## Interview (Part 2)

21 March, 2016

Greg Bernhardt runs an excellent website for discussing physics, math and other topics, called Physics Forums. He recently interviewed me there. Since I used this opportunity to explain a bit about the Azimuth Project and network theory, I thought I’d reprint the interview here. Here is Part 2.

Tell us about your experience with past projects like “This Week’s Finds in Mathematical Physics”.

I was hired by U.C. Riverside back in 1989. I was lonely and bored, since Lisa was back on the other coast. So, I spent a lot of evenings on the computer.

We had the internet back then—this was shortly after stone tools were invented—but the world-wide web hadn’t caught on yet. So, I would read and write posts on “newsgroups” using a program called a “news server”. You have to imagine me sitting in front of an old green­-on­-black cathode ray tube monitor with a large floppy disk drive, firing up the old modem to hook up to the internet.

In 1993, I started writing a series of posts on the papers I’d read. I called it “This Week’s Finds in Mathematical Physics”, which was a big mistake, because I couldn’t really write one every week. After a while I started using it to explain lots of topics in math and physics. I wrote 300 issues. Then I quit in 2010, when I started taking climate change seriously.

Share with us a bit about your current projects like Azimuth and the n­-Café.

The n­-Category Café is a blog I started with Urs Schreiber and the philosopher David Corfield back in 2006, when all three of us realized that n­-categories are the big wave that math is riding right now. We have a bunch more bloggers on the team now. But the n­-Café lost some steam when I quit work in n­-categories and Urs started putting most of his energy into two related projects: a wiki called the nLab and a discussion group called the nForum.

In 2010, when I noticed that global warming was like a huge wave crashing down on our civilization, I started the Azimuth Project. The goal was to create a focal point for scientists and engineers interested in saving the planet. It consists of a team of people, a blog, a wiki and a discussion group. It was very productive for a while: we wrote a lot of educational articles on climate science and energy issues. But lately I’ve realized I’m better at abstract math. So, I’ve been putting more time into working with my grad students.

Around 2004 I started hearing news that sent chills up my spine ­ and what really worried me is how few people were talking about this news, at least in the US.

I’m talking about how we’re pushing the Earth’s climate out of the glacial cycle we’ve been in for over a million years, into brand new territory. I’m talking about things like how it takes hundreds or thousands of years for CO2 to exit the atmosphere after it’s been put in. And I’m talking about how global warming is just part of a bigger phenomenon: the Anthropocene. That’s a new geological epoch, in which the biosphere is rapidly changing due to human influences. It’s not just the temperature:

• About 1/4 of all chemical energy produced by plants is now used by humans.

• The rate of species going extinct is 100­–1000 times the usual background rate.

• Populations of large ocean fish have declined 90% since 1950.

• Humans now take more nitrogen from the atmosphere and convert it into nitrates than all other processes combined.

8­-9 times as much phosphorus is flowing into oceans than the natural background rate.

This doesn’t necessarily spell the end of our civilization, but it is something that we’ll all have to deal with.

So, I felt the need to alert people and try to dream up strategies to do something. That’s why in 2010 I quit work on n­-categories and started the Azimuth Project.

You have life experience on both US coasts. Which do you prefer and why?

There are some differences between the coasts, but they’re fairly minor. The West Coast is part of the Pacific Rim, so there’s more Asian influence here. The seasons are less pronounced here, because winds in the northern hemisphere blow from west to east, and the oceans serve as a temperature control system. Down south in Riverside it’s a semi­-desert, so we can eat breakfast in our back yard in January! But I live here not because I like the West Coast more. This just happens to be where my wife Lisa and I managed to get a job.

What I really like is getting out of the US and seeing the rest of the world. When you’re at cremation ritual in Bali, or a Hmong festival in Laos, the difference between regions of the US starts seeming pretty small.

But I wasn’t a born traveler. When I spent my first summer in England, I was very apprehensive about making a fool of myself. The British have different manners, and their old universities are full of arcane customs and subtle social distinctions that even the British find terrifying. But after a few summers there I got over it. First, all around the world, being American gives you a license to be clueless. If you behave any better than the worst stereotypes, people are impressed. Second, I spend most of my time with mathematicians, who are incredibly forgiving of bad social behavior as long as you know interesting theorems.

By now I’ve gotten to feel very comfortable in England. The last couple of years I’ve spent time at the quantum computation group at Oxford–the group run by Bob Coecke and Samson Abramsky. I like talking to Jamie Vicary about n­categories and physics, and also my old friend Minhyong Kim, who is a number theorist there.

I was also very apprehensive when I first visited Paris. Everyone talks about how the waiters are rude, and so on. But I think that’s an exaggeration. Yes, if you go to cafés packed with boorish tourists, the waiters will treat you like a boorish tourist—so don’t do that. If you go to quieter places and behave politely, most people are friendly. Luckily Lisa speaks French and has some friends in Paris; that opens up a lot of opportunities. I don’t speak French, so I always feel like a bit of an idiot, but I’ve learned to cope. I’ve spent a few summers there working with Paul­-André Melliès on category theory and logic.

Yau Ma Tei Market – Hong Kong

I was also intimidated when I first spent a summer in Hong Kong—and even more so when I spent a summer in Shanghai. Lisa speaks Chinese too: she’s more cultured than me, and she drags me to interesting places. My first day walking around Shanghai left me completely exhausted: everything was new! Walking down the street you see people selling frogs in a bucket, strange fungi and herbs, then a little phone shop where telephone numbers with lots of 8’s cost more, and so on: it’s a kind of cognitive assault.

But again, I came to enjoy it. And coming back to California, everything seemed a bit boring. Why is there so much land that’s not being used? Where are all the people? Why is the food so bland?

I’ve spent the most time outside the US in Singapore. Again, that’s because my wife and I both got job offers there, not because it’s the best place in the world. Compared to China it’s rather sterile and manicured. But it’s still a fascinating place. They’ve pulled themselves up from a British colonial port town to a multi­cultural country that’s in some ways more technologically advanced than the US. The food is great: it’s a mix of Chinese, Indian, Malay and pretty much everything else. There’s essentially no crime: you can walk around in the darkest alley in the worst part of town at 3 am and still feel safe. It’s interesting to live in a country where people from very different cultures are learning to live together and prosper. The US considers itself a melting-pot, but in Singapore they have four national languages: English, Mandarin, Malay and Tamil.

Most of all, it’s great to live in places where the culture and politics is different than where I grew up. But I’m trying to travel less, because it’s bad for the planet.

You’ve gained some fame for your “crackpot index”. What were your motivations for developing it? Any new criteria you’d add?

After the internet first caught on, a bunch of us started using it to talk about physics on the usenet newsgroup sci.physics.

And then, all of a sudden, crackpots around the world started joining in!

Before this, I don’t think anybody realized how many people had their own personal theories of physics. You might have a crazy uncle who spent his time trying to refute special relativity, but you didn’t realize there were actually thousands of these crazy uncles.

As I’m sure you know here at Physics Forums, crackpots naturally tend to drive out more serious conversations. If you have some people talking about the laws of black hole thermodynamics, and some guy jumps in and says that the universe is a black hole, everyone will drop what they’re doing and argue with that guy. It’s irresistible. It reminds me of how when someone brings a baby to a party, everyone will start cooing to the baby. But it’s worse.

When physics crackpots started taking over the usenet newsgroup sci.physics, I discovered that they had a lot of features in common. The Crackpot Index summarizes these common features. Whenever I notice a new pattern, I add it.

For example: if someone starts comparing themselves to Galileo and says the physics establishment is going after them like the Inquisition, I guarantee you that they’re a crackpot. Their theories could be right—but unfortunately, they’ve got delusions of grandeur and a persecution complex.

It’s not being wrong that makes someone a crackpot. Being a full­-fledged crackpot is the endpoint of a tragic syndrome. Someone starts out being a bit too confident that they can revolutionize physics without learning it first. In fact, many young physicists go through this stage! But the good ones react to criticism by upping their game. The ones who become crackpots just brush it off. They come up with an idea that they think is great, and when nobody likes it, they don’t say “okay, I need to learn more.” Instead, they make up excuses: nobody understands me, maybe there’s a conspiracy at work, etc. The excuses get more complicated with each rebuff, and it gets harder and harder for them to back down and say “whoops, I was wrong”.

When I wrote the Crackpot Index, I thought crackpots were funny. Alexander Abian claimed all the world’s ills would be cured if we blew up the Moon. Archimedes Plutonium thinks the Universe is a giant plutonium atom. These ideas are funny. But now I realize how sad it is that someone can start with an passion for physics and end up in this kind of trap. They almost never escape.

Who are some of your math and physics heroes of the past and of today?

Wow, that’s a big question! I think every scientist needs to have heroes. I’ve had a lot.

Marie Curie

When I was a kid, I was in love with Marie Curie. I wanted to marry a woman like her: someone who really cared about science. She overcame huge obstacles to get a degree in physics, discovered not one but two new elements, often doing experiments in her own kitchen—and won not one but two Nobel prizes. She was a tragic figure in many ways. Her beloved husband Pierre, a great physicist in his own right, slipped and was run over by a horse­-drawn cart, dying instantly when the wheels ran over his skull. She herself probably died from her experiments with radiation. But this made me love her all the more.

Later my big hero was Einstein. How could any physicist not have Einstein as a hero? First he came up with the idea that light comes in discrete quanta: photons. Then, two months later, he used Brownian motion to figure out the size of atoms. One month after that: special relativity, unifying space and time! Three months later, the equivalence between mass and energy. And all this was just a warmup for his truly magnificent theory of general relativity, explaining gravity as the curvature of space and time. He truly transformed our vision of the Universe. And then, in his later years, the noble and unsuccessful search for a unified field theory. As a friend of mine put it, what matters here is not that he failed: what matters is that he set physics a new goal, more ambitious than any goal it had before.

Later it was Feynman. As I mentioned, my uncle gave me Feynman’s Lectures on Physics. This is how I first learned Maxwell’s equations, special relativity, quantum mechanics. His way of explaining things with a minimum of jargon, getting straight to the heart of every issue, is something I really admire. Later I enjoyed his books like Surely You Must Be Joking. Still later I learned enough to be impressed by his work on QED.

But when you read his autobiographical books, you can see that he was a bit too obsessed with pretending to be a fun­-loving ordinary guy. A fun­-loving ordinary guy who just happens to be smarter than everyone else. In short, a self­-absorbed showoff. He could also be pretty mean to women—and in that respect, Einstein was even worse. So our heroes should not be admired uncritically.

Alexander Grothendieck

A good example is Alexander Grothendieck. I guess he’s my main math hero these days. To solve concrete problems like the Weil conjectures, he avoided brute force techniques and instead developed revolutionary new concepts that gently dissolved those problems. And these new concepts turned out to be much more important than the problems that motivated him. I’m talking about abelian categories, schemes, topoi, stacks, things like that. Everyone who really wants to understand math at a deep level has got to learn these concepts. They’re beautiful and wonderfully simple—but not easy to master. You have to really change your world view to understand them, just like general relativity or quantum mechanics. You have to rewire your neurons.

At his peak, Grothendieck seemed almost superhuman. It seems he worked almost all day and all night, bouncing his ideas off the other amazing French algebraic geometers. Apparently 20,000 pages of his writings remain unpublished! But he became increasingly alienated from the mathematical establishment and eventually disappeared completely, hiding in a village near the Pyrenees.

Which groundbreaking advances in science and math are you most looking forward to?

I’d really like to see progress in figuring out the fundamental laws of physics. Ideally, I’d like to know the Theory of Everything. Of course, we don’t even know that there is one! There could be an endless succession of deeper and deeper realizations to be had about the laws of physics, with no final answer.

If we ever do discover the Theory of Everything, that won’t be the end of the story. It could be just the beginning. For example, next we could ask why this particular theory governs our Universe. Is it necessary, or contingent? People like to chat about this puzzle already, but I think it’s premature. I think we should find the Theory of Everything first.

Unfortunately, right now fundamental physics is in a phase of being “stuck”. I don’t expect to see the Theory of Everything in my lifetime. I’d be happy to see any progress at all! There are dozens of very basic things we don’t understand.

When it comes to math, I expect that people will have their hands full this century redoing the foundations using ∞-categories, and answering some of the questions that come up when you do this. The crowd working on “homotopy type theory” is making good progress–but so far they’re mainly thinking about ∞-groupoids, which are a very special sort of ∞-category. When we do all of math using ∞-categories, it will be a whole new ballgame.

And then there’s the question of whether humanity will figure out a way to keep from ruining the planet we live on. And the question of whether we’ll succeed in replacing ourselves with something more intelligent—or even wiser.

