A Different Approach (?) Collaboration with D Sudarsky

1. A Different Approach (?)Collaboration with D Sudarsky • Let us take up the notion that space-time contains some granular/discrete aspect with characteristic scale given by MPlanck • The lesson from the previous studies is that such structure, if exists, can not lead to breakdown of Lorentz Invariance. • It is of course hard to envision something like that while thinking classically about a space-time. But what is space-time in a quantum world? We do not know!

2. When faced with total Ignorance proceed by analogy • Consider a physicist trying to get information about the granular structure of a crystal. • Assume that the fundamental crystal lattice has some symmetry, say cubic. • We know that the fundamental granular structure will not become manifest through a breakdown of the symmetry if one studies a macroscopic, but similarly cubic crystal. • However, if the macroscopic crystal does not share the lattice symmetry, we might be able to detect it.

3. Phenomenology • The results so far indicate that the granular structure, if any, would have to respect the Lorentz symmetry. • This would explain why we have not seen the “expected” violation of L.I. : There is none. • So what would be the signature of the discrete structure of space time? • In analogy with the crystal, we consider a macroscopic space-time that is Not Fully Lorentz Invariant.

4. There lies the hope of seeing something • A space-time that is not Minkowski on an extended realm could exhibit the mismatch between the symmetry of the fundamental structure and that of the macroscopic domain. • The departure from Minkowski space-time is characterized by the Riemman. tensor: R .

5. The effects in question should represent a coupling of Riemman to ordinary matter. • Furthermore: part of Riemman is determined by the local T : R - (R/2 )g =8G T and thus would look like a self coupling. That is not the most interesting. • We need to consider couplings of the Weyl tensor W (Riemman without R or R).

6. The direct approach indicates that all such couplings are highly suppressed • A more open minded approach exists: Extract from Weyl some aspects and couple them to matter: One possibility: Find eigenforms and eigenvalues of Weyl: W  X=  X And use these objects to couple to matter fields.

7. Consider the coupling to fermions • The least suppressed term is : L =  (1/MPlanck) ∑a a (a)     Can this ideas be experimentally explored? In the non-relativistic regime the dominant new term in the Hamiltonian for the spin 1/2 particle H=  (1/MPlanck) ∑a a0i(a) i This looks like a coupling of the spin with magnetic field. Can it be distinguished from that ? Could the suppression be 1/M instead of 1/Mplanck ?

8. Comments: • Magnitude: is of the order of the gravitational gradients GM/r3. • A non zero 0iimplies a special direction and a time asymmetry (rotation). • On the other hand the effect should be associated with the gravitational sources and thus susceptible of control. • It should affect even particles with no electromagnetic couplings like Neutrinos.

9. Other comments • Something like this could originate from some QG aspects of holonomic degrees of freedom (i.e. LQG (?)), or from QG bounds in the sectional curvatures (?)... • It is worth noting that no real tests of Quantum Mechanics in the presence of curvature exists up to date!

10. THE CHALLENGE • According to the GR view, COW experiment & the experiments with neutrons at Grenoble are ``just” tests of QM in a non inertial frame… but gravity lies only in the curvature, i.e. in the fact that inertial frames at different events do not coincide. • Other tests, such as those using Quantum Gravity gradiometers, rely only on quantum aspects that are purely local ( no quantum description needed in the domain covered by different Local Inertial frames). • The challenge, from my point of view, is to detect curvature (i.e. tidal forces) using quantum aspects of matter.

11. Further Ideas • Point particles move along geodesics, but how do extended/quantum objects move? • What is the operational definition of geodesic, when we test the world with extended objects? The path of the center of mass? NO! • What are in principle the geodesics of the geometry, or the geometry itself, when all we have to explore it, are quantum particles ( in fact, quantum fields)? • There is too much we do not understand! We need to be open minded when we explore!