Category Archives: Group Theory

Sporadic SICs and exceptional Lie algebras

A while back, I had a bit of a sprawling conversation about certain geometrical oddities over multiple threads at the n-Category Café. I finally got organized enough to gather these notes together, incorporating edits for clarity and recording one construction I haven’t found written in the literature anywhere.

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy.

The Rise of Ironic Physics and/or Machine Physicists?

CONTENT ADVISORY: old-fashioned blog snarkery about broad trends in physics.

Over on his blog, Peter Woit quotes a scene from the imagination of John Horgan, whose The End of Science (1996) visualized physics falling into a twilight:

A few diehards dedicated to truth rather than practicality will practice physics in a nonempirical, ironic mode, plumbing the magical realm of superstrings and other esoterica and fret­ting about the meaning of quantum mechanics. The conferences of these ironic physicists, whose disputes cannot be experimentally resolved, will become more and more like those of that bastion of literary criticism, the Modern Language Association.

OK (*cracks knuckles*), a few points. First, “fretting about the meaning of quantum mechanics” has, historically, been damn important. A lot of quantum information theory came out of people doing exactly that, just with equations. The productive way of “fretting” involves plumbing the meaning of quantum mechanics by finding what new capabilities quantum mechanics can give you. Let’s take one of the least blue-sky applications of quantum information science: securing communications with quantum key distribution. Why trust the security of quantum key distribution? There’s a whole theory behind the idea, one which depends upon the quantum de Finetti theorem. Why is there a quantum de Finetti theorem in a form that physicists could understand and care about? Because Caves, Fuchs and Schack wanted to prove that the phrase “unknown quantum state” has a well-defined meaning for personalist Bayesians.

This example could be augmented with many others. (I selfishly picked one where I could cite my own collaborator.)

It’s illuminating to quote the passage from Horgan’s book just before the one that Woit did:

This is the fate of physics. The vast majority of physicists, those employed in industry and even academia, will continue to apply the knowledge they already have in hand—inventing more versatile lasers and superconductors and computing devices—without worrying about any underlying philosophical issues.

But there just isn’t a clean dividing line between “underlying philosophical issues” and “more versatile computing devices”! In fact, the foundational question of what the nature of “quantum states” really are overlaps with the question of which quantum computations can be emulated on a classical computer, and how some preparations are better resources for quantum computers than others. Flagrantly disregarding attempts to draw a boundary line between “foundations” and “applications” is my day job now, but quantum information was already getting going in earnest during the mid-1990s, so this isn’t a matter of hindsight. (Feynman wasn’t the first to talk about quantum computing, but he was certainly influential, and the motivations he spelled out were pretty explicitly foundational. Benioff, who preceded Feynman, was also interested in foundational matters, and even said as much while building quantum Hamiltonians for Turing machines.) And since Woit’s post was about judging whether a prediction held up or not, I feel pretty OK applying a present-day standard anyway.

In short: Meaning matters.

But then, Horgan’s book gets the Einstein–Podolsky—Rosen thought-experiment completely wrong, and I should know better than to engage with what any book like that on the subject of what quantum mechanics might mean.

To be honest, Horgan is unfair to the Modern Language Association. Their convention program for January 2019 indicates a community that is actively engaged in the world, with sessions about the changing role of journalism, how the Internet has enabled a new kind of “public intellectuals”, how to bring African-American literature into summer reading, the dynamics of organized fandoms, etc. In addition, they plainly advertise sessions as open to the public, which I can only barely imagine a physics conference doing more than a nominal jab at. Their public sessions include a film screening of a documentary about the South African writer and activist Peter Abrahams, as well as workshops on practical skills like how to cite sources. That’s not just valuable training, but also a topic that is actively evolving: How do you cite a tweet, or an archived version of a Wikipedia page, or a post on a decentralized social network like Mastodon?

Dragging the sciences for supposedly resembling the humanities has not grown more endearing since 1996.
Continue reading The Rise of Ironic Physics and/or Machine Physicists?

Recent Advances in Packing

The weekend before last, I overcame my reluctance to travel and went to a mathematics conference, the American Mathematical Society’s Spring Central Sectional Meeting. I gave a talk in the “Recent Advances in Packing” session, spreading the word about SICs. My talk followed those by Steve Flammia and Marcus Appleby, who spoke about the main family of known SIC solutions while I covered the rest (the sporadic SICs). The co-organizer of that session, Dustin Mixon, has posted an overall summary and the speakers’ slides over at his blog.

