , which then generate a lattice .

We now study the doubly periodic meromorphic functions with respect to these periods that are not identically zero. We first observe some constraints on the poles of these functions. Of course, by periodicity, the poles will themselves be periodic, and thus the set of poles forms a finite union of disjoint cosets of the lattice . Similarly, the zeroes form a finite union of disjoint cosets . Using the residue theorem, we can obtain some further constraints:

Lemma 3 (Consequences of residue theorem) Let be a doubly periodic meromorphic function (not identically zero) with periods , poles at , and zeroes at .

• (i) The sum of residues at each (i.e., we sum one residue per coset) is equal to zero.
• (ii) The number of poles (counting multiplicity, but only counting once per coset) is equal to the number of zeroes (again counting multiplicity, and once per coset).
• (iii) The sum of the poles (counting multiplicity, and working in the group ) is equal to the sum of the zeroes .

Proof: For (i), we first apply a translation so that none of the pole cosets intersects the fundamental parallelogram boundary ; this of course does not affect the sum of residues. Then, by the residue theorem, the sum in (i) is equal to the expression

(the sign depends on whether this contour is oriented counter-clockwise or clockwise). But from the double periodicity we see that the integral vanishes (the contribution of parallel pairs of edges cancel each other). For part (ii), apply part (i) to the logarithmic derivative , which is also doubly periodic.

For part (iii), we again translate so that none of the pole or zero cosets intersects , noting from part (ii) that any such translation affects the sum of poles and sum of zeroes by the same amount. By the residue theorem, it now suffices to show that

lies in the lattice . But one can rewrite this using the double periodicity as

so it suffices to show that is an integer for . But (a slight modification of) the argument principle shows that this number is precisely the winding number around the origin of the image of under the map , and the claim follows.

This lemma severely limits the possible number of behaviors for the zeroes and poles of a meromorphic function. To formalise this, we introduce some general notation:

Definition 4 (Divisors)

• (i) A divisor on the torus is a formal integer linear combination , where ranges over a finite collection of points in the torus (i.e., a finite collection of cosets ), and are integers, with the obvious additive group structure; equivalently, the space of divisors is the free abelian group with generators for (with the convention ).
• (ii) The number is the degree of a divisor , the point is the sum of , and each is the order of the divisor at (with the convention that the order is if does not appear in the sum). A divisor is non-negative (or effective) if for all . We write if is non-negative (i.e., the order of is greater than or equal to that of at every point , and if and .
• (iii) Given a meromorphic function (or equivalently, a doubly periodic function ) that is not identically zero, the principal divisor is the divisor , where ranges over the zeroes and poles of , and is the order of the zero (if is a zero) or negative the order of the pole (if is a pole).
• (iv) Given a divisor , we define to be the space of all meromorphic functions that are either zero, or are such that . That is to say, consists of those meromorphic functions that have at most a pole of order at if is positive, or at least zero of order if is negative.

A divisor can be viewed as an abstraction of the concept of a set of zeroes and poles (counting multiplicity). Observe that principal divisors obey the laws , when are meromorphic and non-zero. In particular, the space of principal divisors is a subgroup of the space of all divisors. By Lemma 3(ii), all principal divisors have degree zero, and from Lemma 3(iii), all principal divisors have sum zero as well. Later on we shall establish the converse claim that every divisor of degree and sum zero is a principal divisor; see Exercise 7.

Remark 5 One can define divisors on other Riemann surfaces, such as the complex plane . Observe from the fundamental theorem of algebra that if one has two non-zero polynomials , then if and only if divides as a polynomial. This may give some hint as to the origin of the terminology “divisor”. The machinery of divisors turns out to have a rich algebraic and topological structure when applied to more general Riemann surfaces than tori, for instance enabling one to associate an abelian variety (the Jacobian variety) to every algebraic curve; see these 246C notes for further discussion.

It is easy to see that is always a vector space. All non-zero meromorphic functions belong to at least one of the , namely , so to classify all the meromorphic functions on , it would suffice to understand what all the spaces are.

Liouville’s theorem (in the form of Proposition 2) tells us that all elements of – that is to say, the holomorphic functions on – are constant; thus is one-dimensional. If is a negative divisor, the elements of are thus constant and have at least one zero, thus in these cases is trivial.

Now we gradually work our way up to higher degree divisors . A basic fact, proven from elementary linear algebra, is that every time one adds a pole to , the dimension of the space only goes up by at most one:

Lemma 6 For any divisor and any , is a subspace of of codimension at most one. In particular, is finite-dimensional for any .