The Milky Way and Andromeda Nebula after their first collision, 4 billion years from now

Here’s something cool: red dwarf stars will keep burning for 10 trillion years. If we, or any civilization, can settle down next to one of those, there will be plenty of time to figure things out. That’s what I hope for.

But some of my friends think that life always uses up resources as fast as possible. So one of my big questions is whether intelligent life will develop the patience to sit around and think interesting thoughts, or whether it will burn up red dwarf stars and every other source of energy as fast as it can, as we’re doing now with fossil fuels.

What does the future hold for John Baez? What are your goals?

What the future holds for me, primarily, is death.

That’s true of all of us—or at least most of us. While some hope that technology will bring immortality, or at least a much longer life, I bet most of us are headed for death fairly soon. So I try to make the most of the time I have.

I’m always re­-evaluating what I should do. I used to spend time thinking about quantum gravity and n­-categories. But quantum gravity feels stuck, and n­-category theory is shooting forward so fast that my help is no longer needed.

Climate change is hugely important, and nobody really knows what to do about it. Lots of people are trying lots of different things. Unfortunately I’m no better than the rest when it comes to the most obvious strategies—like politics, or climate science, or safer nuclear reactors, or better batteries and photocells.

The trick is finding things you can do better than other people. Right now for me that means thinking about networks and biology in a very abstract way. I’m inspired by this remark by Patten and Witkamp:

To understand ecosystems, ultimately will be to understand networks.

So that’s my goal for the next five years or so. It’s probably not be the best thing anyone can do to prepare for the Middle Anthropocene. But it may be the best thing I can do: use the math I know to help people understand the biosphere.

It may seem like I keep jumping around: from quantum gravity to n-categories to biology. But I keep wanting to think about networks, and how they change in time.

At some point I hope to retire and become a bit more of a self­-indulgent wastrel. I could write a fun book about group theory in geometry and physics, and a fun book about the octonions. I might even get around to spending more time on music!

John Baez

## Interview (Part 1)

18 March, 2016

Greg Bernhardt runs an excellent website for discussing physics, math and other topics, called Physics Forums. He recently interviewed me there. Since I used this opportunity to explain a bit about the Azimuth Project and network theory, I thought I’d reprint the interview here. Here is Part 1.

Give us some background on yourself.

I’m interested in all kinds of mathematics and physics, so I call myself a mathematical physicist. But I’m a math professor at the University of California in Riverside. I’ve taught here since 1989. My wife Lisa Raphals got a job here nine years later: among other things, she studies classical Chinese and Greek philosophy.

I got my bachelors’s degree in math at Princeton. I did my undergrad thesis on whether you can use a computer to solve Schrödinger’s equation to arbitrary accuracy. In the end, it became obvious that you can. I was really interested in mathematical logic, and I used some in my thesis—the theory of computable functions—but I decided it wasn’t very helpful in physics. When I read the magnificently poetic last chapter of Misner, Thorne and Wheeler’s Gravitation, I decided that quantum gravity was the problem to work on.

I went to math grad school at MIT, but I didn’t find anyone to work with on quantum gravity. So, I did my thesis on quantum field theory with Irving Segal. He was one of the founders of “constructive quantum field theory”, where you try to rigorously prove that quantum field theories make mathematical sense and obey certain axioms that they should. This was a hard subject, and I didn’t accomplish much, but I learned a lot.

I got a postdoc at Yale and switched to classical field theory, mainly because it was something I could do. On the side I was still trying to understand quantum gravity. String theory was bursting into prominence at the time, and my life would have been easier if I’d jumped onto that bandwagon. But I didn’t like it, because most of the work back then studied strings moving on a fixed “background” spacetime. Quantum gravity is supposed to be about how the geometry of spacetime is variable and quantum­-mechanical, so I didn’t want a theory of quantum gravity set on a pre­-existing background geometry!

I got a professorship at U.C. Riverside based on my work on classical field theory. But at a conference on that subject in Seattle, I heard Abhay Ashtekar, Chris Isham and Renate Loll give some really interesting talks on loop quantum gravity. I don’t know why they gave those talks at a conference on classical field theory. But I’m sure glad they did! I liked their work because it was background­-free and mathematically rigorous. So I started work on loop quantum gravity.

Like many other theories, quantum gravity is easier in low dimensions. I became interested in how category theory lets you formulate quantum gravity in a universe with just 3 spacetime dimensions. It amounts to a radical new conception of space, where the geometry is described in a thoroughly quantum­-mechanical way. Ultimately, space is a quantum superposition of “spin networks”, which are like Feynman diagrams. The idea is roughly that a spin network describes a virtual process where particles move around and interact. If we know how likely each of these processes is, we know the geometry of space.

A spin network

Loop quantum gravity tries to do the same thing for full­-fledged quantum gravity in 4 spacetime dimensions, but it doesn’t work as well. Then Louis Crane had an exciting idea: maybe 4­-dimensional quantum gravity needs a more sophisticated structure: a “2­-category”.

I had never heard of 2­-categories. Category theory is about things and processes that turn one thing into another. In a 2­-category we also have “meta-processes” that turn one process into another.

I became very excited about 2-­categories. At the time I was so dumb I didn’t consider the possibility of 3­-categories, and 4­-categories, and so on. To be precise, I was more of a mathematical physicist than a mathematician: I wasn’t trying to develop math for its own sake. Then someone named James Dolan told me about n­-categories! That was a real eye­-opener. He came to U.C. Riverside to work with me. So I started thinking about n­-categories in parallel with loop quantum gravity.

Dolan was technically my grad student, but I probably learned more from him than vice versa. In 1996 we wrote a paper called “Higher­-dimensional algebra and topological quantum field theory”, which might be my best paper. It’s full of grandiose guesses about n-­categories and their connections to other branches of math and physics. We had this vision of how everything fit together. It was so beautiful, with so much evidence supporting it, that we knew it had to be true. Unfortunately, at the time nobody had come up with a good definition of n­-category, except for n < 4. So we called our guesses “hypotheses” instead of “conjectures”. In math a conjecture should be something utterly precise: it’s either true or not, with no room for interpretation.

By now, I think everybody more or less believes our hypotheses. Some of the easier ones have already been turned into theorems. Jacob Lurie, a young hotshot at Harvard, improved the statement of one and wrote a 111-page outline of a proof. Unfortunately he still used some concepts that hadn’t been defined. People are working to fix that, and I feel sure they’ll succeed.

A foam of soap bubbles

Anyway, I kept trying to connect these ideas to quantum gravity. In 1997, I introduced “spin foams”. These are structures like spin networks, but with an extra dimension. Spin networks have vertices and edges. Spin foams also have 2­-dimensional faces: imagine a foam of soap bubbles.

The idea was to use spin foams to give a purely quantum­-mechanical description of the geometry of spacetime, just as spin networks describe the geometry of space. But mathematically, what we’re doing here is going from a category to a 2­-category.

By now, there are a number of different theories of quantum gravity based on spin foams. Unfortunately, it’s not clear that any of them really work. In 2002, Dan Christensen, Greg Egan and I did a bunch of supercomputer calculations to study this question. We showed that the most popular spin foam theory at the time gave dramatically different answers than people had hoped for. I think we more or less killed that theory.

That left me rather depressed. I don’t enjoy programming: indeed, Christensen and Egan did all the hard work of that sort on our paper. I didn’t want to spend years sifting through spin foam theories to find one that works. And most of all, I didn’t want to end up as an old man still not knowing if my work had been worthwhile! To me n­-category theory was clearly the math of the future—and it was easy for me to come up with cool new ideas in that subject. So, I quit quantum gravity and switched to n­-categories.

But this was very painful. Quantum gravity is a kind of “holy grail” in physics. When you work on that subject, you wind up talking to lots of people who believe that unifying quantum mechanics and general relativity is the most important thing in the world, and that nothing else could possibly be as interesting. You wind up believing it. It took me years to get out of that mindset.

Ironically, when I quit quantum gravity, I felt free to explore string theory. As a branch of math, it’s really wonderful. I started looking at how n­-categories apply to string theory. It turns out there’s a wonderful story here: briefly, particles are to categories as strings are to 2­-categories, and all the math of particles can be generalized to strings using this idea! I worked out a bit of this story with Urs Schreiber and John Huerta.

Around 2010, I felt I had to switch to working on environmental issues and math related to engineering and biology, for the sake of the planet. That was another painful renunciation. But luckily, Urs Schreiber and others are continuing to work on n­-categories and string theory, and doing it better than I ever could. So I don’t feel the need to work on those things anymore—indeed, it would be hard to keep up. I just follow along quietly from the sidelines.

It’s quite possible that we need a dozen more good ideas before we really get anywhere on quantum gravity. But I feel confident that n­-categories will have a role to play. So, I’m happy to have helped push that subject forward.

Your uncle, Albert Baez, was a physicist. How did he help develop your interests?

He had a huge effect on me. He’s mainly famous for being the father of the folk singer Joan Baez. But he started out in optics and helped invent the first X­-ray microscope. Later he became very involved in physics education, especially in what were then called third­-world countries. For example, in 1951 he helped set up a physics department at the University of Baghdad.

Albert V. Baez

When I was a kid he worked for UNESCO, so he’d come to Washington D.C. and stay with my parents, who lived nearby. Whenever he showed up, he would open his suitcase and pull out amazing gadgets: diffraction gratings, holograms, and things like that. And he would explain how they work! I decided physics was the coolest thing there is.

When I was eight, he gave me a copy of his book The New College Physics: A Spiral Approach. I immediately started trying to read it. The “spiral approach” is a great pedagogical principle: instead of explaining each topic just once, you should start off easy and then keep spiraling around from topic to topic, examining them in greater depth each time you revisit them. So he not only taught me physics, he taught me about how to learn and how to teach.

Later, when I was fifteen, I spent a couple weeks at his apartment in Berkeley. He showed me the Lawrence Hall of Science, which is where I got my first taste of programming—in BASIC, with programs stored on paper tape. This was in 1976. He also gave me a copy of The Feynman Lectures on Physics. And so, the following summer, when I was working at a state park building trails and such, I was also trying to learn quantum mechanics from the third volume of The Feynman Lectures. The other kids must have thought I was a complete geek—which of course I was.

Give us some insight on what your average work day is like.

During the school year I teach two or three days a week. On days when I teach, that’s the focus of my day: I try to prepare my classes starting at breakfast. Teaching is lots of fun for me. Right now I’m teaching two courses: an undergraduate course on game theory and a graduate course on category theory. I’m also running a seminar on category theory. In addition, I meet with my graduate students for a four­-hour session once a week: they show me what they’ve done, and we try to push our research projects forward.

On days when I don’t teach, I spend a lot of time writing. I love blogging, so I could easily do that all day, but I try to spend a lot of time writing actual papers. Any given paper starts out being tough to write, but near the end it practically writes itself. At the end, I have to tear myself away from it: I keep wanting to add more. At that stage, I feel an energetic glow at the end of a good day spent writing. Few things are so satisfying.

During the summer I don’t teach, so I can get a lot of writing done. I spent two years doing research at the Centre of Quantum Technologies, which is in Singapore, and since 2012 I’ve been working there during summers. Sometimes I bring my grad students, but mostly I just write.

I also spend plenty of time doing things with my wife, like talking, cooking, shopping, and working out at the gym. We like to watch TV shows in the evening, mainly mysteries and science fiction.

We also do a lot of gardening. When I was younger that seemed boring ­ but as you get older, subjective time speeds up, so you pay more attention to things like plants growing. There’s something tremendously satisfying about planting a small seedling, watching it grow into an orange tree, and eating its fruit for breakfast.

I love playing the piano and recording electronic music, but doing it well requires big blocks of time, which I don’t always have. Music is pure delight, and if I’m not listening to it I’m usually composing it in my mind.

If I gave in to my darkest urges and becames a decadent wastrel I might spend all day blogging, listening to music, recording music and working on pure math. But I need other things to stay sane.

What research are you working on at the moment?

Lately I’ve been trying to finish a paper called “Struggles with the Continuum”. It’s about the problems physics has with infinities, due to the assumption that spacetime is a continuum. At certain junctures this paper became psychologically difficult to write, since it’s supposed to include a summary of quantum field theory, which is complicated and sprawling subject. So, I’ve resorted to breaking this paper into blog articles and posting them on Physics Forums, just to motivate myself.

Purely for fun, I’ve been working with Greg Egan on some projects involving the octonions. The octonions are a number system where you can add, subtract, multiply and divide. Such number systems only exist in 1, 2, 4, and 8 dimensions: you’ve got the real numbers, which form a line, the complex numbers, which form a plane, the quaternions, which are 4­-dimensional, and the octonions, which are 8­dimensional. The octonions are the biggest, but also the weirdest. For example, multiplication of octonions violates the associative law: (xy)z is not equal to x(yz). So the octonions sound completely crazy at first, but they turn out to have fascinating connections to string theory and other things. They’re pretty addictive, and if became a decadent wastrel I would spend a lot more time on them.

