Special Relativity and Classical Field Theory

For quite a while Leonard Susskind has been giving some wonderful courses on physics under the name “The Theoretical Minimum”, pitched at a level in between typical popularizations and standard advanced undergraduate courses. This is a great idea, since there is not much else of this kind, while lots of people inspired by a popular book could use something more serious to start learning what is really going on. The courses are available as Youtube lectures here.

Book versions of some of the courses have now appeared, first one (in collaboration with George Hrabovsky) about classical mechanics, then one (with Art Friedman) about quantum mechanics. I wrote a little bit about these here and here, thought they were very well done. When last in Paris I noticed that there’s now a French version of these two books (with a blurb from me for the quantum mechanics one).

The third book in the series (also with Art Friedman) is about to appear. It’s entitled Special Relativity and Classical Field Theory, and is in much the same successful style as the first two books. Robert Crease has a detailed and very positive review in Nature which does a good job of explaining what’s in the book and which I’d mostly agree with.

The basic concept of the book is to cover special relativity and electromagnetism together, getting to the point of understanding the behavior of electric and magnetic fields under Lorentz transformations, and the Lorentz invariance properties of Maxwell’s equation. Along the way, there’s quite a lot of the usual sort of discussion of special relativity in terms of understanding what happens as you change reference frame, a lot of detailed working out of gymnastics with tensors, and some discussion in the Lagrangian language of the Klein-Gordon equation as a simpler case of a (classical) relativistic field theory than the Maxwell theory. Much of what is covered is clearly overkill if you just want to understand E and M, but undoubtedly is motivated by his desire to go on to general relativity in the next volume in this series.

At various points along the way, the book provides a much more detailed and leisurely explanation of crucial topics that a typical textbook would cover all too quickly. This should be very helpful for students (perhaps the majority?) who have trouble following what’s going on in their textbooks or course due to not enough detail or motivation. Besides non-traditional students in a course of self-study, the book may be quite useful for conventional students as a supplement to their textbook.

One of the most annoying things someone can do while reviewing a book is to start going on about their own different take on the material, criticizing the author for not writing a very different book. So, the rest of this posting is no longer a review of the book, it’s now about the very different topic of what I think about this material, nothing to do with Susskind’s valuable and different approach.

This semester I’m teaching a graduate level course on geometry, and by chance the past week have been discussing exactly some of the same material about tensor fields that Susskind covers. The perspective is quite different, starting with trying to explain a coordinate-invariant point of view on what these things are, only then getting to the formalism Susskind discusses. I can’t help thinking that, with all the effort Susskind (and pretty much every other physics textbook…) devotes to endless gymnastics with tensors in coordinates, they could instead be providing an understanding of the geometry behind this story. It’s unfortunate that many if not most of those who study this material in physics don’t ever get exposed to this point of view. Thinking in geometrical terms, the vector potential and field strength have relatively simple interpretations, and using differential forms the equations needed for the part of E and M Susskind covers are pretty much just:

F=dA, dF=0, and d*F=*J

Similarly, for the special relativity material, there’s a danger of the basic simplicity of the story getting lost in calculations of how things appear in coordinates with respect to different reference frames. What you fundamentally need is mainly that objects are described by a (conserved in the absence of forces) energy-momentum p, which satisfies p2= -m2, with Lorentz transformations taking one such p to another. The wider principle is that things are described by solutions to wave equations, with special relativity saying that the Lorentz group takes solutions to solutions.

I’d like to believe that such a very different course and very different book would be possible, quite possibly am very wrong (I’ve never taught special relativity to anyone). Maybe some day someone, inspired by Susskind’s project, might try to do something at a similar level, but from a more geometric point of view.

Posted in Book Reviews | 17 Comments

QCD at $\theta=\pi$

Earlier this week Zohar Komargodski (who is now at the Simons Center) visited Columbia, and gave a wonderful talk on recent work he has been involved in that provides some new insight into a very old question about QCD. Simplifying the problem by ignoring fermions, QCD is a pure SU(3) Yang-Mills gauge theory, a simple to define QFT which has been highly resistant to decades of effort to better understand it.

