2020 Physics Nobel Prize

The 2020 Physics Nobel Prize was announced this morning, with half going to Roger Penrose for his work on black holes, half to two astronomers (Reinhard Genzel and Andrea Ghez) for their work mapping what is going on at the center of our galaxy. I know just about nothing about the astronomy side of this, but am somewhat familiar with Penrose’s work, which very much deserves the prize.

Penrose is a rather unusual choice for a Physics Nobel Prize, in that he’s very much a mathematical physicist, with a Ph.D. in mathematics (are there other physics winners with math Ph.Ds?). In addition, the award is not for a new physical theory, or for anything experimentally testable, but for the rigorous understanding of the implications of Einstein’s general relativity. While I’m a great fan of the importance of this kind of work, I can’t think of many examples of it getting rewarded by the Nobel prize. I had always thought that Penrose was likely to get a Breakthrough Prize rather than a Nobel Prize, still don’t understand why that hasn’t happened already.

Besides the early work on black holes that Penrose is being recognized for, he has worked on many other things which I think are likely to ultimately be of even greater significance. In particular, he’s far and away the person most responsible for twistor theory, a subject which I believe has a great future ahead of it at the core of fundamental physical theory.

In all his work, Penrose has shown a remarkable degree of originality and creativity. He’s not someone who works to make an advance on ideas pioneered by others, but sets out to do something new and different. His book “The Road to Reality” is a masterpiece, an inspiring original and deep vision of the unity of geometry and physics that outshines the mainstream ways of looking at these questions.

Congratulations to Sir Roger, and compliments to the Nobel prize committee for a wonderful choice!

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Quick Links

A few quick links:

  • I was sorry to hear of the recent death of Vaughan Jones. A few things about his life and work have started to appear, see here, here and here.
  • For a wonderful in-depth article about the life of Michael Atiyah written by Nigel Hitchin, see here.
  • There are now many new places where you can find talks about math and physics to listen to. For instance, just for math and just at Harvard, there is a series of Harvard Math Literature talks and Dennis Gaitsgory’s geometric Langlands office hours.
  • Breakthrough Prizes were announced today. There’s an argument to be made that the best policy is to ignore them. Weinberg has another 3 million dollars.
  • For an interview with Avi Loeb about why physics is stuck, see here.
  • For an explanation from John Preskill of why quantum computing is hard (which I’d claim has to do with why the measurement problem is hard), see here.

Update: Last night I watched The Social Dilemma on Netflix, which included some segments with my friend Cathy O’Neil (AKA Mathbabe). Highly recommended, best of the things I’ve read or watched that try and come to grips with the nature of the horror irresponsibly unleashed by Mark Zuckerberg and Facebook in the form of the AI driven News Feed. Comparing to a documentary about Oxycontin from a while back, the effects of the News Feed are arguably more damaging. I’m wondering why the Oxycontin-funded Sackler family donations to cultural organizations and universities have been heavily criticized, unlike the News Feed-funded Zuckerberg/Milner donations to scientists.

Update: Alain Connes has written a short appreciation of Vaughan Jones and his work here.

Update: For another article about Vaughan Jones well-worth reading, see Davide Castelvecchi at Nature.

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Fall Quantum Mechanics Class

I’ll be teaching a course on quantum mechanics this year here at Columbia, from a point of view aimed somewhat at mathematicians, emphasizing the role of Lie groups and their representations. For more details, the course webpage is here.

The course is being taught online using Zoom, with 37 students now enrolled. I’ve set things up in my office to try and teach using the blackboard there, and will be interacting with the students mostly via Zoom. As an experiment, I’ve also set up a Youtube channel. If all goes well you should be able to find a livestream of the class there while it’s happening, which is scheduled for 4:10-5:25 Tuesdays and Thursdays, starting tomorrow, September 8. I’ll also try and make sure the recorded livestreams get uploaded and saved at this playlist. Unfortunately I won’t be able to interact with people watching on Youtube, should have my hands full trying to get to know the students enrolled here in the course, with only this virtual connection.

Posted in Quantum Mechanics | 19 Comments

AMS Open Math Notes

The AMS for the last few years has had a valuable project called AMS Open Math Notes, a site to gather and make available course notes for math classes, documents of the sort that people sometimes make available on their websites. This provides a great place to go to look for worthwhile notes of this kind (many of them are of very high quality), as well as ensuring their availability for the future. They have an advisory board that evaluates whether submitted notes are suitable.

A couple months ago I submitted the course notes I wrote up this past semester for my Fourier Analysis class, and I’m pleased that they were accepted and are now available here at the AMS site (and will remain also available from my website).

