Group Theory in a Nutshell for Physicists

There’s a new book out in the Princeton “Nutshell” series, Tony Zee’s Group Theory in a Nutshell for Physicists. I liked his Quantum Field Theory in a Nutshell quite a lot, it’s packed with all sorts of insights into that subject. Both books are written in a very light, chatty and entertaining style, full of various sorts of worthwhile digressions. Long ago I tried to use the QFT book in a course I was teaching, and there found that it functions better as a supplement to a standard QFT book. Zee’s treatment is just too short, covering too much, to provide the level of detail most students need to learn the subject for the first time. For people who already have gone through a standard book or course, Zee’s QFT book is great as a follow-on, likely to explain a lot of things they found confusing the first time around.

The new book on group theory has a length much better matched to the amount of material (it’s longer than the QFT book, and the material covered is much less complicated). The level of detail for most topics should be a good amount for students encountering the subject for the first time. The main topics covered are:

  • Finite groups and their representations.
  • Unitary and orthogonal groups, their representations, and applications to quantum mechanics.
  • Classification of simple Lie algebras.
  • The Lorentz and Poincare groups and their representations, with a discussion of the Dirac equation and Weyl and Majorana spinors.
  • A grab-bag of some other topics, including a little bit about conformal symmetry and grand unified theories.

While for each of these topics there are other good textbooks out there, this is a great selection for an advanced undergraduate/graduate physics course. I expect this to justifiably become a popular choice for such courses.

While I liked a lot about the book, I have to confess that there were things about it that did put me off. Some of this likely has to do with the fact that I’ve been working for the last few years on a book (see here) that covers some of the same topics, so I’m hyper-aware of both the technicalities involved, and the issues that arise of how best to approach these subjects. In addition, much of these topics is standard core mathematics, but Zee seems to have consulted few if any mathematicians (at least I didn’t recognize any in his acknowledgements). Unlike some others, this is a subject where mathematicians and physicists really can communicate and teach each other a lot.

Some of the choices Zee makes that I don’t think are good ones are things that input from mathematicians probably would have helped with. Maybe the most egregious is his decision to use the same notation for a Lie group and its Lie algebra, on the grounds that physicists sometimes do this, and to notationally distinguish the two in the usual way (upper vs. lowercase letters) is “rather fussy looking”. Using the same notation for two very different things is just asking for confusion, and I remember struggling with this as a student. Zee is well aware of the problem, on page 79 having his interlocutor “Confusio” say:

When I first studied group theory I did not clearly distinguish between Lie group and Lie algebra. That they allow totally different operations did not sink in. I was multiplying the Js together and couldn’t make sense of what I got.

Please, if you’re using this book to teach students about this subject, discourage them from following Zee in this choice.

There are places in the text where Zee gets things wrong in a way that just about any mathematician could likely have saved him from. One minor example is a footnote saying “Mathematicians have listed all possible finite groups up to impressively large values of n” (actually, they’re classified for ALL values of n) (my mistake, I misread and wasn’t looking at the finite group chapters carefully enough. Zee does get this right).

One place Zee gets things wrong is when he writes down the Heisenberg commutation relations, and says this is an “other type of algebra”, off-topic “since this is a textbook on group theory, I talk mostly about Lie algebras”. Actually those are the commutation relations of a Lie algebra, the Heisenberg Lie algebra, and there’s a group too, the Heisenberg group.

This gets into my own prejudices about the subject, with the story of the Heisenberg group to me (and I think to most mathematicians), a central part of the story of quantum mechanics, something little appreciated by most physicists. Another place where I think Zee goes wrong due to current physics prejudices is in ignoring Hamiltonian mechanics in favor of Lagrangian mechanics. As a result, instead of being able to tell the beautiful story of the Lie algebra of functions on phase space and what it has to do with conservation laws, he just mentions that Noether’s theorem leads to conservation laws and refers elsewhere for a discussion. The connection between symmetry and conservation laws is one of the central parts of the connection between Lie groups and physics, and deserves a lot more attention in the context of a course like this.