3, 8, 24, 28, Eureka!

The news has been so unrelentingly bad these past few weeks that I’m taking momentary refuge in good old numerology. I happened to re-read this blog post by John Baez about the free modular lattice on 3 generators. This is a nice bit of pure math that features rather prominently the numbers 3, 8, 24 and 28. The numerological part is that I noticed the same numbers popping up in a problem that I had studied for other reasons, so I figured it would be fun to write about, even if my 28 isn’t exactly equal to Baez’s 28, so to speak.
Continue reading 3, 8, 24, 28, Eureka!

New Paper Dance

B. C. Stacey, “Geometric and Information-Theoretic Properties of the Hoggar Lines” (2016), arXiv:1609.03075 [quant-ph].

We take a tour of a set of equiangular lines in eight-dimensional Hilbert space. This structure defines an informationally complete measurement, that is, a way to represent all quantum states of three-qubit systems as probability distributions. Investigating the shape of this representation of state space yields a pattern of connections among a remarkable spread of mathematical constructions. In particular, studying the Shannon entropy of probabilistic representations of quantum states leads to an intriguing link between the questions of real and of complex equiangular lines. Furthermore, we will find relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.

All things told, SIC-POVMs are just about the damnedest things I’ve ever studied in mathematics.

(Also listed on SciRate.)

On the arXivotubes

Michael Schnabel, Matthias Kaschube, Fred Wolf, “Pinwheel stability, pattern selection and the geometry of visual space” (arXiv:0801.3832).

It has been proposed that the dynamical stability of topological defects in the visual cortex reflects the Euclidean symmetry of the visual world. We analyze defect stability and pattern selection in a generalized Swift-Hohenberg model of visual cortical development symmetric under the Euclidean group E(2). Euclidean symmetry strongly influences the geometry and multistability of model solutions but does not directly impact on defect stability.

Note to self: file alongside Bressloff, Cowan et al. for future reference.

Carnival of Mathematics #24

The twenty-fourth Carnival of Mathematics is online at Ars Mathematica. To Ars, 24 is a special number because the Leech lattice lives in 24 dimensions (and, really, what could be creepier than a lattice of leeches? — methinks that could be the title for the sequel to The Halting Oracle). It’s also interesting to the secretive, cultish cabal of quantum gravity research, since it’s 26 (the critical dimension for bosonic string theory) minus 2 (the dimensionality of the string world-sheet). These two occurrences are actually related, although there’s a reason the relationship falls in a domain called Moonshine theory. . . .

Lent of Physics Blogging

For a while, we had a blog carnival of physics writing, Philosophia Naturalis. However, it looks rather moribund today: the last installment to date was on 4 October (at Dynamics of Cats), and the “next available hosting opportunity” was the first of November, which is already almost a month gone.

Combined with the recent description of the physics blogoweb as “an intellectual wasteland,” and we’ve got plenty of excuses to feel a little depressed.

Oh, and have you noticed that still can’t do math notation? So much for discussing science the way that, you know, actual scientists do. So much for reflecting the increasing quantitative aspect of the life sciences, discussing interdisciplinary work, or doing anything beyond the same old carping over innumeracy. Maybe they’re intimidated by that old “each equation cuts the readership in half” bromide. Or maybe they think that allowing the use of calculus and other such scary mathematics would be “bad framing.”
Continue reading Lent of Physics Blogging

Psychedelic Bibliographies

Advances in the History of Psychology, a blog operated out of York University, has posted annotated bibliographies of psychedelic research, both on general psychological research and on studies focusing specifically on LSD.

(Hah! And you thought I was just trying to make a strange juxtaposition in my title.)

The AHP folks note something which I find interesting but not wholly unexpected: while plenty of papers have been written about LSD and marijuana, the academic literature doesn’t appear to have histories dedicated to the two-carbon phenethylamines like 2C-B or other significant drugs like DMT, DOM or mescaline. These remarkable little molecules sometimes get mentioned in general discussions or in studies of other drugs, but they don’t appear to have peer-reviewed literature of their own. PiHKAL (1991) and TiHKAL (1997) seem to be the end of the line.

