LQG builds on the SU(2) gauge invariant connection introduced by Ashtekar, which in turn is related to the triad [drei bein]. The setting is the ADM 3+1 formulation of general relativity, where the three-metric / triad is the configuration variable and the canonical momentum is made from the corresponding connection.
In the octonionic theory, would-be-gravity is the right-handed sector of the standard model, including three sterile neutrinos, and possessing the symmetry group SU(3)_grav X SU(2)_R X U(1)_grav The standard model is the left-handed sector SU(3)_c X SU(2)_L X U(1)_em Prior to the L-R symmetry breaking, which is also the electroweak symmetry breaking, there is unification of gravity and the standard model [also of electro and weak] in the framework of E_6 and E_8.
It is tempting and plausible that in spirit this SU(2)_R should be thought of as being the same as the SU(2) of LQG. Both describe gravity and Lorentz-invariant spacetime can be built by extending to SL(2,C). The SU(2) rotations are in isospin 3D space for the weak force, and in our familiar physical 3D space in the gravity case. There is a strong suggestion of gravity weak unification in the SU(2)_L X SU(2)_R ~ SO(4) sense, also suggested by our recent split biquaternion construction: SL(2,H) ~ SO(1,5) and writing of Cl(3) as a direct sum of two Cl(2)s - one for weak force, one for gravity. The split quaternion structure makes the weak interaction subtly different from gravity [only space, no time] and yet close enough to suggest gravi-weak unification in 6D spacetime.
If the above thoughts are correct, we see LQG nicely unifying with the standard model, and now also seen as a sub-structure in this revived description of string theory, with chiral fermions nicely added on to LQG. Other researchers have already pointed to the connection between Clifford algebras, fermions, something known as braids, and LQG. The present set of thoughts strengthens that connection. It also suggests that quantum gravity violates parity, and is right-handed, being the RH counterpart of the LH weak force. We also understand that parity violation is a consequence of symmetry breaking, and prior to that the parity symmetry is restored.
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