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Quantum Reality

Jim Baggott’s new book, Quantum Reality, is now out here in US, and I highly recommend it to anyone interested in the issues surrounding the interpretation of quantum mechanics. Starting next week I’ll be teaching a course on quantum mechanics for mathematicians (more about this in a few days when I have a better idea how it’s going to work). I’ll be lecturing about the formalism, and for the topic of how this connects to physical reality I’ll be referring the students to this new book (as well as Philip Ball’s Beyond Weird).

When I was first studying quantum mechanics in the early-mid 1970s, the main popular sources discussing interpretational issues were uniform triumphalist accounts of how physicists had struggled with these issues and finally ended up with the “Copenhagen interpretation” (which no one was sure exactly how to state, due to diversity of opinion among theorists and Bohr’s obscurity of expression). Everyone now says that the reigning ideology of the time was “shut up and calculate”, but that’s not exactly what I remember. The Standard Model had just appeared, offering up a huge advance and a long list of new questions with powerful methods to attack them. In this context it was was hard to justify spending time worrying about the subtleties of what Copenhagen might have gotten wrong.

In recent decades things have changed completely, with the question of what’s wrong with Copenhagen and how to do better getting a lot of attention. By now a huge and baffling literature about alternatives has accumulated, forming somewhat of a tower of Babel confronting anyone trying to learn more about the subject. Some popular accounts have dealt with this complexity by turning the subject into a morality play, with alternative interpretations portrayed as the Rebel Alliance fighting righteous battles against the Copenhagen Empire. Others accounts are pretty much propaganda for a particular alternative, be it Bohmian mechanics or a many-worlds interpretation.

Instead of something like this, Baggott provides a refreshingly sane and sensible survey of the subject, trying to get at the core of what is unsatisfying about the Copenhagen account, while explaining the high points of the many different alternatives that have been pursued. He doesn’t have an ax to grind, sees the subject more as a “Game of Theories” in which one must navigate carefully, avoiding Scylla, Charybdis, and various calls from the Sirens. One thing which is driving this whole subject is the advent of new technologies that allow the experimental study of quantum coherence and decoherence, with great attention being paid as possible quantum computing technology has become the hottest and best-funded topic around. Whatever you think about Copenhagen, what Bohr and others characterized as inaccessible to experiment is now anything but that.

While one of my least favorite aspects of discussions of this subject is the various ways the terms “real” and “reality” get used, I have realized that one has to get over that when trying to follow people’s arguments, since the terms have become standard sign-posts. What’s at issue here are fundamental questions about physical science and reality, including the question of what the words “real” and “reality” might mean. In Quantum Reality, Baggott provides a well-informed, reliable and enlightening tour of the increasingly complex and contentious terrain of arguments over what our best fundamental theory is telling us about what is physically “real”.

Update: For a much better and more detailed review of the book, Sabine Hossenfelder’s is here.

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Funding Priorities

The research that gets done in any field of science is heavily influenced by the priorities set by those who fund the research. For science in the US in general, and the field of theoretical physics in particular, recent years have seen a reordering of priorities that is becoming ever more pronounced. As a prominent example, recently the NSF announced that their graduate student fellowships (a program that funds a large number of graduate students in all areas of science and mathematics) will now be governed by the following language:

Although NSF will continue to fund outstanding Graduate Research Fellowships in all areas of science and engineering supported by NSF, in FY2021, GRFP will emphasize three high priority research areas in alignment with NSF goals. These areas are Artificial Intelligence, Quantum Information Science, and Computationally Intensive Research. Applications are encouraged in all disciplines supported by NSF that incorporate these high priority research areas.

No one seems to know exactly what this means in practice, but it clearly means that if you want the best chance of getting a good start on a career in science, you really should be going into one of

  • Artificial Intelligence
  • Quantum Information Science
  • Computationally Intensive Research

or, even better, trying to work on some intersection of these topics.

Emphasis on these areas is not new; it has been growing significantly in recent years, but this policy change by the NSF should accelerate ongoing changes. As far as fundamental theoretical physics goes, we’ve already seen that the move to quantum information science has had a significant effect. For example, the IAS PiTP summer program that trains students in the latest hot topics in 2018 was devoted to From Qubits to Spacetime. The impact of this change in funding priorities is increased by the fact that the largest source of private funding for theoretical physics research, the Simons Foundation, share much the same emphasis. The new Simons-funded Flatiron Institute here in New York has as mission statement

The mission of the Flatiron Institute is to advance scientific research through computational methods, including data analysis, theory, modeling and simulation.

