- 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 xi, the classical gravitational fields φgr are different and one has a superposition of states with different locations
| I>= ∑i=1,2 | mi ~at~ xi> | φgr,xi> == | L> +|R> .
- Put two such de-localized states with masses mi 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 space-time surfaces representing this situation. Gravitational field of particle is associated with the magnetic body of particle represented as 4-surface and superposition corresponds to a de-localized quantum state in the "world of classical worlds" with xi representing particular WCW coordinates.
- Authors think quantum information theoretically and reduce everything to qubits. The de-localization 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 left-lower 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 V-H and H-V meeting at a transparent mirror located at the upper right corner of interferometer and interfere.
There is detector D0 resp. D1 detecting component of light gone through in vertical resp. horizontal direction of the fourth mirror. Firing of D1 would select the H-V and the firing of D0 the V-H path. This thus would tells what path (V-H or H-V) 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 x1 and x2 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 Di 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 m1 and m2 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
Φi = [m1m2G/ℏ di] Δ t .
Δ t =L/v is the time taken to pass through the path of length L with velocity v. d1 is the smaller distance between upper path for lower mass m2 and lower path for upper mass m1. d2 is the distance between upper path for upper mass m1 and lower m2. See Figure 1 of the article.
- One should have de-localization of massive objects. In atomic scales this is possible. If one has heff/h0>h one could also have zoomed up scale of de-localization 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. Gm1m2 for atomic mass scales is extremely small: since Planck mass mP is something like 1019 proton masses and atomic masses are of order 10-100 atomic masses.
One should have objects with masses not far from Planck mass to make Gm1m2 large enough. Authors suggest using condensed matter objects having masses of order m∼ 10-12 kg, which is about 1015 proton masses 10-4 Planck masses. Authors claim that recent technology allows de-localization 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 Gm1m2> 1 (say). For proposed mass scales this would not be a problem.
- In TGD framework the gravitational Planck hgr= Gm1m2/v0 assignable to the flux tubes mediating interaction between m1 and m2 as macroscopic quantum systems could enter into the game and could reduce in extreme case the value of gravitational fine structure constant from Gm1m2/4π ℏ to Gm1m2/4π ℏeff = β0/4π, β0= v0/c<1. This would make perturbation series convergent even for macroscopic masses behaving like quantal objects. The physically motivated proposal is β0∼ 2-11. This would zoom up the quantum coherence length scales by hgr/h.
- What can one say in TGD framework about the values of phases Φ?
- For ℏ → ℏeff one would have
Φi = [Gm1m2/ℏeff di] Δ t .
For ℏ → ℏeff the phase differences would be reduced for given Δ t. On the other hand, quantum gravitational coherence time is expected to increase like heff 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 ℏeff=ℏgr this would give a universal prediction having no dependence on G or masses mi
Φi = [v0Δ t/di] = [v0/v] [L/di] .
If Planck length is actually equal to CP2 length R∼ 103.5(GNℏ)1/2, one would has GN = R2/ℏeff, ℏeff∼ 107. 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/ℏeff2 scaling for Φ. Also for this option the prediction for the phase difference is universal for heff=hgr.
- What is important is that the universality could be tested by varying the masses mi. This would however require that mi behave as coherent quantum systems gravitationally. It is however possible that the largest quantum systems behaving quantum coherently correspond to much smaller masses.
- For ℏ → ℏeff one would have
See the chapter About the Nottale's formula for hgr and the possibility that Planck length lP and CP2 length R are identical giving G= R2/ℏeff of "Physics in many-sheeted space-time" or the article Is the hierarchy of Planck constants behind the reported variation of Newton's constant?.
For a summary of earlier postings see Latest progress in TGD.