Dear Giovanni,
This is copied on my page as well.
I just started reading Relative locality in a quantum spacetime and the pregeometry of _-Minkowski http://arxiv.org/pdf/1206.3805v1.pd. You seem to be pointing to a similar end. Noncommutative geometry and Hopf algebras are a main tool in the work with Yangians. I will write more when I complete reading your paper.
Equation 1 is interesting, for it proposes a noncommutative relationship between time and the spatial coordinates. This in my opinion harkens back to an old argument by Bohr. In 1930 there was a famous Solvay conference where Einstein and Bohr sparred over the reality of quantum mechanics. Einstein was convinced of reality and locality and argued staunchly for an incompleteness of quantum mechanics. Quantum theory could only be made complete if there are some hidden variables that underlay the probabilistic, nonlocal quirky aspects of quantum mechanics. At the 1930 Solvay conference Einstein proposed an interesting thought experiment. Einstein considered a device which consisted of a box with a door in one of its walls controlled by a clock. The box contains radiation, similar to a high-Q cavity in laser optics. The door opens for some brief period of time t, which is known to the experimenter. The loss of one photon with energy E = ħω reduces the mass of the box-clock system by m = E/c^2, which is weighed. Einstein argued that knowledge of t and the change in weight provides an arbitrarily accurate measurement of both energy and time which may violate the Heisenberg uncertainty principle ΔEΔt ~ ħ.
Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock's new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is 1 - 2GM/rc^2, where a displacement by some δr means the change in the metric term is ~ (GM/c^2r^2)δr. Hence the clock's time intervals T is measured to change by a factor
T --> T sqrt{(1 - 2GM/c^2)δr/r^2} ~ T(1 - GMδr/r^2c^2),
so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the Δp ~ ħΔr < TgΔm, where g = GM/r^2. Similarly the uncertainty in time is found as Δ T = (Tg/c^2)δr. From this ΔT > ħ/Δmc^2 is obtained and the Heisenberg uncertainty relation ΔTΔE > ħ. This demands a Fourier transformation between position and momentum, as well as time and energy.
Consider an example with the Schwarzschild metric terms. The metric change is then ~ 1x10^{-12}m^{-1}δr, which for δr = 10^{-3}m is around 10^{-15}. Thus for a open door time interval of 10^{-2}sec, the time uncertainty is around Δ t ~ 10^{-17}sec. The uncertainty in the energy is further ħΔω, where by Fourier reasoning Δω ~ 10^{17}. Hence the Heisenberg uncertainty is ΔEΔt ~ ħ.
This argument by Bohr is one of those things which I find myself re-reading. This argument by Bohr is in my opinion on of these spectacular brilliant events in physics.
This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein's box as a black hole with mass m. The quantum periodicity of this black hole is given by some multiple of Planck masses. For a black hole of integer number n of Planck masses the time it takes a photon to travel across the event horizon is t ~ Gm/c^3 = nT_p, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is
ΔT ~ Tg/c(δr - GM/c^2),
as measured by a distant observer. Similarly the change in the energy is given by E_2/E_1 = sqrt{(1 - 2M/r_1)/(1 - 2M/r_2)}, which gives an energy uncertainty of
ΔE ~ (ħ/T_1)g/c^2(δr - GM/c^2)^{-1}.
Consequently the Heisenberg uncertainty principle still holds ΔEΔT ~ ħ. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies Δω ~ sqrt{c^5/Għ}, which is the Planck frequency.
The uncertainty in the ΔE ~ ħ/Δ t does have a funny situation, where if the energy is Δ E is larger than the Planck mass there is the occurrence of an event horizon. The horizon has a radius R ~ 2GΔE/c^4, which is the uncertainty in the radial position R = Δr associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have
ΔrΔt ~ (2Għ)/c^4 = L^2_{Planck}/c.
So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.
Cheers LC
