Cohl Furey Understands Neither Plain Theory Nor Octonions

Her mathematical masturbations are physically meaningless

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 yesteryear Ms Cohl Furey that is claimed to live on based on the octonions \({\mathbb O}\). If yous read it, the human purpose of the story as good as the spirit of the mathematics used for physics sounds virtually isomorphic to Garrett Lisi in addition to his would-be theories of everything based on the exceptional grouping \(E_8\). So people are told almost some ingenious outsider who has some rattling non-intellectual hobbies but who tin shipping away even so acquire closer to a theory of everything than all the professional person physicists combined yesteryear insisting that an exceptional mathematical construction underlies the patterns of particles in addition to fields inwards Nature.

The primary departure is that Lisi is just a "surfer dude" piece Furey is a "ski-accordion-yoga-mat-rented-car gal trained inwards 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 pregnant of the mathematical structures they claim to love – in addition to they are trying to usage them inwards patch theory inwards ways that destroy the relationships that brand the honey mathematical construction what it should be; or that are impossible inwards quantum patch theory.




Needless to say, both Lisi in addition to Furey are parts of a much larger crackpot movement. When such positive articles almost meaningless claims are published e.g. inwards the Quanta Magazine, yous may live on sure that at that topographic point volition live on positive reactions because lots of other numerologists in addition to crackpots arrive, praise each other, in addition to promote their ain twists on the crackpottery.




Recall that Lisi wanted to claim that he had constitute something of import almost \(E_8\) or fifty-fifty discovered it – he's done neither. And the largest exceptional simple Lie grouping was used inwards his model almost as a guess grouping inwards the grand unification but non quite. He wants to live on to a greater extent than ambitious so the 248-dimensional adjoint representation should include non solely guess bosons but also all the quarks in addition to fermions.

It's non really possible in addition to as a result, his construction is nil else than a deceitful lipstick masking a wishful thinking that cannot live on backed yesteryear anything except for pure numerology. The \(E_8\) adjoint representation only has a high plenty issue of components which is why he may house the fields of the Standard Model "somewhere into it". But do the fields experience comfortable (thanks, Andrei)? Is the adjoint of \(E_8\) the right description of the usual relationships betwixt the fields? It's not.

In existent physics, the guess groups that allow chiral (left-right-asymmetric) fermions demand to guide hold complex representations – which agency representations that tin shipping away live on complex conjugated to acquire unequivalent but analogous ones (the complex conjugation is used for the switch from particles to antiparticles). \(E_8\) solely has existent representations (the complex conjugation does nil to them at all) so it can't perchance depict left-right-asymmetric phenomena such as the left-handed (chiral) neutrinos that differ from the right-handed (chiral) antineutrinos. On superlative of that, the spousal human relationship of bosons in addition to fermions within the same representation is impossible. He claims to unite the bosons in addition to fermions through a symmetry but the solely symmetry that may relate bosons in addition to fermions is a Grassmann-odd i.e. fermionic symmetry i.e. supersymmetry. The \(E_8\) symmetry is a bosonic i so it only cannot relate the properties of bosons to properties of fermions.

In heterotic string theory in addition to its duals, i tin shipping away outset alongside the \(E_8\) guess grouping in addition to acquire downward to the Standard Model but that's because the \(E_8\) is broken to \(E_6\) or \(SO(10)\) etc. – in addition to those do acknowledge complex representations – yesteryear the guess fields on the compactification manifolds, something that is non available inwards the purely four-dimensional model edifice such as Lisi's framework. So Lisi's constructions don't move yesteryear fifty-fifty the most basic tests. You just can't unify all the fields into an \(E_8\) representation inwards 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 live on something similar representations of some algebra related to octonions inwards some incoherent way. Note that \(\RR,\CC,\HHH,\OO\) are 1-, 2-, 4-, 8-dimensional segmentation algebras (where yous tin shipping away add together in addition to multiply numbers alongside i or to a greater extent than existent components, in addition to where the inverse exists for every nonzero number). She isn't satisfied alongside the octonions themselves so she also uses the Dixon algebra\[

\RR \otimes \CC \otimes \HHH \otimes \OO.

