Schrödinger and Einstein

5 January, 2020

  

Schrödinger and Einstein helped invent quantum mechanics. But they didn’t really believe in its implications for the structure of reality, so in their later years they couldn’t get themselves to simply use it like most of their colleagues. Thus, they were largely sidelined. While others made rapid progress in atomic, nuclear and particle physics, they spent a lot of energy criticizing and analyzing quantum theory.

They also spent a lot of time on ‘unified field theories’: theories that sought to unify gravity and electromagnetism, without taking quantum mechanics into account.

After he finally found his equations describing gravity in November 1915, Einstein spent years working out their consequences. In 1917 he changed the equations, introducing the ‘cosmological constant’ Λ to keep the universe from expanding. Whoops.

In 1923, Einstein got excited about attempts to unify gravity and electromagnetism. He wrote to Niels Bohr:

I believe I have finally understood the connection between electricity and gravitation. Eddington has come closer to the truth than Weyl.

You see, Hermann Weyl and Arthur Eddington had both tried to develop unified field theories—theories that unified gravity and electromagnetism. Weyl had tried a gauge theory—indeed, he invented the term ‘gauge transformations’ at this time. In 1918 he asked Einstein to communicate a paper on it to the Berlin Academy. Einstein did, but pointed out a crushing physical objection to it in a footnote!

In 1921, Eddington tried a theory where the fundamental field was not the spacetime metric, but a torsion-free connection. He tried to show that both electromagnetism and gravity could be described by such a theory. But he didn’t even get as far as writing down field equations.

Einstein wrote three papers on Eddington’s ideas in 1923. He was so excited that he sent the first to the Berlin Academy from a ship sailing from Japan! He wrote down field equations and sought to connect them to Maxwell’s equations and general relativity. He was very optimistic at this time, concluding that

Eddington’s general idea in context with the Hamiltonian principle leads to a theory almost free of ambiguities; it does justice to our present knowledge about gravitation and electricity and unifies both kinds of fields in a truly accomplished manner.

Later he noticed the flaws in the theory. He had an elaborate approach to getting charged particles from singular solutions of the equation, though he wished they could be described by nonsingular solutions. He was stumped by the fact that the negatively and positively charged particles he knew—the electron and proton—had different masses. The same problem afflicted Dirac later, until the positron was discovered. But there were also problems even in getting Maxwell’s equations and general relativity from this framework, even approximately.

By the 1925 his enthusiasm had faded. He wrote to his friend Besso:

Regrettably, I had to throw away my work in the spirit of Eddington. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl–Eddington.

So, he started work on another unified field theory. And another.

And another.

Einstein worked obsessively on unified field theories until his death in 1955. He lost touch with his colleagues’ discoveries in particle physics. He had an assistant, Valentine Bargmann, try to teach him quantum field theory—but he lost interest in a month. All he wanted was a geometrical explanation of gravity and electromagnetism. He never succeeded in this quest.

But there’s more to this story!

The other side of the story is Schrödinger. In the 1940s, he too became obsessed with unified field theories. He and Einstein became good friends—but also competitors in their quest to unify the forces of nature.

But let’s back up a bit. In June 1935, after the famous Einstein-Podolsky-Rosen paper arguing that quantum mechanics was incomplete, Schrödinger wrote to Einstein:

I am very happy that in the paper just published in P.R. you have evidently caught dogmatic q.m. by the coat-tails.

Einstein replied:

You are the only person with whom I am actually willing to come to terms.

They bonded over their philosophical opposition to the Bohr–Heisenberg attitude to quantum mechanics. In November 1935, Schrödinger wrote his paper on ‘Schrödinger’s cat‘.



Schrödinger fled Austria after the Nazis took over. In 1940 he got a job at the brand-new Dublin Institute for Advanced Studies.

In 1943 he started writing about unified field theories, corresponding with Einstein. He worked on some theories very similar to those of Einstein and Straus, who were trying to unify gravity and electromagnetism in a theory involving a connection with torsion, whose Levi-Civita symbol was therefore non-symmetric. He wrote 8 papers on this subject.

