Cohl Furey understands neither field theory nor octonions

Luboš Motl, July 21, 2018

Quanta Magazine’s Natalie Wolchover wrote a cheesy celebration

The Peculiar Math That Could Underlie the Laws of Nature

of a would-be theory of everything by Ms Cohl Furey that is claimed to be based on the octonions \({\mathbb O}\). If you read it, the human part of the story as well as the spirit of the mathematics used for physics sounds virtually isomorphic to Garrett Lisi and his would-be theories of everything based on the exceptional group \(E_8\). So people are told about some ingenious outsider who has some very non-intellectual hobbies but who can still get closer to a theory of everything than all the professional physicists combined by insisting that an exceptional mathematical structure underlies the patterns of particles and fields in Nature.

The main difference is that Lisi is just a «surfer dude» while Furey is a «ski-accordion-yoga-mat-rented-car gal trained in martial arts, as her muscular physique betrays». Cool: the surfer dude is clearly a second-rate genius when compared to Furey now. 😉

But both of these people misunderstand the meaning of the mathematical structures they claim to love – and they are trying to use them in field theory in ways that destroy the relationships that make the beloved mathematical structure what it should be; or that are impossible in quantum field theory.

Needless to say, both Lisi and Furey are parts of a much larger crackpot movement. When such positive articles about meaningless claims are published e.g. in the Quanta Magazine, you may be sure that there will be positive reactions because lots of other numerologists and crackpots arrive, praise each other, and promote their own twists on the crackpottery.

Recall that Lisi wanted to claim that he had found something important about \(E_8\) or even discovered it – he’s done neither. And the largest exceptional simple Lie group was used in his model almost as a gauge group in the grand unification but not quite. He wants to be more ambitious so the 248-dimensional adjoint representation should include not only gauge bosons but also all the quarks and fermions.

It’s not really possible and as a result, his construction is nothing else than a deceitful lipstick masking a wishful thinking that cannot be backed by anything except for pure numerology. The \(E_8\) adjoint representation simply has a high enough number of components which is why he may place the fields of the Standard Model «somewhere into it». But do the fields feel comfortable (thanks, Andrei)? Is the adjoint of \(E_8\) the right description of the mutual relationships between the fields? It’s not.

In real physics, the gauge groups that allow chiral (left-right-asymmetric) fermions need to have complex representations – which means representations that can be complex conjugated to get unequivalent but analogous ones (the complex conjugation is used for the switch from particles to antiparticles). \(E_8\) only has real representations (the complex conjugation does nothing to them at all) so it can’t possibly describe left-right-asymmetric phenomena such as the left-handed (chiral) neutrinos that differ from the right-handed (chiral) antineutrinos. On top of that, the union of bosons and fermions within the same representation is impossible. He claims to unite the bosons and fermions through a symmetry but the only symmetry that may relate bosons and fermions is a Grassmann-odd i.e. fermionic symmetry i.e. supersymmetry. The \(E_8\) symmetry is a bosonic one so it simply cannot relate the properties of bosons to properties of fermions.

In heterotic string theory and its duals, one can start with the \(E_8\) gauge group and get down to the Standard Model but that’s because the \(E_8\) is broken to \(E_6\) or \(SO(10)\) etc. – and those do admit complex representations – by the gauge fields on the compactification manifolds, something that is not available in the purely four-dimensional model building such as Lisi’s framework. So Lisi’s constructions don’t pass even the most basic tests. You just can’t unify all the fields into an \(E_8\) representation in this way.

Furey’s would-be theory of everything using the octonions \({\mathbb O}\) is analogously flawed.

She claims the fields of the Standard Model to be something like representations of some algebra related to octonions in some incoherent way. Note that $$\mathbb R,\mathbb C,\mathbb H,\mathbb O$$ are 1-, 2-, 4-, 8-dimensional division algebras (where you can add and multiply numbers with one or more real components, and where the inverse exists for every nonzero number). She isn’t satisfied with the octonions themselves so she also uses the Dixon algebra

$$\mathbb R \otimes \mathbb C \otimes \mathbb H \otimes \mathbb O$$

Let me tell you a secret. The Dixon algebra isn’t named after a great physicist (Lance Dixon) but after a complete crackpot (Geoffrey Dixon) and it makes absolutely no sense. There is no interesting algebraic structure that could be described in this way. In fact, the sequence of symbols may be seen to be silly very easily. In particular, \(\mathbb R \otimes\) is a trivial factor because the tensor multiplication of a vector space \(V\). with the 1-dimensional space of real numbers changes nothing about \(V\).