So, in summary, the book is highly recommended, with the caveats that you absolutely shouldn’t use the same notation for Lie groups and Lie algebras, and you should supplement Zee’s treatment with that of a certain more mathematically-minded blogger…

Posted in Book Reviews | 30 Comments

For Your Reading Pleasure

Since I just spent some of the morning not doing what I should have been doing, but reading about other things, in case you also want to do this, here are some options:

  • I’m very excited to see an article at Smithsonian Magazine with the title Can Physicists Ever Prove the Multiverse is Real? (remember, answers always no to headlines). Unlike just about every other effort of this kind, the author (Sarah Scoles) brings up the obvious problems, quoting Carlo Rovelli:

    Some theoretical physicists say their field needs more cold, hard evidence and worry about where the lack of proof leads. “It is easy to write theories,” says Carlo Rovelli of the Center for Theoretical Physics in Luminy, France. Here, Rovelli is using the word colloquially, to talk about hypothetical explanations of how the universe, fundamentally, works. “It is hard to write theories that survive the proof of reality,” he continues. “Few survive. By means of this filter, we have been able to develop modern science, a technological society, to cure illness, to feed billions. All this works thanks to a simple idea: Do not trust your fancies. Keep only the ideas that can be tested. If we stop doing so, we go back to the style of thinking of the Middle Ages.”

  • John Horgan has a wonderful, very long, interview with Scott Aaronson. Highly recommended as a way to avoid work and learn all sort of interesting things from and about Scott, whose blog you should be reading anyway. If you want to discuss this, you likely can do so with the man himself here.
  • If you just can’t get enough of the multiverse, there’s something else quite long available, a podcast of Sam Harris in conversation with Max Tegmark.
Posted in Multiverse Mania | 8 Comments

This Week’s Hype

It occurred today that the past year or so there haven’t been as many editions here of This Week’s Hype, with in particular the previously common “Scientists finally find a way to test string theory!” stories now less common than they used to be. What is often replacing bogus claims about testability though is just unvarnished hype about string theory, minus any claims that it can ever be tested. Examples I’ve run into today include Cumrun Vafa’s Fundamental Lessons From String Theory presentation at the April APS Meeting, which tells us that

String theory is leading to a revolutionary revision of many fundamental and long held principles of physics

despite a lack of any connection to experiment, either now or in the future.

In the local bookstore I took a look at Christophe Galfard’s The Universe in Your Hand, which builds up to a final chapter with some sort of rather incomprehensible voyage with a robot to the string theory multiverse. Nothing anywhere to be seen there about whether this might be science or fantasy. Jennifer Ouellette has a review in the New York Times here.

While mulling over these thoughts about the new prediction-free environment for string theory, I noticed that an article has just appeared that seemed to contradict such thoughts, Natalie Wolchover’s Physicists Hunt for the Big Bang’s Triangles, with a headline claiming that “evidence for string theory” could be found in the sky. It’s by far the best popular piece I’ve seen about “string cosmology”, giving an excellent idea of what people in that field are up to these days (which includes large amounts of hype, coming from the scientists, not the journalist).

In summary, here’s what we learn about current string cosmology. One of the main targets is a “prediction” of the level of non-gaussianity in the CMB, something which all observations so far have shown to be unobservably small:

  • Matthew Kleban and Eva Silverstein are described as “cosmological clocksmiths”, working out the non-Gaussianity “predictions” of a large range of string cosmology models. It seems that you can get any number you want this way by an appropriately complicated model. Kleban likes unwinding inflation and we’re told:

    “I think it’s pretty plausible that some version of this happens,” he said.

    Though Kleban acknowledges that it’s too soon to tell whether he or anyone else is on to something, plans are under way to find out.

    Silverstein has “many string inflationary models” of all sorts, so can get you one with any amount of non-gaussianity you might want. As far as she is concerned, having plenty of complicated models with zero evidence for any of them, such that, no matter what you see or don’t see, there’s always lots more and more “predictions”, is a perfectly traditional kind of science. Her reaction to people pointing out the problems with this?