One aspect of the theory is that it can be studied as a function of an angular parameter, the so-called $\theta$-angle. Most information about the theory comes from simplifying by taking $\theta=0$, which seems to be the physically relevant value, one at which the theory is time reversal invariant. There is however another value for which the theory is time reversal invariant, $\theta=\pi$, and what happens there has always been rather mysterious.

The new ideas about this question that Komargodski talked about are in the paper Theta, Time Reversal and Temperature from earlier this year, joint work with Gaiotto, Kapustin and Seiberg. Much of the talk was taken up with going over the details of the toy model described in Appendix D of this paper. This is an extremely simple quantum mechanical model, that of a particle moving on a circle, where you add to the Lagrangian a term proportional to the velocity, which is where the angle $\theta$ appears. You can also think of this as a coupling to an electromagnetic field describing flux through the circle.

Even if you’re put off by the difficulty of questions about quantum field theories such as QCD, I strongly recommend reading their Appendix. It’s a simple and straightforward quantum mechanics story, with the new feature of a beautiful interpretation of the model in terms of a projective representation of the group O(2), or equivalently, a representation of Pin(2), a central extension of O(2). In the analogy to SU(N) Yang-Mills, it is the $\mathbf Z_N$ symmetry of the theory that gets realized projectively.

Komargodski himself commented at the beginning of the talk on the reasons that people are returning to look again at old, difficult problems about QCD. The new ideas he described are closely related to ones that are part of the recent hot topic of symmetry protected phases in condensed matter theory. It’s great to see that this QFT research may not just have condensed matter applications, but seems to be leading to a renewal of interest in long-standing problems about QCD itself.

Besides the paper mentioned above, there are now quite a few others. One notable one is very recent work of Komargodski and collaborators, Time-Reversal Breaking in QCD4, Walls and Dualities in 2+1 Dimensions.

Posted in Uncategorized | 11 Comments

Modern Theories of Quantum Gravity

Quanta magazine today has a column by Robbert Dijkgraaf that comes with the abstract:

Reductionism breaks the world into elementary building blocks. Emergence finds the simple laws that arise out of complexity. These two complementary ways of viewing the universe come together in modern theories of quantum gravity.

It struck me that at this point I don’t know what a “modern theory of quantum gravity” is. Much of the article is a clear explanation of the usual story of the renormalization group and effective field theory, but towards the end, when quantum gravity comes up, I have trouble following. String theory has gone from being an exciting new idea to being part of historical tradition:

Traditional approaches to quantum gravity, such as perturbative string theory, try to find a fully consistent microscopic description of all particles and forces. Such a “final theory” necessarily includes a theory of gravitons, the elementary particles of the gravitational field.

That “reductionist” tradition is opposed to a new “emergent” holographic theory, and we’re told that

The present point of view thinks of space-time not as a starting point, but as an end point, as a natural structure that emerges out of the complexity of quantum information, much like the thermodynamics that rules our glass of water. Perhaps, in retrospect, it was not an accident that the two physical laws that Einstein liked best, thermodynamics and general relativity, have a common origin as emergent phenomena.

In some ways, this surprising marriage of emergence and reductionism allows one to enjoy the best of both worlds. For physicists, beauty is found at both ends of the spectrum.

Dijkgraaf seems to be saying that a viable emergent theory of four-dimensional quantum gravity based on the complexity of quantum information has been found, but I seem to have missed this. Can someone point me to a paper describing it?

Posted in Uncategorized | 20 Comments

Modern Geometry

This semester I’m teaching the first semester of Modern Geometry, our year-long course on differential geometry aimed at our first-year Ph.D. students. A syllabus and some other information about the course is available here.