\] Let me tell yous a secret. The Dixon algebra isn't named later a groovy physicist (Lance Dixon) but later a consummate crackpot (Geoffrey Dixon) in addition to it makes absolutely no sense. There is no interesting algebraic construction that could live on described inwards this way. In fact, the sequence of symbols may live on seen to live on airheaded rattling easily. In particular, \(\RR\otimes\) is a niggling factor because the tensor multiplication of a vector infinite \(V\). alongside the 1-dimensional infinite of existent numbers changes nil almost \(V\).

It's rattling clear what Geoffrey Dixon, Cohl Furey, in addition to other crackpots think almost these segmentation algebras. They think it must live on a groovy thought to guide hold generalizations of the existent numbers whose dimensions are powers of two, in addition to they tin shipping away do fifty-fifty meliorate yesteryear picking higher powers. For example, the tensor production higher upwards is supposed to guide hold the dimension \(1\times 2 \times 4 \times eight = 64\).

But that's a totally incorrect lesson that someone may extract from the segmentation algebras. In fact, the octonions are connected alongside the "highest might of two" where a segmentation algebra exists in addition to there's nil genuinely analogous inwards higher power-of-two dimensions. And it's also meaningless to "tensor multiply" quaternions alongside octonions. You know, the footing vectors of \(\HHH\otimes \OO\) should live on proportional to products of imaginary units from \(\HHH\) in addition to imaginary units from \(\OO\). But there's no mathematically interesting dominion to define the products from 2 copies of quaternions or octonions (or i quaternion in addition to i octonion). Interesting multiplication tables solely be on i re-create of quaternions, or i re-create of octonions. And if yous solely define the tensor production formally, similar \(j\otimes C\), in addition to assume that the quaternions human activity from i side in addition to octonions from the other, it's no goodness because quaternions in addition to octonions solely deserve the cry if they tin shipping away live on multiplied from both sides. On superlative of that, it's just incorrect to verbalise almost octonionic representations of groups in addition to algebras because the activity of groups in addition to algebras must live on associative in addition to octonions are not! One may create algebras of octonionic matrices – the \(3\times 3\) "Hermitian" octonionic matrices alongside the anticommutator guide hold the \(F_4\) automorphism grouping – but because of the non-associativity of \(\OO\), they don't guide hold representations that are "columns of octonions" inwards whatsoever sense!

The total multiplication table is the cool laid of gems that defines the beauty of quaternions in addition to octonions; I hash out the tables at the bottom of this text ("the bonus"). If yous don't attention what the multiplication tabular array is or if yous advise a incorrect one, yous won't acquire the beauty of the quaternions in addition to octonions! Crackpots clearly neglect to acquire the niggling indicate but it's rattling important, anyway:

Quaternions in addition to octonions are much to a greater extent than than powers of two.

If yous claim that yous guide hold made a revolution using octonions without using the multiplication table, it's precisely similar if yous claim that yous are a Formula One practiced or champion because you've noticed that the issue of a formula's wheels is a might of 2 (and yous may also give-up the ghost a super-champion if the automobile has 64 wheels instead). The Formula One – in addition to the fine art of its driving – is much to a greater extent than than the issue four. It matters how the wheels are connected to the engine, what the driver does inwards the vehicle, in addition to so on. I think that most laymen empathize this niggling declaration inwards the representative of Formula One but for some reason, the as obvious claim almost the segmentation algebras seems incomprehensible to many.