    I find it surreal, because we are currently doing some traditional science with string theory.

  • Her Stanford colleagues Andre Linde and Renate Kallosh are taking a different approach, promoting theories with no observable non-gaussianity, and, it seems, no observable effects at all:

    Linde isn’t bothered by this. In supporting the alpha-attractor models, he and Kallosh are staking a position in favor of simplicity and theoretical beauty, at the expense of ever knowing for sure whether their cosmological origin story is correct. An alpha-attractor universe, Linde said, is like one of the happy families in the famous opening line of Anna Karenina. As he paraphrased Tolstoy: “Any happy family, well, they look in a sense alike. But all unhappy families — they’re unhappy for different reasons.”

  • On the East Coast, there’s Arkani-Hamed and Maldacena, with a paper last year on Cosmological Collider Physics. It described observable signatures of new physics at the inflationary scale, but the only comment I could find about how you would actually observe such things was

    In terms of measurability, if the couplings are Planck suppressed, then it seems impossible to measure this through the CMB or large scale structure. (See e.g. [58, 59, 60, 61] for a discussion of measuring these effects via large scale structure.) But it might be possible using the 21cm tomography [50].

    Arkani-Hamed explains to Wolchover that there’s a “cosmic variance” problem making things inherently unobservable, but he removed most discussion of this from the paper, hoping to get around it by changing the laws of quantum mechanics:

    In his paper with Maldacena, Arkani-Hamed initially included a discussion of this issue, but he removed most of it. He finds the possibility of a limit to knowledge “tremendously disturbing” and sees it as evidence that quantum mechanics must be extended. One possible way to do this is suggested by his work on the amplituhedron, which casts quantum mechanical probabilities (and with them, unitarity) as emergent consequences of an underlying geometry. He plans to discuss this possibility in a forthcoming paper that will relate an analogue of the amplituhedron to non-Gaussianities in the sky.

  • There has been further work on this by Kamionkowski and collaborators, and we’re told:

    Observing the signals predicted by Arkani-Hamed, Maldacena and Kamionkowski would be like striking gold, but the gold is buried deep: Their strength is probably near the gravitational floor and will require at least 1,000 times the sensitivity of current equipment to detect.

    No word on prospects for more than 1,000 times more sensitive experiments.

For about as long as I can remember, string cosmologists have been promising that their ideas would be tested by the Planck experiment. Now that negative results are in from that, we’re told to instead look forward to a new generation of experiments. The ones mentioned in the article are SPHEREx and the LSST, with results next decade. SPHEREx claims that they may be able to push down bounds on non-gaussianity by a factor of 10, which would be an impressive result. This would rule out lots of string cosmology models, but of course there would still be plenty more.

The next experiment all involved are talking about is looking for signals in the 21cm hydrogen line, see Sabine Hossenfelder here for more about that. No time estimates on that one. As far as I can tell, the plan for this starts off with “First, build a base on the other side of the moon…”

Posted in This Week's Hype | 9 Comments

The Man Who Knew Infinity

Last night I went to see a showing at the Tribeca Film Festival of the new movie about Ramanujan, The Man Who Knew Infinity. It was extremely good, infinitely better than the most recent high profile film about a mathematician, the one about Turing (see here).

The story of Ramanujan is one of the great romantic stories of mathematics, with a large part played in it by the Cambridge mathematician G. H. Hardy. The filmmaker was inspired by Robert Kanigel’s excellent 1991 biography of the same name (he says his mother gave it to him to read, she had it through her book club). The book is an excellent source for the story of Ramanujan’s life, and Hardy’s A Mathematician’s Apology is something everyone should read (for one thing, it’s short). For some more about the film from an expert on Ramanujan’s work, the AMS Notices have this from George Andrews.

Some dramatic license was taken, for instance in having Jeremy Irons play Hardy as a much older man than he actually was when he met Ramanujan. After the film there was a panel discussion, with filmmaker and screenwriter Matt Brown explaining that it took 10 years to get the film made, largely because of the difficulty of financing it. He claimed that he could have gotten the financing much earlier if he had been willing to go along with certain plot changes the financiers wanted: in particular they wanted the story to revolve around a love affair of Ramanujan with a (white) nurse, to be played by a high-profile starlet.