In the latest development on this front, the White House announced today \$1 billion in funding for artificial intelligence and quantum information science research institutes:

“Thanks to the leadership of President Trump, the United States is accomplishing yet another milestone in our efforts to strengthen research in AI and quantum. We are proud to announce that over $1 billion in funding will be geared towards that research, a defining achievement as we continue to shape and prepare this great Nation for excellence in the industries of the future,” said Advisor to the President Ivanka Trump.

This includes an NSF component of \$100 million dollars in new funding for five Artificial Intelligence research institutes. One of these will largely be a fundamental theoretical physics institute, to be called the NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI). The theory topics the institute will concentrate on will be

  • Accelerating Lattice Field Theory with AI
  • Exploring the Multiverse with AI
  • Classifying Knots with AI
  • Astrophysical Simulations with AI
  • Towards an AI Physicist
  • String Theory Conjectures via AI

As far as trying to get beyond the Standard Model, the IAIFI plan is to

work to understand physics beyond the SM in the frameworks of string and knot theory.

I’m rather mystified by how knot theory is going to give us beyond the SM physics, perhaps the plan is to revive Lord Kelvin’s vortex theory.

Update: Some more here about the knots. No question that you can study knots with a computer, but I’m still mystified by their supposed connection to beyond SM physics.

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Guys and Their Theories of Everything

I’m a big fan of Sabine Hossenfelder’s music videos, the latest of which, Theories of Everything, has recently appeared. I also agree with much of the discussion of this at her latest blog posting where Steven Evans writes

nobody wants to see Peter Woit sing.

and Terry Bollinger chimes in:

Please, under no circumstances and in no situations, should folks like Peter Woit, Lee Smolin, Garrett Lisi, Sean Carroll, or even John Baez try to spice up their blogs or tweets by adding clips of themselves singing self-composed physics songs.

Trust me, fellow males of the species: However tempted you may be by Sabine’s spectacular success in this arena, it just ain’t gonna work for you!

The chorus of Sabine’s song goes:

All you guys with theories of everything
Who follow me wherever I am traveling
Your theories are neat
I hope they will succeed
But please, don’t send them to me

One reason for her bursting into song like this was probably her recent participation in this discussion. I’d like to think (for no good reason) that it had nothing to do with my recently sending her a copy of this.

Today brought a new discussion of theories of everything, by Brian Greene and Cumrun Vafa. When asked by Greene to give a grade to string theory, Vafa said that he would give it a grade of A+, although its grade was less than A on the experimental verification front.

While I’m enthusiastic about new ideas involving twistors and happily continuing to work on them, it’s pretty clear that this is not a good time to be bringing them to market. The elite academic world of Harvard and Princeton theorists that I was trained in has been doing an excellent job of convincing everyone that even the smartest people in the world could not make any progress towards a TOE, and that all claims for such progress from the most respected experts around are not very credible. Best to ignore not just the cranks who fill up your inbox with such claims, but all of them, judging the whole concept to be doomed until the point in the far distant future when an experiment finally provides the clue to the correct way forward.

Be warned though, if people don’t pay some more attention, I’m going to start writing songs and singing them here.

Update: Note, an ill-advised attempt at humor referring to identity politics was obviously a mistake and has been deleted (along with some references to it in the comments). The threat to start singing is also a joke.

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Twistors and the Standard Model

For the past few months I’ve been working on writing up some ideas I’m quite excited about, and the pandemic has helped move things along by removing distractions and forcing me to mostly stay home. There’s now something written that I’d like to publicize, a draft manuscript entitled Twistor Geometry and the Standard Model in Euclidean Space, which at some point soon I’ll put on the arXiv. My long experience with both hype about unification in physics as well as theorist’s huge capacity for self-delusion on the topic of their own ideas makes me wary, but I’m very optimistic that these ideas are a significant step forward on the unification front. I believe they provide a remarkable possibility for how internal and space-time symmetries become integrated at short distances, without the usual problem of introducing a host of new degrees of freedom.

Twistor theory has a long history going back to the 1960s, and it is such a beautiful idea that there always has been a good argument that there is something very right about it. But it never seemed to have any obvious connection to the Standard Model and its pattern of internal symmetries. The main idea I’m writing about is that one can get such a connection, as long as one looks at what is happening not just in Minkowski space, but also in Euclidean space. One of the wonderful things about twistor theory is that it includes both Minkowski and Euclidean space as real slices of a complex, holomorphic, geometry. The points in these spaces are best understood as complex lines in another space, projective twistor space. It is on projective twistor space that the internal symmetries of the Standard Model become visible.