The 240 unit integral octonions, projected onto a plane

There’s a concept of “integer” for the octonions, and integral octonions form a lattice, a repeating pattern of points, in 8 dimensions. This is called the E8 lattice. There’s another lattices that lives in in 24 dimensions, called the “Leech lattice”. Both are connected to string theory. Notice that 8+2 equals 10, the dimension superstrings like to live in, and 24+2 equals 26, the dimension bosonic strings like to live in. That’s not a coincidence! The 2 here comes from the 2­-dimensional world-sheet of the string.

Since 3×8 is 24, Egan and I became interested in how you could built the Leech lattice from 3 copies of the E8 lattice. People already knew a trick for doing it, but it took us a while to understand how it worked—and then Egan showed you could do this trick in exactly 17,280 ways! I want to write up the proof. There’s a lot of beautiful geometry here.

There’s something really exhilarating about struggling to reach the point where you have some insight into these structures and how they’re connected to physics.

My main work, though, involves using category theory to study networks. I’m interested in networks of all kinds, from electrical circuits to neural networks to “chemical reaction networks” and many more. Different branches of science and engineering focus on different kinds of networks. But there’s not enough communication between researchers in different subjects, so it’s up to mathematicians to set up a unified theory of networks.

I’ve got seven grad students working on this project—or actually eight, if you count Brendan Fong: I’ve been helping him on his dissertation, but he’s actually a student at Oxford.

Brendan was the first to join the project. I wanted him to work on electrical circuits, which are a nice familiar kind of network, a good starting point. But he went much deeper: he developed a general category­-theoretic framework for studying networks. We then applied it to electrical circuits, and other things as well.

Blake Pollard and Brendan Fong at the Centre for Quantum Technologies

Blake Pollard is a student of mine in the physics department here at U. C. Riverside. Together with Brendan and me, he developed a category­-theoretic approach to Markov processes: random processes where a system hops around between different states. We used Brendan’s general formalism to reduce Markov processes to electrical circuits. Now Blake is going further and using these ideas to tackle chemical reaction networks.

My other students are in the math department at U. C. Riverside. Jason Erbele is working on “control theory”, a branch of engineering where you try to design feedback loops to make sure processes run in a stable way. Control theory uses networks called “signal flow diagrams”, and Jason has worked out how to understand these using category theory.

Signal flow diagram for an inverted pendulum on a cart

Jason isn’t using Brendan’s framework: he’s using a different one, called PROPs, which were developed a long time ago for use in algebraic topology. My student Franciscus Rebro has been developing it further, for use in our project. It gives a nice way to describe networks in terms of their basic building blocks. It also illuminates the similarity between signal flow diagrams and Feynman diagrams! They’re very similar, but there’s a big difference: in signal flow diagrams the signals are classical, while Feynman diagrams are quantum­-mechanical.

My student Brandon Coya has been working on electrical circuits. He’s sort of continuing what Brendan started, and unifying Brendan’s formalism with PROPs.

My student Adam Yassine is starting to work on networks in classical mechanics. In classical mechanics you usually consider a single system: you write down the Hamiltonian, you get the equations of motion, and you try to solve them. He’s working on a setup where you can take lots of systems and hook them up into a network.

My students Kenny Courser and Daniel Cicala are digging deeper into another aspect of network theory. As I hinted earlier, a category is about things and processes that turn one thing into another. In a 2­-category we also have “meta-processes” that turn one process into another. We’re starting to bring 2­-categories into network theory.

For example, you can use categories to describe an electrical circuit as a process that turns some inputs into some outputs. You put some currents in one end and some currents come out the other end. But you can also use 2­-categories to describe “meta-processes” that turn one electrical circuit into another. An example of a meta-process would be a way of simplifying an electrical circuit, like replacing two resistors in series by a single resistor.

Ultimately I want to push these ideas in the direction of biochemistry. Biology seems complicated and “messy” to physicists and mathematicians, but I think there must be a beautiful logic to it. It’s full of networks, and these networks change with time. So, 2­-categories seem like a natural language for biology.

It won’t be easy to convince people of this, but that’s okay.

## Information Geometry (Part 16)

14 January, 2016

joint with Blake Pollard

Lately we’ve been thinking about open Markov processes. These are random processes where something can hop randomly from one state to another (that’s the ‘Markov process’ part) but also enter or leave the system (that’s the ‘open’ part).

The ultimate goal is to understand the nonequilibrium thermodynamics of open systems—systems where energy and maybe matter flows in and out. If we could understand this well enough, we could understand in detail how life works. That’s a difficult job! But one has to start somewhere, and this is one place to start.

We have a few papers on this subject:

• Blake Pollard, A Second Law for open Markov processes. (Blog article here.)

• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes. (Blog article here.)

• Blake Pollard, Open Markov processes: A compositional perspective on non-equilibrium steady states in biology. (Blog article here.)

However, right now we just want to show you three closely connected results about how relative entropy changes in open Markov processes.

### Definitions

An open Markov process consists of a finite set $X$ of states, a subset $B \subseteq X$ of boundary states, and an infinitesimal stochastic operator $H: \mathbb{R}^X \to \mathbb{R}^X,$ meaning a linear operator with

$H_{ij} \geq 0 \ \ \text{for all} \ \ i \neq j$

and

$\sum_i H_{ij} = 0 \ \ \text{for all} \ \ j$

For each state $i \in X$ we introduce a population $p_i \in [0,\infty).$ We call the resulting function $p : X \to [0,\infty)$ the population distribution.

Populations evolve in time according to the open master equation:

$\displaystyle{ \frac{dp_i}{dt} = \sum_j H_{ij}p_j} \ \ \text{for all} \ \ i \in X-B$

$p_i(t) = b_i(t) \ \ \text{for all} \ \ i \in B$

So, the populations $p_i$ obey a linear differential equation at states $i$ that are not in the boundary, but they are specified ‘by the user’ to be chosen functions $b_i$ at the boundary states. The off-diagonal entry $H_{ij}$ for $i \neq j$ describe the rate at which population transitions from the $j$th to the $i$th state.

A closed Markov process, or continuous-time discrete-state Markov chain, is an open Markov process whose boundary is empty. For a closed Markov process, the open master equation becomes the usual master equation:

$\displaystyle{ \frac{dp}{dt} = Hp }$

In a closed Markov process the total population is conserved:

$\displaystyle{ \frac{d}{dt} \sum_{i \in X} p_i = \sum_{i,j} H_{ij}p_j = 0 }$

This lets us normalize the initial total population to 1 and have it stay equal to 1. If we do this, we can talk about probabilities instead of populations. In an open Markov process, population can flow in and out at the boundary states.

For any pair of distinct states $i,j,$ $H_{ij}p_j$ is the flow of population from $j$ to $i.$ The net flux of population from the $j$th state to the $i$th state is the flow from $j$ to $i$ minus the flow from $i$ to $j$:

$J_{ij} = H_{ij}p_j - H_{ji}p_i$

A steady state is a solution of the open master equation that does not change with time. A steady state for a closed Markov process is typically called an equilibrium. So, an equilibrium obeys the master equation at all states, while for a steady state this may not be true at the boundary states. The idea is that population can flow in or out at the boundary states.

We say an equilibrium $p : X \to [0,\infty)$ of a Markov process is detailed balanced if all the net fluxes vanish:

$J_{ij} = 0 \ \ \text{for all} \ \ i,j \in X$

or in other words:

$H_{ij}p_j = H_{ji}p_i \ \ \text{for all} \ \ i,j \in X$

Given two population distributions $p, q : X \to [0,\infty)$ we can define the relative entropy

$\displaystyle{ I(p,q) = \sum_i p_i \ln \left( \frac{p_i}{q_i} \right)}$

When $q$ is a detailed balanced equilibrium solution of the master equation, the relative entropy can be seen as the ‘free energy’ of $p.$ For a precise statement, see Section 4 of Relative entropy in biological systems.

The Second Law of Thermodynamics implies that the free energy of a closed system tends to decrease with time, so for closed Markov processes we expect $I(p,q)$ to be nonincreasing. And this is true! But for open Markov processes, free energy can flow in from outside. This is just one of several nice results about how relative entropy changes with time.

### Results

Theorem 1. Consider an open Markov process with $X$ as its set of states and $B$ as the set of boundary states. Suppose $p(t)$ and $q(t)$ obey the open master equation, and let the quantities

$\displaystyle{ \frac{Dp_i}{Dt} = \frac{dp_i}{dt} - \sum_{j \in X} H_{ij}p_j }$

$\displaystyle{ \frac{Dq_i}{Dt} = \frac{dq_i}{dt} - \sum_{j \in X} H_{ij}q_j }$

measure how much the time derivatives of $p_i$ and $q_i$ fail to obey the master equation. Then we have

$\begin{array}{ccl} \displaystyle{ \frac{d}{dt} I(p(t),q(t)) } &=& \displaystyle{ \sum_{i, j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right)} \\ \\ && \; + \; \displaystyle{ \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} + \frac{\partial I}{\partial q_i} \frac{Dq_i}{Dt} } \end{array}$

This result separates the change in relative entropy change into two parts: an ‘internal’ part and a ‘boundary’ part.

It turns out the ‘internal’ part is always less than or equal to zero. So, from Theorem 1 we can deduce a version of the Second Law of Thermodynamics for open Markov processes:

Theorem 2. Given the conditions of Theorem 1, we have

$\displaystyle{ \frac{d}{dt} I(p(t),q(t)) \; \le \; \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} + \frac{\partial I}{\partial q_i} \frac{Dq_i}{Dt} }$

Intuitively, this says that free energy can only increase if it comes in from the boundary!

There is another nice result that holds when $q$ is an equilibrium solution of the master equation. This idea seems to go back to Schnakenberg:

Theorem 3. Given the conditions of Theorem 1, suppose also that $q$ is an equilibrium solution of the master equation. Then we have

$\displaystyle{ \frac{d}{dt} I(p(t),q) = -\frac{1}{2} \sum_{i,j \in X} J_{ij} A_{ij} \; + \; \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} }$

where

$J_{ij} = H_{ij}p_j - H_{ji}p_i$

is the net flux from $j$ to $i,$ while

$\displaystyle{ A_{ij} = \ln \left(\frac{p_j q_i}{p_i q_j} \right) }$

is the conjugate thermodynamic force.

The flux $J_{ij}$ has a nice meaning: it’s the net flow of population from $j$ to $i.$ The thermodynamic force is a bit subtler, but this theorem reveals its meaning: it says how much the population wants to flow from $j$ to $i.$

More precisely, up to that factor of $1/2,$ the thermodynamic force $A_{ij}$ says how much free energy loss is caused by net flux from $j$ to $i.$ There’s a nice analogy here to water losing potential energy as it flows downhill due to the force of gravity.