Or, as Peter Morgan puts it inwards his most meaningful comment nether Wolchover's article (I added this quote half a twenty-four hours later this weblog postal service was written down):

The newspaper linked to above, having introduced the non-associative octonions, takes a page to present that the left activity of \(\CC\otimes \OO\) on itself is isomorphic as an associative composition algebra to the Clifford algebra \(C \ell(6)\), in addition to then the non-associativity of the octonions plays no farther part.
To me the displace from \(\CC\otimes\HHH\) to \(C \ell(4)\) seems artificial, to a greater extent than justified yesteryear it beingness possible to usage \(C\ell(6)\otimes C\ell(4)\) to acquire to \(C \ell(10)\) in addition to thence to \(SU(5)\) than yesteryear whatsoever really principled argument.

With apologies, the whole seems to a greater extent than a marketing attempt for \(C\ell(10)\) in addition to Georgi's \(SU(5)\) than, for example, a principled introduction of non-associativity.

Incidentally, inwards a video, she also repeats the widespread persuasion 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 yoke of spacetime points so it makes no feel to stand upwards for the four-vectors as quaternions. When the four-vectors are represented yesteryear quaternions, it's some other mere bookkeeping device (which is possible because both guide hold iv components) in addition to the production – the primary mathematical construction that makes quaternions quaternions – remains unused.

Also, just similar Lisi's work, her way to get together the Standard Model fields is pure numerology. She has a high plenty issue of components, so she throws the quark in addition to lepton fields somewhere inwards them. All the actual octonionic construction is totally broken yesteryear this treatment. You know, when yous claim to guide hold a construction based on octonions, yous should notice that the octonions guide hold the 14-dimensional \(G_2\) automorphism grouping (a symmetry renaming the octonions into others, a subgroup of \(SO(7)\)).

But if your construction has no line of the \(G_2\) symmetry, it doesn't guide hold anything to do alongside octonions.

There's no \(G_2\) symmetry left inwards her construction which is why it's deceitful to say it has something to do alongside octonions. Well, it doesn't fifty-fifty guide hold the \(SO(3)\) automorphism grouping of the quaternions. So all the claims almost connecting octonions alongside physics of the Standard Model are totally spurious. (Some of these claims are hers, most of them are copied from before crackpots of the same form plus some semi-legit researchers trying a incorrect track, starting inwards the 1970s alongside Günaydin – so much of this criticism is primarily directed against these older authors on whose shoulders she is standing.)

Octonions in addition to other exceptional structures are groovy in addition to I love them – in addition to usage them inwards my inquiry rattling often. But the query whether the octonions guide hold something to do alongside the guess grouping of the Standard Model in addition to the representations of quarks in addition to leptons is a question. I think that the obvious comparisons of fingerprints, an analysis nosotros tin shipping away make, makes it almost sure that the respond is No, the Standard Model fields just don't guide hold anything to do alongside octonions.

Crackpots similar Furey don't ever seek to respond questions impartially. The relevance of the octonions is treated as a dogma in addition to Furey in addition to others are ready to destroy the inner workings of both the octonions in addition to the Standard Model inwards their futile efforts to salve the dogma. The dogma is almost sure enough wrong.

You may await at her papers. Physicists don't read them much, inwards the pocket-sized issue of followups, most of them are her ain piece Geoffrey Dixon, Lee Smolin, in addition to a few other crackpots dominate the rest. Physicists don't read them because they don't really brand sense.

OK, accept the latest i from June 2018, so far alongside 0 citations. The championship combines the Standard Model group, segmentation algebras, in addition to ladder operators. If yous don't think almost the content, it looks similar a perfectly legitimate newspaper alongside the right grammar, right form of mathematical symbols in addition to jargon, in addition to a professional person ratio betwixt words in addition to formulae, amidst other things.

But if yous guide hold the expertise, read it, in addition to think almost it, yous similar a shot run into that it's just a pile of hogwash. (It has gotten into a magazine – similar authors persistently shipping their nonsensical papers everywhere in addition to it's statistically guaranteed that a referee who doesn't desire to bother or struggle emerges in addition to such papers acquire occasionally published. This referee's blunder is clearly the primary argue why the newspaper was hyped inwards the Quanta Magazine, too.) She basically claims to derive the 12-dimensional Standard Model grouping as some purpose of the 24-dimensional \(SU(5)\) grand unified grouping of Georgi in addition to Glashow – it is supposed to live on the purpose that respects some incoherent rules mentioning tensor products of segmentation algebras, Clifford algebras, tensor products of segmentation algebras, left ideals, ladder operators, in addition to other things.