It’s very clear what Geoffrey Dixon, Cohl Furey, and other crackpots think about these division algebras. They think it must be a great idea to have generalizations of the real numbers whose dimensions are powers of two, and they can do even better by picking higher powers. For example, the tensor product above is supposed to have the dimension \(1\times 2 \times 4 \times 8 = 64\).

But there’s no mathematically interesting rule to define the products from two copies of quaternions or octonions (or one quaternion and one octonion). Interesting multiplication tables only exist on one copy of quaternions, or one copy of octonions. And if you only define the tensor product formally, like \(j\otimes C\), and assume that the quaternions act from one side and octonions from the other, it’s no good because quaternions and octonions only deserve the name if they can be multiplied from both sides. On top of that, it’s just wrong to talk about octonionic representations of groups and algebras because the action of groups and algebras must be associative and octonions are not! One may construct algebras of octonionic matrices – the \(3\times 3\) «Hermitian» octonionic matrices with the anticommutator have the \(F_4\) automorphism group – but because of the non-associativity of \(\mathbb O\), they don’t have representations that are «columns of octonions» in any sense!

The full multiplication table is the cool set of gems that defines the beauty of quaternions and octonions; I discuss the tables at the bottom of this text («the bonus»). If you don’t care what the multiplication table is or if you propose a wrong one, you won’t get the beauty of the quaternions and octonions! Crackpots clearly fail to get the trivial point but it’s very important, anyway:

Quaternions and octonions are much more than powers of two.

If you claim that you have made a revolution using octonions without using the multiplication table, it’s exactly like if you claim that you are a Formula One expert or champion because you’ve noticed that the number of a formula’s wheels is a power of two (and you may also become a super-champion if the car has 64 wheels instead). The Formula One – and the art of its driving – is much more than the number four. It matters how the wheels are connected to the engine, what the driver does in the vehicle, and so on. I think that most laymen understand this trivial statement in the case of Formula One but for some reason, the equally obvious claim about the division algebras seems incomprehensible to many.

Or, as Peter Morgan puts it in his most meaningful comment under Wolchover’s article (I added this quote half a day after this blog post was written down):

The paper linked to above, having introduced the non-associative octonions, takes a page to show that the left action of \(\mathbb C \otimes \mathbb O\) on itself is isomorphic as an associative composition algebra to the Clifford algebra \(C \ell(6)\), then the non-associativity of the octonions plays no further part.
To me the move from \(\mathbb C\otimes\mathbb H\) to \(C \ell(4)\) seems artificial, more justified by it being possible to use \(C\ell(6)\otimes C\ell(4)\) to get to \(C \ell(10)\) and thence to \(SU(5)\) than by any really principled argument.

With apologies, the whole seems more a marketing effort for \(C\ell(10)\) and Georgi’s \(SU(5)\) than, for example, a principled introduction of non-associativity.

Incidentally, in a video, she also repeats the widespread view that quaternions underlie special relativity which is really a misconception for the same reason: the spacetime is 4D but there’s no natural «product» defined on a pair of spacetime points so it makes no sense to represent the four-vectors as quaternions. When the four-vectors are represented by quaternions, it’s another mere bookkeeping device (which is possible because both have four components) and the product – the main mathematical structure that makes quaternions quaternions – remains unused.

Also, just like Lisi’s work, her way to assemble the Standard Model fields is pure numerology. She has a high enough number of components, so she throws the quark and lepton fields somewhere in them. All the actual octonionic structure is totally broken by this treatment. You know, when you claim to have a construction based on octonions, you should notice that the octonions have the 14-dimensional \(G_2\) automorphism group (a symmetry renaming the octonions into others, a subgroup of \(SO(7)\)).

But if your construction has no trace of the \(G_2\) symmetry, it doesn’t have anything to do with octonions.