Also at the panel discussion were two mathematicians: Princeton’s Manjul Bhargava and my Columbia colleague Ina Petkova. One reason the film is so true to the real story of the mathematics and mathematicians involved in it is the involvement of Ken Ono and Bhargava. Ono was heavily involved in the filming (and he has a memoir from Springer, My Search for Ramanujan, about to appear). Bhargava was involved in the editing, in particular in helping choose among the many takes of the actors acting out a mathematical discussion those which seemed true to life.

The film is supposed to be released here in the US on April 29, I can’t recommend it enough.

Update: Scott Aaronson has a far better review of the film than mine here.

Posted in Uncategorized | 29 Comments

Various and Sundry

Some quick links:

  • Via my Columbia colleague at Mathematics Without Apologies, a documentary about Perelman that I was unaware of.
  • I learned something yesterday about another math department colleague, Mikhail Khovanov: he has games called Ringsanity and Ringiana available as apps.
  • Took a quick look at a Stephon Alexander’s new book The Jazz of Physics, which is now in bookstores. While I enjoyed reading some of his account of his career in theoretical physics, I’m afraid that his two main topics, jazz and models of the big bang, both are things that pretty much leave me cold. For those with more interest in either topic, you probably should take a closer look.
  • I’m sorry to see that there’s some sort of fight developing over Grothendieck’s papers. A shame.
  • Previous attempts to figure out what “ER=EPR” is supposed to mean have left me baffled. Trying to read Susskind’s write up of his IAS lectures on the topic hasn’t helped, I’m afraid.

: Mikhail Khovanov tells me that Ringiana for Android can be found here. It was developed jointly with Nikolay Gromov.

He also comments that “Thompson groups V and T act on the states of Ringiana and Ringsanity, correspondingly. These groups were the first two infinite, finitely-presented, simple groups discovered by mathematicians.”

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I don’t know what this is about, but Stephen Hawking and Yuri Milner are here in New York today, with a press conference downtown scheduled at noon, supposedly to announce a mysterious new space exploration initiative, to be called Starshot. There’s also evidently some sort of Columbia connection, with a reception scheduled up here. More later, when we find out what this is all about. Last year there was a Hawking/Milner announcement of Breakthrough Listen and Breakthrough Message, projects related to communication with alien civilizations.

Update: Livestream will be available here.

Update: According to this source (and Google translate…), this is a project to send a small space probe to Alpha Centauri.

Update: “Breakthrough Starshot” is a $100 million research program, hoping to develop (ultimate cost of order $10 billion) very small probes attached to light-sails, pushed towards Alpha Centauri by ground-based lasers (a “Silicon Valley approach to spaceflight”). The claim is that such things could travel at a significant fraction of the speed of light, get there in 20 years or so. One thing I’m not seeing is how you get a signal back to earth.

Update: They have a website here, there’s a story at the New York Times here, at Scientific American here.

One thought about this is that if you really could accelerate probes this way with lasers, sending them out to solar system planets in days would seem to be a more interesting application.

Update: A more detailed story is here.

Posted in Uncategorized | 37 Comments

Local Debates

I noticed that tomorrow (Tuesday, April 5) evening here in New York City there will be not one, but two debates involving theoretical physicists:

  • At 7 pm the American Museum of Natural History will host the 2016 Asimov Debate, with this year the topic Is the Universe a Simulation?. You can watch a livestream at that site.

    I confess that if this were a few days earlier, I would be convinced it was definitely a joke. But, it seems not, that instead this “has become a serious line of theoretical and experimental investigation among physicists, astrophysicists, and philosophers” and that it’s a “provocative and revolutionary idea”. One thing this is not is new. Nearly nine years ago it got a lot of media attention, and I wrote about it here (and here, where quite possibly my Message to Our Overlords kept them from turning us off). Sadly, the “blink” feature of html no longer seems to be supported, so the red text there won’t blink. Maybe it annoyed the overlords and they had it turned off.

