Is it possible to determine experimentally whether gravitation is quantal interaction?
Marletto and Vedral have proposed (thanks for link to Ulla) an interesting method for measuring whether gravitation is quantal interaction (see this). I tried to understand what the proposal suggests and how it translates to TGD language.
 If gravitational field is quantum it makes possible entanglement between two states. This is the intuitive idea but what it means in TGD picture? Feynman interpreted this as entanglement of gravitational field of an objects with the state of object. If object is in a state, which is superposition of states localized at two different points x_{i}, the classical gravitational fields φ_{gr} are different and one has a superposition of states with different locations
 I>= ∑_{i=1,2 } m_{i} ~at~ x_{i}>  φ_{gr},x_{i}> ==  L> +R> .
 Put two such delocalized states with masses m_{i} at some distance d to get state I1>I2>,
 i> = L>_{i} + R>_{i}. The 4 components pairs of the states interact gravitationally and since there are different gravitational fields between different states the states get different phases, one can obtain entangled state.
Gravitational field would entangle the masses. If one integrates over the degrees of freedom associated with gravitational field one obtains density matrix and the density matrix is not pure if gravitational field is quantum in the sense that it entangles with the particle position.
That gravitation is able to entangle the masses would be a proof for the quantum nature of gravitational field. It is not however easy to detect this. If gravitation only serves as a parameter in the interaction Hamiltonian of the two masses, entanglement can be generated but does not prove that gravitational interaction is quantal. It is required that the only interaction between the systems is gravitational so that other interactions do not generate entanglement. Certainly, one should use masses having no em charges.
 In TGD framework the view of Feynman is natural. One has superposition of spacetime surfaces representing this situation. Gravitational field of particle is associated with the magnetic body of particle represented as 4surface and superposition corresponds to a delocalized quantum state in the "world of classical worlds" with x_{i} representing particular WCW coordinates.
I am not specialist in quantum information theory nor as quantum gravity experimentalist, and hereafter I must proceed keeping fingers crossed and I can only hope that I have understood correctly. To my best understanding, the general idea of the experiment would be to use interferometer to detect phase differences generated by gravitational interaction and inducing the entanglement. Not for photons but for gravitationally interacting masses m_{1} and m_{2} assumed to be in quantum coherent state and be describable by wave function analogous to em field. It is assumed that gravitational interact can be describe classically and this is also the case in TGD by quantumclassical correspondence.
 Authors think quantum information theoretically and reduce everything to qubits. The delocalization of masses to a superposition of two positions correspond to a qubit analogous to spin or a polarization of photon.
 One must use and analog of interferometer to measure the phase difference between different values of this "polarization".
In the normal interferometer is a flattened square like arrangement. Photons in superpositions of different spin states enter a beam splitter at the leftlower corner of interferometer dividing the beam to two beams with different polarizations: horizontal (H) and vertical (V). Vertical (horizontal) beam enters to a mirror which reflects it to horizontal (vertical beam). One obtains paths VH and HV meeting at a transparent mirror located at the upper right corner of interferometer and interfere.
There is detector D_{0} resp. D_{1} detecting component of light gone through in vertical resp. horizontal direction of the fourth mirror. Firing of D_{1} would select the HV and the firing of D_{0} the VH path. This thus would tells what path (VH or HV) the photon arrived. The interference and thus also the detection probabilities depend on the phases of beams generated during the travel: this is important.
 If I have understood correctly, this picture about interferometer must be generalized. Photon is replaced by mass m in quantum state which is superposition of two states with polarizations corresponding to the two different positions. Beam splitting would mean that the components of state of mass m localized at positions x_{1} and x_{2} travel along different routes. The wave functions must be reflected in the first mirrors at both path and transmitted through the mirror at the upper right corner. The detectors D_{i} measure which path the mass state arrived and localize the mass state at either position. The probabilities for the positions depend on the phase difference generated during the path. I can only hope that I have understood correctly: in any case the notion of mirror and transparent mirror in principle make sense also for solutions of Schrödinger eequation.
