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.

Posted in Uncategorized | 15 Comments

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.
Posted in BRST | Comments Off on Quantization and Dirac Cohomology

(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

Yesterday’s Hype

Every summer CERN runs a summer student programme, designed to bring in a group of students to participate in scientific activities at CERN and provide lectures for them about the basics and latest state of the field of high energy physics. Because of the COVID situation, this summer they have not been able to bring students in, but are providing instructional lectures and Q and A’s. This year’s sessions are based on having students follow materials from last year’s lectures, followed by a Q and A to answer their questions.

One of the topics the students are presented is What is String Theory?, and you can watch the 2019 video or look at the slides. Timo Weigand’s presentation can be accurately described as pure, unadulterated hype, with not a hint of the existence of any significant problem with ideas presented. In the Q and A yesterday, Weigand did come up with a new piece of “evidence for string theory”: it “predicts” no continuous spin representations.

I can’t begin to understand why anyone thinks it’s all right for CERN to subject impressionable students to this kind of thing. Someone, not me, should be complaining to the organizers and to CERN management.

This is unfortunately now an all too common example of what passes for “Sci Comm” in much of the field of fundamental physics: endless repetition of old discredited arguments in favor of a failed theory, coupled with pretending not to know about what is wrong with these arguments. The field that was once one of the greatest examples of the power of the human mind and the strength of the scientific method has become something very different and quite dangerous: all-too-visible ammunition for those who want to make the case that scientists are as deluded and tribalistic as anyone else, so not to be trusted.

Posted in This Week's Hype | 28 Comments

What is “Spin”?

The explanation for the lack of blogging here the past month is mostly that I haven’t seen any news worth blogging about. It took only a little bit of self-control to not do things like make snarky comments about recent conferences on string theory and quantum gravity.

Today I noticed a discussion on Twitter of the perennial question about what “spin” means in quantum theory, with some of the tweets included this highly appropriate meme:

I thought it might be worth while to make a stab at explaining what “spin” really is. For a much more detailed version, I wrote a book. But this post is much shorter…

Picking a particular point and a particular direction (say the z-direction), the angular momentum $J_z$ is defined to be the “generator” of rotations about that point, around the z-axis. This means that when you do such a rotation by an angle $\theta$, for any observable (function of position and momentum) $F$
$$\frac{dF}{d\theta}|_{\theta=0}=\{F, J_z\}$$
where the bracket is the Poisson bracket. A short calculation shows
$$J_z=r_xp_y-r_yp_x$$
which is often given as the definition. $J_z$ is itself an observable, which you can say is the angular momentum about the z-axis of a point particle with x,y coordinates of its position and momentum given by $r_x,r_y,p_x,p_y$. In classical physics $J_z$ can take on any values.

In quantum mechanics, observables are operators acting on states, and $J_z$ becomes the operator $\widehat{J}_z$ which (with an additional factor of $-i$ to get unitary transformations) generates rotations on states. This means (using units such that $\hbar=1$)
$$\frac{d}{d\theta}\ket{\psi(\theta)}=-i\widehat J_z \ket{\psi(\theta)}$$
You can solve this differential equation and see that if you rotate a state by an angle $\theta$ about the $z$ axis, you get
$$\ket{\psi(\theta)}=e^{-i{\widehat J}_z\theta}\ket{\psi(0)}$$
States that are eigenvectors of ${\widehat J}_z$ are supposed to be the ones with a well-defined value of the classical observable $J_z$, given by the eigenvalue.

One finds experimentally that the observed values of $J_z$ are given by
$$\frac{n}{2}$$
Unlike the classical case, as expected this number is quantized (that’s why they call it quantum mechanics…), but the factor of $2$ is unexpected. Since a rotation by $2\pi$ should bring the state back to itself, one expects that
$$e^{-iJ_z2\pi}=1$$
so $J_z$ should be an integer. If one finds a state with $J_z=\frac{1}{2}$, rotating it by an angle $2\pi$ changes its sign. This is weird, but the sign of a state isn’t itself something you can measure.