### Proofs

Proof of Theorem 1. We begin by taking the time derivative of the relative information:

$\begin{array}{ccl} \displaystyle{ \frac{d}{dt} I(p(t),q(t)) } &=& \displaystyle{ \sum_{i \in X} \frac{\partial I}{\partial p_i} \frac{dp_i}{dt} + \frac{\partial I}{\partial q_i} \frac{dq_i}{dt} } \end{array}$

We can separate this into a sum over states $i \in X - B,$ for which the time derivatives of $p_i$ and $q_i$ are given by the master equation, and boundary states $i \in B,$ for which they are not:

$\begin{array}{ccl} \displaystyle{ \frac{d}{dt} I(p(t),q(t)) } &=& \displaystyle{ \sum_{i \in X-B, \; j \in X} \frac{\partial I}{\partial p_i} H_{ij} p_j + \frac{\partial I}{\partial q_i} H_{ij} q_j }\\ \\ && + \; \; \; \displaystyle{ \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{dp_i}{dt} + \frac{\partial I}{\partial q_i} \frac{dq_i}{dt}} \end{array}$

For boundary states we have

$\displaystyle{ \frac{dp_i}{dt} = \frac{Dp_i}{Dt} + \sum_{j \in X} H_{ij}p_j }$

and similarly for the time derivative of $q_i.$ We thus obtain

$\begin{array}{ccl} \displaystyle{ \frac{d}{dt} I(p(t),q(t)) } &=& \displaystyle{ \sum_{i,j \in X} \frac{\partial I}{\partial p_i} H_{ij} p_j + \frac{\partial I}{\partial q_i} H_{ij} q_j }\\ \\ && + \; \; \displaystyle{ \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} + \frac{\partial I}{\partial q_i} \frac{Dq_i}{Dt}} \end{array}$

To evaluate the first sum, recall that

$\displaystyle{ I(p,q) = \sum_{i \in X} p_i \ln (\frac{p_i}{q_i})}$

so

$\displaystyle{\frac{\partial I}{\partial p_i}} =\displaystyle{1 + \ln (\frac{p_i}{q_i})} , \qquad \displaystyle{ \frac{\partial I}{\partial q_i}}= \displaystyle{- \frac{p_i}{q_i} }$

Thus, we have

$\displaystyle{ \sum_{i,j \in X} \frac{\partial I}{\partial p_i} H_{ij} p_j + \frac{\partial I}{\partial q_i} H_{ij} q_j = \sum_{i,j\in X} (1 + \ln (\frac{p_i}{q_i})) H_{ij} p_j - \frac{p_i}{q_i} H_{ij} q_j }$

We can rewrite this as

$\displaystyle{ \sum_{i,j \in X} H_{ij} p_j \left( 1 + \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right) }$

Since $H_{ij}$ is infinitesimal stochastic we have $\sum_{i} H_{ij} = 0,$ so the first term drops out, and we are left with

$\displaystyle{ \sum_{i,j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right) }$

as desired.   █

Proof of Theorem 2. Thanks to Theorem 1, to prove

$\displaystyle{ \frac{d}{dt} I(p(t),q(t)) \; \le \; \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} + \frac{\partial I}{\partial q_i} \frac{Dq_i}{Dt} }$

it suffices to show that

$\displaystyle{ \sum_{i,j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right) \le 0 }$

or equivalently (recalling the proof of Theorem 1):

$\displaystyle{ \sum_{i,j} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) + 1 - \frac{p_i q_j}{p_j q_i} \right) \le 0 }$

The last two terms on the left hand side cancel when $i = j.$ Thus, if we break the sum into an $i \ne j$ part and an $i = j$ part, the left side becomes

$\displaystyle{ \sum_{i \ne j} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) + 1 - \frac{p_i q_j}{p_j q_i} \right) \; + \; \sum_j H_{jj} p_j \ln(\frac{p_j}{q_j}) }$

Next we can use the infinitesimal stochastic property of $H$ to write $H_{jj}$ as the sum of $-H_{ij}$ over $i$ not equal to $j,$ obtaining

$\displaystyle{ \sum_{i \ne j} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) + 1 - \frac{p_i q_j}{p_j q_i} \right) - \sum_{i \ne j} H_{ij} p_j \ln(\frac{p_j}{q_j}) } =$

$\displaystyle{ \sum_{i \ne j} H_{ij} p_j \left( \ln(\frac{p_iq_j}{p_j q_i}) + 1 - \frac{p_i q_j}{p_j q_i} \right) }$

Since $H_{ij} \ge 0$ when $i \ne j$ and $\ln(s) + 1 - s \le 0$ for all $s > 0,$ we conclude that this quantity is $\le 0.$   █

Proof of Theorem 3. Now suppose also that $q$ is an equilibrium solution of the master equation. Then $Dq_i/Dt = dq_i/dt = 0$ for all states $i,$ so by Theorem 1 we need to show

$\displaystyle{ \sum_{i, j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right) \; = \; -\frac{1}{2} \sum_{i,j \in X} J_{ij} A_{ij} }$

We also have $\sum_{j \in X} H_{ij} q_j = 0,$ so the second
term in the sum at left vanishes, and it suffices to show

$\displaystyle{ \sum_{i, j \in X} H_{ij} p_j \ln(\frac{p_i}{q_i}) \; = \; - \frac{1}{2} \sum_{i,j \in X} J_{ij} A_{ij} }$

By definition we have

$\displaystyle{ \frac{1}{2} \sum_{i,j} J_{ij} A_{ij}} = \displaystyle{ \frac{1}{2} \sum_{i,j} \left( H_{ij} p_j - H_{ji}p_i \right) \ln \left( \frac{p_j q_i}{p_i q_j} \right) }$

This in turn equals

$\displaystyle{ \frac{1}{2} \sum_{i,j} H_{ij}p_j \ln \left( \frac{p_j q_i}{p_i q_j} \right) - \frac{1}{2} \sum_{i,j} H_{ji}p_i \ln \left( \frac{p_j q_i}{p_i q_j} \right) }$

and we can switch the dummy indices $i,j$ in the second sum, obtaining

$\displaystyle{ \frac{1}{2} \sum_{i,j} H_{ij}p_j \ln \left( \frac{p_j q_i}{p_i q_j} \right) - \frac{1}{2} \sum_{i,j} H_{ij}p_j \ln \left( \frac{p_i q_j}{p_j q_i} \right) }$

or simply

$\displaystyle{ \sum_{i,j} H_{ij} p_j \ln \left( \frac{p_j q_i}{p_i q_j} \right) }$

But this is

$\displaystyle{ \sum_{i,j} H_{ij} p_j \left(\ln ( \frac{p_j}{q_j}) + \ln (\frac{q_i}{p_i}) \right) }$

and the first term vanishes because $H$ is infinitesimal stochastic: $\sum_i H_{ij} = 0.$ We thus have

$\displaystyle{ \frac{1}{2} \sum_{i,j} J_{ij} A_{ij}} = \sum_{i,j} H_{ij} p_j \ln (\frac{q_i}{p_i} )$

as desired.   █

## Information Geometry (Part 15)

11 January, 2016

It’s been a long time since you’ve seen an installment of the information geometry series on this blog! Before I took a long break, I was explaining relative entropy and how it changes in evolutionary games. Much of what I said is summarized and carried further here:

• John Baez and Blake Pollard, Relative entropy in biological systems, Entropy 18 (2016), 46. (Blog article here.)

But now Blake has a new paper, and I want to talk about that:

• Blake Pollard, Open Markov processes: a compositional perspective on non-equilibrium steady states in biology, Entropy 18 (2016), 140.

I’ll focus on just one aspect: the principle of minimum entropy production. This is an exciting yet controversial principle in non-equilibrium thermodynamics. Blake examines it in a situation where we can tell exactly what’s happening.

Life exists away from equilibrium. Left isolated, systems will tend toward thermodynamic equilibrium. However, biology is about open systems: physical systems that exchange matter or energy with their surroundings. Open systems can be maintained away from equilibrium by this exchange. This leads to the idea of a non-equilibrium steady state—a state of an open system that doesn’t change, but is not in equilibrium.

A simple example is a pan of water sitting on a stove. Heat passes from the flame to the water and then to the air above. If the flame is very low, the water doesn’t boil and nothing moves. So, we have a steady state, at least approximately. But this is not an equilibrium, because there is a constant flow of energy through the water.

Of course in reality the water will be slowly evaporating, so we don’t really have a steady state. As always, models are approximations. If the water is evaporating slowly enough, it can be useful to approximate the situation with a non-equilibrium steady state.

There is much more to biology than steady states. However, to dip our toe into the chilly waters of non-equilibrium thermodynamics, it is nice to start with steady states. And already here there are puzzles left to solve.

### Minimum entropy production

Ilya Prigogine won the Nobel prize for his work on non-equilibrium thermodynamics. One reason is that he had an interesting idea about steady states. He claimed that under certain conditions, a non-equilibrium steady state will minimize entropy production!

There has been a lot of work trying to make the ‘principle of minimum entropy production’ precise and turn it into a theorem. In this book:

• G. Lebon and D. Jou, Understanding Non-equilibrium Thermodynamics, Springer, Berlin, 2008.

the authors give an argument for the principle of minimum entropy production based on four conditions:

time-independent boundary conditions: the surroundings of the system don’t change with time.

linear phenomenological laws: the laws governing the macroscopic behavior of the system are linear.

constant phenomenological coefficients: the laws governing the macroscopic behavior of the system don’t change with time.

symmetry of the phenomenological coefficients: since they are linear, the laws governing the macroscopic behavior of the system can be described by a linear operator, and we demand that in a suitable basis the matrix for this operator is symmetric: $T_{ij} = T_{ji}.$

The last condition is obviously the subtlest one; it’s sometimes called Onsager reciprocity, and people have spent a lot of time trying to derive it from other conditions.

However, Blake goes in a different direction. He considers a concrete class of open systems, a very large class called ‘open Markov processes’. These systems obey the first three conditions listed above, and the ‘detailed balanced’ open Markov processes also obey the last one. But Blake shows that minimum entropy production holds only approximately—with the approximation being good for steady states that are near equilibrium!

However, he shows that another minimum principle holds exactly, even for steady states that are far from equilibrium. He calls this the ‘principle of minimum dissipation’.

We actually discussed the principle of minimum dissipation in an earlier paper:

• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes. (Blog article here.)

But one advantage of Blake’s new paper is that it presents the results with a minimum of category theory. Of course I love category theory, and I think it’s the right way to formalize open systems, but it can be intimidating.

Another good thing about Blake’s new paper is that it explicitly compares the principle of minimum entropy to the principle of minimum dissipation. He shows they agree in a certain limit—namely, the limit where the system is close to equilibrium.

Let me explain this. I won’t include the nice example from biology that Blake discusses: a very simple model of membrane transport. For that, read his paper! I’ll just give the general results.

### The principle of minimum dissipation

An open Markov process consists of a finite set $X$ of states, a subset $B \subseteq X$ of boundary states, and an infinitesimal stochastic operator $H: \mathbb{R}^X \to \mathbb{R}^X,$ meaning a linear operator with

$H_{ij} \geq 0 \ \ \text{for all} \ \ i \neq j$

and

$\sum_i H_{ij} = 0 \ \ \text{for all} \ \ j$

I’ll explain these two conditions in a minute.

For each $i \in X$ we introduce a population $p_i \in [0,\infty).$ We call the resulting function $p : X \to [0,\infty)$ the population distribution. Populations evolve in time according to the open master equation:

$\displaystyle{ \frac{dp_i}{dt} = \sum_j H_{ij}p_j} \ \ \text{for all} \ \ i \in X-B$

$p_i(t) = b_i(t) \ \ \text{for all} \ \ i \in B$

So, the populations $p_i$ obey a linear differential equation at states $i$ that are not in the boundary, but they are specified ‘by the user’ to be chosen functions $b_i$ at the boundary states.

The off-diagonal entries $H_{ij}, \ i \neq j$ are the rates at which population hops from the $j$th to the $i$th state. This lets us understand the definition of an infinitesimal stochastic operator. The first condition:

$H_{ij} \geq 0 \ \ \text{for all} \ \ i \neq j$

says that the rate for population to transition from one state to another is non-negative. The second:

$\sum_i H_{ij} = 0 \ \ \text{for all} \ \ j$

says that population is conserved, at least if there are no boundary states. Population can flow in or out at boundary states, since the master equation doesn’t hold there.

A steady state is a solution of the open master equation that does not change with time. A steady state for a closed Markov process is typically called an equilibrium. So, an equilibrium obeys the master equation at all states, while for a steady state this may not be true at the boundary states. Again, the reason is that population can flow in or out at the boundary.

We say an equilibrium $q : X \to [0,\infty)$ of a Markov process is detailed balanced if the rate at which population flows from the $i$th state to the $j$th state is equal to the rate at which it flows from the $j$th state to the $i$th:

$H_{ji}q_i = H_{ij}q_j \ \ \text{for all} \ \ i,j \in X$

Suppose we’ve got an open Markov process that has a detailed balanced equilibrium $q$. Then a non-equilibrium steady state $p$ will minimize a function called the ‘dissipation’, subject to constraints on its boundary populations. There’s a nice formula for the dissipation in terms of $p$ and $q.$

Definition. Given an open Markov process with detailed balanced equilibrium $q$ we define the dissipation for a population distribution $p$ to be

$\displaystyle{ D(p) = \frac{1}{2}\sum_{i,j} H_{ij}q_j \left( \frac{p_j}{q_j} - \frac{p_i}{q_i} \right)^2 }$

This formula is a bit tricky, but you’ll notice it’s quadratic in $p$ and it vanishes when $p = q.$ So, it’s pretty nice.

Using this concept we can formulate a principle of minimum dissipation, and prove that non-equilibrium steady states obey this principle:

Definition. We say a population distribution $p: X \to \mathbb{R}$ obeys the principle of minimum dissipation with boundary population $b: X \to \mathbb{R}$ if $p$ minimizes $D(p)$ subject to the constraint that

$p_i = b_i \ \ \text{for all} \ \ i \in B.$

Theorem 1. A population distribution $p$ is a steady state with $p_i = b_i$ for all boundary states $i$ if and only if $p$ obeys the principle of minimum dissipation with boundary population $b$.

Proof. This follows from Theorem 28 in A compositional framework for Markov processes.

### Minimum entropy production versus minimum dissipation

How does dissipation compare with entropy production? To answer this, first we must ask: what really is entropy production? And: how does the equilibrium state $q$ show up in the concept of entropy production?