Now consider the space for some point . Lemma 6 tells us that the dimension of this space is either one or two, since was one-dimensional. The space consists of functions that possibly have a simple pole at most at , and no other poles. But Lemma 3(i) tells us that the residue at has to vanish, and so is in fact in and thus is constant. (One could also argue here using the other two parts of Lemma 2; how?) So is no larger than , and is thus also one-dimensional.

Now let us study the space – the space of meromorphic functions that have at most a double pole at and no other poles. Again, Lemma 6 tells us that this space is one or two dimensional. To figure out which, we can normalise to be the origin coset . The question is now whether there is a doubly periodic meromorphic function that has a double pole at each point of . A naive candidate for such a function would be the infinite series

Now we show that is doubly periodic, thus and for . We just prove the first identity, as the second is analogous. From (6) we have

By construction, lies in , and is clearly non-constant. Thus is two-dimensional, being spanned by the constant function and . By translation, we see that is two-dimensional for any other point as well.

From (6) it is also clear that the function is even: . In particular, for any avoiding the half-lattice (so that and occupy different locations in the torus ), the function has a zero at both and . By Lemma 3(ii) there are no other zeroes of this function (and this claim is also consistent with Lemma 3(iii)); thus the divisor of this function is given by

Exercise 7 (Classification of principal divisors)

• (i) Let be four points such that . Show that the divisor is a principal divisor. (Hint: if are all distinct, use the function

  

  

  If some of the coincide, use some transformed version of the Weierstrass elliptic function instead.)
• (ii) Show that every divisor of degree zero and sum zero is a principal divisor.
• (iii) Two divisors are said to be equivalent if their difference is a principal divisor. Show that two divisors are equivalent if and only if they have the same degree and same sum.
• (iv) Show that the quotient group (known as the divisor class group or Picard group) is isomorphic (as a group) to , and that the subgroup arising from degree zero divisors (also known as the Jacobian variety of ) is isomorphic to .

Now let us study the space , where we again normalise for sake of discussion. Lemma 6 tells us that this space is two or three dimensional, being spanned by , , and possibly one other function. Note that the derivative of the meromorphic function is also doubly periodic with a triple pole at , so it lies in and is not a linear combination of or (as these have a lower order singularity at ). Thus is three-dimensional, being spanned by . A formal term-by-term differentiation of (6) gives (7). To justify (7), observe that the arguments that demonstrated the meromorphicity of the right-hand side of (6) also show the meromorphicity of (7). From Fubini’s theorem, the fundamental theorem of calculus, and (6) we see that

Turning now to , we could differentiate yet again to generate a doubly periodic function with a fourth order pole at the origin, but we can also work with the square of the Weierstrass function. From Lemma 6 we conclude that is four-dimensional and is spanned by . In a similar fashion, is a five-dimensional space spanned by .

Something interesting happens though at . Lemma 6 tells us that this space is the span of , and possibly one other function, which will have a pole of order six at the origin. Here we have two natural candidates for such a function: the cube of the Weierstrass function, and the square of its derivative. Both have a pole of order exactly six and lie in , and so must be a linear combination of . But since are even and are odd, must in fact just be a linear combination of . To work out the precise combination, we see by repeating the derivation of (7) that

Exercise 8 Derive (8) directly from Proposition 2 by showing that the difference between the two sides is doubly periodic and holomorphic after removing singularities.

Exercise 9 (Classification of doubly periodic meromorphic functions)

• (i) For any , show that has dimension , and every element of this space is a polynomial combination of .
• (ii) Show that every doubly periodic meromorphic function is a rational function of .

We have an alternate form of (8):

Exercise 10 Define the roots , , .

• (i) Show that are distinct, and that

  

  for all . (Hint: use (10).) Conclude in particular that , , and .
• (ii) Show that the modular discriminant

  

  is equal to , and is in particular non-zero.

If we now define the elliptic curve

Lemma 11 The map defined by (11) is a bijection between and .

Among other things, this lemma implies that the elliptic curve is topologically equivalent (i.e., homeomorphic to) a torus, which is not an entirely obvious fact (though if one squints hard enough, the real analogue of an elliptic curve does resemble a distorted slice of a torus embedded in ).

Proof: Clearly is the only point that maps to , and (from (10)) the half-periods are the only points that map to . It remains to show that all the other points arise via from exactly one element of . The function has exactly two zeroes by Lemma 3(ii), which lie at for some as is even; since , is not equal to , hence is not a half-period. As is odd, the map (11) must therefore map to the two points of the elliptic curve that lie above , and the claim follows.