Looking more closely at the operator $\widehat{J}_z$ for quantum systems, one finds that for some states it has exactly the same relation to position and momentum as in classical physics
$$\widehat{J}_z=\widehat{r}_x\widehat{p}_y-\widehat{r}_y\widehat{p}_x$$
When states are given by a wavefunction depending on spatial coordinates, one can show that this is just the expected action by infinitesimal rotation of the spatial coordinates. In this case rotation by $2\pi$ doesn’t change the state, and $J_z$ has integral (not half-integral) values.

For many quantum systems though, there is an extra term:
$$\widehat{J}_z=\widehat{r}_x\widehat{p}_y-\widehat{r}_y\widehat{p}_x+\widehat{S}_z$$
and it is this extra term $\widehat {S}_z$ that is the “spin” observable. When states are given by wavefunctions, what the equation above is telling you is that when you act on a state by a rotation, you get not just the expected induced action from the rotation on spatial coordinates, but also an extra term. A natural guess is that, as in the meme, a point particle is really a ball of some new stuff, with $\widehat {S}_z$ the effective extra term caused by the positions and momenta of the new stuff.

For an elementary particle such as an electron, experimentally one finds that $\widehat S_z$ has eigenvalues $\pm 1/2$, which explains why one sees half-integral quantization. As the meme says, there is no viable physical model of rotating stuff that would give this result. Something very different is going on.

So far I’ve stuck to talking about rotations about the z-axis, but one also should consider rotations about other axes. The problem is that more sophisticated mathematics is needed, since the generators of rotations around different axes don’t commute (doing the rotations in the opposite order gives a different result). The mathematics needed is that of the representation theory of the rotation group $SO(3)$ and its double-cover $SU(2)$. From this representation theory one learns that the only consistent possibilities are given by putting together copies of a “spin n/2” representation for $n=0,1,2,\cdots$. These are $n+1$-dimensional vector spaces, on which $\widehat{S}_z$ acts with eigenvalues
$$\frac{-n}{2}, \frac{-n +2}{2},\cdots,\frac{n-2}{2},\frac{n}{2}$$
The case $n=0$ is that of $\widehat{S}_z=0$, and the simplest non-trivial case is the $n=1$ case which gives $\widehat{S}_z$ for the electron.

So, the “spin 1/2” characteristic of the electron is something completely new, unrelated to anything in classical mechanics. If you describe the electron by a wavefunction, it will take values not in the complex numbers, but in pairs of complex numbers, with rotations acting on the pairs by the spin-1/2 representation (also known as the “spinor” representation). Besides the non-classical physical behavior, the geometry is also non-classical, with the spinor representation something that cannot be described by the usual formalism of vectors and tensors.

Another reason I haven’t been writing much on the blog this past month is that I’ve been working on writing up something about twistors. I’ll write about twistors in detail here when this is done, but one thing they do is give a picture of space-time geometry in which spinors are fundamental, not vectors. A fundamental idea of twistor theory is that a point in space-time is a complex two-plane inside complex four-space. In twistor theory the answer to the question of where the spinor degree of freedom at a point comes from is tautological: the two complex dimensional spinor degree of freedom at a point IS the point.

Bonus link for those who have gotten this far. A presentation by CERN director Fabiola Gianotti, which comes off a bit differently than news reports saying CERN is going ahead with FCC. On page 5

  • Strategy gives a direction for future collider(s) at CERN (FCC). Prudent: feasibility study first.
  • Intensified accelerator R&D to prepare alternatives if FCC feasibility study fails.
  • No consensus in European community on which type of Higgs factory(linear or circular).

Page 9 lists three “first priorities” for the feasibility study:

  • find funds for the tunnel
  • [Be sure] no show-stoppers for ~100 km tunnel in Geneva region
  • magnet technology [are the FCC-hh magnets feasible?]; how to minimise environmental impact
Posted in Uncategorized | 28 Comments

HEP News

The CERN Council is meeting today and tomorrow, and should approve the long-awaited 2020 update of the European Strategy for Particle Physics. There will be a live webcast of the open part of the Council meeting on Friday.