There’s no \(G_2\) symmetry left in her construction which is why it’s deceitful to say it has something to do with octonions. Well, it doesn’t even have the \(SO(3)\) automorphism group of the quaternions. So all the claims about connecting octonions with physics of the Standard Model are totally spurious. (Some of these claims are hers, most of them are copied from earlier crackpots of the same kind plus some semi-legit researchers trying a wrong track, starting in the 1970s with Günaydin – so much of this criticism is primarily directed against these older authors on whose shoulders she is standing.)

Octonions and other exceptional structures are great and I love them – and use them in my research very often. But the question whether the octonions have something to do with the gauge group of the Standard Model and the representations of quarks and leptons is a question. I think that the obvious comparisons of fingerprints, an analysis we can make, makes it almost certain that the answer is No, the Standard Model fields just don’t have anything to do with octonions.

Crackpots like Furey don’t ever try to answer questions impartially. The relevance of the octonions is treated as a dogma and Furey and others are ready to destroy the inner workings of both the octonions and the Standard Model in their futile efforts to save the dogma. The dogma is almost certainly wrong.

You may look at her papers. Physicists don’t read them much, in the small number of followups, most of them are her own while Geoffrey Dixon, Lee Smolin, and a few other crackpots dominate the rest. Physicists don’t read them because they don’t really make sense.

OK, take the latest one from June 2018, so far with 0 citations. The title combines the Standard Model group, division algebras, and ladder operators. If you don’t think about the content, it looks like a perfectly legitimate paper with the correct grammar, right kind of mathematical symbols and jargon, and a professional ratio between words and formulae, among other things.

But if you have the expertise, read it, and think about it, you immediately see that it’s just a pile of hogwash. (It has gotten into a journal – similar authors persistently send their nonsensical papers everywhere and it’s statistically guaranteed that a referee who doesn’t want to bother or fight emerges and such papers get occasionally published. This referee’s blunder is clearly the main reason why the paper was hyped in the Quanta Magazine, too.) She basically claims to derive the 12-dimensional Standard Model group as some part of the 24-dimensional \(SU(5)\) grand unified group of Georgi and Glashow – it is supposed to be the part that respects some incoherent rules mentioning tensor products of division algebras, Clifford algebras, tensor products of division algebras, left ideals, ladder operators, and other things.

Well, one could say she’s an actual great example of someone who is «lost in math». She uses lots of these phrases from algebra but her mixture of these words doesn’t make any sense. She constantly pretends to have found some new laws of physics but there are none. In fact, one could argue that none of these things ever appears in physics of quantum field theory. If you want to organize fields,

fields just form representations of groups.

To make it even more constraining, generators of Lie groups correspond to gauge fields and these Lie groups should better be compact (because the norms on the Lie algebra are linked to probabilities that should better be positively definite). That’s the actual framework we have. Operators (such as fields) form algebras under the multiplication (and the commutator, that from Lie algebras, is even more widespread than the product itself) and that’s it. There’s nothing else and if you added some different algebraic structure to the organization of fields in quantum field theories, that would be a technical yet far-reaching development, indeed. If you could meaningfully add semigroups, ideals that aren’t just representations of groups, algebras that aren’t Lie algebras etc., quantum fields that have to transform as (not just under) Clifford algebras by themselves, physicists would care. But it would have to work.

Furey’s construction doesn’t work and she doesn’t care.

By the way, her isolation of the 12 generators out of 24 generators of \(SU(5)\) – yes, it’s been known for a while that it’s exactly one-half – isn’t a new observation (you can find it on this blog and in other papers, I guess) – but the justification is just nonsense. One may define a parity on the adjoint representation of \(SU(5)\) and define the generators of the Standard Model group to be positive (even) and the remaining one to be negative (odd). The parity will correctly (multiplicatively) behave under the commutator.

So there’s some way to semi-naturally segregate the \(SU(5)\) generators into the Standard Model (desirable) ones and the unwanted ones. But that’s still far from having a physical mechanism that actually breaks \(SU(5)\) to the Standard Model group. She claims to have and not have \(SU(5)\) at the same moment. And she claims to break \(SU(5)\) to the Standard Model without the Higgs fields. The Standard Model group is picked because it’s compatible with some of her ideas or constructions. In actual field theory, the field content and its consistency follows its own rules. She should either obey these rules, or justify some replacement for these rules and/or the replacement for the Higgs, symmetry-breaking mechanism etc.

