- Is a tabletop search for Planck scale signals feasible?
Jacob D. Bekenstein
The idea is roughly the following: Take a single photon and spread its wavefunction by suitable lenses, then let it hit some macroscopic solid block, for example a crystal. Focus the photon and detect it.
Since the crystal has a refractive index, the photon has to discard momentum into it. This momentum will be evenly spread into the crystal, distributed by phonons, and be returned to the photon upon exit. Essentially, the block reacts not like single atoms but in one piece (though it cannot instantly do so, the distribution of momentum must take a finite amount of time).
If you give the crystal a momentum for the duration of the photon’s passage, it will move, but since it’s macroscopically heavy, it will move only a tiny distance. If you look at the shift of its center-of-mass, the distance it will move scales with the energy of the incoming photon over the mass of the block.
Bekenstein puts in the numbers and finds that with presently available technology, the energy of a single photon could be so tiny that the distance the crystal moves would have to be smaller than the Planck length. This, he argues would “occasionally be at odds with the non-smooth texture of spacetime on Planck scale.” If that is so, the photon would not be able to transverse the crystal, leading to an unexpected, and observable, decrease in the transmission probability.
He also estimates sources of noise that could move the block oh-so slightly and affect the probability of the photon trespassing, thus rendering the outcome inconclusive. Bekenstein argues that by cooling the block to some Kelvin, which is cold indeed but still feasible, the noise could be kept under control. This might seem implausible at first sight, but note that the thermal noise for the motion of the center-of-mass itself is not the problem because the photon spends only a very short time inside the crystal. The relevant question is whether the center-of-mass moves in that short duration.
So far, so brilliant. The proposed experiment is an excellent example for a model-independent test. It is so model-independent in fact that I don’t know which model could be constrained by it.
The usual expectation from Planck-scale fluctuations is that they lead to a position uncertainty that cannot become smaller than the Planck length. This does not forbid you to move an item by distances less than the Planck length, it just tells you that the position of the crystal wasn’t defined to a precision better than the Planck length to begin with.
Now, if space-time was a discrete regular lattice with Planck-length spacing then you could not move the crystal, as a rigid block, by anything shorter than the Planck length. Already if the lattice isn’t regular, this is no longer true. But even if the lattice was regular, the crystal would have to be very rigid indeed, so as to not allow any relative shift among atoms that could account for the motion of the center-of-mass. For example, if your block has a number of particles about Avogadro’s number, 1023, and you move one out of 1015 of these atoms by a distance of 10-20 m (that’s less than the size of a proton and less than the LHC can probe), you’d move the center of mass by about a Planck length. Now I don’t know much about crystals, but it seems quite implausible to me that the effective description of phonons on the lattice should be sensitive to such tiny shifts at all (even worse if the block is not a crystal but some amorphous solid).
Besides this, I don’t understand how the “rejection” of the photon should come about if one took the path integral of all possible trajectories and scatterings in the crystal, none of which is sensitive to Planck scale effects.In summary: The proposed table-top argument tests a quantity, the shift of the crystal’s center-of-mass, which is of the order Planck length. It is unclear however if there is any plausible model for the phenomenology of quantum gravity that would be constrained by this a measurement. Is a tabletop search for Planck scale signals feasible? Maybe. Is it possible with the proposed experiment? Probably not. Does it have to do anything with Planck mass black holes? No.