Einstein even sent Schrödinger two of his unpublished papers on these ideas!

In late 1946, Schrödinger had a new insight. He was thrilled.

By 1947 Schrödinger thought he’d made a breakthrough. He presented a paper on January 27th at the Dublin Institute of Advanced Studies. He even called a press conference to announce his new theory!

He predicted that a rotating mass would generate a magnetic field.

The story of the great discovery was quickly telegraphed around the world, and the science editor of the New York Times interview Einstein to see what he thought.

Einstein was not impressed. In a carefully prepared statement he shot Schrödinger down:

Einstein was especially annoyed that Schrödinger had called a press conference to announce his new theory before there was any evidence supporting it.

Wise words. I wish people heeded them!

Schrödinger apologized in a letter to Einstein, claiming that he’d done the press conference just to get a pay raise. Einstein responded curtly, saying “your theory does not really differ from mine”.

They stopped writing to each other for 3 years.

I’d like to understand Schrödinger’s theory using the modern tools of differential geometry. I don’t think it’s promising. I just want to know what it actually says, and what it predicts! Go here for details:

Schrödinger’s unified field theory, The n-Category Café, December 26, 2019.

For more on Schrödinger’s theory, try his book:

• Erwin Schrödinger, Space-Time Structure, Cambridge U. Press, Cambridge, 1950. Chapter XII: Generalizations of Einstein’s theory.

and his first paper on the theory:

• Erwin Schödinger, The final affine field laws I, Proceedings of the Royal Irish Academy A 51 (1945–1948), 163–171.

For a wonderfully detailed analysis of the history of unified field theories, including the work of Einstein and Schrödinger, read these:

• Hubert F. M. Goenner, On the history of unified field theories, Living Reviews in Relativity 7 (2004), article no. 2. On the history of unified field theories II (ca. 1930–ca. 1965), Living Reviews in Relativity 17 (2014), article no. 5.

especially Section 6 of the second paper. For more on the story of Einstein and Schrödinger, I recommend this wonderful book:

• Walter Moore, Schrödinger: Life and Thought, Cambridge U. Press, Cambridge, 1989.

This is where I got most of my quotes.


Compositionality — First Issue

30 December, 2019


Compositionality

Yay! The first volume of Compositionality has been published! You can read it here:

https://compositionality-journal.org

“Compositionality” is about how complex things can be assembled out of simpler parts. Compositionality is a journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Compositionality is a diamond open access journal. That means it’s free to publish in and free to read.

The executive board consists of Brendan Fong, Nina Otter and Joshua Tan. I thank them for all their work making this dream a reality!

The coordinating editors are Aleks Kissinger and Joachim Kock. The steering board consists of John Baez, Bob Coecke, Kathryn Hess, Steve Lack and Valeria de Paiva.

The editors are:

Corina Cirstea, University of Southampton, UK
Ross Duncan, University of Strathclyde, UK
Andrée Ehresmann, University of Picardie Jules Verne, France
Tobias Fritz, Perimeter Institute, Canada
Neil Ghani, University of Strathclyde, UK
Dan Ghica, University of Birmingham, UK
Jeremy Gibbons, University of Oxford, UK
Nick Gurski, Case Western Reserve University, USA
Helle Hvid Hansen, Delft University of Technology, Netherlands
Chris Heunen, University of Edinburgh, UK
Martha Lewis, University of Amsterdam, Netherlands
Samuel Mimram, École Polytechnique, France
Simona Paoli, University of Leicester, UK
Dusko Pavlovic, University of Hawaii, USA
Christian Retoré, Université de Montpellier, France
Mehrnoosh Sadrzadeh, Queen Mary University, UK
Peter Selinger, Dalhousie University, Canada
Pawel Sobocinski, University of Southampton, UK
David Spivak, MIT, USA
Jamie Vicary, University of Birmingham and University of Oxford, UK
Simon Willerton, University of Sheffield, UK


How to Solve Climate Change

28 December, 2019

Happy New Year!

This podcast of an interview with Saul Griffith is a great way to start your year:

• Ezra Klein, How to solve climate change and make life more awesome.