Analogously to the Riemann sphere , the elliptic curve can be given the structure of a Riemann surface, by prescribing the following charts:

• (i) When is a point in other than or , then locally is the graph of a holomorphic branch of the square root of near , and one can use as a coordinate function in a sufficiently small neighbourhood of .
• (ii) In the neighbourhood of for some , the function has a simple zero at and so has a local inverse that maps a neighbourhood of to a neighbourhood of , and a point sufficiently near can be parameterised by . One can then use as a coordinate function in a neighbourhood of .
• (iii) A neighbourhood of consists of and the points in the remaining portion of with sufficiently large; then is asymptotic to a square root of , so in particular and should both go to zero as goes to infinity in . We rewrite the defining equation of the curve in terms of and as . The function has a simple zero at zero and thus has a holomorphic local inverse that maps to , and we have in a neighbourhood of infinity. We can then use as a coordinate function in a neighbourhood of , with the convention that this coordinate function vanishes at infinity.

It is then a tedious but routine matter to check that has the structure of a Riemann surface. We then claim that the bijection defined by (11) is holomorphic, and thus a complex diffeomorphism of Riemann surfaces. In the neighbourhood of any point of the torus other than the origin , maps to a neighbourhood of finite point of , including the three points , the holomorphicity is a routine consequence of composing together the various local holomorphic functions and their inverses. In the neighbourhood of the origin , maps for small to a point of with a Laurent expansion

While we have shown that all tori are complex diffeomorphic to elliptic curves, the converse statement that all elliptic curves are diffeomorphic to tori will have to wait until the next section for a proof, once we have set up the machinery of modular forms.

Exercise 12 (Group law on elliptic curves)

• (i) Let be three distinct elements of the torus that are not equal to the origin . Show that if and only if the three points , , are collinear in , in the sense that they lie on a common complex line for some complex numbers with not both zero.
• (ii) What happens in (i) if (say) and agree? What about if ?
• (iii) Using (i), (ii), give a purely geometric definition of a group addition law on the elliptic curve which is compatible with the group addition law on the torus via (11). (We remark that the associativity property of this law is not obvious from a purely geometric perspective, and is related to the Cayley-Bacharach theorem in classical geometry; see this previous blog post.)

Exercise 13 (Addition law) Show that for any lying in distinct cosets of , one has

Exercise 14 (Special case of Riemann-Roch)

• (i) Show that if two divisors are equivalent (in the sense of Exercise 7(iii)), then the vector spaces and are isomorphic (in particular, they have the same dimension).
• (ii) If is a divisor of some degree , show that the dimension of the space is zero if , equal to if , equal to if and has non-zero sum, and equal to if and has zero sum. (Hint: use Exercise 7(iii) and part (i) to replace with an equivalent divisor of a simple form.)
• (iii) Verify the identity

  

  for any divisor . This is a special case of the more general Riemann-Roch theorem, discussed in these 246C notes.

Exercise 15 (Elliptic integrals)

• (i) Show that is a covering map from to the thrice-punctured plane .
• (ii) Let be a contour in from some complex number to another complex number , and suppose that there is a holomorphic branch of the square root of in a neighbourhood of . Show that there exists complex numbers with , such that

  

Remark 16 The integral is an example of an elliptic integral; many other elliptic integrals (such as the integral arising when computing the perimeter of an ellipse) can be transformed into this form (or into a closely related integral) by various elementary substitutions. Thus the Weierstrass elliptic function can be employed to evaluate elliptic integrals, which may help explain the terminology “elliptic” that occurs throughout these notes. In 246C notes we will introduce the notion of a meromorphic -form on a Riemann surface. The identity (12) can then be interpreted in this language as the differential form identity , where are the standard coordinates on the elliptic curve ; the meromorphic -form is initially only defined on outside of the four points , but this identity in fact reveals that the form extends holomorphically to all of ; it is an example of what is known as an Abelian differential of the first kind.

Remark 17 The elliptic curve (for various choices of parameters ) can be defined in other fields than the complex numbers (though some technicalities arise in characteristic two and three due to the pathological behaviour of the discriminant in those cases). On the other hand, the Weierstrass elliptic function is a transcendental function which only exists in complex analysis and does not have a direct analogue in other fields. So this connection between elliptic curves and tori is specific to the complex field. Nevertheless, many facts about elliptic curves that were initially discovered over the complex numbers through this complex-analytic link to tori, were then reproven by purely algebraic means, so that they could be extended without much difficulty to many other fields than the complex numbers, such as finite fields. (For instance, the role of the complex torus can be replaced by the Jacobian variety, which was briefly introduced in Exercise 7.) Elliptic curves over such fields are of major importance in number theory (and cryptography), but we will not discuss these topics further here.