The relative entropy of two population distributions $p,q$ is given by

$\displaystyle{ I(p,q) = \sum_i p_i \ln \left( \frac{p_i}{q_i} \right) }$

It is well known that for a closed Markov process with $q$ as a detailed balanced equilibrium, the relative entropy is monotonically decreasing with time. This is due to an annoying sign convention in the definition of relative entropy: while entropy is typically increasing, relative entropy typically decreases. We could fix this by putting a minus sign in the above formula or giving this quantity $I(p,q)$ some other name. A lot of people call it the Kullback–Leibler divergence, but I have taken to calling it relative information. For more, see:

• John Baez and Blake Pollard, Relative entropy in biological systems. (Blog article here.)

We say ‘relative entropy’ in the title, but then we explain why ‘relative information’ is a better name, and use that. More importantly, we explain why $I(p,q)$ has the physical meaning of free energy. Free energy tends to decrease, so everything is okay. For details, see Section 4.

Blake has a nice formula for how fast $I(p,q)$ decreases:

Theorem 2. Consider an open Markov process with $X$ as its set of states and $B$ as the set of boundary states. Suppose $p(t)$ obeys the open master equation and $q$ is a detailed balanced equilibrium. For any boundary state $i \in B,$ let

$\displaystyle{ \frac{Dp_i}{Dt} = \frac{dp_i}{dt} - \sum_{j \in X} H_{ij}p_j }$

measure how much $p_i$ fails to obey the master equation. Then we have

$\begin{array}{ccl} \displaystyle{ \frac{d}{dt} I(p(t),q) } &=& \displaystyle{ \sum_{i, j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right)} \\ \\ && \; + \; \displaystyle{ \sum_{i \in B} \frac{\partial I}{\partial p_i} \frac{Dp_i}{Dt} } \end{array}$

Moreover, the first term is less than or equal to zero.

Proof. For a self-contained proof, see Information geometry (part 16), which is coming up soon. It will be a special case of the theorems there.   █

Blake compares this result to previous work by Schnakenberg:

• J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48 (1976), 571–585.

The negative of Blake’s first term is this:

$\displaystyle{ K(p) = - \sum_{i, j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right) }$

Under certain circumstances, this equals what Schnakenberg calls the entropy production. But a better name for this quantity might be free energy loss, since for a closed Markov process that’s exactly what it is! In this case there are no boundary states, so the theorem above says $K(p)$ is the rate at which relative entropy—or in other words, free energy—decreases.

For an open Markov process, things are more complicated. The theorem above shows that free energy can also flow in or out at the boundary, thanks to the second term in the formula.

Anyway, the sensible thing is to compare a principle of ‘minimum free energy loss’ to the principle of minimum dissipation. The principle of minimum dissipation is true. How about the principle of minimum free energy loss? It turns out to be approximately true near equilibrium.

For this, consider the situation in which $p$ is near to the equilibrium distribution $q$ in the sense that

$\displaystyle{ \frac{p_i}{q_i} = 1 + \epsilon_i }$

for some small numbers $\epsilon_i.$ We collect these numbers in a vector called $\epsilon.$

Theorem 3. Consider an open Markov process with $X$ as its set of states and $B$ as the set of boundary states. Suppose $q$ is a detailed balanced equilibrium and let $p$ be arbitrary. Then

$K(p) = D(p) + O(\epsilon^2)$

where $K(p)$ is the free energy loss, $D(p)$ is the dissipation, $\epsilon_i$ is defined as above, and by $O(\epsilon^2)$ we mean a sum of terms of order $\epsilon_i^2.$

Proof. First take the free energy loss:

$\displaystyle{ K(p) = -\sum_{i, j \in X} H_{ij} p_j \left( \ln(\frac{p_i}{q_i}) - \frac{p_i q_j}{p_j q_i} \right)}$

Expanding the logarithm to first order in $\epsilon,$ we get

$\displaystyle{ K(p) = -\sum_{i, j \in X} H_{ij} p_j \left( \frac{p_i}{q_i} - 1 - \frac{p_i q_j}{p_j q_i} \right) + O(\epsilon^2) }$

Since $H$ is infinitesimal stochastic, $\sum_i H_{ij} = 0,$ so the second term in the sum vanishes, leaving

$\displaystyle{ K(p) = -\sum_{i, j \in X} H_{ij} p_j \left( \frac{p_i}{q_i} - \frac{p_i q_j}{p_j q_i} \right) \; + O(\epsilon^2) }$

or

$\displaystyle{ K(p) = -\sum_{i, j \in X} \left( H_{ij} p_j \frac{p_i}{q_i} - H_{ij} q_j \frac{p_i}{q_i} \right) \; + O(\epsilon^2) }$

Since $q$ is a equilibrium we have $\sum_j H_{ij} q_j = 0,$ so now the last term in the sum vanishes, leaving

$\displaystyle{ K(p) = -\sum_{i, j \in X} H_{ij} \frac{p_i p_j}{q_i} \; + O(\epsilon^2) }$

Next, take the dissipation

$\displaystyle{ D(p) = \frac{1}{2}\sum_{i,j} H_{ij}q_j \left( \frac{p_j}{q_j} - \frac{p_i}{q_i} \right)^2 }$

and expand the square, getting

$\displaystyle{ D(p) = \frac{1}{2}\sum_{i,j} H_{ij}q_j \left( \frac{p_j^2}{q_j^2} - 2\frac{p_i p_j}{q_i q_j} + \frac{p_i^2}{q_i^2} \right) }$

Since $H$ is infinitesimal stochastic, $\sum_i H_{ij} = 0.$ The first term is just this times a function of $j,$ summed over $j,$ so it vanishes, leaving

$\displaystyle{ D(p) = \frac{1}{2}\sum_{i,j} H_{ij}q_j \left(- 2\frac{p_i p_j}{q_i q_j} + \frac{p_i^2}{q_i^2} \right) }$

Since $q$ is an equilibrium, $\sum_j H_{ij} q_j = 0.$ The last term above is this times a function of $i,$ summed over $i,$ so it vanishes, leaving

$\displaystyle{ D(p) = - \sum_{i,j} H_{ij}q_j \frac{p_i p_j}{q_i q_j} = - \sum_{i,j} H_{ij} \frac{p_i p_j}{q_i} }$

This matches what we got for $K(p),$ up to terms of order $O(\epsilon^2).$   █

In short: detailed balanced open Markov processes are governed by the principle of minimum dissipation, not minimum entropy production. Minimum dissipation agrees with minimum entropy production only near equilibrium.

## The Inverse Cube Force Law

30 August, 2015

Here you see three planets. The blue planet is orbiting the Sun in a realistic way: it’s going around an ellipse.

The other two are moving in and out just like the blue planet, so they all stay on the same circle. But they’re moving around this circle at different rates! The green planet is moving faster than the blue one: it completes 3 orbits each time the blue planet goes around once. The red planet isn’t going around at all: it only moves in and out.

What’s going on here?

In 1687, Isaac Newton published his Principia Mathematica. This book is famous, but in Propositions 43–45 of Book I he did something that people didn’t talk about much—until recently. He figured out what extra force, besides gravity, would make a planet move like one of these weird other planets. It turns out an extra force obeying an inverse cube law will do the job!

Let me make this more precise. We’re only interested in ‘central forces’ here. A central force is one that only pushes a particle towards or away from some chosen point, and only depends on the particle’s distance from that point. In Newton’s theory, gravity is a central force obeying an inverse square law:

$F(r) = - \displaystyle{ \frac{a}{r^2} }$

for some constant $a.$ But he considered adding an extra central force obeying an inverse cube law:

$F(r) = - \displaystyle{ \frac{a}{r^2} + \frac{b}{r^3} }$

He showed that if you do this, for any motion of a particle in the force of gravity you can find a motion of a particle in gravity plus this extra force, where the distance $r(t)$ is the same, but the angle $\theta(t)$ is not.

In fact Newton did more. He showed that if we start with any central force, adding an inverse cube force has this effect.

Newton’s theorem of revolving orbits, Wikipedia.

I haven’t fully understood all of this, but it instantly makes me think of three other things I know about the inverse cube force law, which are probably related. So maybe you can help me figure out the relationship.

The first, and simplest, is this. Suppose we have a particle in a central force. It will move in a plane, so we can use polar coordinates $r, \theta$ to describe its position. We can describe the force away from the origin as a function $F(r).$ Then the radial part of the particle’s motion obeys this equation:

$\displaystyle{ m \ddot r = F(r) + \frac{L^2}{mr^3} }$

where $L$ is the magnitude of particle’s angular momentum.

So, angular momentum acts to provide a ‘fictitious force’ pushing the particle out, which one might call the centrifugal force. And this force obeys an inverse cube force law!

Furthermore, thanks to the formula above, it’s pretty obvious that if you change $L$ but also add a precisely compensating inverse cube force, the value of $\ddot r$ will be unchanged! So, we can set things up so that the particle’s radial motion will be unchanged. But its angular motion will be different, since it has a different angular momentum. This explains Newton’s observation.

It’s often handy to write a central force in terms of a potential:

$F(r) = -V'(r)$

Then we can make up an extra potential responsible for the centrifugal force, and combine it with the actual potential $V$ into a so-called effective potential:

$\displaystyle{ U(r) = V(r) + \frac{L^2}{2mr^2} }$

The particle’s radial motion then obeys a simple equation:

$\ddot{r} = - U'(r)$

For a particle in gravity, where the force obeys an inverse square law and $V$ is proportional to $-1/r,$ the effective potential might look like this:

This is the graph of

$\displaystyle{ U(r) = -\frac{4}{r} + \frac{1}{r^2} }$

If you’re used to particles rolling around in potentials, you can easily see that a particle with not too much energy will move back and forth, never making it to $r = 0$ or $r = \infty.$ This corresponds to an elliptical orbit. Give it more energy and the particle can escape to infinity, but it will never hit the origin. The repulsive ‘centrifugal force’ always overwhelms the attraction of gravity near the origin, at least if the angular momentum is nonzero.

On the other hand, suppose we have a particle moving in an attractive inverse cube force! Then the potential is proportional to $1/r^2,$ so the effective potential is

$\displaystyle{ U(r) = \frac{c}{r^2} + \frac{L^2}{mr^2} }$

where $c$ is negative for an attractive force. If this attractive force is big enough, namely

$\displaystyle{ c < -\frac{L^2}{m} }$

then this force can exceed the centrifugal force, and the particle can fall in to $r = 0.$

If we keep track of the angular coordinate $\theta,$ we can see what’s really going on. The particle is spiraling in to its doom, hitting the origin in a finite amount of time!

This should remind you of a black hole, and indeed something similar happens there, but even more drastic:

For a nonrotating uncharged black hole, the effective potential has three terms. Like Newtonian gravity it has an attractive $-1/r$ term and a repulsive $1/r^2$ term. But it also has an attractive term $-1/r^3$ term! In other words, it’s as if on top of Newtonian gravity, we had another attractive force obeying an inverse fourth power law! This overwhelms the others at short distances, so if you get too close to a black hole, you spiral in to your doom.

For example, a black hole can have an effective potential like this:

But back to inverse cube force laws! I know two more things about them. A while back I discussed how a particle in an inverse square force can be reinterpreted as a harmonic oscillator:

Planets in the fourth dimension, Azimuth.

There are many ways to think about this, and apparently the idea in some form goes all the way back to Newton! It involves a sneaky way to take a particle in a potential

$\displaystyle{ V(r) \propto r^{-1} }$

and think of it as moving around in the complex plane. Then if you square its position—thought of as a complex number—and cleverly reparametrize time, you get a particle moving in a potential

$\displaystyle{ V(r) \propto r^2 }$

This amazing trick can be generalized! A particle in a potential

$\displaystyle{ V(r) \propto r^p }$

can transformed to a particle in a potential

$\displaystyle{ V(r) \propto r^q }$

if

$(p+2)(q+2) = 4$

A good description is here:

• Rachel W. Hall and Krešimir Josić, Planetary motion and the duality of force laws, SIAM Review 42 (2000), 115–124.

This trick transforms particles in $r^p$ potentials with $p$ ranging between $-2$ and $+\infty$ to $r^q$ potentials with $q$ ranging between $+\infty$ and $-2.$ It’s like a see-saw: when $p$ is small, $q$ is big, and vice versa.

But you’ll notice this trick doesn’t actually work at $p = -2,$ the case that corresponds to the inverse cube force law. The problem is that $p + 2 = 0$ in this case, so we can’t find $q$ with $(p+2)(q+2) = 4.$

So, the inverse cube force is special in three ways: it’s the one that you can add on to any force to get solutions with the same radial motion but different angular motion, it’s the one that naturally describes the ‘centrifugal force’, and it’s the one that doesn’t have a partner! We’ve seen how the first two ways are secretly the same. I don’t know about the third, but I’m hopeful.