My understanding is that the most difficult and contentious decision, that of how and whether to go forward with a new energy frontier collider, has been put off until 2026, when there will be a new update. In the meantime, design work will emphasize studies for the leading contender: a new large circular electron-positron machine. Studies of a linear collider design (CLIC) will continue at a reduced rate. New work will begin on the possibility of a muon collider, as well as other advanced accelerator technologies that might someday be usable.

There will be some move in the direction of the US program, which has abandoned the energy frontier, including more participation in the US and Japanese neutrino programs. A “scientific diversity program”, Physics Beyond Colliders, will receive new support. This program will try and come up with new experiments that don’t require a new energy frontier machine. For more about it, see this CERN report and this article in Nature.

In other news from CERN, work on the LHC should start resuming this summer, with the ongoing LS2 extended by a few months because of the COVID shutdown, so beams back in the LHC late next summer. There likely will be no significant new data coming from the LHC during 2021. The extended shutdown may provide the time for magnet quench training needed to bring the machine to its design energy of 7 TeV/beam.

Update: The CERN strategy report is here, also see here, here and a press release here. There is press coverage here, here and here.

The headline news is that this backs the FCC plan: a 100km new ring, first run as an electron-positron collider, then as a much higher energy proton-proton collider. There are however a whole bunch of very significant caveats:

  • No plan for how to finance this very expensive proposal.
  • The press release mentions a construction start timescale of “less than 10 years after the full exploitation of the HL-LHC, which is expected to complete operations in 2038”. This is twenty years or so away, a very long time.
  • The main near-term goal mentioned is work on designing the magnets needed for the proton-proton machine, to know by 2026 whether a pp machine is feasible. If the design of appropriate magnets with an acceptable cost for the pp machine is not possible, the implication is that there would be no point in building the large ring and ee machine.
  • The main competitor to the FCC plan, CLIC, is not at all canceled, but work will continue on it.
  • A new project to try and design a muon collider will be funded, with a planned 2026 decision about whether to move forward on a test facility for that. The technology for this still does not exist (muons decay very quickly…) but if such a collider were feasible, it would be much smaller and likely much cheaper than something like the FCC project.

So, those who want to argue one way or another about whether it’s a good idea to spend a lot of money on building a new collider should rest assured that the future holds many, many more years in which to conduct such arguments…

Update: I find it very frustrating to see that the online discussion of this is dominated by a pointless argument about whether, as reported, CERN should be going ahead and spending more than \$20 billion or so on a new machine. THEY ARE NOT DOING THIS. What has happened is that, after a lot of work, they have identified the best possible way forward at the energy frontier (the FCC proposal) and decided not to go ahead with it now but to keep studying it and the required technologies. If the cost of this proposal had been a few billion dollars, they likely would have tried to come up with a plan to allocate much of the over billion \$/year CERN budget in future years to the project and start construction. Instead, for the next six years they are allocating .1 – .2% of the CERN budget to further studies of the proposal. Those who have been loudly complaining that this is too expensive a proposal for the HEP community to afford should declare victory, not go to war over this.

Update: The CERN press release has been changed, with “construction” starting within ten years after 2038 changed to “operation” starting within ten years after 2038. This makes more sense, the earlier version seemed absurdly far in the future. My understanding is that the current plan is essentially to put off to 2026 a decision about going ahead with FCC. By 2027 the HL-LHC will be in place, freeing up some money for a new project, possibly the FCC. A 2027 start to FCC construction would allow a start of operations within ten years after the 2038 HL-LHC end date.

Update: Adrian Cho at Science magazine has a report on this that gets it right, headlined European physicists boldly take small step toward 100-kilometer-long atom smasher. It includes the crucial:

However, CERN Director-General Fabiola Gianotti emphasizes that no commitment has been made to build a new mammoth collider, which could cost $20 billion. “There is no recommendation for the implementation of any project,” she says. “This is coming in a few years.”