— 2. Modular functions and modular forms —

In Exercise 32 of 246A Notes 5, it was shown that two tori and are complex diffeomorphic if and only if one has

Let us write for the set of all tori quotiented by the equivalence relation of complex diffeomorphism; this is the (classical, level one, noncompactified) modular curve. By the above discussion, this set can also be identified with the set of pairs of linearly independent (over ) complex numbers quotiented by the equivalence relation given implicitly by (13). One can simplify this a little by observing that any pair is equivalent to for some in the upper half-plane , namely either or depending on the relative phases of and ; this quantity is known as the period ratio. From (13) (swapping the roles of as necessary), we then see that two pairs are equivalent if one has

If we use the relation to write

Exercise 18 Suppose that is an element of which is fixed by some element of which is not the identity or negative identity. Let be the lattice .

• (i) Show that obeys a dilation invariance for some complex number which is not real.
• (ii) Show that the dilation in part (i) must have magnitude one. (Hint: look at a non-zero element of of minimal magnitude.)
• (iii) Show that there is no rotation invariance with . (Hint: again, work with a non-zero element of of minimal magnitude, and use the fact that is closed under addition and subtraction. It may help to think geometrically and draw plenty of pictures.)
• (iv) Show that is equivalent to either the Gaussian lattice or the Eisenstein lattice , and conclude that the period ratio is equivalent to either or .

Remark 19 The conformal map on the complex numbers preserves the Gaussian integers and thus descends to a conformal map from the Gaussian torus to itself; similarly the conformal map preserves the Eisenstein integers and thus descends to a conformal map from the Eisenstein torus to itself. These rare examples of complex tori equipped with additional conformal automorphisms are examples of tori (or elliptic curves) endowed with complex multiplication. There are additional examples of elliptic curves endowed with conformal endomorphisms that are still considered to have complex multiplication, and have a particularly nice algebraic number theory structure, but we will not pursue this topic further here.

Remark 20 The fact that the action of on lattices contains fixed points is somewhat annoying, as it prevents one from immediately viewing the modular curve as a Riemann surface. However by passing to a suitable finite index subgroup of , one can remove these fixed points, leading to a theory that is cleaner in some respects. For instance, one can work with the congruence group , which roughly speaking amounts to decorating the lattices (or their tori ) with an additional “-marking” that eliminates the fixed points. This leads to a modification of the theory which is for instance well suited for studying theta functions; the role of the -invariant in the discussion below is then played by the modular lambda function , which also gives a uniformisation of the twice-punctured complex plane . However we will not develop this parallel theory further here.

If we let be the elements of not equivalent to or , and the equivalence class of tori not equivalent to the Gaussian torus or the Eisenstein torus , then can be viewed as the quotient of the Riemann surface by the free and proper action of , so it has the structure of a Riemann surface; can thus be thought of as the Riemann surface with two additional points added. Later on we will also add a third point (known as the cusp) to the Riemann surface to compactify it to .

A function on the modular curve can be thought of, equivalently, as a function that is -invariant in the sense that for all and , or equivalently that one has the identity

We define a modular function to be a meromorphic function on that obeys the condition (15), and which also has at most polynomial growth at the cusp in the sense that one has a bound of the form

Exercise 21

• (i) Let be two elements of with . Show that it is possible to transform the quadruplet to the quadruplet after a finite number of applications of the moves

  

  

  and

  

  ({Hint: use the principle of infinite descent, applying the moves in a suitable order to decrease the lengths of and when the dot product is not too small, taking advantage of the Lagrange identity to determine when this procedure terminates. It may help to think geometrically and draw plenty of pictures.) Conclude that the two matrices (17) generate all of .
• (ii) Show that a function obeys (15) if and only if it obeys both (18) and (19).

Exercise 22 (Standard fundamental domain) Define the standard fundamental domain for to be the set

• (i) Show that every lattice is equivalent (up to dilations) to a lattice with , with unique except when it lies on the boundary of , in which case the lack of uniqueness comes either from the pair for some , or from the pair for some . (Hint: arrange so that is a non-zero element of of minimal magnitude.)
• (ii) Show that can be identified with the fundamental domain after identifying with for , and with for . Show also that the set is then formed the same way, but first deleting the points from .

We will give some examples of modular functions (beyond the trivial example of constant functions) shortly, but let us first observe that when one differentiates a modular function one gets a more general class of function, known as a modular form. In more detail, observe from (14) that the derivative of the Möbius transformation is , and hence by the chain rule and (15) the derivative of a modular function would obey the variant law

Exercise 23 Let be a natural number. Show that a function obeys (20) if and only if it is -periodic in the sense of (18) and obeys the law

for all .

Exercise 24 (Lattice interpretation of modular forms) Let be a modular form of weight . Show that there is a unique function from lattices to complex numbers such that

for all , and such that one has the homogeneity relation

for any lattice and non-zero complex number .