### Quantum aspects

Finally, here’s a fourth way in which the inverse cube law is special. This shows up most visibly in quantum mechanics… and this is what got me interested in this business in the first place.

You see, I’m writing a paper called ‘Struggles with the continuum’, which discusses problems in analysis that arise when you try to make some of our favorite theories of physics make sense. The inverse square force law poses interesting problems of this sort, which I plan to discuss. But I started wanting to compare the inverse cube force law, just so people can see things that go wrong in this case, and not take our successes with the inverse square law for granted.

Unfortunately a huge digression on the inverse cube force law would be out of place in that paper. So, I’m offloading some of that material to here.

In quantum mechanics, a particle moving in an inverse cube force law has a Hamiltonian like this:

$H = -\nabla^2 + c r^{-2}$

The first term describes the kinetic energy, while the second describes the potential energy. I’m setting $\hbar = 1$ and $2m = 1$ to remove some clutter that doesn’t really affect the key issues.

To see how strange this Hamiltonian is, let me compare an easier case. If $p < 2,$ the Hamiltonian

$H = -\nabla^2 + c r^{-p}$

is essentially self-adjoint on $C_0^\infty(\mathbb{R}^3 - \{0\}),$ which is the space of compactly supported smooth functions on 3d Euclidean space minus the origin. What this means is that first of all, $H$ is defined on this domain: it maps functions in this domain to functions in $L^2(\mathbb{R}^3)$. But more importantly, it means we can uniquely extend $H$ from this domain to a self-adjoint operator on some larger domain. In quantum physics, we want our Hamiltonians to be self-adjoint. So, this fact is good.

Proving this fact is fairly hard! It uses something called the Kato–Lax–Milgram–Nelson theorem together with this beautiful inequality:

$\displaystyle{ \int_{\mathbb{R}^3} \frac{1}{4r^2} |\psi(x)|^2 \,d^3 x \le \int_{\mathbb{R}^3} |\nabla \psi(x)|^2 \,d^3 x }$

for any $\psi\in C_0^\infty(\mathbb{R}^3).$

If you think hard, you can see this inequality is actually a fact about the quantum mechanics of the inverse cube law! It says that if $c \ge -1/4,$ the energy of a quantum particle in the potential $c r^{-2}$ is bounded below. And in a sense, this inequality is optimal: if $c < -1/4$, the energy is not bounded below. This is the quantum version of how a classical particle can spiral in to its doom in an attractive inverse cube law, if it doesn’t have enough angular momentum. But it’s subtly and mysteriously different.

You may wonder how this inequality is used to prove good things about potentials that are ‘less singular’ than the $c r^{-2}$ potential: that is, potentials $c r^{-p}$ with $p < 2.$ For that, you have to use some tricks that I don’t want to explain here. I also don’t want to prove this inequality, or explain why its optimal! You can find most of this in some old course notes of mine:

• John Baez, Quantum Theory and Analysis, 1989.

See especially section 15.

But it’s pretty easy to see how this inequality implies things about the expected energy of a quantum particle in the potential $c r^{-2}$. So let’s do that.

In this potential, the expected energy of a state $\psi$ is:

$\displaystyle{ \langle \psi, H \psi \rangle = \int_{\mathbb{R}^3} \overline\psi(x)\, (-\nabla^2 + c r^{-2})\psi(x) \, d^3 x }$

Doing an integration by parts, this gives:

$\displaystyle{ \langle \psi, H \psi \rangle = \int_{\mathbb{R}^3} |\nabla \psi(x)|^2 + cr^{-2} |\psi(x)|^2 \,d^3 x }$

The inequality I showed you says precisely that when $c = -1/4,$ this is greater than or equal to zero. So, the expected energy is actually nonnegative in this case! And making $c$ greater than $-1/4$ only makes the expected energy bigger.

Note that in classical mechanics, the energy of a particle in this potential ceases to be bounded below as soon as $c < 0.$ Quantum mechanics is different because of the uncertainty principle! To get a lot of negative potential energy, the particle’s wavefunction must be squished near the origin, but that gives it kinetic energy.

It turns out that the Hamiltonian for a quantum particle in an inverse cube force law has exquisitely subtle and tricky behavior. Many people have written about it, running into ‘paradoxes’ when they weren’t careful enough. Only rather recently have things been straightened out.

For starters, the Hamiltonian for this kind of particle

$H = -\nabla^2 + c r^{-2}$

has different behaviors depending on $c.$ Obviously the force is attractive when $c > 0$ and repulsive when $c < 0,$ but that’s not the only thing that matters! Here’s a summary:

$c \ge 3/4.$ In this case $H$ is essentially self-adjoint on $C_0^\infty(\mathbb{R}^3 - \{0\}).$ So, it admits a unique self-adjoint extension and there’s no ambiguity about this case.

$c < 3/4.$ In this case $H$ is not essentially self-adjoint on $C_0^\infty(\mathbb{R}^3 - \{0\}).$ In fact, it admits more than one self-adjoint extension! This means that we need extra input from physics to choose the Hamiltonian in this case. It turns out that we need to say what happens when the particle hits the singularity at $r = 0.$ This is a long and fascinating story that I just learned yesterday.

$c \ge -1/4.$ In this case the expected energy $\langle \psi, H \psi \rangle$ is bounded below for $\psi \in C_0^\infty(\mathbb{R}^3 - \{0\}).$ It turns out that whenever we have a Hamiltonian that is bounded below, even if there is not a unique self-adjoint extension, there exists a canonical ‘best choice’ of self-adjoint extension, called the Friedrichs extension. I explain this in my course notes.

$c < -1/4.$ In this case the expected energy is not bounded below, so we don’t have the Friedrichs extension to help us choose which self-adjoint extension is ‘best’.

To go all the way down this rabbit hole, I recommend these two papers:

• Sarang Gopalakrishnan, Self-Adjointness and the Renormalization of Singular Potentials, B.A. Thesis, Amherst College.

• D. M. Gitman, I. V. Tyutin and B. L. Voronov, Self-adjoint extensions and spectral analysis in the Calogero problem, Jour. Phys. A 43 (2010), 145205.

The first is good for a broad overview of problems associated to singular potentials such as the inverse cube force law; there is attention to mathematical rigor the focus is on physical insight. The second is good if you want—as I wanted—to really get to the bottom of the inverse cube force law in quantum mechanics. Both have lots of references.

Also, both point out a crucial fact I haven’t mentioned yet: in quantum mechanics the inverse cube force law is special because, naively, at least it has a kind of symmetry under rescaling! You can see this from the formula

$H = -\nabla^2 + cr^{-2}$

by noting that both the Laplacian and $r^{-2}$ have units of length-2. So, they both transform in the same way under rescaling: if you take any smooth function $\psi$, apply $H$ and then expand the result by a factor of $k,$ you get $k^2$ times what you get if you do those operations in the other order.

In particular, this means that if you have a smooth eigenfunction of $H$ with eigenvalue $\lambda,$ you will also have one with eigenfunction $k^2 \lambda$ for any $k > 0.$ And if your original eigenfunction was normalizable, so will be the new one!

With some calculation you can show that when $c \le -1/4,$ the Hamiltonian $H$ has a smooth normalizable eigenfunction with a negative eigenvalue. In fact it’s spherically symmetric, so finding it is not so terribly hard. But this instantly implies that $H$ has smooth normalizable eigenfunctions with any negative eigenvalue.

This implies various things, some terrifying. First of all, it means that $H$ is not bounded below, at least not on the space of smooth normalizable functions. A similar but more delicate scaling argument shows that it’s also not bounded below on $C_0^\infty(\mathbb{R}^3 - \{0\}),$ as I claimed earlier.

This is scary but not terrifying: it simply means that when $c \le -1/4,$ the potential is too strongly negative for the Hamiltonian to be bounded below.

The terrifying part is this: we’re getting uncountably many normalizable eigenfunctions, all with different eigenvalues, one for each choice of $k.$ A self-adjoint operator on a countable-dimensional Hilbert space like $L^2(\mathbb{R}^3)$ can’t have uncountably many normalizable eigenvectors with different eigenvalues, since then they’d all be orthogonal to each other, and that’s too many orthogonal vectors to fit in a Hilbert space of countable dimension!

This sounds like a paradox, but it’s not. These functions are not all orthogonal, and they’re not all eigenfunctions of a self-adjoint operator. You see, the operator $H$ is not self-adjoint on the domain we’ve chosen, the space of all smooth functions in $L^2(\mathbb{R}^3).$ We can carefully choose a domain to get a self-adjoint operator… but it turns out there are many ways to do it.

Intriguingly, in most cases this choice breaks the naive dilation symmetry. So, we’re getting what physicists call an ‘anomaly’: a symmetry of a classical system that fails to give a symmetry of the corresponding quantum system.

Of course, if you’ve made it this far, you probably want to understand what the different choices of Hamiltonian for a particle in an inverse cube force law actually mean, physically. The idea seems to be that they say how the particle changes phase when it hits the singularity at $r = 0$ and bounces back out.

(Why does it bounce back out? Well, if it didn’t, time evolution would not be unitary, so it would not be described by a self-adjoint Hamiltonian! We could try to describe the physics of a quantum particle that does not come back out when it hits the singularity, and I believe people have tried, but this requires a different set of mathematical tools.)

For a detailed analysis of this, it seems one should take Schrödinger’s equation and do a separation of variables into the angular part and the radial part:

$\psi(r,\theta,\phi) = \Psi(r) \Phi(\theta,\phi)$

For each choice of $\ell = 0,1,2,\dots$ one gets a space of spherical harmonics that one can use for the angular part $\Phi.$ The interesting part is the radial part, $\Psi.$ Here it is helpful to make a change of variables

$u(r) = \Psi(r)/r$

At least naively, Schrödinger’s equation for the particle in the $cr^{-2}$ potential then becomes

$\displaystyle{ \frac{d}{dt} u = -iH u }$

where

$\displaystyle{ H = -\frac{d^2}{dr^2} + \frac{c + \ell(\ell+1)}{r^2} }$

Beware: I keep calling all sorts of different but related Hamiltonians $H,$ and this one is for the radial part of the dynamics of a quantum particle in an inverse cube force. As we’ve seen before in the classical case, the centrifugal force and the inverse cube force join forces in an ‘effective potential’

$\displaystyle{ U(r) = kr^{-2} }$

where

$k = c + \ell(\ell+1)$

So, we have reduced the problem to that of a particle on the open half-line $(0,\infty)$ moving in the potential $kr^{-2}.$ The Hamiltonian for this problem:

$\displaystyle{ H = -\frac{d^2}{dr^2} + \frac{k}{r^2} }$

is called the Calogero Hamiltonian. Needless to say, it has fascinating and somewhat scary properties, since to make it into a bona fide self-adjoint operator, we must make some choice about what happens when the particle hits $r = 0.$ The formula above does not really specify the Hamiltonian.

This is more or less where Gitman, Tyutin and Voronov begin their analysis, after a long and pleasant review of the problem. They describe all the possible choices of self-adjoint operator that are allowed. The answer depends on the values of $k,$ but very crudely, the choice says something like how the phase of your particle changes when it bounces off the singularity. Most choices break the dilation invariance of the problem. But intriguingly, some choices retain invariance under a discrete subgroup of dilations!

So, the rabbit hole of the inverse cube force law goes quite deep, and I expect I haven’t quite gotten to the bottom yet. The problem may seem pathological, verging on pointless. But the math is fascinating, and it’s a great testing-ground for ideas in quantum mechanics—very manageable compared to deeper subjects like quantum field theory, which are riddled with their own pathologies. Finally, the connection between the inverse cube force law and centrifugal force makes me think it’s not a mere curiosity.

### In four dimensions

It’s a bit odd to study the inverse cube force law in 3-dimensonal space, since Newtonian gravity and the electrostatic force would actually obey an inverse cube law in 4-dimensional space. For the classical 2-body problem it doesn’t matter much whether you’re in 3d or 4d space, since the motion stays on the plane. But for quantum 2-body problem it makes more of a difference!

Just for the record, let me say how the quantum 2-body problem works in 4 dimensions. As before, we can work in the center of mass frame and consider this Hamiltonian:

$H = -\nabla^2 + c r^{-2}$

And as before, the behavior of this Hamiltonian depends on $c.$ Here’s the story this time:

$c \ge 0.$ In this case $H$ is essentially self-adjoint on $C_0^\infty(\mathbb{R}^4 - \{0\}).$ So, it admits a unique self-adjoint extension and there’s no ambiguity about this case.

