So, I've been thinking about, and reading up on the history of, this RCHO business on and off for a few months, and I'd like to take a new look at it...
First of all, Geoffrey Dixon has a new book out ('
Division Algebras, Lattices, Physics, Windmill Tilting'), but I haven't gotten my hands on it (yet).
Also, it's been said that it's hard to get the Higgs in the division algebra approach, but it seems that it was actually there, in some sense, in one of the earliest attempts (which I don't think has been mentioned in this thread so far) by Finkelstein, Jauch, Schiminovich and Speiser: they used a '
principle of general q-covariance', i.e. invariance under transformations of the form [itex]\Psi\to q\Psi q^{-1}[/itex], and find a massless boson (photon), and two additional massive, charged vector bosons ([itex]W^{±}[/itex]). Tony Smith calls it:
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the first paper [...] that used Quaternionic SU(2) symmetry to describe the mechanism whereby two charged SU(2) bosons get mass, and the electromagnetic field is unified with the SU(2) bosons. Their paper effectively did the "Higgs Mechanism" before Higgs, and did ElectroWeak Unification before Glashow,Salam, and Weinberg
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They don't find the [itex]Z^0[/itex], perhaps related to the fact that they only use SU(2), rather than the full U(1)xSU(2) (i.e. [itex]ℂ\otimes \mathbb{H}[/itex]).
The paper '
Octonionic Structures in Particle Physics' by Gürsey gives some fascinating insight into the history of the subject, in particular how octonionic extensions of quantum mechanics were originally looked into in order to incorporate the new phenomena of nuclear physics before the advent of gauge theory. Some other interesting papers (some of which will probably have been mentioned already):
'
Remark on the Algebra of Interactions' - A. Pais (an early one, 1961)
'
Octonionic Quark Confinement' - H. Ruegg (related to the idea that the non-associativity of the octonions ensures the unobservability of quarks; in a sense, a special imaginary unit is picked out, which gives us our usual quantum mechanics, and has the side effects of breaking [itex]G_2[/itex] to SU(3), and bringing [itex]SL(2,\mathbb{O}) \simeq SO(9,1)[/itex] down to [itex]SO(3,1)[/itex]. Under this SU(3), the split-octonion units [itex]u_0 = \frac{1}{2}(e_0+ie_7)[/itex] and [itex]u_0^* = \frac{1}{2}(e_0-ie_7)[/itex] transform as a singlet and antisinglet, while [itex]u_i = \frac{1}{2}(e_i+ie_7)[/itex] and [itex]u_i^* = \frac{1}{2}(e_i-ie_7)[/itex] (i = 1,2,3) transform as a triplet and antitriplet -- a lepton and quark, together with antiparticles. (This is of course the familiar Günaydin-Gürsey scheme.) So this gives us 'half a generation' of fermions living in Minkowski space.)
'
SO(8) Color as a Possible Origin of Generations' - Z.K. Silagadze (discusses a possible extension of the above to incorporate a full generation, then uses SO(8)'s triality to find the observed three.)
'
Algebraic Realization of Quark-Diquark Supersymmetry' - S. Catto (related to my question above; 'composite' SUSY from octonionic color algebra --
arivero, I think, is familiar with this)
'
Quaternion Higgs and the Electroweak Gauge Group' - DeLeo, Rotelli (another look at Higgsing from a div. alg. perspective)
'
Derivation of the Standard Model' - Dixon (I
think this is the first paper where Dixon lays out his model, or its basics, completely; of course, the full treatment is available in his first book (most of which I don't understand, unfortunately))
'
Algebraic Approach to the Quark Problem' - Casalbuoni et al (a somewhat alternative approach to that of Günaydin/Gürsey, explaining the quark confinement through the realization that in a path-integral quantized version of their theory, only color singlet states can propagate)
There's many more, but this post is already getting rather lengthy, and there's one last, perhaps somewhat too out-there, thing that I've come across that I wanted to share. In entanglement theory, there's an interesting connection between two- and three-qubit entanglement and the Hopf fibrations, and hence, the division algebras, laid out in the paper '
Geometry of the Three-Qubit State, Entanglement, and Division Algebras' by Bernevig and Chen, and summarized in the slides to
this talk by Chen. The idea is, basically, that the Hopf map is sensitive to the entanglement properties of two- and three-qubit systems: the state space of one qubit is given by the first Hopf fibration, [itex]S^1 \hookrightarrow S^3 \to S^2[/itex], where the [itex]S^3[/itex] is the qubit state space, the [itex]S^1[/itex] fiber is the global phase, and [itex]S^2[/itex] is the Bloch sphere. Analogously, the state spaces of two- and three-qubit systems, [itex]S^7[/itex] and [itex]S^{15}[/itex] can be related to the second and third Hopf fibrations, [itex]S^3 \hookrightarrow S^7 \to S^4[/itex] and [itex]S^7 \hookrightarrow S^{15} \to S^8[/itex]. Of course, there are no more Hopf fibrations after that, because there are no more division algebras beyond the octonions. In each case, the fiber is the unit sphere of the complexes, quaternions, or octonions respectively, while the base spaces are the respective projective lines. This map is entanglement sensitive in the sense that if, for instance, the three-qubit stat is biseparable, it maps only into the complex subspace of the octonionic projective line. Furthermore, each base space can be seen as containing one qubit, plus the entanglement degrees of freedom, while the rest of the state (two qubits for the three qubit case, one for two qubits) lives in the fiber.
Now this is quite a surprising way for the division algebras to turn up in entanglement! In particular, this appears to allow us to consider a two-qubit state as a single quaternionic qubit, and similarly, a three-qubit state as a single octonionic qubit (essentially via the Cayley-Dickson construction: if two qubits are parametrized by the complex numbers [itex]\alpha_k = a_k + ib_k[/itex], k = 1...4, then it can be parametrized by the quaternions [itex]q_1 = (a_1 + ib_1) + j(a_2 + ib_2)[/itex] and [itex]q_2 = (a_3 + ib_3) + j(a_4 + ib_4)[/itex], where [itex]j^2=-1[/itex] and [itex]|q_1|^2+|q_2|^2=1[/itex]; an analogous construction works for the three-qubit case). (The connection between Hopf fibrations and qubits over division algebras was also noticed in the paper '
Extremal Black Holes as Qudits', by M. Rios who I think posts here occasionally.)
But this brings us right to the quaternionic and octonionic extensions of quantum mechanics discussed earlier! So, could there be a connection? (Probably not, but it's just the kind of '...but what if?'-thing that sometimes goes through my head at night...) In particular, since I've always liked the 'spacetime is made of qubits'-idea from Weizsäcker's ur-theory, to look for 'inner space' in entanglement between urs seems somehow appealing to me... But I realize this is just far-out speculation.
In any case, this has gotten somewhat lengthy, so thanks to anyone who persisted to this point; I'd be very glad for any comments, especially elucidations of things I don't quite grasp yet (my background is not in particle physics, so this is not really my home field...).
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of the sBootstrap, but it is fringe physics.
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