Tests

There are some relevant differences to the Penrose-Diosi (hereafter PD) model. ´

In the PD model, the deviation from the standard Schrodinger equation comes from a noise-kernel that scales ¨ with the gravitational self-energy of the mass-density.

The strength of this effect first grows for small separation, as long as the wave-packets overlap. But once the wave-packets are to good approximation orthogonal, the strength of the deviation drops with the inverse of the distance between them. (If the branches are not each internally coherent, a residue remains for each branch.)

In the model proposed here, in contrast, the contribution to the Penrose phase grows with the dislocation of the wavepackets, but once the wave-packets are orthogonal it becomes constant and to good approximation independent of the distance.

This is basically because the residual that I am using measures distance in Hilbert-space and not in spacetime. The other difference between the models is what I already mentioned in III A: The effect in this new model scales with the sum of the gravitational potentials, whereas in the PDmodel it scales with the variance of the gravitational selfenergy. In both cases though one has to avoid integrating over

point-like sources because that would bring in a divergence from the Newtonian potential. I will therefore here, as usual, assume that the mass-density is smeared out over a radius that roughly scales like the width of the wave-packet.

A. Estimates

The great benefit of this model is that since the collapse is caused by a known entity—gravity—the process has no free parameters. Let us therefore make some order-of-magnitude estimates for when the effects of the model proposed here should become relevant. In the best case (near a maximally localised particle), we can estimate Φ12 ∼ (m/mp) 2 , where mp is the Planck mass. This gives us a Penrose phase ∼ τm3/m2 p .

Our task is now to find cases where the collapse induced by this phase is not masked by decoherence. For elementary particles, the Penrose phase is ridiculously small. For an electron, for example, its effect will become noticeable at a time of approximately

τe ∼ m2 p m3 ∼ 7 × 1023 sec . (45)

Even for a heavy nucleus with, say, 100 nucleons, we have m ∼ 100 GeV and τ ∼ 108 seconds. These effects will be masked by environmental decoherence easily. What about quantum computers? At first, this sounds like a promising idea because quantum computers are designed to produce massive amounts of entanglement while keeping decoherence at bay. However, in a quantum computer the masses that move are tiny. Consider for example a superconducting circuit that will move something like Ne = 106 electrons. To get a collapse time of ∼ 1 second, we would thus optimistically need ∼ 1017 qubits, if we were to create a fully entangled state. To take the other extreme, if we just have a product of qubits, each of which is in a superposition (of electron locations), then we have to use the √ n scaling from subsection II D and we need ∼ 1035 qubits. In a realistic setting the number would be somewhere in between but probably closer to the latter estimate. In any case, clearly, this is not going to happen any time soon. The situation is even worse for other types of qubits, such as ion traps, neutral atoms, or photon states, because they dislocate even smaller masses. This makes my estimate considerably less optimistic than those put forward in [28, 29]. What we need is instead a lot of mass that moves coherently, so that we can witness its collapse. This is the bad news. The good news is that this mass does not need to move by a lot, it just needs to move enough so that we can consider the wavefunctions as sufficiently displaced for our estimate to hold. This is not much. We just need to displace e.g. atomic nuclei by more than the typical diameter of the nucleus, that is, some femtometres.

B. Existing Proposals

Experimental setups which are in the parameter range of testing the model proposed here (maybe not so surprisingly) are attempts to probe quantum gravity by bringing small objects (typically made of silicon) into a superposition of two coherent oscillation states [30–32]. To get a collapse time of ∼ 1 second, we would need to displace a total mass of about 0.2 nanogram (or, equivalently, a mass of about 1 ng with a coherent fraction of 0.2). The model proposed here predicts that superpositions which exceed this bound from the Penrose phase do not exist, or they cannot stay coherent, respectively.

The decoherence time of these objects is currently in the ms range [33], so either the masses of the oscillators need to further increase or the coherence be improved, but we are not so far away from being able to test this model. I consider this to be the currently most promising experimental avenue. A completely different way to test this model would be to see whether matter and gravity actually can be entangled. There are some experiments gearing up to look for this, see e.g. [34]. However, most of the experiments currently underway that use “entanglement witnesses” [35, 36] to probe the quantisation of gravity actually measure the entanglement indirectly through the matter sector, so they are not sensitive to the product state constraint.

The product state requirement will not affect real graviton emission provided one treats the graviton also as a particle5 . It will set constraints on the possible matter states created in scattering processes. However, since we have zero evidence that gravitons exist, and measuring them is far outside experimental reach for the foreseeable future, it is rather moot to discuss this point. Another possibility to test this model would be to investigate closer just which type of in-medium interactions result in a sufficient accumulation of ||R|| to induce a collapse, and whether there are cases when that would happen before decoherence makes the effect undetectable. However, for all in-medium interactions that I have looked at so far, the accumulated residual and the inverse decoherence time are pretty much the same, so this does not seem to be promising. It is worth noting that the model proposed here cannot be tested by looking for dispersive effects like those induced by the noise-kernel of the Penrose-Diosi model [37]. ´