So apparently the “Integral” gamma-ray observatory has put some pretty strong limits on predictions of a “grain size” for spacetime, like in Loop Quantum Gravity, or other theories predicting similar violations of Lorentz invariants which would be detectable in higher- and lower-energy photons coming from distant sources. (Original paper.) I didn’t actually hear much about such predictions when I was the conference “Quantum Theory and Gravitation” last month in Zurich, though partly that was because it was focused on bringing together people from a variety of different approaches , so the Loop QG and String Theory camps were smaller than at some other conferences on the same subject. It was a pretty interesting conference, however (many of the slides and such material can be found here). As one of the organizers, Jürg Fröhlich, observed in his concluding remarks, it gave grounds for optimism about physics, in that it was clear that we’re nowhere near understanding everything about the universe. Which seems like a good attitude to have to the situation – and it informs good questions: he asked questions in many of the talks that went right to the heart of the most problematic things about each approach.
Often after attending a conference like that, I’d take the time to do a blog about all the talks – which I was tempted to do, but I’ve been busy with things I missed while I was away, and now it’s been quite a while. I will probably come back at some point and think about the subject of conformal nets, because there were some interesting talks by Andre Henriques at one workshop I was at, and another by Roberto Longo at this one, which together got me interested in this subject. But that’s not what I’m going to write about this time.
This time, I want to talk about a different kind of topic. Talking in Zurich with various people – John Barrett, John Baez, Laurent Freidel, Derek Wise, and some others, on and off – we kept coming back to kept coming back to various seemingly strange algebraic structures. One such structure is a “loop“, also known (maybe less confusingly) as a “quasigroup” (in fact, a loop is a quasigroup with a unit). This was especially confusing, because we were talking about these gadgets in the context of gauge theory, where you might want to think about assigning an element of one as the holonomy around a LOOP in spacetime. Limitations of the written medium being what they are, I’ll just avoid the problem and say “quasigroup” henceforth, although actually I tend to use “loop” when I’m speaking.
The axioms for a quasigroup look just like the axioms for a group, except that the axiom of associativity is missing. That is, it’s a set with a “multiplication” operation, and each element has a left and a right inverse, called and . (I’m also assuming the quasigroup is unital from here on in). Of course, in a group (which is a special kind of quasigroup where associativity holds), you can use associativity to prove that , but we don’t assume it’s true in a quasigroup. Of course, you can consider the special case where it IS true: this is a “quasigroup with two-sided inverse”, which is a weaker assumption than associativity.
In fact, this is an example of a kind of question one often asks about quasigroups: what are some extra properties we can suppose which, if they hold for a quasigroup , make life easier? Associativity is a strong condition to ask, and gives the special case of a group, which is a pretty well-understood area. So mostly one looks for something weaker than associativity. Probably the most well-known, among people who know about such things, is the Moufang axiom, named after Ruth Moufang, who did a lot of the pioneering work studying quasigroups.
There are several equivalent ways to state the Moufang axiom, but a nice one is:
Which you could derive from the associative law if you had it, but which doesn’t imply associativity. With associators, one can go from a fully-right-bracketed to a fully-left-bracketed product of four things: . There’s no associator here (a quasigroup is a set, not a category – though categorifying this stuff may be a nice thing to try), but the Moufang axiom says this is an equation when . One might think of the stronger condition that says it’s true for all , but the Moufang axiom turns out to be the more handy one.
One way this is so is found in the division algebras. A division algebra is a (say, real) vector space with a multiplication for which there’s an identity and a notion of division – that is, an inverse for nonzero elements. We can generalize this enough that we allow different left and right inverses, but in any case, even if we relax this (and the assumption of associativity), it’s a well-known theorem that there are still only four finite dimensional ones. Namely, they are , , , and : the real numbers, complex numbers, quaternions, and octonions, with real dimensions 1, 2, 4, and 8 respectively.
So the pattern goes like this. The first two, and , are commutative and associative. The quaternions are noncommutative, but still associative. The octonions are neither commutative nor associative. They also don’t satisfy that stronger axiom . However, the octonions do satisfy the Moufang axiom. In each case, you can get a quasigroup by taking the nonzero elements – or, using the fact that there’s a norm around in the usual way of presenting these algebras, the elements of unit norm. The unit quaternions, in fact, form a group – specifically, the group . The unit reals and complexes form abelian groups (respectively, , and ). These groups all have familiar names. The quasigroup of unit octonions doesn’t have any other more familiar name. If you believe in the fundamental importance of this sequence of four division algebras, though, it does suggest that a natural sequence in which to weaken axioms for “multiplication” goes: commutative-and-associative, associative, Moufang.
The Moufang axiom does imply some other commonly suggested weakenings of associativity, as well. For instance, a quasigroup that satisfies the Moufang axiom must also be alternative (a restricted form of associativity when two copies of one element are next to each other: i.e. the left alternative law , and right alternative law ).
Now, there are various ways one could go with this; the one I’ll pick is toward physics. The first three entries in that sequence of four division algebras – and the corresponding groups – all show up all over the place in physics. is the simplest nontrivial group, so this could hardly fail to be true, but at any rate, it appears as, for instance, the symmetry group of the set of orientations on a manifold, or the grading in supersymmetry (hence plays a role distinguishing bosons and fermions), and so on. is, among any number of other things, the group in which action functionals take their values in Lagrangian quantum mechanics; in the Hamiltonian setup, it’s the group of phases that characterizes how wave functions evolve in time. Then there’s , which is the (double cover of the) group of rotations of 3-space; as a consequence, its representation theory classifies the “spins”, or angular momenta, that a quantum particle can have.