In the spring semester Simon Brendle will be covering Riemannian geometry, so this gives me an excuse to spend a lot of time on aspects of differential geometry that don’t use a metric. In particular, I’ll cover in detail the general theory of connections and curvature, rather than starting with the Levi-Civita connection that shows up in Riemannian geometry. I’ll be starting with connections on principal bundles, only later getting to connections on vector bundles. Most books do this in the other order, although Kobayashi and Nomizu does principal bundles first. In some sense a lot of what I’ll be doing is just explicating Kobayashi and Nomizu, which is a great book, but not especially user-friendly.

A major goal of the course is to get to the point of writing down the main geometrically-motivated equations of fundamental physics and a few of their solutions as examples. This includes the Einstein eqs. of general relativity, although I’ll mostly be leaving that topic to the second semester course.

Ideally I think every theoretical physicist should know enough about geometry to appreciate the geometrical basis of gauge theories and general relativity. In addition, any geometer should know about how geometry gets used in these two areas of physics. I’ve off and on thought about writing an outline of the subject aimed at these two audiences, and thought about writing something this semester. Thinking more about it though, at this point I’m pretty sick of expository writing (proofs of my QM book are supposed to arrive any moment…). In addition, I just took a look again at the 1980 review article by Eguchi, Gilkey and Hanson (see here or here) from which I first learned a lot of this material. It really is very good, and anything I’d write would spend a lot of time just reproducing that material.

Posted in Uncategorized | 26 Comments

This and That

  • The Stacks Project (see an earlier post here) had a very successful workshop in Ann Arbor earlier this month. This is a remarkable effort pioneered by Johan de Jong to produce a high quality open source reference for the field of algebraic geometry. It now is over 6000 pages, with an increasingly large number of papers citing it (according to data from Pieter Belmans, 85 citations in the arXiv so far in 2017 alone). During the workshop plans were discussed for the future of the project, with work on a new version of the project infrastructure underway (see slides and a blog post from Belmans).
  • The latest AMS Notices has a wonderful article by my Barnard/Columbia colleague Dusa McDuff about her remarkable family history and reflecting on her equally remarkable mathematical career. A post earlier this year discussed a Quanta article about her recent work with Katrin Wehrheim on technical issues in the foundations of symplectic topology. Kenji Fukaya has recently written something for the Simons Center website (see here) explaining his take on this story.
  • The Stanford Encyclopedia of Philosophy has a new entry about the fine-tuning problem, by Simon Friedrich.
  • The LHC operators have run into some difficulty in recent weeks (reflected in the accumulated luminosity plots here and here), with problems centered around an unknown source of gas in the beam pipe at a specific location, leading to losses of the beam. Some information about this is available here. The past few days they seem to be having success running the machine with around 1500 bunches, much less than the 2500 or so of earlier in the summer. The target for the year is 40 inverse fb which may still be achieved, while more optimistic numbers that looked plausible earlier now seem less likely.

Update: Joe Polchinski has put on the arXiv a long autobiographical document, with a detailed discussion of his scientific career.

Update: As mentioned in the comments, Go Yamashita has posted a long document surveying Mochizuki’s claimed proof of the abc conjecture. Experts may find that this makes it more possible to understand and check the claimed proof, we’ll see.

Update: Also at the Simons Center website, there’s an interview with Michael Green. It’s interesting to see that in recent years his research interests have led him to getting closer to mathematics and to an appreciation of what mathematicians do. As for his claim about string theory that

I don’t think there is a substantial antagonism to it among those who have studied it, other than a few individuals who enjoy publicizing their views.

I think he’s quite wrong if you properly take “it” to refer to the aspect of string theory there is widespread antagonism to among physicists, the overhyped claims about a unified theory based on string theory.

Posted in Uncategorized | 20 Comments

Road Trip

Blogging will be light to non-existent for the next ten days or so, as I head out west on a road trip to see next Monday’s solar eclipse. Current plan is to fly to Denver tomorrow, pick up a vehicle, and head up to Wyoming the next day. If weather projections look good for the Wyoming/Idaho part of the track, that’s where we’ll plan to end up, likely camping out somewhere (accommodations along the track have long been booked up).