  • Much further downtown, at the New York Academy of Sciences, at the same time there will be a panel discussion on a much more sensible and interesting topic What Does the Future Hold for Physics: Is There a Limit to Human Knowledge?. Also at 7 pm, livestream here.

    If I’d been asked (actually I was asked, and then unasked, a rather mystifying situation) for my views on this, I’d make the point that there’s no way to know what the limits will be to human understanding of physical laws. It has however become all too clear what the danger is of what will happen when we reach those limits. Instead of prominent theorists frankly admitting “we don’t know”, there will be an attempt to sell the story to the public that theorists have a wonderful, successful theory which describes everything, which sadly has the unfortunate feature of not making any falsifiable predictions. The string landscape/multiverse scenario now is being very aggressively sold as exactly this kind of endpoint to physics, to a large degree by people unwilling to admit the failure of string theory-based unification. There’s a very real danger that this will enter the textbooks, and that we will in our lifetimes see the end of fundamental physics as a human endeavor. The limit we will have hit will be due not to the nature of our minds, but instead the nature of our sociology.

    I suppose one other way of seeing if we’ve reached the end of physics would be if physicists started spending their time debating things like whether we live in a simulation. Oh, wait…

: At the NYAS evidently there was some discussion of the multiverse, with the audience told “The multiverse hypothesis is no more speculative than the universe hypothesis”.

Update: Clara Moskowitz at Scientific American has a report from the AMNH debate. At least there is one participant I agree with:

And the statistical argument that most minds in the future will turn out to be artificial rather than biological is also not a given, said Lisa Randall, a theoretical physicist at Harvard University. “It’s just not based on well-defined probabilities. The argument says you’d have lots of things that want to simulate us. I actually have a problem with that. We mostly are interested in ourselves. I don’t know why this higher species would want to simulate us.” Randall admitted she did not quite understand why other scientists were even entertaining the notion that the universe is a simulation. “I actually am very interested in why so many people think it’s an interesting question.” She rated the chances that this idea turns out to be true “effectively zero.”

One thing I’ve noticed about these kinds of things: they often feature physicists going on about mathematics, but mathematicians are never invited…

Update: The Asimov debate is available here, the NYAS one here.

Posted in Multiverse Mania | 66 Comments

Is String Theory Scientific?

Among the various April Fool’s things on the web, the most subtle one I’ve found is by the people at James Madison University, who are advertising an April 1 event discussing the question of Is String Theory Scientific?

Part of the joke surely is Betteridge’s Law or Hinchcliffe’s Rule, which assure us that that answer to the question is “No”.

Update: Among today’s other April Fool’s efforts, Kyle Cranmer updates an oldie but goodie, supersplit supersymmetry.

Update: Another April 1 effort, this one an essay on the multiverse by Robert Lawrence Kuhn. Kuhn claims that the majority of cosmologists disagree with George Ellis about the problems with the multiverse, and that Andrei Linde (with Steve Weinberg agreeing with him) represents the consensus viewpoint. Very funny.

: INTO THE MULTIVERSE: God’s Voice in String Theory is labeled March 31, but surely it too is an April 1 effort.

Posted in Multiverse Mania, Uncategorized | 6 Comments

Short Items

A few short items:

  • Beams are back in the LHC. You can follow what is going on here real-time, or here for details of this year’s beam commissioning. Physics runs scheduled to start last week of April.
  • There’s a wonderful interview with John Baez in two parts, here and here.
  • For news of US HEP, take a look at the HEPAP presentations as they appear here. Reports from the LHC experiments are scheduled for April Fool’s Day.
  • In the last month the omnipresent Nima Arkani-Hamed was giving talks at the David Gross Fest (video here, nice comments about Gross at the beginning, presentation here), a series of lectures in Trieste (see here), and a colloquium talk at Fermilab.