One unfortunate consequence of this lack is our inability to judge the universality of neurological reactions to chemical stimuli. In this context, I’d like to bring up the paper by Bressloff, Cowan, Golubitsky and Thomas in Neural Computation (2002), “What geometric visual hallucinations tell us about the visual cortex.”
Continue reading Psychedelic Bibliographies

Einstein Summation and Levi-Civita Symbols

PUBLIC SERVICE ANNOUNCEMENT: if any of you saw me wearing black corduroy pants and a purple T-shirt emblazoned with a picture of my friend Mike wearing a squid on his head, yes, it was laundry day. Rest assured, the reality disruption was only temporary, and normal service should be resumed shortly.

Now, to the business of the day. Earlier, we took a look at rotations and found a way to summarize their behavior using commutator relations. Recall that the commutator of A and B is defined to be

[tex]\{A,B\} = AB – BA.[/tex]

For real or complex numbers, the commutator vanishes, but as we saw, the commutators of matrices can be non-zero and quite interesting. We recognized that this would have to be the case, since we used matrices to describe rotations in three-dimensional space, and rotations about different axes in 3D do not commute. Looking at very small rotations, we also found that the commutators of rotation generators were tied up together in a way which involved cyclic permutations. Today, we’ll express this discovery more neatly, using the Einstein summation convention and a mathematical object called the Levi-Civita tensor.
Continue reading Einstein Summation and Levi-Civita Symbols

Rotation Matrices

My group theory teacher, Prof. Daniel Freedman, had some interesting professorial habits. When invoking some bit of background knowledge with which we were all supposed to have been familiar, he would say, “As you learned in high school. . . .” Typically, this would make a lecture sound a bit like the following:

“To finish the proof, note that we’re taking the trace of a product of matrices. As you learned in high school, the trace is invariant under cyclic permutations. . . .”

Prof. Freedman also said “seventeen” for “zero” from time to time. After working out a long series of mathematical expressions on the blackboard, showing that this and that cancel so that the overall result should be nothing, with the students alternating their glances between the board and their notes, he would complete the equation and proclaim, “Equals seventeen!” At which point, all the students look up and wonder, momentarily, what they just missed.

“Here, we’re summing over the indices of an antisymmetric tensor, so by exchanging i and j here and relabeling there, we can show that the quantity has to equal the negative of itself. The contraction of the tensor is therefore, as you learned in high school — seventeen!”

One day, I managed to best his line. I realized that the formula currently on the board had to work out to one, not zero, so when he wrote the equals sign, paused and turned to the class with an inquiring eye, I quickly raised my hand and said, “Eighteen!”

Incidentally, truly simple topics like Euler’s formula and trigonometric identities were supposed to have been learned in middle or elementary school.

Today, we’ll talk about one of the things Prof. Freedman said we should have covered in high school: the rotation matrices for two- and three-dimensional rotations. This will give us the quantitative, symbolic tools necessary to talk about commutativy and non-commutativity, the topic we explored in an earlier post.
Continue reading Rotation Matrices

Group Theory Homework

One reason I call this site a “blag” and not a “blog” is that I’m always late.

For example, I’m finally typing up the group-theory homework assignment which Ben gave last Monday (and which will be due next Monday). During our seminar over in BU territory, we discussed the relations among the Lie groups SU(2) and SO(3) and the manifolds S3 and RP3. Problems will be given below the fold.

Also, Eric will be discussing statistical physics this afternoon at NECSI.
Continue reading Group Theory Homework

Group Theory Tonight at BU

Just a reminder:

This evening at 17 o’clock, or shortly thereafter, we will be meeting in Boston University mathematics territory to discuss group theory. Ben will be leading the discussion, poking us to consider the various transformations of the type

[tex] f: \mathbb{R}^2 \rightarrow \mathbb{R}^2. [/tex]

What are the different structures of interest which these mappings can preserve? What types of maps preserve distances, angles, areas, orientation (chirality), or topological properties? Notes for past group-theory sessions are available in PDF format, or as a gzipped tarball for those who wish to play with the original LaTeX source.

(Don’t forget your stat-mech homework!)