 One must however have two interferometers. One for each mass. Masses m_{1} and m_{2} interact quantum gravitationally and the phases generated for different polarization states differ. The phase is generated by the gravitational interaction. Authors estimate that phases generate along the paths are of form
Φ_{i} = [m_{1}m_{2}G/ℏ d_{i}] Δ t .
Δ t =L/v is the time taken to pass through the path of length L with velocity v. d_{1} is the smaller distance between upper path for lower mass m_{2} and lower path for upper mass m_{1}. d_{2} is the distance between upper path for upper mass m_{1} and lower m_{2}. See Figure 1 of the article.
What one needs for the experiment?
 One should have delocalization of massive objects. In atomic scales this is possible. If one has h_{eff}/h_{0}>h one could also have zoomed up scale of delocalization and this might be very relevant. Fountain effect of superfluidity pops up in mind.
 The gravitational fields created by atomic objects are extremely weak and this is an obvious problem. Gm_{1}m_{2} for atomic mass scales is extremely small: since Planck mass m_{P} is something like 10^{19} proton masses and atomic masses are of order 10100 atomic masses.
One should have objects with masses not far from Planck mass to make Gm_{1}m_{2} large enough. Authors suggest using condensed matter objects having masses of order m∼ 10^{12} kg, which is about 10^{15} proton masses 10^{4} Planck masses. Authors claim that recent technology allows delocalization of masses of this scale at two points. The distance d between the objects would be of order micron.
 For masses larger than Planck mass one could have difficulties since quantum gravitational perturbation series need not converge for Gm_{1}m_{2}> 1 (say). For proposed mass scales this would not be a problem.
What can one say about the situation in TGD framework?
 In TGD framework the gravitational Planck h_{gr}= Gm_{1}m_{2}/v_{0} assignable to the flux tubes mediating interaction between m_{1} and m_{2} as macroscopic quantum systems could enter into the game and could reduce in extreme case the value of gravitational fine structure constant from Gm_{1}m_{2}/4π ℏ to Gm_{1}m_{2}/4π ℏ_{eff} = β_{0}/4π, β_{0}= v_{0}/c<1. This would make perturbation series convergent even for macroscopic masses behaving like quantal objects. The physically motivated proposal is β_{0}∼ 2^{11}. This would zoom up the quantum coherence length scales by h_{gr}/h.
 What can one say in TGD framework about the values of phases Φ?
 For ℏ → ℏ_{eff} one would have
Φ_{i} = [Gm_{1}m_{2}/ℏ_{eff} d_{i}] Δ t .
For ℏ → ℏ_{eff} the phase differences would be reduced for given Δ t. On the other hand, quantum gravitational coherence time is expected to increase like h_{eff} so that the values of phase differences would not change if Δ t is increased correspondingly. The time of 10^{6} seconds could be scaled up but this would require the increase of the total length L of interferometer arms and/or slowing down of the velocity v.
 For ℏ_{eff}=ℏ_{gr} this would give a universal prediction having no dependence on G or masses m_{i}
Φ_{i} = [v_{0}Δ t/d_{i}] = [v_{0}/v] [L/d_{i}] .
If Planck length is actually equal to CP_{2} length R∼ 10^{3.5}(G_{N}ℏ)^{1/2}, one would has G_{N} = R^{2}/ℏ_{eff}, ℏ_{eff}∼ 10^{7}. One can consider both smaller and larger values of G and for larger values the phase difference would be larger. For this option one would obtain 1/ℏ_{eff}^{2} scaling for Φ. Also for this option the prediction for the phase difference is universal for h_{eff}=h_{gr}.
 What is important is that the universality could be tested by varying the masses m_{i}. This would however require that m_{i} behave as coherent quantum systems gravitationally. It is however possible that the largest quantum systems behaving quantum coherently correspond to much smaller masses.
See the chapter About the Nottale's formula for h_{gr} and the relation between Planck length and CP_{2} length R or the article Is the hierarchy of Planck constants behind the reported variation of Newton's constant?.