Somewhat analogously to how we used Lemma 3 to investigate the spaces for divisors on a torus, we will investigate the space of modular forms via the following basic formula:

Theorem 25 (Valence formula) Let be a modular form of weight , not identically zero. Then we have

where is the order of vanishing of at , is the order of vanishing of (i.e., viewed as a function of the nome ) at , and ranges over the zeroes of that are not equivalent to , with just one zero counted per equivalence class. (This will be a finite sum.)

Informally, this formula asserts that the point only “deserves” to be counted in with multiplicity due to its order stabiliser, while the point only “deserves” to be counted in with multiplicity due to its order stabiliser. (The cusp has an infinite stabiliser, but this is compensated for by taking the order with respect to the nome variable rather than the period ratio variable .) The general philosophy of weighting points by the reciprocal of the order of their stabiliser occurs throughout mathematics; see this blog post for more discussion.

Proof: Firstly, from Exercise 22, we can place all the zeroes in the fundamental domain . When parameterised in terms of the nome , this domain is compact, hence has only finitely many zeros, so the sum in (22) is finite.

As in the proof of Lemma 3(ii), we use the residue theorem. For simplicity, let us first suppose that there are no zeroes on the boundary of the fundamental domain except possibly at the cusp . Then for large enough, we have from the residue theorem that

Suppose now that there is a zero on the right edge of , and hence also on the left edge by periodicity, for some . One can account for this zero by perturbing the contour to make a little detour to the right of (e.g., by a circular arc), and a matching detour to the right of . One can then verify that the same argument as before continues to work, with this boundary zero being counted exactly once. Similarly, if there is a zero on the left arc for some , and hence also at by modularity, one can make a detour slightly above and slightly below (with the two detours being related by the transform to ensure cancellation), and again we can argue as before. If instead there is a zero at , one makes an (approximately) semicircular detour above ; in this case the detour does not cancel out, but instead contributes a factor of in the limit as the radius of the detour goes to zero. Finally, if there is a zero at (and hence also at ), one makes detours by two arcs of angle approximately at these two points; these two (approximate) sixth-circles end up contributing a factor of in the limit, giving the claim.

Exercise 26 (Quick applications of the valence formula)

• (i) Let be a modular form of weight , not identically zero. Show that is equal to or an even number that is at least .
• (ii) (Liouville theorem for ) If is a modular form of weight zero, show that it is constant. (Hint: apply the valence theorem to various shifts of by constant.)
• (iii) For , show that the vector space of modular forms of weight is at most one dimensional. (Hint: in these cases, there are a very limited number of solutions to the equation with natural numbers.)
• (iv) Show that there are no cusp forms of weight when or , and for the space of cusp forms of weight is at most one dimensional.
• (v) Show that for any , the space of cusp forms of weight is a subspace of the space of modular forms of weight of codimension at most one, and that both spaces are finite-dimensional.

A basic example of modular forms are provided by the Eisenstein series

Exercise 27 Give an alternate proof that is a cusp form, not using the valence identity, by first establishing that and .

We can now create our first non-trivial modular function, the -invariant

Lemma 28 We have and .

Proof: Using the rotation symmetry we see that , hence which implies that and hence . Similarly, using the rotation symmetry we have , hence . (One can also use the valence formulae to get the vanishing ).

Being modular, we can think of as a map from to . We have the following fundamental fact:

Proposition 29 The map is a bijection.

Proof: Note that for any , if and only if is a zero of . It thus suffices to show that for every , the zeroes of the function in consist of precisely one orbit of . This function is a modular form of weight that does not vanish at infinity (since does not vanish while does). By the valence formula, we thus have

As the orders are all natural numbers, some case checking reveals that there are now only three possibilities:

• has a simple zero at precisely one -orbit, not equivalent to or .
• has a double zero at (and equivalent points), and no other zeroes.
• has a triple zero at (and equivalent points), and no other zeroes.

In any of these three cases, the claim follows.

Note that this proof also shows that has a double zero at and has a triple zero at , but that has a simple zero for any not equivalent to or .

We can now give the entire modular curve the structure of a Riemann surface by declaring to be the coordinate function. This is compatible with the existing Riemann surface structure on since was already holomorphic on this portion of the curve. Any modular function can then factor as for some meromorphic function that is initially defined on the punctured complex plane ; but from meromorphicity of on and at infinity we see that blows up at an at most polynomial rate as one approaches , , or , and so is in fact a meromorphic function on the entire Riemann sphere and is thus a rational function (Exercise 19 of 246A Notes 4). We conclude

Proposition 30 Every modular function is a rational function of the -invariant .

Conversely, it is clear that every rational function of is modular, thus giving a satisfactory description of the modular functions.

Exercise 31 Show that every modular function is the ratio of two modular forms of equal weight (with the denominator not identically zero).