This will be the ninth eclipse I’ve traveled to see, and I urge anyone thinking of making a trip to the eclipse track to do so. A total solar eclipse is something quite different than a partial one, and this is a very rare opportunity to see this in the US. Besides the eclipse, a major motivation for these trips has always been that of getting to visit a more or less random place on Earth that one wouldn’t otherwise have any excuse to see. I’ve driven quickly through Idaho and Wyoming a few times over the years, look forward to spending more time in that part of the country this coming week (unless the weather there looks bad, in which case maybe we’ll end up in Oregon or Nebraska).

Some other random advice about eclipses:

  • Be very careful about use of binoculars or telescopes, improper use of these at any time other than the period of totality is what can cause serious eye damage (by itself the eye is pretty good about automatically protecting itself).
  • Don’t put a lot of effort into photography during totality, since that’s likely to lead to you spending the time you should be enjoying the experience fiddling with camera equipment (and not getting a good result anyway…). A simple thing to do is to set up a camera to take video of the overall eclipse scene as it happens, turn it on at some point then ignore it.

If you miss this one, next couple are far south in South America, there will be another chance in the US relatively soon, April 2024.

Update: Now back in New York. Had a very good view of the eclipse from a spectacular location: Stanley, Idaho, up in the Sawtooth mountains. Only not quite optimal part of the plan was camping out not not well-equipped for the the unexpected fact that it gets down to about freezing at night in that part of Idaho, even in August…

Posted in Uncategorized | 19 Comments

GR=QM?

In recent years a hot topic in some theoretical physics circles has been the 2013 “ER=EPR” conjecture first discussed by Maldacena and Susskind here. Every so often I try and read something explaining what this is about, but all such efforts have left me unenlightened. I’m left thinking it best to wait for this to be better understood and for someone to then produce a readable exposition.

Instead of that happening, it seems that the field is moving ever forward in a post-modern direction I can’t follow. Tonight the arXiv has something new from Susskind about this, where he argues that one should go beyond “ER=EPR”, to “GR=QM”. While the 2013 paper had very few equations, this one has none at all, and is actually written in the form not of a scientific paper, but of a letter to fellow “Qubitzers”. On some sort of spectrum of precision of statements, with Bourbaki near one end, this paper is way at the other end.

Susskind starts out:

It is said that general relativity and quantum mechanics are separate subjects that don’t fit together comfortably. There is a tension, even a contradiction between them—or so one often hears. I take exception to this view. I think that exactly the opposite is true. It may be too strong to say that gravity and quantum mechanics are exactly the same thing, but those of us who are paying attention, may already sense that the two are inseparable, and that neither makes sense without the other.

I just finished writing a book about quantum mechanics, and it all seemed to me to make perfect sense without invoking gravity, but as explained above I guess I’m one of those who is not (successfully) paying attention. Another route to understanding would be to focus on the new experimental implications of the ideas. In the abstract Susskind claims that his ideas imply that we’ll observe quantum gravity using quantum computers in a lab “sometime in the next decade or so”. When that happens maybe this will all become clearer.

Update: Sabine Hossenfelder has a commentary on the paper here.

Posted in Uncategorized | 31 Comments

Cosmology for the Curious

There’s a new college-level textbook out, Cosmology for the Curious, targeted at physics courses designed to explain basics of cosmology to non-physics majors. The authors are Delia Perlov and Alex Vilenkin. Back in 2006 Vilenkin published a popular book promoting the multiverse, Many Worlds in One, which I wrote about at the time, making the obvious comment that there was nothing like a testable experimental prediction to be found in the book. It seemed to me then that the physics community would never take seriously an inherently untestable theory, recognizing such a thing as pseudo-science. I thought that the only reason claims like those of Vilenkin were getting any attention was that they had some novelty. Surely after a few more years of attempts to extract a prediction of some sort led to nothing, the emptiness of this sort of idea would become clear to all and everyone would lose interest.