But there must still be a replacement. If it makes any sense to talk about \(SU(5)\) at any stage of the construction, there has to be something that breaks it to the Standard Model. In quantum field theory-based model building, it’s always the Higgs fields. In string theory, one has new tools such as the Wilson lines around cycles of the compactification manifold. Indeed, those are new physical – and characteristically stringy – mechanisms that may break \(SU(5)\) to the Standard Model. But she has nothing. So either she talks about theories with the \(SU(5)\) symmetry or she doesn’t. Her own answer really makes it clear that she doesn’t have any \(SU(5)\). So why is she mentioning it at all?

Lisi’s and Furey’s are efforts that belong to a much more widespread subcommunity of the crackpot movement – whose members also write papers about the «graviweak» unification. If you forgot about «graviweak» folks, those claim that they may embed the Lorentz group and the Standard Model group into a larger, simple group. Lisi is basically an example of that, too. However, the Lorentz group acts on the spacetime while the gauge groups don’t – they act inside a point. So they are clearly qualitatively different and cannot be related by any symmetry to each other. In practice, the graviweak people misunderstand the difference between the diffeomorphism symmetry of general relativity (which moves points to other places) and the local Lorentz group (that doesn’t). But those are completely different things. The diffeomorphism group of general relativity is an example of a generalized «gauge symmetry» but it’s simply not an example of the gauge symmetry of the Yang-Mills type.

So Lisi, Furey, and many others just don’t understand the mathematical structures (they’re using) too well – and they misunderstand their actual and possible relationship with physics completely. But we read many more articles lionizing these crackpots than we read about the actual exciting physics research. John Baez actually and surprisingly gave a rather reasonable feedback for Wolchover’s article, including some technicalities. On the other hand, Pierre Ramond and especially Michael Duff made it sound as her 0-citation meaningless article really revolutionizes particle physics.

In particular, Mike Duff said that it could be revolutionary and have other adjectives and so on. Oh, really, Mike? Haven’t you noticed that the paper is pure crackpottery? Have you noticed that when you praise such a thing that makes no sense, you’re just absolutely full of šit?

I believe that Duff should understand it and he has some political reasons why he praises this crackpottery. But most of the laymen – even those who have dedicated some time to trying to superficially follow theoretical or particle physics – just don’t distinguish real physics from crackpottery of Furey’s type. So everything that is controlled by the laymen is more or less guaranteed to gradually replace physicists with crackpots. Those who flatter the laymen and those who own windsurfing boards and yoga mats will clearly be preferred (and get all the Nobel prizes when the movement conquers the Scandinavian institutions).

The Quanta Magazine is a worrisome borderline example. It’s funded by Jim Simons who has been an excellent mathematician close to theoretical physics – exactly the type of person who used to understand (and maybe still understands?) the criticism I wrote above. But even the Quanta Magazine which is funded by Simons ended up being a medium that – whenever it writes about fundamental physics – lionizes crackpots and attacks actual top physicists most of the time. Crackpottery is so much more attractive for a larger number of readers – and the number of crackpots is vastly higher than the number of physicists.

Much of the serious research still deserves the allocation of time and energy of the researcher and the funding. In a world increasingly controlled by stupid laymen, is the top-tier serious research in pure science sustainable at all? Isn’t it guaranteed that crackpots teamed up with other crackpots are going to overtake not only the Quanta Magazines but the universities such as Cambridge as well?

I am grateful to have spent at least a part of my life in an epoch when this wasn’t the case, in a world where physicists and crackpots knew their places, the places weren’t the same, and where it was possible for a physicist to explain why crackpots’ ideas don’t work. I am afraid the mankind is going to deteriorate into a bunch of stupid animals again.

Bonus: the actual beauty of the division algebras

\(\mathbb R,\mathbb C,\mathbb H,\mathbb O\) have dimensions 1,2,4,8, respectively. But if you just know the dimensions, you’re extremely far from understanding why these are the only four division algebras. I won’t prove that they are the only ones here but I will sketch it so that you understand some of the beauty if you focus.