Exercise 32 (All elliptic curves are tori) Let be two complex numbers with . Show that there is a lattice such that and , so in particular the elliptic curve

is complex diffeomorphic to a torus .

Remark 33 By applying some elementary algebraic geometry transformations one can show that any (smooth, irreducible) cubic plane curve generated by a polynomial of degree is a Riemann surface complex diffeomorphic to a torus after adding some finite number of points at infinity; also, some degree curves such as

can also be placed in this form. However we will not detail the required transformations here.

A famous application of the theory of the -invariant is to give a short Riemann surface-based proof of the the little Picard theorem (first proven in Theorem 55 of 246A Notes 4):

Theorem 34 (Little Picard theorem) Let be entire and non-constant. Then omits at most one point of .

Proof: Suppose for contradiction that omits at least two points of . By applying a linear transformation, we may assume that omits the points and . Then is a holomorphic function from to . Since the domain is simply connected, lifts to a holomorphic function from to . Since is complex diffeomorphic to a disk, this lift must be constant by Liouville’s theorem, hence is constant as required. (This is essentially Picard’s original proof of this theorem.)

The great Picard theorem can also be proven by a more sophisticated version of these methods, but it requires some study of the possible behavior of elements of ; see Exercise 37 below.

All modular forms are -periodic, and hence by Proposition 1 should have a Fourier expansion, which is also a Laurent expansion in the nome. As it turns out, the Fourier coefficients often have a highly number-theoretic interpretation. This can be illustrated with the Eisenstein series ; here we follow the treatment in Stein-Shakarchi. To compute the Fourier coefficients we first need a computation:

Exercise 35 Let and , and let be the nome. Establish the identity

in two different ways:

• (i) By applying the Poisson summation formula (Proposition 3(v) of Notes 2).
• (ii) By first establishing the identity

  

  by applying Proposition 1 to the difference of the two sides, and differentiating in . (It is also possible to establish (27) from differentiating and then manipulating the identities in Exercises 25 or 27 of Notes 1.)

From (25), (26) (and symmetry) one has

Remark 36 If one expands out a few more terms in the above expansions, one can calculate

The various coefficients in here have several remarkable algebraic properties. For instance, applying this expansion at for a natural number , so that , one obtains the approximation

Now for certain values of , most famously , the torus admits a complex multiplication that allows for computation of the -invariant by algebraic means (think of this as a more advanced version of Lemma 28; it is closely related to the fact that the ring of algebraic integers in admit unique factorisation, see these previous notes for some related discussion). For instance, one can eventually establish that

which eventually leads to the famous approximation

(first observed by Hermite, but also attributed to Ramanujan via an April Fools’ joke of Martin Gardner) which is accurate to twelve decimal places. The remaining coefficients have a remarkable interpretation as dimensions of components of a certain representation of the monster group known as the moonshine module, a phenomenon known as monstrous moonshine. For instance, the smallest irreducible representation of the monster group has dimension , precisely one less than the coefficient of . The Fourier coefficients of the (normalised) modular discriminant,

form a sequence known as the Ramanujan function and obeys many remarkable properties. For instance, there is the totally non-obvious fact that this function is multiplicative in the sense that whenever are coprime; see Exercise 43.

Exercise 37 (Great Picard theorem)

• (i) Show that every fractional linear transformation on with , is either of finite order (elliptic case), conjugate to a translation for some after conjugating by another fractional linear transformation (parabolic case), or conjugate to a dilation for some after conjugating by another fractional linear transformation (hyperbolic case). (Hint: study the eigenvalues and eigenvectors of , based on the value of the trace and in particular whether the magnitude of the trace is less than two, equal to two, or greater than two. Note that the trace also has to be an integer.)
• (ii) Let be holomorphic. Show that there exists a holomorphic function such that for all , as well as a fractional linear transformation with and such that for all .
• (iii) If the transformation in (ii) is in the elliptic case of (i), show that is bounded in a neighbourhood of , and hence has a removable singularity at the origin. (Hint: will have some finite period and can thus be studied using Proposition 1 after applying a Möbius transform to map to a disk.)
• (iv) If the transformation in (ii) is in the hyperbolic case of (i), show that is bounded in a neighbourhood of , and hence has a removable singularity at the origin. (Hint: The standard branch of maps to an annulus, and is invariant with respect to the dilation action . Use this to create a bounded -periodic holomorphic function on .)
• (v) If the transformation in (ii) is in the parabolic case of (i), show that exhibits at most polynomial growth as one approaches , and hence has at most a pole at the origin. (Hint: If for instance , then is -periodic and takes values in , and one can now repeat the arguments of (iii). Also use the expansion (28).)
• (vi) Use the previous parts of this exercise to give another proof of the great Picard theorem (Theorem 56 of 245A Notes 4): the image of a holomorphic function in a punctured disk with an essential singularity at omits at most one value of .