Eleven years later I’m as baffled by what has happened to the field of fundamental physics as I’m baffled by what has happened to democracy in the US. As all attempts to extract a testable prediction from the multiverse have failed, instead of going away, pseudo-science has become ever more dominant, with a hugely successful publicity campaign (including a lot of “Fake Physics”) overcoming scientific failure. Now this sort of thing is moving from speculative pop science to getting the status of accepted science, taught as such to undergraduates.

Many are worried about the status of science in our society, as it faces new challenges. I don’t see how the physics community is going to continue to have any credibility with the rest of society if it sits back and allows multiverse mania to enter the canon. Non-scientists taking science classes need to be taught about the importance of always asking: what would it take to show that this theory is wrong? how do I know this is science not ideology?

Any student who reads this textbook and looks for answers to these questions in it will find just two “tests” of the multiverse proposed:

  • Look for evidence of bubble collisions.
  • Believe this paper, and then if you find a black hole population with a certain kind of mass spectrum, that would be evidence for the multiverse.

Of course there is no evidence for bubble collisions or such a black hole population, but these are no-lose “tests”: no matter what you observe or don’t observe, the multiverse “theory” can only win, it can never lose. Is it really a good idea to teach courses telling college students that this is how science works?

Posted in Book Reviews, Multiverse Mania | 26 Comments

Quick Links

  • For representation theory aficionados, George Lusztig has put on the arXiv a long document with comments on his papers (for a bit more about him, see this).
  • For a new idea exemplifying the potential grand unification of mathematics and physics, Minhyong Kim and others have been developing arithmetic Chern-Simons theory (see here and here). There was a recent workshop on the topic, videos of talks available here.
  • The editors of the Journal of Algebraic Combinatorics are leaving the Springer journal, setting up a new journal, Algebraic Combinatorics. For more about this, there’s a press release, a story at Inside Higher Education, and a blog entry by Timothy Gowers.
  • Landon Clay, founder of the Clay Mathematics Insitute, passed away last week, more about him available here.
  • There’s a profile and interview with Carlo Rovelli here.
  • While Google continues to develop our new machine overlords, Google money will fund a new IAS program to address their theoretical underpinnings.
Posted in Uncategorized | 2 Comments

A Few Items

Just a few items:

  • The Simons Foundation has announced a new Origins of the Universe initiative, which will fund efforts to “develop testable predictions about string theory, quantum gravity and a cosmological ‘Big Bounce.'” I don’t think even all of Jim Simons’ money will be enough to fund a real “prediction of string theory”, but the fake kind can be had rather cheaply. I was interested to see in this presentation from the NSF that grants from Simons and other private sources are starting to change the way they do business:

    One major challenge affecting Theory is the entrance of non-traditional (private philanthropic) funding sources. NSF has developed new procedures for evaluating overlapping sources of funding and introducing such evaluations into the proposal review process.

    I’m curious how they are dealing with this. If someone is being funded by Simons, will the NSF/DOE also fund them? Will the NSF/DOE stop funding fields that are being heavily financed by Simons/Templeton/Kavli? Does this have anything to with the NSF/DOE cuts in HEP theory funding of recent years?

  • The latest AMS Notices has a couple of articles about gravitational radiation, see here.
  • Yesterday a two week graduate summer school on automorphic forms and the Langlands program started. Lectures are being given by Kevin Buzzard and are on video here. Buzzard has set up a web-site for the lectures here.

    In his first lecture he explained that Richard Taylor’s CalTech lectures in 1992 (scans of Buzzard’s notes here) had a huge effect on him, and the plan of his lectures is to cover an updated version of some of the same material, ending up by getting to the latest developments, now available solely on a blog here and here. Buzzard also explained that in 1992 he devoted his time in LA to working on understanding the lectures during the week, going to raves on weekends. No news on whether MSRI is making similar arrangements for weekend activities of students in the summer school.

  • Two new articles from Michael Harris: The Perfectoid Concept: Test Case for an Absent Theory and Do Mathematicians Have Responsibilities?.
Posted in Langlands, Uncategorized | 10 Comments