Well, i could say she's an actual groovy representative of someone who is "lost inwards math". She uses lots of these phrases from algebra but her mixture of these words doesn't brand whatsoever sense. She constantly pretends to guide hold constitute some novel laws of physics but at that topographic point are none. In fact, i could debate that none of these things ever appears inwards physics of quantum patch theory. If yous desire to organize fields,

fields just grade representations of groups.

To brand it fifty-fifty to a greater extent than constraining, generators of Lie groups fit to guess fields in addition to these Lie groups should meliorate live on compact (because the norms on the Lie algebra are linked to probabilities that should meliorate live on positively definite). That's the actual framework nosotros have. Operators (such as fields) grade algebras nether the multiplication (and the commutator, that from Lie algebras, is fifty-fifty to a greater extent than widespread than the production itself) in addition to that's it. There's nil else in addition to if yous added some dissimilar algebraic construction to the organisation of fields inwards quantum patch theories, that would live on a technical yet far-reaching development, indeed. If yous could meaningfully add together semigroups, ideals that aren't just representations of groups, algebras that aren't Lie algebras etc., quantum fields that guide hold to transform as (not just under) Clifford algebras yesteryear themselves, physicists would care. But it would guide hold to work.

Furey's construction doesn't run in addition to 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 piece that it's precisely half – isn't a novel observation (you tin shipping away observe it on this weblog in addition to inwards other papers, I guess) – but the justification is just nonsense. One may define a parity on the adjoint representation of \(SU(5)\) in addition to define the generators of the Standard Model grouping to live on positive (even) in addition to the remaining i to live on negative (odd). The parity volition correctly (multiplicatively) acquit nether the commutator.

So there's some way to semi-naturally segregate the \(SU(5)\) generators into the Standard Model (desirable) ones in addition to the unwanted ones. But that's even so far from having a physical machinery that really breaks \(SU(5)\) to the Standard Model group. She claims to guide hold in addition to non guide hold \(SU(5)\) at the same moment. And she claims to suspension \(SU(5)\) to the Standard Model without the Higgs fields. The Standard Model grouping is picked because it's compatible alongside some of her ideas or constructions. In actual patch theory, the patch content in addition to its consistency follows its ain rules. She should either obey these rules, or justify some replacement for these rules and/or the replacement for the Higgs, symmetry-breaking machinery etc.

But at that topographic point must even so live on a replacement. If it makes whatsoever feel to verbalise almost \(SU(5)\) at whatsoever stage of the construction, at that topographic point has to live on something that breaks it to the Standard Model. In quantum patch theory-based model building, it's ever the Higgs fields. In string theory, i has novel tools such as the Wilson lines roughly cycles of the compactification manifold. Indeed, those are novel physical – in addition to characteristically stringy – mechanisms that may suspension \(SU(5)\) to the Standard Model. But she has nothing. So either she talks almost theories alongside the \(SU(5)\) symmetry or she doesn't. Her ain respond really makes it clear that she doesn't guide hold whatsoever \(SU(5)\). So why is she mentioning it at all?

Lisi's in addition to Furey's are efforts that belong to a much to a greater extent than widespread subcommunity of the crackpot displace – whose members also write papers almost the "graviweak" unification. If yous forgot almost "graviweak" folks, those claim that they may embed the Lorentz grouping in addition to the Standard Model grouping into a larger, simple group. Lisi is basically an representative of that, too. However, the Lorentz grouping acts on the spacetime piece the guess groups don't – they human activity within a point. So they are clearly qualitatively different in addition to cannot live on related yesteryear whatsoever symmetry to each other. In practice, the graviweak people misunderstand the departure betwixt the diffeomorphism symmetry of full general relativity (which moves points to other places) in addition to the local Lorentz grouping (that doesn't). But those are completely dissimilar things. The diffeomorphism grouping of full general relativity is an representative of a generalized "gauge symmetry" but it's only non an representative of the guess symmetry of the Yang-Mills type.