Exercise 38 (Dimension of space of modular forms)

• (i) If is an even natural number, show that the dimension of the space of modular forms of weight is equal to except when is equal to mod , in which case it is equal to . (Hint: for this follows from Exercise 26; to cover the larger ranges of , use the modular discriminant to show that the space of cusp forms of weight is isomorphic to the space of modular forms of weight .
• (ii) If is an even natural number, show that a basis for the space of modular forms of weight is provided by the powers where range over natural numbers (including zero) with .

Thus far we have constructed modular forms and modular functions starting from Eisenstein series . There is another important, and seemingly quite different, way to generate modular forms coming from theta functions. Typically these functions are not quite modular in the sense given in these notes, but are close enough that after some manipulation one can transform theta functions into modular forms. The simplest example of a theta function is the Jacobi theta function

Exercise 39 Define the Dedekind eta function by the formula

or in terms of the nome

where is one of the roots of .

• (i) Establish the modified -periodicity

  

  and the modified modularity

  

  using the standard branch of the square root. (Hint: a direct application of Poisson summation applied to gives a sum that looks somewhat like but with different numerical constants (in particular, one sees terms like instead of arising). Split the index of summation into three components , , based on the residue classes modulo and rearrange each component separately.)
• (ii) Establish the identity

  

  (Hint: show that both sides are cusp forms of weight that vanish like near the cusp.)

Remark 40 The relationship between and the power of the eta function can be interpreted (after some additional effort) as a relation between the modular discriminant and the theta function of a certain highly symmetric -dimensional lattice known as the Leech lattice, but we will not pursue this connection further here.

The function has a remarkable factorisation coming from Euler’s pentagonal number theorem

Theorem 41 (Jacobi triple product identity) For any and , one has

Observe that by replacing by and with we have

Proof: Let us denote the left-hand side and right-hand side of (33) by and respectively. For fixed , both sides are clearly holomorphic in , with . Our strategy in showing that and agree (following Stein-Shakarchi) is to first observe that they have many of the same periodicity properties. We clearly have -periodicity

Remark 42 Another equivalent form of (32) is

where is the partition function of – the number of ways to represent as the sum of positive integers (up to rearrangement). Among other things, this formula can be used to ascertain the asymptotic growth of (which turns out to roughly be of the order of , as famously established by Hardy and Ramanujan).

Theta functions can be used to encode various number-theoretic quantities involving quadratic forms, such as sums of squares. For instance, from (30) and collecting terms one obtains the formula

Exercise 43 (Hecke operators) Let be a natural number.

• (i) If is a modular form of weight , and is the corresponding function on lattices given by Exercise 24, and is a positive natural number, show that there is a unique modular form of weight whose corresponding function on lattices is related to by the formula

  

  where the sum ranges over all sublattices of whose index is equal to . Show that is a linear operator on the space of weight modular forms that also maps the space of weight cusp forms to itself; this operator is known as a Hecke operator.
• (ii) Give the more explicit formula

  

• (iii) Show that the Hecke operators all commute with each other, thus whenever is a modular form of weight and are positive natural numbers. Furthermore show that if are coprime.
• (iv) If is a modular form of weight with Fourier expansion , show that for any coprime positive integers that the coefficient of is equal to .
• (v) Establish the multiplicativity of the Ramanujan tau function (the Fourier coefficients of the modular discriminant). (Hint: use the one-dimensionality of the space of cusp forms of weight to conclude that is a simultaneous eigenfunction of the Hecke operators.)

Simultaneous eigenfunctions of the Hecke operators are known as Hecke eigenfunctions and are of major importance in number theory.

The craziest challenge I’ve undertaken hasn’t been skydiving; sailing the Amazon on a homemade raft; scaling Mt. Everest; or digging for artifacts atop a hill in a Middle Eastern desert, near midday, during high summer.¹ The craziest challenge has been to study the possibility that quantum phenomena affect cognition significantly. 

Most physicists agree that quantum phenomena probably don’t affect cognition significantly. Cognition occurs in biological systems, which have high temperatures, many particles, and watery components. Such conditions quash entanglement (a relationship that quantum particles can share and that can produce correlations stronger than any produceable by classical particles). 

Yet Matthew Fisher, a condensed-matter physicist, proposed a mechanism by which entanglement might enhance coordinated neuron firing. Phosphorus nuclei have spins (quantum properties similar to angular momentum) that might store quantum information for long times when in Posner molecules. These molecules may protect the information from decoherence (leaking quantum information to the environment), via mechanisms that Fisher described.