Posted in Experimental HEP News | 37 Comments

HEP Theory Job Situation

Way back in the 1980s and 1990s I was, for obvious personal reasons, paying close attention to the job situation for young HEP theorists. They were not good at all: way more talented young theorists than jobs, many if not most Ph.D.s who wanted to continue in the field unhappily spending many years in various postdocs before giving up and doing something else. By the later part of the 1990s I had found a satisfying permanent position in math, so this problem seemed much less interesting. When I was writing “Not Even Wrong” I did spend quite a bit of time gathering numbers to try and quantify the problem, and wrote about them in the book.

Since then I haven’t paid a lot of attention to the HEP theory job situation, hoped that it might have gotten a bit better as the wave of physicists hired during the 1960s hit retirement age, opening up some permanent positions. Today someone sent me a link to a personal statement on Facebook (sorry, but you need to login to a Facebook account to see this) from a young theorist (Angnis Schmidt-May) who has recently decided to leave the field, for reasons that she explains. These include:

We are put in competition with each other from day one, and only very few of us will be given prestigious positions in the end. Most of us never see a permanent contract, keep jumping from place to place and eventually need to find a second career after having sacrificed our entire 20s and 30s to academia. After having made it through the worst part of this and more or less securing my career, it still made me sick to see young physicists entering this spiral. I felt terrible about encouraging them to continue on this path because it is impossible to tell who will make it in the end and who will end up miserable with regrets…

Science itself is severely suffering from the poor working conditions and lack of genuine career prospects. I personally found it extremely hard to focus on the science while constantly being worried about the duration and location of my next contract. #PublishOrPerish. Interactions with and among colleagues are often dominated by the drive to “show off”. Very few people focus on removing misunderstandings or ask honest questions in order to fill their knowledge gaps. The general atmosphere is dominated by doubt instead of trust. We constantly need to outshine our peers. Better to demonstrate superficial knowledge of broad subjects than to focus on the details of a deep problem. Your next result needs to be “groundbreaking”, otherwise you’re out of a job. But produce it and have it published at least one year before your contract ends because that’s when you need to apply for a new one. Science has become a show…

I see absolutely no chance that any of the above will change any time soon.

She also makes important points about the personal cost of this system:

During the last 10 years, I was forced to constantly move around, losing contact to people who meant a lot to me and not being able to establish new lasting relationships.

Sadly, it seems pretty much nothing at all has changed in the last 30-40 years, and I continue to believe this is one reason the subject has been intellectually stagnant during this period. About the only positive suggestion I can make for anyone who wants to try and do anything about this is to take a look at the analogous job situation in mathematics. My knowledge of this is mostly anecdotal, but my impression is that while, like most academic fields, the career path for a new math Ph.D. is not easy, the situation is not at all as bad as the one in HEP theory described above.

Completely Off-Topic: Xenon1T has reported new results today. This seems to me unlikely to be new physics (extraordinary claims require extraordinary evidence), so if you want to follow this story, you should be consulting Jester, not me.

Posted in Uncategorized | 30 Comments

Feynman Lectures on the Strong Interactions

Available at the arXiv this evening is something quite fascinating. Jim Cline has posted course notes from Feynman’s last course, given in 1987-88 on QCD. There are also some audio files of a few of the lectures available here. The course was interrupted by Feynman’s final illness, with the last lecture given just a couple weeks before Feynman’s death in February of 1988. There’s an introduction to the notes by Cline in which he explains more about the course and how the notes came to be.

The course was given over thirty years ago, and many textbooks have appeared since then, but it seems to me this has held up well as an excellent place for a student to go to learn the subject.

Posted in Uncategorized | 2 Comments

An Advertisement for Representation Theory

There’s a new article at Quanta today promoting representation theory, Kevin Hartnett’s The ‘Useless’ Perspective that Transformed Mathematics. Representation theory is a central, unifying theme in modern mathematics, one that deserves a lot more attention than it usually gets, with undergraduate math majors often not exposed to the subject at all. My book on quantum mechanics is very much based on the idea that the subject is best understood in terms of representation theory. Unfortunately, physics students typically get even less exposure to representation theory than math students.