So Lisi, Furey, in addition to many others just don't empathize the mathematical structures (they're using) besides good – in addition to they misunderstand their actual in addition to possible human relationship alongside physics completely. But nosotros read many to a greater extent than articles lionizing these crackpots than nosotros read almost the actual exciting physics research. John Baez really in addition to surprisingly gave a rather reasonable feedback for Wolchover's article, including some technicalities. On the other hand, Pierre Ramond in addition to peculiarly Michael Duff made it audio as her 0-citation meaningless article really revolutionizes particle physics.

In particular, Mike Duff said that it could live on revolutionary in addition to guide hold other adjectives in addition to so on. Oh, really, Mike? Haven't yous noticed that the newspaper is pure crackpottery? Have yous noticed that when yous praise such a thing that makes no sense, you're just absolutely total of šit?

I believe that Duff should empathize it in addition to he has some political reasons why he praises this crackpottery. But most of the laymen – fifty-fifty those who guide hold dedicated some time to trying to superficially follow theoretical or particle physics – just don't distinguish existent physics from crackpottery of Furey's type. So everything that is controlled yesteryear the laymen is to a greater extent than or less guaranteed to gradually supervene upon physicists alongside crackpots. Those who flatter the laymen in addition to those who ain windsurfing boards in addition to yoga mats volition clearly live on preferred (and acquire all the Nobel prizes when the displace conquers the Scandinavian institutions).

The Quanta Magazine is a worrisome borderline example. It's funded yesteryear Jim Simons who has been an fantabulous mathematician closed to theoretical physics – precisely the type of someone who used to empathize (and maybe even so understands?) the criticism I wrote above. But fifty-fifty the Quanta Magazine which is funded yesteryear Simons ended upwards beingness a medium that – whenever it writes almost fundamental physics – lionizes crackpots in addition to attacks actual superlative physicists most of the time. Crackpottery is so much to a greater extent than attractive for a larger issue of readers – in addition to the issue of crackpots is vastly higher than the issue of physicists.

Much of the serious inquiry even so deserves the allotment of fourth dimension in addition to release energy of the researcher in addition to the funding. In a earth increasingly controlled yesteryear stupid laymen, is the top-tier serious inquiry inwards pure scientific discipline sustainable at all? Isn't it guaranteed that crackpots teamed upwards alongside other crackpots are going to overtake non solely the Quanta Magazines but the universities such as Cambridge as well?

I am grateful to guide hold spent at to the lowest degree a purpose of my life inwards an epoch when this wasn't the case, inwards a earth where physicists in addition to crackpots knew their places, the places weren't the same, in addition to where it was possible for a physicist to explicate 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 segmentation algebras

\(\RR,\CC,\HHH,\OO\) guide hold dimensions 1,2,4,8, respectively. But if yous just know the dimensions, you're extremely far from agreement why these are the solely iv segmentation algebras. I won't evidence that they are the solely ones hither but I volition sketch it so that yous empathize some of the beauty if yous focus.

\(\RR\) are the existent numbers such as \(-20.18\). You may add together them, subtract them, multiply them, in addition to split upwards them – unless the denominator is zero. Everyone should larn how to do it from the uncomplicated schoolhouse or elsewhere as a kid.

