There is some renewed activity in the field. Dray, Manogue and Wilson have a paper out on the arXiv few days back: Octions, an E_8 description of the standard model. They will give a talk on this coming Monday, you can find out more about it here:
The typical goal here is to show that the standard model gauge symmetries, and the quarks and leptons, are embedded in the larger of the exceptional groups. Most of the time, the attention is focused on a GUTS type scheme, with Lorentz symmetry included, but not gravitation.
I have landed in this octonion space from very different considerations. Those of Quantum Foundations.
There must exist a reformulation of quantum theory which does not make any reference to classical space-time labelled by four real numbers. Such a space-time exists if and only if the universe is dominated by classical macroscopic objects, just as today's low energy universe is. Such classical objects are however a limiting case of quantum systems and therefore, in invoking spacetime, quantum theory depends on its own classical limit. This is approximate, no matter how well it agrees with current experiments. We can in principle imagine a low energy universe devoid of classical objects, and hence devoid of classical spacetime. How to describe the dynamics then?
We can also ask what is the ground state of low energy quantum gravity? It is not 4D classical spacetime. This latter is the ground state of classical general relativity sourced by classical matter fields. When the sources are quantum fermions, they have a non-trivial ground state and a zero point energy - this cannot give rise to a spacetime with a point structure labelled by real numbers. We must resort to a coordinate geometry with non-commuting coordinates. And write the rules of a (pre-) quantum theory on this non-commuting coordinate geometry. On general grounds, the coordinates themselves can be expected to be complex, not real. From here the familiar quantum theory on classical spacetime must emerge as an approximation.
One way to construct the pre-dynamics is to raise the classical dynamical variables to the status of operators/matrices, but not impose the Heisenberg commutation variables. Instead we have a matrix-valued Lagrangian dynamics, and an accompanying Hamiltonian dynamics, from which the Heisenberg algebra and Heisenberg equations of motion are emergent in an approximation.
In this Lagrangian dynamics, it should be possible to define spin as a dynamical variable corresponding to an angular canonical variable. Such an angle clearly cannot be a part of the familiar 4D spacetime nor of its non-commuting version. The doubling of the non-commuting spacetime coordinates from four dimensions to eight is strongly indicated. This is (one of the many reasons) as to why we land up with the octonions. Octonions label the (non-commuting) coordinate geometry of the pre-quantum theory, and all the work being done on octonions and the standard model is seen from this light. The standard model forces as well as gravitation are symmetries of this physical space, which is equivalent to ten dimensional Minkowski spacetime SO(9,1). The octonionic space is like a square-root of this 10D Minkowski spacetime, analogous to tetrads, only now non-commuting. The ground state of quantum gravity is this octonionic space. Quantum gravity and quantum standard model forces are switched on as the curving of this octonion geometry. In the same way that the curving of 4D classical Minkowski spacetime is switched on by classical gravity through the metric and the Riemann tensor.
However, the geometry of octonions has a very fundamental difference from that of Minkowski spacetime. Unlike in the latter, fermions cannot have arbitrary properties in octonion space. Even before the curving of octonion geometry is turned on by switching on interactions, the non-commutative geometry already dictates properties of fermions. There are correct number of quarks and leptons. Electric charge is quantised, just as observed. These properties have been proved earlier by other researchers, but viewed by them in an algebraic context [SM and GUTSs in terms of the algebra of the octonions]. They do not interpret their results as consequences of this new spacetime geometry of the octonions. But such a spacetime interpretation is most essential - it adds dynamics and quantum theory to the algebraic approach. Not only does spacetime tell matter how to move; it also tells matter what to be. This is a natural extension of general relativity, and also of quantum theory - in the latter theory, quantisation of energy levels e.g. is dictated by the dynamics. So it is not a surprise if a non-trivial coordinate geometry dictates particle properties.
The standard model is a mystery when looked at from a 4D Minkowski spacetime labelled by four real numbers. But not when viewed from the `square' root of 10D Minkowski spacetime - the octonionic space.
The extra dimensions are complex and never compactified. Quantum systems probe these extra dimensions, always. Only classical systems live in 4D. The curving of the higher dimensions represents the standard model forces - equivalence principle is not obeyed because the curving is proportional to electric charge, not to mass.
Excellent.
ReplyDeleteThank you.
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