Quaternion Higgs and the LHC

[S.Daedalus]

As everyone knows, since the fourth of July, the family of elementary particles has been re-united with its long-lost son, the Higgs boson. Of course, as every discovery, so this one, too, serves to open up further questions. The first one that presents itself is certainly: So, is this the Higgs? Really? I mean, really really? It could always be something else, something strange, something no-one had yet on the cards. But the agreement with standard predictions is too good (almost conspicuously so) to really make this a likely option.

I'm more interested in the question: so, what kind of Higgs is this? And to this end, I would like to review an old, almost forgotten, proposal by Finkelstein, Jauch, Schiminovich, and Speiser, dating from 1963 (yes, the year before the famous papers by Higgs, Guralnik, Hagen and Kibble, and Brout and Englert; but a year after Anderson's).

Already in 1962, they had proposed a concrete realization of quaternion quantum mechanics (QQM) [1]. The problem of formulating a theory of quantum mechanics using quaternions essentially is that due to their non-commutativity, the definition of the tensor product becomes ambiguous, and hence, treatment of multi-particle systems is difficult to do in a consistent way. Essentially, they solved this problem by requiring that the physics should be unaffected by this ambiguity, i.e. invariant under transformation of the form $\psi \to q \psi q^{-1}$, where q is some arbitrary unit quaternion. This makes any theory of QQM into an SU(2)-gauge theory, since the unit quaternions is just $S^3$, and the quaternion multiplication (under which the set is closed) gives it the requisite group structure. They also showed that the appropriate Schrödinger equation takes the form $H\psi = \eta\dot{\psi}$, with $\eta$ being a pure imaginary ($\eta^2=-1$), but otherwise arbitrary, quaternion. However, this Schrödinger equation is only 'q-covariant' if $\eta$ itself transforms like a dynamical field.

In order to formulate the theory, in [2] they devise the theory of quaternion parallel transport to find a suitable covariant derivative, which proceeds much as in the Yang-Mills case (see also [3], [4] for a more recent explanation of the techniques, albeit in a different context). From this, they build a 'q-curvature' $K_{\mu \nu}$, and propose the Lagrangian:
$$L=\frac{1}{4\alpha} K_{\mu \nu} K^{\mu \nu} + \frac{1}{2\beta}(D^\mu \cdot \eta)(D_\mu \cdot \eta)$$

From this, the equation of motion for the $\eta$ follows: $\frac{1}{\beta}D^\mu D_\mu \eta + \lambda\eta = 0$, where $\lambda$ is a Lagrange multiplier arising from the constraint $\eta^2=-1$. This looks a lot like the Klein-Gordon equation for a particle with mass $m=\sqrt{\beta\lambda}$, though they don't explicitly say so. This field naturally has a VEV, as it is constrained to be of unit modulus.

Their further analysis of the $\frac{1}{4\alpha} K_{\mu \nu} K^{\mu \nu}$ term shows that it yields a massless, uncharged boson that fulfills Maxwell's equations, and two charged bosons of mass $m=\sqrt{\frac{\alpha}{\beta}}$ (which I'll tentatively identify as the W bosons). Singling out one special direction for $\eta$ effectively breaks the symmetry from SU(2) to the U(1) of electromagnetism.

It seems to me that they get very much for very little -- symmetry breaking, massive vector bosons, a massive scalar with a nonvanishing VEV, all just from their principle of general q-covariance. They don't get the $Z^0$, but I think that could be fixed by considering an extra U(1) added in by local complex transformations.

The question is, of course, could their $\eta$ field be what the LHC's seeing? Or does it lead to some unacceptable phenomenology? Are there ideas even correct, or is there something obviously wrong with them that I'm just not seeing?

I'm very much not a high energy theorist, so it might be that these things are obviously obsolete for some reason, and my understanding is just insufficient to see this. But any way, I'd be very thankful for any and all input!

[1] D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, J. Math. Phys. 3, 207 (1962)
[2] D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, J. Math. Phys. 4, 788 (1963)
[3] S.P. Brumby, G.C. Joshi, Global Effects in Quaternionic Quantum Field Theory, arXiv:hep-th/9610033v1
[4] S.P. Brumby, B.E. Hanlon, G.C. Joshi, Implications of Quaternionic Dark Matter, arXiv:hep-th/9610210v1

 

[arivero]

Also, it predates Connes :-D

 

[Hans de Vries]

From this, the equation of motion for the $\eta$ follows: $\frac{1}{\beta}D^\mu D_\mu \eta + \lambda\eta = 0$, where $\lambda$ is a Lagrange multiplier arising from the constraint $\eta^2=-1$. This looks a lot like the Klein-Gordon equation for a particle with mass $m=\sqrt{\beta\lambda}$, though they don't explicitly say so. This field naturally has a VEV, as it is constrained to be of unit modulus.

Makes me wonder...

They get the extra Lagrange multiplier term $\lambda\eta$ in the equation of motion
because of the Euler Lagrange differentiation of a field with a constraint.

Now, the Dirac field has a constraint as well because the four-momentum
is encoded twice, in the co-variant derivatives as well as in the vector
current. Has anyone ever tried to figure out a corresponding Lagrange
multiplier term in the corresponding equation of motion?

The Q-covariance paper can be found here also.

In the acknowledgments it says: "C.N. Yang contributed much constructive
criticism" so I assume the non-Abelian connection they worked out is sound.

Hans.