\(\CC\) are complex numbers of the grade \(x+iy\). The multiplication inverse of that issue is\[

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

\] If yous usage \(i^2=-1\), yous may banking concern check that the right mitt side times \(x+iy\) is equal to one. The complex numbers are groovy e.g. because the \(n\)-th floor algebraic equation has precisely \(n\) roots \(x_i\in\CC\) – some of them may coincide. You may evidence it e.g. yesteryear studying the stage of the polynomial for \(|x|\to\infty\) inwards the complex airplane \(x=r\cdot \exp(i\phi)\in\CC\). The stage \(\phi_p\) of the polynomial winds along a circle roughly the beginning \(n\) times. Because \(\phi_p\) is ill-defined when the polynomial is zero, the points \(x\in\CC\) where the polynomial vanishes allow yous to alter the winding issue yesteryear one, so at that topographic point must live on \(n\) of them.

Sorry if I were besides concise.

Complex numbers are really to a greater extent than natural than the existent numbers. They are absolutely needed as probability amplitudes inwards quantum mechanics. You know, the total probability has to live on fixed but the amplitudes must oscillate. The solely way how they (think almost the release energy eigenstates to brand it simple) tin shipping away oscillate yet save an invariant is for them to live on complex so that the stage oscillates piece the absolute value stays the same.

Also, inwards representation theory of groups, all the representations are complex yesteryear default. The existent or quaternionic/pseudoreal representations may live on defined as initially complex representations alongside some extra construction – a complex conjugation defined yesteryear the antilinear "structure map" \(j\) that doesn't spoil the remaining operations. In this sense, existent in addition to quaternionic representations are as far from the most fundamental in addition to simplest representations – those are the complex ones. This "centrality" of the complex numbers is misunderstood yesteryear all the members of that displace – they either think that \(\RR\) are the most fundamental amidst the three, or the highest-dimensional algebras such as \(\HHH,\OO\) are the fundamental edifice blocks.

The quaternion \(z\in\HHH\) is a issue of the form\[

a+ib+jc+kd

\] where \(i,j,k\) are 3 imaginary units obeying\[

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

\] So the unit of measurement \(i\) may live on identified alongside the complex imaginary unit, but so tin shipping away \(j\) or \(k\). And \(ij=k=-ji\) in addition to cyclic permutations define the totally associative but maximally non-commutative multiplication tabular array of the 3 imaginary units. The three-dimensional infinite generated yesteryear \(i,j,k\) may live on rotated yesteryear \(SO(3)\) transformations – the quaternions may live on \(SO(3)\)-renamed – so that the multiplicative relationships betwixt them remain the same. We say that \(SO(3)\) is the automorphism grouping of the quaternions.

Why is it a segmentation 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" \(b,c,d\) in addition to divided it yesteryear 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 damage \(a^2,b^2,c^2,d^2\) are patch at that topographic point if yous usage the distributive police draw – because \(i^2=-1\) in addition to the minus sign cancels because at that topographic point are contrary signs inwards forepart of \(b\) inwards the 2 damage etc.

And all the mixed damage 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 damage cancelled because at that topographic point were explicitly contrary signs inwards the 2 terms. The \(bc\)-like damage cancelled because \(ij=-ji\) – because the imaginary units anticommute. There are 3 imaginary units because they may live on used as the 2 factors in addition to i lawsuit inwards a multiplication table. You won't observe a larger associative segmentation algebra.

If yous 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 even so live on inverse but such an inverse wouldn't live on unique or the multiplication wouldn't live on associative. An fifty-fifty simpler sick example: if yous defined all the products of imaginary units to live on \(\pm 1\), piece preserving the anticommutativity, whole families of such "broken quaternions" would acquit as the same issue nether 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 segmentation algebra. They guide hold 7 imaginary units, allow me telephone telephone them \(i,j,k,A,B,C,D\). These 7 units may live on written using products of 3 basic ones, \(i,j,A\), in addition to parentheses. The multiplication tabular array of the 7 units is such that the squares such as \(B^2\) are equal to minus one; in addition to the production of 2 dissimilar units is anti-commutative, e.g. \(CD=-DC\), just similar for the quaternions.

On superlative of that, the production of 3 units such as \((ij)A\) is maximally non-associative whenever it differs from \(\pm 1\). It agency that e.g. \[

(ij)A = -i(jA).