\(\mathbb R\) are the real numbers such as \(-20.18\). You may add them, subtract them, multiply them, and divide them – unless the denominator is zero. Everyone should learn how to do it from the elementary school or elsewhere as a kid.

\(\mathbb C\) are complex numbers of the form \(x+iy\). The multiplication inverse of that number is

$$\frac{1}{x+iy} = \frac{x-iy}{x^2+y^2}$$

If you use \(i^2=-1\), you may check that the right hand side times \(x+iy\) is equal to one. The complex numbers are great e.g. because the \(n\)-th degree algebraic equation has exactly \(n\) roots \(x_i\in\mathbb C\) – some of them may coincide. You may prove it e.g. by studying the phase of the polynomial for \(|x|\to\infty\) in the complex plane \(x=r\cdot \exp(i\phi)\in\mathbb C\). The phase \(\phi_p\) of the polynomial winds along a circle around the origin \(n\) times. Because \(\phi_p\) is ill-defined when the polynomial is zero, the points \(x\in\mathbb C\) where the polynomial vanishes allow you to change the winding number by one, so there must be \(n\) of them.

Sorry if I were too concise.

Complex numbers are really more natural than the real numbers. They are absolutely needed as probability amplitudes in quantum mechanics. You know, the total probability has to be fixed but the amplitudes must oscillate. The only way how they (think about the energy eigenstates to make it simple) can oscillate yet preserve an invariant is for them to be complex so that the phase oscillates while the absolute value stays the same.

Also, in representation theory of groups, all the representations are complex by default. The real or quaternionic/pseudoreal representations may be defined as initially complex representations with some extra structure – a complex conjugation defined by the antilinear «structure map» \(j\) that doesn’t spoil the remaining operations. In this sense, real and quaternionic representations are equally far from the most fundamental and simplest representations – those are the complex ones. This «centrality» of the complex numbers is misunderstood by all the members of that movement – they either think that \(\mathbb R\) are the most fundamental among the three, or the highest-dimensional algebras such as \(\mathbb H,\mathbb O\) are the fundamental building blocks.

The quaternion \(z\in\mathbb H\) is a number of the form

$$a+ib+jc+kd$$

where \(i,j,k\) are three imaginary units obeying

$$i^2=j^2=k^2=ijk=jki=kij=-1$$

So the unit \(i\) may be identified with the complex imaginary unit, but so can \(j\) or \(k\). And \(ij=k=-ji\) and cyclic permutations define the totally associative but maximally non-commutative multiplication table of the three imaginary units. The three-dimensional space generated by \(i,j,k\) may be rotated by \(SO(3)\) transformations – the quaternions may be \(SO(3)\)-renamed – so that the multiplicative relationships between them remain the same. We say that \(SO(3)\) is the automorphism group of the quaternions.

Why is it a division algebra? It’s because the inverse is defined analogously as for the complex numbers

$$\frac{1}{a+ib+jc+kd} = \frac{a-ib-jc-kd}{a^2+b^2+c^2+d^2}$$

I just changed the signs of the imaginary «coordinates» \(b,c,d\) and divided it by the squared Euclidean length of the four-dimensional vector. Why are the numbers inverse to each other? Well, it’s because

$$(a\!+\!ib\!+\!jc\!+\!kd)(a\!-\!ib\!-\!jc\!-\!kd) = a^2+b^2+c^2+d^2$$

Why is it so? The terms \(a^2,b^2,c^2,d^2\) are obviously there if you use the distributive law – because \(i^2=-1\) and the minus sign cancels because there are opposite signs in front of \(b\) in the two terms etc.

And all the mixed terms cancel because they are either of the form

$$a\cdot (-ib) + (+ib) \cdot a = 0$$

or of the form

$$(ib)\cdot (-jc) + (+jc)\cdot (-jb) = 0$$

The \(ab\)-like terms cancelled because there were explicitly opposite signs in the two terms. The \(bc\)-like terms cancelled because \(ij=-ji\) – because the imaginary units anticommute. There are three imaginary units because they may be used as the two factors and one result in a multiplication table. You won’t find a larger associative division algebra.

If you defined a «simpler» but «uglier» multiplication table, e.g. if all the products of the imaginary units were \(\pm k\), the «complex conjugate over the squared norm» would still be inverse but such an inverse wouldn’t be unique or the multiplication wouldn’t be associative. An even simpler sick example: if you defined all the products of imaginary units to be \(\pm 1\), while preserving the anticommutativity, whole families of such «broken quaternions» would behave as the same number under multiplication.