Skip straight down to the bottom and listen to the interview! Or right click on the link below and

download the .mp3.

I usually prefer reading stuff, but this is only available in audio form—and it’s worth it.

One important thing he says:

We have not had anyone stand up and espouse a vision of the future that could sound like success.

I think it’s time to start doing that. I think I’m finally figuring out how. But this interview with Saul Griffith does it already!




Applied Category Theory 2020 — Adjoint School

23 December, 2019

Like last year and the year before, there will be a school associated to this year’s conference on applied category theory! If you’re trying to get into applied category theory, this is the best possible way.

Applied Category Theory 2020 — Adjoint School.

The school will consist of online meetings from February to June 2020, followed by a research week June 29–July 3, 2020 at MIT in Cambridge Massachusetts. The conference follows on July 6–10, 2020, and if you attend the school you should also go to the conference.

The deadline to apply is January 15 2020; apply here.

There will be 4 mentors teaching courses at the school:

• Michael Johnson, Categories of maintainable relations.

• Nina Otter, Diagrammatic and algebraic approaches to distances between persistence modules.

• Valeria de Paiva, Dialectica categories of Petri nets.

• Michael Shulman, A practical type theory for symmetric monoidal categories.

Click on the links for more detailed information!

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of minorities, and of any groups which are underrepresented in the mathematics and computer science communities, are especially encouraged to apply.

Structure of the school

Every participant will be assigned to one of the groups above, according to their preference (and to the availability of places within the groups). Each group will consist of a mentor, a TA, and 4-5 students.

Online meetings

Between February and June 2020 there will be an online reading seminar. Each group will have a reading list of two papers, which they will study, and then present to the rest of the school during weekly online meetings. Every member of the school is encouraged to take part in the discussion of every paper, first during the meeting via live chat, and then, in written form, on an online forum. After the presentation and the forum discussion the students of each group will write a blog post about their assigned paper on the n-Category Café.

During this period, the TAs will be there to help the students, answer any question they might have, and moderate the discussions. This way, all the participants will build the necessary background to take part in the research activities during the week at MIT.

Research week

After the online meetings, there will be a two-week event at MIT, from June 29th to July 10th 2020. The first week is dedicated exclusively to the participants of the school. They will work in groups on the research projects outlined above, led by their mentors, with the help of their TAs.

During the second week the ACT 2020 Conference will take place, which is open to a wider audience. The member of each group of the school will have the possibility to present their activity to the audience of the conference, and share their ideas. The conference is not technically part of the school, but is about very similar topics, and participation is very much encouraged. The online meetings should prepare students to be able to follow some of the conference presentations to a reasonable degree, and introduce them to the main problems and techniques of the field.

Questions?

For any questions or doubts please write us at the address act adjoint school at gmail dot com.

Organizers

Carmen Constantin

Eliana Lorch

Paolo Perrone


Applied Category Theory Postdocs at NIST

13 December, 2019

An advertisement:

We are looking to expand our group of applied category theorists at the National Institute of Standards and Technology (NIST). Our group develops use cases, tools and methodology to apply category theory and related methods in a broad range of disciplines centered around the design, implementation, operation and evolution of engineered systems.

We encourage those eligible and interested to apply for the National Research Council Research Associateship Program. The upcoming deadline is February 1st, for those looking to start by December 2020.

The relevant postdoctoral opportunities can be found here:

Mathematical Foundations for System Interoperability
Research in Cyber-Physical Systems

These 2-year postdoctoral positions are only open to US citizens, come with a base stipend around $72k (12 month), great benefits, and travel support.

For non-US citizens, NIST has mechanisms to host foreign guest researchers (undergrad through professor). Typically, such researchers propose their own projects to be completed in collaboration with researchers and use of facilities at NIST.

For more information, contact Spencer Breiner (spencer.breiner@nist.gov), Blake Pollard (blake.pollard@nist.gov), and/or Eswaran Subrahmanian (sub@cmu.edu).


Applied Category Theory Meeting at UCR (Part 3)

15 November, 2019

 

We had a special session on applied category theory here at UCR:

Applied category theory, Fall Western Sectional Meeting of the AMS, 9–10 November 2019, U.C. Riverside.