I can’t check how correct Fisher’s proposal is; I’m not a biochemist. But I’m a quantum information theorist. So I can identify how Posners could process quantum information if Fisher were correct. I undertook this task with my colleague Elizabeth Crosson, during my PhD. 

Experimentalists have begun testing elements of Fisher’s proposal. What if, years down the road, they find that Posners exist in biofluids and protect quantum information for long times? We’ll need to test whether Posners can share entanglement. But detecting entanglement tends to require control finer than you can exert with a stirring rod. How could you check whether a beakerful of particles contains entanglement?

I asked that question of Adam Bene Watts, a PhD student at MIT, and John Wright, then an MIT postdoc and now an assistant professor in Texas. John gave our project its codename. At a meeting one day, he reported that he’d watched the film Avengers: Endgame. Had I seen it? he asked.

No, I replied. The only superhero movie I’d seen recently had been Ant-Man and the Wasp—and that because, according to the film’s scientific advisor, the movie riffed on research of mine. 

Go on, said John.

Spiros Michalakis, the Caltech mathematician in charge of this blog, served as the advisor. The film came out during my PhD; during a meeting of our research group, Spiros advised me to watch the movie. There was something in it “for you,” he said. “And you,” he added, turning to Elizabeth. I obeyed, to hear Laurence Fishburne’s character tell Ant-Man that another character had entangled with the Posner molecules in Ant-Man’s brain.² 

John insisted on calling our research Project Ant-Man.

John and Adam study Bell tests. Bell test sounds like a means of checking whether the collar worn by your cat still jingles. But the test owes its name to John Stewart Bell, a Northern Irish physicist who wrote a groundbreaking paper in 1964. 

Say you’d like to check whether two particles share entanglement. You can run an experiment, described by Bell, on them. The experiment ends with a measurement of the particles. You repeat this experiment in many trials, using identical copies of the particles in subsequent trials. You accumulate many measurement outcomes, whose statistics you calculate. You plug those statistics into a formula concocted by Bell. If the result exceeds some number that Bell calculated, the particles shared entanglement.

We needed a variation on Bell’s test. In our experiment, every trial would involve hordes of particles. The experimentalists—large, clumsy, classical beings that they are—couldn’t measure the particles individually. The experimentalists could record only aggregate properties, such as the intensity of the phosphorescence emitted by a test tube.

Adam, MIT physicist Aram Harrow, and I concocted such a Bell test, with help from John. Physical Review A published our paper this month—as a Letter and an Editor’s Suggestion, I’m delighted to report.

For experts: The trick was to make the Bell correlation function nonlinear in the state. We assumed that the particles shared mostly pairwise correlations, though our Bell inequality can accommodate small aberrations. Alas, no one can guarantee that particles share only mostly pairwise correlations. Violating our Bell inequality therefore doesn’t rule out hidden-variables theories. Under reasonable assumptions, though, a not-completely-paranoid experimentalist can check for entanglement using our test. 

One can run our macroscopic Bell test on photons, using present-day technology. But we’re more eager to use the test to characterize lesser-known entities. For instance, we sketched an application to Posner molecules. Detecting entanglement in chemical systems will require more thought, as well as many headaches for experimentalists. But our paper broaches the cask—which I hope to see flow in the next Ant-Man film. Due to debut in 2022, the movie has the subtitle Quantumania. Sounds almost as crazy as studying the possibility that quantum phenomena affect cognition.

¹Of those options, I’ve undertaken only the last.

²In case of any confusion: We don’t know that anyone’s brain contains Posner molecules. The movie features speculative fiction.

Like a lot of people I’m watching WandaVision, the latest Marvel show. CJ is an MCU fanatic and this show, well-acted, imaginatively shot, and legible without extreme knowledge of Marvel lore, is a good one for us to watch together.

It has settled, on the surface, into being a more “normal” MCU show after doing a lot of really interesting stuff in the first half of the season. But weirdness remains, under the surface. For example (and now the rest of this is spoilers) — the scene where Wanda magically blasts a new rendition of her dead husband Vision out of her own abdomen is clearly shot as a childbirth scene, which makes Vision both her son and her husband, so the whole thing has suddenly taken on a Freudian cast which I don’t think is from the comics. And this explains the shock of the old expert witch Agatha Harkness, who tells Wanda she’s something that isn’t supposed to exist; she is “chaos magic,” a witch with the power to spontaneously create. Witches, traditionally, are supposed to be infertile, but Wanda is not. (This is complicated, I guess, by the fact that Harkness herself apparently has a son in comics continuity but she’s presented as married and childless here.)

Isn’t the Mind Stone placed in the middle of Vision’s forehead a little like a third eye? And isn’t death by getting that eye ripped out kind of Vision’s thing?

I know, I know, sometimes a synthezoid is just a synthezoid.