What about the octonions – or indeed the quasigroup of unit octonions? This is a little less clear, but I will mention this: John Baez has been interested in octonions for a long time, and in Zurich, gave a talk about what kind of role they might play in physics. This is supposed to partially explain what’s going on with the “special dimensions” that appear in string theory – these occur where the dimension of a division algebra (and a Clifford algebra that’s associated to it) is the same as the codimension of a string worldsheet. J.B.’s student, John Huerta, has also been interested in this stuff, and spoke about it here in Lisbon in February – it’s the subject of his thesis, and a couple of papers they’ve written. The role of the octonions here is not nearly so well understood as elsewhere, and of course whether this stuff is actually physics, or just some interesting math that resembles it, is open to experiment – unlike those other examples, which are definitely physics if anything is!
So at this point, the foregoing sets us up to wonder about two questions. First: are there any quasigroups that are actually of some intrinsic interest which don’t satisfy the Moufang axiom? (This might be the next step in that sequence of successively weaker axioms). Second: are there quasigroups that appear in genuine, experimentally tested physics? (Supposing you don’t happen to like the example from string theory).
Well, the answer is yes on both counts, with one common example – a non-Moufang quasigroup which is of interest precisely because it has a direct physical interpretation. This example is the composition of velocities in Special Relativity, and was pointed out to me by Derek Wise as a nice physics-based example of nonassociativity. That it’s also non-Moufang is also true, and not too surprising once you start trying to check it by a direct calculation: in each case, the reason is that the interpretation of composition is very non-symmetric. So how does this work?
Well, if we take units where the speed of light is 1, then Special Relativity tells us that relative velocities of two observers are vectors in the interior of . That is, they’re 3-vectors with length less than 1, since the magnitude of the relative velocity must be less than the speed of light. In any elementary course on Relativity, you’d learn how to compose these velocities, using the “gamma factor” that describes such things as time-dilation. This can be derived from first principles, nor is it too complicated, but in any case the end result is a new “addition” for vectors:
where is the reciprocal of the aforementioned “gamma” factor. The vectors and are the components of the vector which are parallel to, and perpendicular to, , respectively.
The way this is interpreted is: if is the velocity of observer B as measured by observer A, and is the velocity of observer C as measured by observer B, then is the velocity of observer C as measured by observer A.
Clearly, there’s an asymmetry in how and are treated: the first vector, , is a velocity as seen by the same observer who sees the velocity in the final answer. The second, , is a velocity as seen by an observer who’s vanished by the time we have in hand. Just looking at the formular, you can see this is an asymmetric operation that distinguishes the left and right inputs. So the fact (slightly onerous, but not conceptually hard, to check) that it’s noncommutative, and indeed nonassociative, and even non-Moufang, shouldn’t come as a big shock.
The fact that it makes into a quasigroup is a little less obvious, unless you’ve actually worked through the derivation – but from physical principles, is closed under this operation because the final relative velocity will again be less than the speed of light. The fact that this has “division” (i.e. cancellation), is again obvious enough from physical principles: if we have , the relative velocity of A and C, and we have one of or – the relative velocity of B to either A or C – then the relative velocity of B to the other one of these two must exist, and be findable using this formula. That’s the “division” here.
So in fact this non-Moufang quasigroup, exotic-sounding algebraic terminology aside, is one that any undergraduate physics student will have learned about and calculated with.
One point that Derek was making in pointing this example out to me was as a comment on a surprising claim someone (I don’t know who) had made, that mathematical abstractions like “nonassociativity” don’t really appear in physics. I find the above a pretty convincing case that this isn’t true.
In fact, physics is full of Lie algebras, and the Lie bracket is a nonassociative multiplication (except in trivial cases). But I guess there is an argument against this: namely, people often think of a Lie algebra as living inside its universal enveloping algebra. Then the Lie bracket is defined as , using the underlying (associative!) multiplication. So maybe one can claim that nonassociativity doesn’t “really” appear in physics because you can treat it as a derived concept.
An even simpler example of this sort of phenomenon: the integers with subtraction (rather than addition) are nonassociative, in that . But this only suggests that subtraction is the wrong operation to use: it was derived from addition, which of course is commutative and associative.
In which case, the addition of velocities in relativity is also a derived concept. Because, of course, really in SR there are no 3-space “velocities”: there are tangent vectors in Minkowski space, which is a 4-dimensional space. Adding these vectors in is again, of course, commutative and associative. The concept of “relative velocity” of two observers travelling along given vectors is a derived concept which gets its strange properties by treating the two arguments asymmetrically, just like like “commutator” and “subtraction” do: you first use one vector to decide on a way of slicing Minkowski spacetime into space and time, and then use this to measure the velocity of the other.
Even the octonions, seemingly the obvious “true” example of nonassociativity, could be brushed aside by someone who really didn’t want to accept any example: they’re constructed from the quaternions by the Cayley-Dickson construction, so you can think of them as pairs of quaternions (or 4-tuples of complex numbers). Then the nonassociative operation is built from associative ones, via that construction.
So are there any “real” examples of “true” nonassociativity (let alone non-Moufangness) that can’t simply be dismissed as not a fundamental operation by someone sufficiently determined? Maybe, but none I know of right now. It may be quite possible to consistently hold that anything nonassociative can’t possibly be fundamental (for that matter, elements of noncommutative groups can be represented by matrices of commuting real numbers). Maybe it’s just my attitude to fundamentals, but somehow this doesn’t move me much. Even if there are no “fundamentals” examples, I think those given above suggest a different point: these derived operations have undeniable and genuine meaning – in some cases more immediate than the operations they’re derived from. Whether or not subtraction, or the relative velocity measured by observers, or the bracket of (say) infinitesimal rotations, are “fundamental” ideas is less important than that they’re practical ones that come up all the time.