$c < 0.$ In this case $H$ is not essentially self-adjoint on $C_0^\infty(\mathbb{R}^4 - \{0\}).$

$c \ge -1.$ In this case the expected energy $\langle \psi, H \psi \rangle$ is bounded below for $\psi \in C_0^\infty(\mathbb{R}^3 - \{0\}).$ So, there is exists a canonical ‘best choice’ of self-adjoint extension, called the Friedrichs extension.

$c < -1.$ In this case the expected energy is not bounded below, so we don’t have the Friedrichs extension to help us choose which self-adjoint extension is ‘best’.

I’ve been assured these are correct by Barry Simon, and a lot of this material will appear in Section 7.4 of his book:

• Barry Simon, A Comprehensive Course in Analysis, Part 4: Operator Theory, American Mathematical Society, Providence, RI, 2015.

• Barry Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Analysis 52 (1973), 44–48.

### Notes

The animation was made by ‘WillowW’ and placed on Wikicommons. It’s one of a number that appears in this Wikipedia article:

Newton’s theorem of revolving orbits, Wikipedia.

I made the graphs using the free online Desmos graphing calculator.

The picture of a spiral was made by ‘Anarkman’ and ‘Pbroks13’ and placed on Wikicommons; it appears in

Hyperbolic spiral, Wikipedia.

The hyperbolic spiral is one of three kinds of orbits that are possible in an inverse cube force law. They are vaguely analogous to ellipses, hyperbolas and parabolas, but there are actually no bound orbits except perfect circles. The three kinds are called Cotes’s spirals. In polar coordinates, they are:

• the epispiral:

$\displaystyle{ \frac{1}{r} = A \cos\left( k\theta + \varepsilon \right) }$

• the hyperbolic spiral:

$\displaystyle{ \frac{1}{r} = A \cosh\left( k\theta + \varepsilon \right) }$

• the Poinsot spiral:

$\displaystyle{ \frac{1}{r} = A \theta + \varepsilon }$

## The Physics of Butterfly Wings

11 August, 2015

Some butterflies have shiny, vividly colored wings. From different angles you see different colors. This effect is called iridescence. How does it work?

It turns out these butterfly wings are made of very fancy materials! Light bounces around inside these materials in a tricky way. Sunlight of different colors winds up reflecting off these materials in different directions.

We’re starting to understand the materials and make similar substances in the lab. They’re called photonic crystals. They have amazing properties.

Here at the Centre for Quantum Technologies we have people studying exotic materials of many kinds. Next door, there’s a lab completely devoted to studying graphene: crystal sheets of carbon in which electrons can move as if they were massless particles! Graphene has a lot of potential for building new technologies—that’s why Singapore is pumping money into researching it.

Some physicists at MIT just showed that one of the materials in butterfly wings might act like a 3d form of graphene. In graphene, electrons can only move easily in 2 directions. In this new material, electrons could move in all 3 directions, acting as if they had no mass.

The pictures here show the microscopic structure of two materials found in butterfly wings:

The picture at left actually shows a sculpture made by the mathematical artist Bathsheba Grossman. But it’s a piece of a gyroid: a surface with a very complicated shape, which repeats forever in 3 directions. It’s called a minimal surface because you can’t shrink its area by tweaking it just a little. It divides space into two regions.

The gyroid was discovered in 1970 by a mathematician, Alan Schoen. It’s a triply periodic minimal surfaces, meaning one that repeats itself in 3 different directions in space, like a crystal.

Schoen was working for NASA, and his idea was to use the gyroid for building ultra-light, super-strong structures. But that didn’t happen. Research doesn’t move in predictable directions.

In 1983, people discovered that in some mixtures of oil and water, the oil naturally forms a gyroid. The sheets of oil try to minimize their area, so it’s not surprising that they form a minimal surface. Something else makes this surface be a gyroid—I’m not sure what.

Butterfly wings are made of a hard material called chitin. Around 2008, people discovered that the chitin in some iridescent butterfly wings is made in a gyroid pattern! The spacing in this pattern is very small, about one wavelength of visible light. This makes light move through this material in a complicated way, which depends on the light’s color and the direction it’s moving.

So: butterflies have naturally evolved a photonic crystal based on a gyroid!

The universe is awesome, but it’s not magic. A mathematical pattern is beautiful if it’s a simple solution to at least one simple problem. This is why beautiful patterns naturally bring themselves into existence: they’re the simplest ways for certain things to happen. Darwinian evolution helps out: it scans through trillions of possibilities and finds solutions to problems. So, we should expect life to be packed with mathematically beautiful patterns… and it is.

The picture at right above shows a ‘double gyroid’. Here it is again:

This is actually two interlocking surfaces, shown in red and blue. You can get them by writing the gyroid as a level surface:

$f(x,y,z) = 0$

and taking the two nearby surfaces

$f(x,y,z) = \pm c$

for some small value of $c.$.

It turns out that while they’re still growing, some butterflies have a double gyroid pattern in their wings. This turns into a single gyroid when they grow up!

The new research at MIT studied how an electron would move through a double gyroid pattern. They calculated its dispersion relation: how the speed of the electron would depend on its energy and the direction it’s moving.

An ordinary particle moves faster if it has more energy. But a massless particle, like a photon, moves at the same speed no matter what energy it has. The MIT team showed that an electron in a double gyroid pattern moves at a speed that doesn’t depend much on its energy. So, in some ways this electron acts like a massless particle.

But it’s quite different than a photon. It’s actually more like a neutrino! You see, unlike photons, electrons and neutrinos are spin-1/2 particles. Neutrinos are almost massless. A massless spin-1/2 particle can have a built-in handedness, spinning in only one direction around its axis of motion. Such a particle is called a Weyl spinor. The MIT team showed that a electron moving through a double gyroid acts approximately like a Weyl spinor!

How does this work? Well, the key fact is that the double gyroid has a built-in handedness, or chirality. It comes in a left-handed and right-handed form. You can see the handedness quite clearly in Grossman’s sculpture of the ordinary gyroid:

Beware: nobody has actually made electrons act like Weyl spinors in the lab yet. The MIT team just found a way that should work. Someday someone will actually make it happen, probably in less than a decade. And later, someone will do amazing things with this ability. I don’t know what. Maybe the butterflies know!

### References and more

For a good introduction to the physics of gyroids, see:

• James A. Dolan, Bodo D. Wilts, Silvia Vignolini, Jeremy J. Baumberg, Ullrich Steiner and Timothy D. Wilkinson, Optical properties of gyroid structured materials: from photonic crystals to metamaterials, Advanced Optical Materials 3 (2015), 12–32.

For some of the history and math of gyroids, see Alan Schoen’s webpage:

• Alan Schoen, Triply-periodic minimal surfaces.

For more on gyroids in butterfly wings, see:

• K. Michielsen and D. G. Stavenga, Gyroid cuticular structures in butterfly wing scales: biological photonic crystals.

• Vinodkumar Saranathana et al, Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales, PNAS 107 (2010), 11676–11681.

The paper by Michielsen and Stavenga is free online! They say the famous ‘blue Morpho’ butterfly shown in the picture at the top of this article does not use a gyroid; it uses a “two-dimensional photonic crystal slab consisting of arrays of rectangles formed by lamellae and microribs.” But they find gyroids in four other species: Callophrys rubi, Cyanophrys remus, Pardes sesostris and Teinopalpus imperialis. It compares tunnelling electron microscope pictures of slices of their iridescent patches with computer-generated slices of gyroids. The comparison looks pretty good to me:

For the evolution of iridescence, see:

• Melissa G. Meadows et al, Iridescence: views from many angles, J. Roy. Soc. Interface 6 (2009).

For the new research at MIT, see:

• Ling Lu, Liang Fu, John D. Joannopoulos and Marin Soljačić, Weyl points and line nodes in gapless gyroid photonic crystals.

• Ling Lu, Zhiyu Wang, Dexin Ye, Lixin Ran, Liang Fu, John D. Joannopoulos and Marin Soljačić, Experimental observation of Weyl points, Science 349 (2015), 622–624.

Again, the first is free online. There’s a lot of great math lurking inside, most of which is too mind-blowing too explain quickly. Let me just paraphrase the start of the paper, so at least experts can get the idea:

Two-dimensional (2d) electrons and photons at the energies and frequencies of Dirac points exhibit extraordinary features. As the best example, almost all the remarkable properties of graphene are tied to the massless Dirac fermions at its Fermi level. Topologically, Dirac cones are not only the critical points for 2d phase transitions but also the unique surface manifestation of a topologically gapped 3d bulk. In a similar way, it is expected that if a material could be found that exhibits a 3d linear dispersion relation, it would also display a wide range of interesting physics phenomena. The associated 3D linear point degeneracies are called “Weyl points”. In the past year, there have been a few studies of Weyl fermions in electronics. The associated Fermi-arc surface states, quantum Hall effect, novel transport properties and a realization of the Adler–Bell–Jackiw anomaly are also expected. However, no observation of Weyl points has been reported. Here, we present a theoretical discovery and detailed numerical investigation of frequency-isolated Weyl points in perturbed double-gyroid photonic crystals along with their complete phase diagrams and their topologically protected surface states.

Also a bit for the mathematicians:

Weyl points are topologically stable objects in the 3d Brillouin zone: they act as monopoles of Berry flux in momentum space, and hence are intimately related to the topological invariant known as the Chern number. The Chern number can be defined for a single bulk band or a set of bands, where the Chern numbers of the individual bands are summed, on any closed 2d surface in the 3d Brillouin zone. The difference of the Chern numbers defined on two surfaces, of all bands below the Weyl point frequencies, equals the sum of the chiralities of the Weyl points enclosed in between the two surfaces.

This is a mix of topology and physics jargon that may be hard for pure mathematicians to understand, but I’ll be glad to translate if there’s interest.

For starters, a ‘monopole of Berry flux in momentum space’ is a poetic way of talking about a twisted complex line bundle over the space of allowed energy-momenta of the electron in the double gyroid. We get a twist at every ‘Weyl point’, meaning a point where the dispersion relations look locally like those of a Weyl spinor when its energy-momentum is near zero. Near such a point, the dispersion relations are a Fourier-transformed version of the Weyl equation.

## A Compositional Framework for Passive Linear Networks

28 April, 2015

Here’s a new paper on network theory:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

While my paper with Jason Erbele studies signal flow diagrams, this one focuses on circuit diagrams. The two are different, but closely related.

I’ll explain their relation at the Turin workshop in May. For now, let me just talk about this paper with Brendan. There’s a lot in here, but let me just try to explain the main result. It’s all about ‘black boxing’: hiding the details of a circuit and only remembering its behavior as seen from outside.

### The idea

In late 1940s, just as Feynman was developing his diagrams for processes in particle physics, Eilenberg and Mac Lane initiated their work on category theory. Over the subsequent decades, and especially in the work of Joyal and Street in the 1980s, it became clear that these developments were profoundly linked: monoidal categories have a precise graphical representation in terms of string diagrams, and conversely monoidal categories provide an algebraic foundation for the intuitions behind Feynman diagrams. The key insight is the use of categories where morphisms describe physical processes, rather than structure-preserving maps between mathematical objects.

In work on fundamental physics, the cutting edge has moved from categories to higher categories. But the same techniques have filtered into more immediate applications, particularly in computation and quantum computation. Our paper is part of a new program of applying string diagrams to engineering, with the aim of giving diverse diagram languages a unified foundation based on category theory.

Indeed, even before physicists began using Feynman diagrams, various branches of engineering were using diagrams that in retrospect are closely related. Foremost among these are the ubiquitous electrical circuit diagrams. Although less well-known, similar diagrams are used to describe networks consisting of mechanical, hydraulic, thermodynamic and chemical systems. Further work, pioneered in particular by Forrester and Odum, applies similar diagrammatic methods to biology, ecology, and economics.

As discussed in detail by Olsen, Paynter and others, there are mathematically precise analogies between these different systems. In each case, the system’s state is described by variables that come in pairs, with one variable in each pair playing the role of ‘displacement’ and the other playing the role of ‘momentum’. In engineering, the time derivatives of these variables are sometimes called ‘flow’ and ‘effort’.

 displacement:    $q$ flow:      $\dot q$ momentum:      $p$ effort:           $\dot p$ Mechanics: translation position velocity momentum force Mechanics: rotation angle angular velocity angular momentum torque Electronics charge current flux linkage voltage Hydraulics volume flow pressure momentum pressure Thermal Physics entropy entropy flow temperature momentum temperature Chemistry moles molar flow chemical momentum chemical potential

In classical mechanics, this pairing of variables is well understood using symplectic geometry. Thus, any mathematical formulation of the diagrams used to describe networks in engineering needs to take symplectic geometry as well as category theory into account.

While diagrams of networks have been independently introduced in many disciplines, we do not expect formalizing these diagrams to immediately help the practitioners of these disciplines. At first the flow of information will mainly go in the other direction: by translating ideas from these disciplines into the language of modern mathematics, we can provide mathematicians with food for thought and interesting new problems to solve. We hope that in the long run mathematicians can return the favor by bringing new insights to the table.