The draft paper contains the details, but I should make clear what some of the arguments are for taking this seriously:

  • Unlike other ideas about unification out there, it’s beautiful. The failure of string theory unification has caused a backlash against the idea of using beauty as a criterion for judging unification proposals. I won’t repeat here my usual rant about this. As an example of what I mean about “beauty”, the fundamental spinor degree of freedom appears here tautologically: a point is by definition exactly the $\mathbf C^2$ spinor degree of freedom at that point.
  • Conformal invariance is built-in. The simplest and most highly symmetric possibility for what fundamental physics does at short distances is that it’s conformally invariant. In twistor geometry, conformal invariance is a basic property, realized in a simple way, by the linear $SL(4,\mathbf C)$ group action on the twistor space $\mathbf C^4$. This is a complex group action with real forms $SU(2,2)$ (Minkowski) and $SL(2,\mathbf H)$ (Euclidean).
  • The electroweak $SU(2)$ is inherently chiral. For many other ideas about unification, it’s hard to get chiral interactions. In twistor theory one problem has always been the inherent chiral nature of the theory. Here this becomes not a problem but a solution.

At the same time I should also make clear that what I’m describing here is very incomplete. Two of the main problems are:

  • The degrees of freedom naturally live not on space-time but on projective twistor space $PT$, with space-time points complex projective lines in $PT$. Standard quantum field theory with fields parametrized by space-time points doesn’t apply and how to work instead on $PT$ is unclear. There has been some work on formulating QFT on $PT$ as a holomorphic Chern-Simons theory, and perhaps that work can be applied here.
  • There is no idea for where generations come from. Instead of $PT$ perhaps the theory should be formulated on $S^7$ (space of unit length twistors) and other aspects of the geometry there exploited. In some sense, the incarnations of twistors as four complex number or two quaternions are getting used, but maybe the octonions are relevant.

What I think is probably most important here is that this picture gives a new and compelling idea about how internal and space-time symmetries are related. The conventional argument has always been that the Coleman-Mandula no-go theorem says you can’t combine internal and space-time symmetries in a non-trivial way. Coleman-Mandula does not seem to apply here: these symmetries live on $PT$, not space-time. To really show that this is all consistent, one needs a full theory formulated on $PT$, but I don’t see a Coleman-Mandula argument that a non-trivial such thing can’t exist.

What is most bizarre about this proposal is the way in which, by going to Euclidean space-time, you change what is a space-time and what is an internal symmetry. The argument (see a recent posting) is that, formulated in Euclidean space, the 4d Euclidean symmetry is broken to 3d Euclidean symmetry by the very definition of the theory’s state space, and one of the 4d $SU(2)$s give an internal symmetry, not just analytic continuation of the Minkowski boost symmetry. There is still a lot about how this works I don’t understand, but I don’t see anything inconsistent, i.e. any obstruction to things working out this way. If the identification of the direction of the Higgs field with a choice of imaginary time direction makes sense, perhaps a full theory will give Higgs physics in some way observably different from the usual Standard Model.

One thing not discussed in this paper is gravity. Twistor geometry can also describe curved space-times and gravitational degrees of freedom, and since the beginning, there have been attempts to use it to get a quantum theory of gravity. Perhaps the new ideas described here, including especially the Euclidean point of view with its breaking of Euclidean rotational invariance, will indicate some new way forward for a twistor-based quantum gravity.

Bonus (but related) links: For the last few months the CMSA at Harvard has been hosting a Math-Science Literature Lecture Series of talks. Many worth watching, but one in particular features Simon Donaldson discussing The ADHM construction of Yang-Mills instantons (video here, slides here). This discusses the Euclidean version of the twistor story, in the context it was used back in the late 1970s to relate solutions of the instanton equations to holomorphic bundles.

Update: After looking through the literature, I’ve decided to add some more comments about gravity to the draft paper. The chiral nature of twistor geometry fits naturally with a long tradition going back to Plebanski and Ashtekar of formulating gravity theories using just the self-dual part of the spin connection. For a recent discussion of the sort of gravity theory that appears naturally here, see Kirill Krasnov’s Self-Dual Gravity. For a discussion of the relation of this to twistors, see Yannick Herfray’s Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity.