While I think the article is a great idea, and well-worth reading, I do have two quibbles, one minor and one major. The minor quibble is that one example given of a group, the real numbers with multiplication, is not quite right: you need to remove the element 0, since it has no inverse. If the group law is the additive one, then the real number line with nothing removed truly is a group.

The major quibble is with the theme of the article that a group representation can be thought of as a simplification of something more complicated, the group itself. This is a good way of thinking about one aspect of the use of representation theory in number theory, where representations provide a tractable way to get at the much more complicated structure of the absolute Galois group of a number field. The talk by Geordie Williamson linked to in the article (slides here) explains this well, but Williamson also gets right the more general context, where the group can be easy to understand, the representations complicated. For a simple example of this, in the case of the circle group $S^1$ the group is very easy to understand, its representation theory (the theory of Fourier series) is much more complicated (and much more interesting).

As Williamson explains, a good way to think about what is going on is that representation theory does simplify something by linearizing it, but it’s not the group, it’s a group action. When people talk about the importance of the study of “symmetry” in mathematics, physics, and elsewhere, they often make the mistake of only paying attention to the symmetry groups. The structure you actually have is not just a group (the abstract “symmetries”), but an action of that group on some other object, the thing that has symmetries. When you talk about “rotational symmetry” you have a rotation group, but also something else: the thing that is getting rotated. Representation theory is the linearization of this situation, often achieved by going from the group action on an object to the corresponding group action on some version of functions on the object. Once linearized, the group action becomes a problem in linear algebra, with the group elements represented as matrices, which act on the vectors of the linearization.

To further add to the confusion, “symmetry” is often described in popular accounts as meaning “invariance”. In typical examples given, “invariance” just means that you have a group action, since the group is taking elements of the set to other elements of the set (e.g. rotations not of an arbitrary object, but of a sphere). In representation theory, you have a different notion of invariance. For instance, for the representation of rotations on functions on the sphere, the constant functions are a one-dimensional invariant subspace, giving a trivial representation. But, there are lots of more interesting invariant subspaces of higher dimensions. These are the irreducible representations on the sets of spherical harmonics.

Posted in Uncategorized | 10 Comments

The Week’s Anti-Hype

I never thought I would see this happen: a university PR department correcting media hype about its research. You might have noticed this comment here a week ago, about a flurry of media hype about neutrinos and parallel universes. A new CNN story does a good job of explaining where the nonsense came from. The main offender was New Scientist, which got the parallel universe business somehow from Neil Turok and from here.

The ANITA scientists and their institution’s PR people were not exactly blameless, having participated in a 2018 publicity campaign to promote the idea that they had discovered not a parallel universe, but supersymmetry. They reported an observation here, which led to lots of dubious speculative theory papers, such as this one about staus. The University of Hawaii in December 2018 put out a press release announcing that UH professor’s Antarctica discovery may herald new model of physics. One can find all sorts of stories from this period about how this was evidence for supersymmetry, see for instance here, or here.

It’s great to see that the University of Hawaii has tried to do something at least about the latest “parallel universe” nonsense, putting out last week a press release entitled Media incorrectly connects UH research to parallel universe theory. CNN quotes a statement from NASA (I haven’t seen a public source for this), which includes:

Tabloids have misleadingly connected NASA and Gorham’s experimental work, which identified some anomalies in the data, to a theory proposed by outside physicists not connected to the work. Gorham believes there are more plausible, easier explanations to the anomalies.

The public understanding of fundamental physics research and the credibility of the subject have suffered a huge amount of damage over the past few decades, due to the overwhelming amount of misleading, self-serving BS about parallel universes and failed speculative ideas put out by physicists, university PR departments and the journalists who mistakenly take them seriously. I hope this latest is the beginning of a new trend of people in all these categories starting to fight hype, not spread it.

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