\] The permutation of 2 dissimilar units inwards a production flips the sign; in addition to the rearrangement of the parentheses inwards products such as i higher upwards flips the sign, besides (whenever the production differs from \(\pm 1\)). In some binary counting, the basic imaginary units \(i,j,A\) may live on associated alongside binary numbers \(100,010,001\) in addition to the 7 units, given yesteryear 3 bits, are multiplied yesteryear adding the 3 bits modulo two. However, the precise sign matters in addition to it's such that yous impose the "maximal non-commutativity" in addition to "maximum non-associativity".

That's a groovy algebraic construction where the inverse of a nonzero issue exists in addition to is unique. The automorphism grouping isn't the whole 21-dimensional \(SO(7)\) inwards this case. It's just a grouping that remembers some relationships betwixt the 7 imaginary units that are encoded inwards the 3 bits – bits that may live on added separately. So the symmetry grouping ends upwards beingness the 14-dimensional \(G_2\) only. Only 2/3 of the generators are even so symmetries of the multiplication table. Why 2/3?

Pick i of the 7 units, it doesn't affair which one, for representative \(i\). The remaining half-dozen units \(j,k,A,B,C,D\), may live on divided to pairs that, along alongside the \(i\), may build the quaternionic triplets:\[

ijk, iAB, iCD

\] Now, \(SO(7)\) is generated yesteryear 21 generators of rotations of 2-planes, such as the rotation of \(jk\) into each other, \(AB\) into each other, in addition to \(CD\) into each other. In \(SO(7)\), at that topographic point would live on 3 parameters (angles) for these 3 generators. But solely if \(\phi_1+\phi_2+\phi_3=0\), the octonionic construction – including the right multiplicative human relationship alongside \(i\) – is preserved. You may run into it e.g. yesteryear realizing that \(i,j,A\) are "fundamental" but \(C,D\) may already live on written inwards damage of products of \(i,j,A\) alongside parentheses – so whether \(C,D\) guide hold to rotate to save the construction of the octonions is already determined yesteryear thee rotations involving \(i,j,k=ij,A,B=iA\).

The octonionic multiplication tabular array may live on reconstructed from remembering 7 "triplets that embed quaternions into octonions". These 7 triplets incorporate \(7\times 3=21\) pairs of octonionic imaginary units which is all unordered pairs of octonionic units. That's why the issue of imaginary units inwards a construction that industrial plant must live on \(1+2\times 3\) because the combination issue "7 select 2" is \(7\times half-dozen / 2\times 1\) in addition to it must live on the same as \(7\times 3\). That's why at that topographic point is no 10-dimensional, 16-dimensional, 32-dimensional, or 64-dimensional extension of the octonions.

Even if yous don't fully empathize what I just wrote, I desire to guide a to a greater extent than full general point: the exceptional traits of the quaternions in addition to peculiarly octonions require that yous analyze the multiplication tabular array of the imaginary units, including the precise signs of the products (which cry upwards whether the structures are commutative and/or associative). If yous don't analyze the multiplication tables of your 4- or 8- or 16- or 32- or 64-component "numbers", in addition to then yous are non working alongside anything similar quaternions or octonions!

You are just talking almost some generic vector spaces whose dimensions are powers of 2 in addition to yous are doing so because yous guide hold heard that it's deep. But without the multiplication tabular array (and/or without the anticommutators of the SUSY generators – powers of 2 also appear inwards supersymmetry) – in addition to without the additional constraints that the tabular array imposes on the usage of these segmentation algebras – at that topographic point is absolutely nil deep almost the vector spaces alongside these dimensions! And that's the representative of Furey et al. She just doesn't acquire what makes the octonions deep. She solely uses the give-and-take "octonion" to brand her claims audio sexy – but her observation is nil to a greater extent than than "the issue of fields inwards the Standard Model is smaller than a might of two".