Although we keep the broad applicability of network diagrams in the back of our minds, our paper talks in terms of electrical circuits, for the sake of familiarity. We also consider a somewhat limited class of circuits. We only study circuits built from ‘passive’ components: that is, those that do not produce energy. Thus, we exclude batteries and current sources. We only consider components that respond linearly to an applied voltage. Thus, we exclude components such as nonlinear resistors or diodes. Finally, we only consider components with one input and one output, so that a circuit can be described as a graph with edges labeled by components. Thus, we also exclude transformers. The most familiar components our framework covers are linear resistors, capacitors and inductors.

While we want to expand our scope in future work, the class of circuits made from these components has appealing mathematical properties, and is worthy of deep study. Indeed, these circuits has been studied intensively for many decades by electrical engineers. Even circuits made exclusively of resistors have inspired work by mathematicians of the caliber of Weyl and Smale!

Our work relies on this research. All we are adding is an emphasis on symplectic geometry and an explicitly ‘compositional’ framework, which clarifies the way a larger circuit can be built from smaller pieces. This is where monoidal categories become important: the main operations for building circuits from pieces are composition and tensoring.

Our strategy is most easily illustrated for circuits made of linear resistors. Such a resistor dissipates power, turning useful energy into heat at a rate determined by the voltage across the resistor. However, a remarkable fact is that a circuit made of these resistors always acts to minimize the power dissipated this way. This ‘principle of minimum power’ can be seen as the reason symplectic geometry becomes important in understanding circuits made of resistors, just as the principle of least action leads to the role of symplectic geometry in classical mechanics.

Here is a circuit made of linear resistors:

The wiggly lines are resistors, and their resistances are written beside them: for example, $3\Omega$ means 3 ohms, an ‘ohm’ being a unit of resistance. To formalize this, define a circuit of linear resistors to consist of:

• a set $N$ of nodes,
• a set $E$ of edges,
• maps $s,t : E \to N$ sending each edge to its source and target node,
• a map $r: E \to (0,\infty)$ specifying the resistance of the resistor
labelling each edge,
• maps $i : X \to N,$ $o : Y \to N$ specifying the inputs and outputs of the circuit.

When we run electric current through such a circuit, each node $n \in N$ gets a potential $\phi(n).$ The voltage across an edge $e \in E$ is defined as the change in potential as we move from to the source of $e$ to its target, $\phi(t(e)) - \phi(s(e)).$ The power dissipated by the resistor on this edge is then

$\displaystyle{ \frac{1}{r(e)}\big(\phi(t(e))-\phi(s(e))\big)^2 }$

The total power dissipated by the circuit is therefore twice

$\displaystyle{ P(\phi) = \frac{1}{2}\sum_{e \in E} \frac{1}{r(e)}\big(\phi(t(e))-\phi(s(e))\big)^2 }$

The factor of $\frac{1}{2}$ is convenient in some later calculations.

Note that $P$ is a nonnegative quadratic form on the vector space $\mathbb{R}^N.$ However, not every nonnegative definite quadratic form on $\mathbb{R}^N$ arises in this way from some circuit of linear resistors with $N$ as its set of nodes. The quadratic forms that do arise are called Dirichlet forms. They have been extensively investigated, and they play a major role in our work.

We write

$\partial N = i(X) \cup o(Y)$

for the set of terminals: that is, nodes corresponding to inputs or outputs. The principle of minimum power says that if we fix the potential at the terminals, the circuit will choose the potential at other nodes to minimize the total power dissipated. An element $\psi$ of the vector space $\mathbb{R}^{\partial N}$ assigns a potential to each terminal. Thus, if we fix $\psi,$ the total power dissipated will be twice

$Q(\psi) = \min_{\substack{ \phi \in \mathbb{R}^N \\ \phi\vert_{\partial N} = \psi}} \; P(\phi)$

The function $Q : \mathbb{R}^{\partial N} \to \mathbb{R}$ is again a Dirichlet form. We call it the power functional of the circuit.

Now, suppose we are unable to see the internal workings of a circuit, and can only observe its ‘external behavior’: that is, the potentials at its terminals and the currents flowing into or out of these terminals. Remarkably, this behavior is completely determined by the power functional $Q.$ The reason is that the current at any terminal can be obtained by differentiating $Q$ with respect to the potential at this terminal, and relations of this form are all the relations that hold between potentials and currents at the terminals.

The Laplace transform allows us to generalize this immediately to circuits that can also contain linear inductors and capacitors, simply by changing the field we work over, replacing $\mathbb{R}$ by the field $\mathbb{R}(s)$ of rational functions of a single real variable, and talking of impedance where we previously talked of resistance. We obtain a category $\mathrm{Circ}$ where an object is a finite set, a morphism $f : X \to Y$ is a circuit with input set $X$ and output set $Y,$ and composition is given by identifying the outputs of one circuit with the inputs of the next, and taking the resulting union of labelled graphs. Each such circuit gives rise to a Dirichlet form, now defined over $\mathbb{R}(s),$ and this Dirichlet form completely describes the externally observable behavior of the circuit.

We can take equivalence classes of circuits, where two circuits count as the same if they have the same Dirichlet form. We wish for these equivalence classes of circuits to form a category. Although there is a notion of composition for Dirichlet forms, we find that it lacks identity morphisms or, equivalently, it lacks morphisms representing ideal wires of zero impedance. To address this we turn to Lagrangian subspaces of symplectic vector spaces. These generalize quadratic forms via the map

$\Big(Q: \mathbb{F}^{\partial N} \to \mathbb{F}\Big) \longmapsto$

$\mathrm{Graph}(dQ) = \{(\psi, dQ_\psi) \mid \psi \in \mathbb{F}^{\partial N} \} \; \subseteq \; \mathbb{F}^{\partial N} \oplus (\mathbb{F}^{\partial N})^\ast$

taking a quadratic form $Q$ on the vector space $\mathbb{F}^{\partial N}$ over the field $\mathbb{F}$ to the graph of its differential $dQ.$ Here we think of the symplectic vector space $\mathbb{F}^{\partial N} \oplus (\mathbb{F}^{\partial N})^\ast$ as the state space of the circuit, and the subspace $\mathrm{Graph}(dQ)$ as the subspace of attainable states, with $\psi \in \mathbb{F}^{\partial N}$ describing the potentials at the terminals, and $dQ_\psi \in (\mathbb{F}^{\partial N})^\ast$ the currents.

This construction is well-known in classical mechanics, where the principle of least action plays a role analogous to that of the principle of minimum power here. The set of Lagrangian subspaces is actually an algebraic variety, the Lagrangian Grassmannian, which serves as a compactification of the space of quadratic forms. The Lagrangian Grassmannian has already played a role in Sabot’s work on circuits made of resistors. For us, its importance it that we can find identity morphisms for the composition of Dirichlet forms by taking circuits made of parallel resistors and letting their resistances tend to zero: the limit is not a Dirichlet form, but it exists in the Lagrangian Grassmannian.

Indeed, there exists a category $\mathrm{LagrRel}$ with finite dimensional symplectic vector spaces as objects and Lagrangian relations as morphisms: that is, linear relations from $V$ to $W$ that are given by Lagrangian subspaces of $\overline{V} \oplus W,$ where $\overline{V}$ is the symplectic vector space conjugate to $V$—that is, with the sign of the symplectic structure switched.

To move from the Lagrangian subspace defined by the graph of the differential of the power functional to a morphism in the category $\mathrm{LagrRel}$—that is, to a Lagrangian relation— we must treat seriously the input and output functions of the circuit. These express the circuit as built upon a cospan:

Applicable far more broadly than this present formalization of circuits, cospans model systems with two ‘ends’, an input and output end, albeit without any connotation of directionality: we might just as well exchange the role of the inputs and outputs by taking the mirror image of the above diagram. The role of the input and output functions, as we have discussed, is to mark the terminals we may glue onto the terminals of another circuit, and the pushout of cospans gives formal precision to this gluing construction.

One upshot of this cospan framework is that we may consider circuits with elements of $N$ that are both inputs and outputs, such as this one:

This corresponds to the identity morphism on the finite set with two elements. Another is that some points may be considered an input or output multiple times, like here:

This lets to connect two distinct outputs to the above double input.

Given a set $X$ of inputs or outputs, we understand the electrical behavior on this set by considering the symplectic vector space $\mathbb{F}^X \oplus {(\mathbb{F}^X)}^\ast,$ the direct sum of the space $\mathbb{F}^X$ of potentials and the space ${(\mathbb{F}^X)}^\ast$ of currents at these points. A Lagrangian relation specifies which states of the output space $\mathbb{F}^Y \oplus {(\mathbb{F}^Y)}^\ast$ are allowed for each state of the input space $\mathbb{F}^X \oplus {(\mathbb{F}^X)}^\ast.$ Turning the Lagrangian subspace $\mathrm{Graph}(dQ)$ of a circuit into this information requires that we understand the ‘symplectification’

$Sf: \mathbb{F}^B \oplus {(\mathbb{F}^B)}^\ast \to \mathbb{F}^A \oplus {(\mathbb{F}^A)}^\ast$

and ‘twisted symplectification’

$S^tf: \mathbb{F}^B \oplus {(\mathbb{F}^B)}^\ast \to \overline{\mathbb{F}^A \oplus {(\mathbb{F}^A)}^\ast}$

of a function $f: A \to B$ between finite sets. In particular we need to understand how these apply to the input and output functions with codomain restricted to $\partial N$; abusing notation, we also write these $i: X \to \partial N$ and $o: Y \to \partial N.$

The symplectification $Sf$ is a Lagrangian relation, and the catch phrase is that it ‘copies voltages’ and ‘splits currents’. More precisely, for any given potential-current pair $(\psi,\iota)$ in $\mathbb{F}^B \oplus {(\mathbb{F}^B)}^\ast,$ its image under $Sf$ consists of all elements of $(\psi', \iota')$ in $\mathbb{F}^A \oplus {(\mathbb{F}^A)}^\ast$ such that the potential at $a \in A$ is equal to the potential at $f(a) \in B,$ and such that, for each fixed $b \in B,$ collectively the currents at the $a \in f^{-1}(b)$ sum to the current at $b.$ We use the symplectification $So$ of the output function to relate the state on $\partial N$ to that on the outputs $Y.$

As our current framework is set up to report the current out of each node, to describe input currents we define the twisted symplectification:

$S^tf: \mathbb{F}^B \oplus {(\mathbb{F}^B)}^\ast \to \overline{\mathbb{F}^A \oplus {(\mathbb{F}^A)}^\ast}$

almost identically to the above, except that we flip the sign of the currents $\iota' \in (\mathbb{F}^A)^\ast.$ This again gives a Lagrangian relation. We use the twisted symplectification $S^ti$ of the input function to relate the state on $\partial N$ to that on the inputs.

The Lagrangian relation corresponding to a circuit then comprises exactly a list of the potential-current pairs that are possible electrical states of the inputs and outputs of the circuit. In doing so, it identifies distinct circuits. A simple example of this is the identification of a single 2-ohm resistor:

with two 1-ohm resistors in series:

Our inability to access the internal workings of a circuit in this representation inspires us to call this process black boxing: you should imagine encasing the circuit in an opaque black box, leaving only the terminals accessible. Fortunately, this information is enough to completely characterize the external behavior of a circuit, including how it interacts when connected with other circuits!

Put more precisely, the black boxing process is functorial: we can compute the black-boxed version of a circuit made of parts by computing the black-boxed versions of the parts and then composing them. In fact we shall prove that $\mathrm{Circ}$ and $\mathrm{LagrRel}$ are dagger compact categories, and the black box functor preserves all this extra structure:

Theorem. There exists a symmetric monoidal dagger functor, the black box functor

$\blacksquare: \mathrm{Circ} \to \mathrm{LagrRel}$

mapping a finite set $X$ to the symplectic vector space $\mathbb{F}^X \oplus (\mathbb{F}^X)^\ast$ it generates, and a circuit $\big((N,E,s,t,r),i,o\big)$ to the Lagrangian relation

$\bigcup_{v \in \mathrm{Graph}(dQ)} S^ti(v) \times So(v) \subseteq \overline{\mathbb{F}^X \oplus (\mathbb{F}^X)^\ast} \oplus \mathbb{F}^Y \oplus (\mathbb{F}^Y)^\ast$

where $Q$ is the circuit’s power functional.

The goal of this paper is to prove and explain this result. The proof is more tricky than one might first expect, but our approach involves concepts that should be useful throughout the study of networks, such as ‘decorated cospans’ and ‘corelations’.

Give it a read, and let us know if you have questions or find mistakes!