I was bowled over by the large number of cool ideas. I’ll have to blog about some of them. A bunch of people stayed for a few days afterwards, and we had lots of great conversations.

The biggest news was that Brendan Fong and David Spivak definitely want to set up an applied category theory in the San Francisco Bay Area, which they’re calling the Topos Institute. They are now in the process of raising funds for this institute! I plan to be involved, so I’ll be saying more about this later.

But back to the talks. We didn’t make videos, but here are the slides. Click on talk titles to see abstracts of the talks. For a multi-author talk, the person whose name is in boldface is the one who gave the talk. You also might enjoy comparing the 2017 talks.

Saturday November 9, 2019

8:00 a.m.
Fibrations as generalized lens categoriestalk slides.
David I. Spivak, Massachusetts Institute of Technology

9:00 a.m.
Supplying bells and whistles in symmetric monoidal categoriestalk slides.
Brendan Fong, Massachusetts Institute of Technology
David I. Spivak, Massachusetts Institute of Technology

9:30 a.m.
Right adjoints to operadic restriction functorstalk slides.
Philip Hackney, University of Louisiana at Lafayette
Gabriel C. Drummond-Cole, IBS Center for Geometry and Physics

10:00 a.m.
Duality of relationstalk slides.
Alexander Kurz, Chapman University

10:30 a.m.
A synthetic approach to stochastic maps, conditional independence, and theorems on sufficient statisticstalk slides.
Tobias Fritz, Perimeter Institute for Theoretical Physics

3:00 p.m.
Constructing symmetric monoidal bicategories functoriallytalk slides.
Michael Shulman, University of San Diego
Linde Wester Hansen, University of Oxford

3:30 p.m.
Structured cospanstalk slides.
Kenny Courser, University of California, Riverside
John C. Baez, University of California, Riverside

4:00 p.m.
Generalized Petri netstalk slides.
Jade Master, University of California, Riverside

4:30 p.m.
Formal composition of hybrid systemstalk slides and website.

Paul Gustafson, Wright State University
Jared Culbertson, Air Force Research Laboratory
Dan Koditschek, University of Pennsylvania
Peter Stiller, Texas A&M University

5:00 p.m.
Strings for cartesian bicategoriestalk slides.
M. Andrew Moshier, Chapman University

5:30 p.m.
Defining and programming generic compositions in symmetric monoidal categoriestalk slides.
Dmitry Vagner, Los Angeles, CA

Sunday November 10, 2019

8:00 a.m.
Mathematics for second quantum revolutiontalk slides.
Zhenghan Wang, UCSB and Microsoft Station Q

9:00 a.m.
A compositional and statistical approach to natural languagetalk slides.
Tai-Danae Bradley, CUNY Graduate Center

9:30 a.m.
Exploring invariant structure in neural activity with applied topology and category theorytalk slides.
Brad Theilman, UC San Diego
Krista Perks, UC San Diego
Timothy Q Gentner, UC San Diego

10:00 a.m.
Of monks, lawyers and villages: new insights in social network science — talk cancelled due to illness.
Nina Otter, Mathematics Department, UCLA
Mason A. Porter, Mathematics Department, UCLA

10:30 a.m.
Functorial cluster embeddingtalk slides.

Steve Huntsman, BAE Systems FAST Labs

2:00 p.m.
Quantitative equational logictalk slides.
Prakash Panangaden, School of Computer Science, McGill University
Radu Mardare, Strathclyde University
Gordon D. Plotkin, University of Edinburgh

3:00 p.m.
Brakes: an example of applied category theorytalk slides in PDF and Powerpoint.
Eswaran Subrahmanian, Carnegie Mellon University / National Institute of Standards and Technology

3:30 p.m.
Intuitive robotic programming using string diagramstalk slides.
Blake S. Pollard, National Institute of Standards and Technology

4:00 p.m.
Metrics on functor categoriestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

4:30 p.m.
Hausdorff and Wasserstein metrics on graphs and other structured datatalk slides.
Evan Patterson, Stanford University


Why Is Category Theory a Trending Topic?