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Quantization and Dirac Cohomology

For many years I’ve been fascinated by the topic of “Dirac cohomology” and its possible relations to various questions about quantization and quantum field theory. At first I was mainly trying to understand the relation to BRST, and wrote some things here on the blog about that. As time has gone on, my perspective on the subject has kept changing, and for a long time I’ve been wanting to write something here about these newer ideas. Last year I gave a talk at Dartmouth, explaining some of my point of view at the time. Over the last few months I’ve unfortunately yet again changed direction on where this is going. I’ll write about this new direction here in some detail next week, but in the meantime, have decided to make available the slides from the Dartmouth talk, and a version of the document I was writing on Quantization and Dirac Cohomology.

Some warnings:

  • Best to ignore the comments at the end of the slides about applications to Poincaré group representations and BRST. Both of these applications require getting the Dirac cohomology machinery to work in cases of non-reductive Lie algebras. As far as Poincaré goes, I’ve recently come to the conclusion that doing things with the conformal group (which is reductive) is both more interesting and works better. I’ll write more about this next week. For BRST, there is a lot one can say, but I likely won’t get back to writing more about that for a while.
  • The Quantization and Dirac Cohomology document is kind of a mess. It’s an amalgam of various pieces written from different perspectives, and some lecture notes from a course on representation theory. Some day I hope to find the time for a massive rewrite from a new perspective, but maybe some people will find interesting what’s there now.
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(Imaginary) Time Asymmetry

When people write down a list of axioms for quantum mechanics, they typically neglect to include a crucial one: positivity (or more generally, boundedness below) of the energy. This is equivalent to saying that something very different happens when you Fourier transform with respect to time versus with respect to space. If $\psi(t,x)$ is a wavefunction depending on time and space, and you Fourier transform with respect to both time and space
$$\widetilde{\psi}(E,p)=\frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty \psi(t,x)e^{iEt}e^{-ipx}dtdx$$
(the difference in sign for $E$ and $p$ is just a convention) a basic axiom of the theory is that, while $\widetilde{\psi}(E,p)$ can be non-zero for all values of $p$, it must be zero for negative values of $E$.

This fundamental asymmetry in the theory also becomes very apparent if you want to “Wick rotate” the theory. This involves formulating the theory for complex time and exploiting holomorphicity in the time variable. One way to do this is to inverse Fourier transform $\widetilde{\psi}(E,p)$ in $E$, using a complex variable $z=t+i\tau$:
$$\widehat{\psi}(z,p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \widetilde{\psi}(E,p)e^{-iEz} dE$$
The exponential term in the integral will be
$$e^{-iE(t+i\tau)}=e^{-iEt}e^{E\tau}$$
which (since $E$ is non-negative) will only have good behavior for $\tau <0$, i.e. in the lower-half $z$-plane. Thinking of Wick rotation as involving analytic continuation of wave-functions from $z=t$ to $z=t+i\tau$, this will only work for $\tau <0$: there is a fundamental asymmetry in the theory for (imaginary) time.

If you decide to define a quantum theory starting with imaginary time and Wick rotating (analytically continuing) back to real, physical time at the end of a calculation, then you need to build in $\tau$ asymmetry from the beginning. One way this shows up in any formalism for doing this is in the necessity of introducing a $\tau$-reflection operation into the definition of physical states, with the Osterwalder-Schrader positivity condition then needed in order to ensure unitarity of the theory.

Why does one want to formulate the theory in imaginary time anyway? A standard answer to this question is that path integrals don’t actually make any sense in real time, but in imaginary time often become perfectly well-defined objects that can be thought of as expectation values in a statistical mechanical system. For a somewhat different answer, note that even for the simplest free particle theory, when you start calculating things like propagators you immediately run into integrals that involve integrating a function with a pole, for instance integrating over $E$ integrals with a term
$$\frac{1}{E-\frac{p^2}{2m}}$$
Every quantum mechanics and quantum field theory textbook has a discussion of what to do to make sense of such calculations, by defining the integral involved as a specific limit. The imaginary time formalism has the advantage of being based on integrals that are well-defined, with the ambiguities showing up only when one analytically continues to real time. Whether or not you use imaginary time methods, the real time objects getting computed are inherently not functions, but boundary values of holomorphic functions, defined of necessity as limits as one approaches the real axis.

A mathematical formalism for handling such objects is the theory of hyperfunctions. I’ve started writing up some notes about this, see here. As I find time, these should get significantly expanded.

One reason I’ve been interested in this is that I’ve never found a convincing explanation of how to deal with Euclidean spinor fields. Stay tuned, soon I’ll write something here about some ideas that come from thinking about that problem.

Posted in Quantum Mechanics | 22 Comments