    In the FNAL talk Arkani-Hamed advertises a “Modern S-matrix program”, based on recent work on amplitudes. He’s a reliable source for what the conventional wisdom is among the most influential people in the field, and he has this to say about the current situation (right at the end of the talk):

    String theory killed QFT, then QFT killed string theory back, now QFT is king. We’re in a situation where most people think QFT is king and string theory a derivative thing in some limits.

    His own opinion is that we need “something else”, neither QFT nor string theory, but he doesn’t know what it is.

  • I seem to have missed this paper on String theory and general methodology (arxiv version here) when it came out quite a few years ago. At the time the authors felt that

    the majority has not been convinced [by criticisms of string theory] and they continue to believe that string theory is the right way to go.

    I’m not sure if the authors had any data to back that up, curious if anyone would still make this claim now. Arkani-Hamed seems to think the majority opinion has changed, with QFT killing string theory.

  • For some interesting talks at one of the few conferences not featuring Arkani-Hamed, see the Nambu Memorial Symposium.
  • I was sorry to hear recently that Rudolf Haag passed away back in January. For a short biography, see here. For an earlier posting linking to an autobiographical piece, see here. His book Local Quantum Physics is well worth reading. For a discussion of perhaps his most famous result, Haag’s theorem, see here.
  • Multiverse mania shows no signs of slowing down, with a long BBC article here. Despite the length of the article, the author doesn’t seem to have been able to locate anyone who could add a note of skepticism amidst the usual thick layers of hype.

Update: A lot of data on recent DOE funding trends is available here and here. From FY 13 – FY 17, theory funding is down %20 at universities, 2% at labs. A bit over $20 million/year is now being spent by the DOE on HEP theory and computational research. For the most recent round of reviews, 23 groups were funded, and of these two were ones not previously funded, with two previously funded turned down.

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Two Book Reviews

Blogging has been light here, trying to finish a complete draft of the book I’m working on, this should be done very soon. Here are a couple all-too-short reviews of books with some relation to math or physics.

A Doubter’s Almanac

The main character of Ethan Canin’s new novel A Doubter’s Almanac is a mathematician, one who solves a great problem early on in his career (as a graduate student in Berkeley, then a faculty member at Princeton). It’s a beautifully written work, with a remarkably convincing portrayal of a talented young mathematician struggling with a difficult problem and making his way through life. I wouldn’t have guessed that anyone who hadn’t lived and worked in this kind of environment would be able to describe it so realistically.There are only a couple false notes in the many details of the part of the story set in academia. In particular, I don’t think anyone would consider a “subchairmanship” to be much of an inducement, even at Princeton, and they don’t give Abel Prizes to young geniuses. Besides getting the details right, the characters come up with some quite insightful remarks about mathematics, including some that deal with the way talent and immersion in a mathematical problem may alienate one from the rest of the humanity.

While I greatly enjoyed the first half of the book, I have to admit that the later part held less interest, turning away from academia to a long story of family relations and the ravages of alcoholism. Not at all an upbeat book, if that’s what you’re looking for, but I can’t think of another novel as good that so deeply engages with some aspects of mathematics and the mathematical life.

Black Hole Blues

Janna Levin’s Black Hole Blues has just been published, with excellent timing for anyone who wants to know more about the story of LIGO and its first observation of gravitational waves. The main strength of the book derives from her interviews with some of the people crucial to building LIGO (in particular Kip Thorne and Rainer Weiss). Together with research and other interviews she has put together a rich version of the history of the project and the roles of the three physicists (Drever, Thorne and Weiss) whose vision and dedication made it successful. LIGO has been a very long term project, with its beginnings going back 40 years. It’s remarkable that it didn’t get abandoned or defunded at some point, with the NSF playing a very important role in supporting the project over many years.

Drever, Thorne and Weiss will likely soon be the recipients of all sorts of well-deserved honors and prizes (I’d bet on this year’s Breakthrough Prize and probably the Nobel too). I was sad to learn that this is coming too late for one of them, Ronald Drever, who is ill and suffering from dementia. The physics of LIGO has a bright future, it’s great to have the story told of the people who made it happen.

Posted in Book Reviews | 8 Comments