8 November, 2019

I wrote something for the Spanish newspaper El País, which has a column on mathematics called “Café y Teoremas”. Ágata Timón helped me a lot with writing this, and she also translated it into Spanish:

• John Baez, Qué es la teoría de categorías y cómo se ha convertido en tendencia, El País, 8 November 2019.

Here’s the English-language version I wrote. It’s for a general audience so don’t expect hard-core math!

Why has “category theory” become a trending topic?

Recently, various scientific media have been paying attention to a branch of mathematics called “category theory” that has become pretty popular inside the mathematical community in recent years. Some mathematicians are even starting to complain on Twitter that more people are tweeting about category theory than their own specialties. But what is this branch of mathematics, and why is it becoming so fashionable?

Category theory was invented in 1945 as a general technique to transform problems in one field of pure mathematics into problems in another field, where they could be solved. For example, we know that at any moment there must be a location on the surface of the Earth there where the wind velocity is zero. This is a marvelous result—but to prove this result, we must translate it into a fact about algebra, and a bit of category theory is very helpful here. More difficult results often require more category theory. The proof of Fermat’s Last Theorem, for example, builds on a vast amount of 20th-century mathematics, in which category theory plays a crucial role.

Category theory is sometimes called “the mathematics of mathematics”, since it stands above many other fields of mathematics, connecting and uniting them. Unfortunately even mathematicians have a limited tolerance for this high level of abstraction. So, for a long time many mathematicians called category theory “abstract nonsense”—using it reluctantly when it was necessary for their work, but not really loving it.

On the other hand, other mathematicians embraced the beauty and power of category theory. Thus, its influence has gradually been spreading. Since the 1990s, it has been infiltrating computer science: for example, new programming languages like Haskell and Scala use ideas from this subject. But now we are starting to see people apply category theory to chemistry, electrical engineering, and even the design of brakes in cars! “Applied category theory”, once an oxymoron, is becoming a real subject.

To understand this we need a little taste of the ideas. A category consists of a set of “objects” together with “morphisms”—some kind of processes, or paths—going between these objects. For example, we could take the objects to be cities, and the morphisms to be routes from one city to another. The key requirement is that if we have a morphism from an object x to an object y and a morphism from y to an object z, we can “compose” them and get a morphism from x to z. For example, if you have a way to drive from Madrid to Seville and a way to drive from Seville to Faro, that gives a way to drive from Madrid to Faro. Thus there is a category of cities and routes between them.

In mathematics, this focus on morphisms represented a radical shift of viewpoint. Starting around 1900, logicians tried to build the whole of mathematics on solid foundations. This turned out to be a difficult and elusive task, but their best attempt at the time involved “set theory”. A set is simply a collection of elements. In set theory as commonly practiced by mathematicians, these elements are also just sets. In this worldview, everything is just a set. It is a static worldview, as if we had objects but no morphisms. On the other hand, category theory builds on set theory by emphasizing morphisms—ways of transforming things—as equal partners to things themselves. It is not incompatible with set theory, but it offers new ways of thinking.

The idea of a category is simple. Exploiting it is harder. A loose group of researchers are starting to apply category theory to subjects beyond pure mathematics. The key step is to focus a bit less on things and a bit more on morphisms, which are ways to go between things, or ways to transform one thing into another. This is attitude is well suited to computer programming: a program is a way to transform input data into output data, and composing programs is the easiest way to build complicated programs from simpler ones. But personally, I am most excited by applications to engineering and the natural sciences, because these are newer and more surprising.

I was very pleased when two of my students got internships at the engineering firm Siemens, applying category theory to industrial processes. The first, Blake Pollard, now has a postdoctoral position at the National Institute of Standards and Technology in the USA. Among other things, he has used a programming method based on category theory to help design a “smart grid”—an electrical power network that is flexible enough to handle the ever-changing power generated by thousands of homes equipped with solar panels.

Rumors say that soon there may even be an institute of applied category theory, connecting mathematicians to programmers and businesses who need this way of thinking. It is too early to tell if this is the beginning of a trend, but my friends and colleagues on Twitter are very excited.