So the existence of the quaternions is really linked to the fact that a three-dimensional vector is also a two-form, something that determines the rotation of the remaining two, orthogonal vectors.

Now, the octonions are also a division algebra. They have seven imaginary units, let me call them \(i,j,k,A,B,C,D\). These seven units may be written using products of three basic ones, \(i,j,A\), and parentheses. The multiplication table of the seven units is such that the squares such as \(B^2\) are equal to minus one; and the product of two different units is anti-commutative, e.g. \(CD=-DC\), just like for the quaternions.

On top of that, the product of three units such as \((ij)A\) is maximally non-associative whenever it differs from \(\pm 1\). It means that e.g.

$$(ij)A = -i(jA)$$

The permutation of two different units in a product flips the sign; and the rearrangement of the parentheses in products such as one above flips the sign, too (whenever the product differs from \(\pm 1\)). In some binary counting, the basic imaginary units \(i,j,A\) may be associated with binary numbers \(100,010,001\) and the 7 units, given by 3 bits, are multiplied by adding the three bits modulo two. However, the precise sign matters and it’s such that you impose the «maximal non-commutativity» and «maximum non-associativity».

That’s a great algebraic structure where the inverse of a nonzero number exists and is unique. The automorphism group isn’t the whole 21-dimensional \(SO(7)\) in this case. It’s just a group that remembers some relationships between the 7 imaginary units that are encoded in the 3 bits – bits that may be added separately. So the symmetry group ends up being the 14-dimensional \(G_2\) only. Only 2/3 of the generators are still symmetries of the multiplication table. Why 2/3?

Pick one of the 7 units, it doesn’t matter which one, for example \(i\). The remaining 6 units \(j,k,A,B,C,D\), may be divided to pairs that, along with the \(i\), may build the quaternionic triplets:

$$ijk, iAB, iCD$$

Now, \(SO(7)\) is generated by 21 generators of rotations of 2-planes, such as the rotation of \(jk\) into each other, \(AB\) into each other, and \(CD\) into each other. In \(SO(7)\), there would be three parameters (angles) for these three generators. But only if \(\phi_1+\phi_2+\phi_3=0\), the octonionic structure – including the right multiplicative relationship with \(i\) – is preserved. You may see it e.g. by realizing that \(i,j,A\) are «fundamental» but \(C,D\) may already be written in terms of products of \(i,j,A\) with parentheses – so whether \(C,D\) have to rotate to preserve the structure of the octonions is already determined by thee rotations involving \(i,j,k=ij,A,B=iA\).

The octonionic multiplication table may be reconstructed from remembering 7 «triplets that embed quaternions into octonions». These 7 triplets contain \(7\times 3=21\) pairs of octonionic imaginary units which is all unordered pairs of octonionic units. That’s why the number of imaginary units in a construction that works must be \(1+2\times 3\) because the combination number «7 choose 2» is \(7\times 6 / 2\times 1\) and it must be the same as \(7\times 3\). That’s why there is no 10-dimensional, 16-dimensional, 32-dimensional, or 64-dimensional extension of the octonions.

Even if you don’t fully understand what I just wrote, I want to convey a more general point: the exceptional traits of the quaternions and especially octonions require that you analyze the multiplication table of the imaginary units, including the precise signs of the products (which remember whether the structures are commutative and/or associative). If you don’t analyze the multiplication tables of your 4- or 8- or 16- or 32- or 64-component «numbers», then you are not working with anything like quaternions or octonions!

You are just talking about some generic vector spaces whose dimensions are powers of two and you are doing so because you have heard that it’s deep. But without the multiplication table (and/or without the anticommutators of the SUSY generators – powers of two also appear in supersymmetry) – and without the additional constraints that the table imposes on the usage of these division algebras – there is absolutely nothing deep about the vector spaces with these dimensions! And that’s the case of Furey et al. She just doesn’t get what makes the octonions deep. She only uses the word «octonion» to make her claims sound sexy – but her observation is nothing more than «the number of fields in the Standard Model is smaller than a power of two».