Entropic gravity (EG) introduced by Verlinde has stimulated a great interest. One of the most interesting reactions is the commentary of Sabine Hossenfelder. The article of Kobakhidze relies on experiments supporting the existence of Schrödinger amplitudes of neutron in the gravitational field of Earth develops an argument suggesting that EG hypothesis in the form in which it excludes gravitons is wrong. Indeed, the mere existence of gravitational bound states suggests strongly the existence of transitions between them by graviton emission. The following arguments represent TGD inspired view about what entropic gravity (EG) could be if one throws out the unnecessary assumptions such as the emerging dimensions and absence of gravitons. Also the GRT limit of TGD is discussed leading to
rather strong implications concerning the TGD counterparts of blackholes.
If one does not believe in TGD, one could start from the idea that stochastic quantization">stochastic quantization or something analogous to it might imply something analogous to entropic gravity (EG). What is required is the replacement of the path integral with functional integral. More precisely, one has functional integral
in which the real contribution to Kähler action of the preferred extremal from Euclidian regions of the space-time surface to the exponent represents Kähler function and the imaginary contribution from Minkowskian regions serves as a Morse function so that the counterpart of Morse theory in WCW is obtained on stationary phase approximation in accordance with the vision about TGD as almost topological QFT. The exponent of Kähler function is the new element making the functional integral well-defined and the presence of phase factor gives rise to the interference effects characteristic for quantum field theories although one does not integrate over all space-time surfaces. In zero energy ontology one has however pairs of 3-surfaces at the opposite light-like boundaries of CD so that something very much analogous to path integral is obtained.
Holography requires that everything reduces to the level of 3-metrics and more generally, to the level of 3-D field configurations. Something like this happens if one can approximate path integral integral with the integral over small deformations for the minima of the action. This also happens in completely integral quantum field theories.
The basic vision behind quantum TGD is that this approximation is much nearer to reality than the original theory. In other words, holography is realized in the sense that to a given 3-surface the metric of WCW assigns a unique space-time and this space-time serves as the analog of Bohr orbit and allows to realize 4-D general coordinate invariance in the space of 3-surfaces so that classical theory becomes an exact part of quantum theory. This point of view will be adopted in the following also in the framework of general relativity where one considers abstract 4-geometries instead of 4-surfaces: functional integral should be over 3-geometries with the definition of Kähler metric assigning to 3-geometry a unique 4-geometry.
A powerful constraint is that the functional integral is free of divergences. Both 4-D path integral and stochastic quantization for gravitation fail in this respect due to the local divergences (in super-gravity situation might be different). The TGD inspired approach reducing quantum TGD to almost topological QFT with Chern-Simons term and a constraint term depending on metric associated with preferred 3-surfaces allows to circumvent this difficulty. This picture will applied to the quantization of GRT and one could see the resulting theory as a guess for what GRT limit of TGD could be. The first guess that Kähler function corresponds to Einstein-Maxwell action for this kind of preferred extremal turns out to be correct. An essential and radically new element of TGD is the possibility of space-time regions with Euclidian signature of the induced metric replacing the interiors of blackholes: this element will be assumed also now. The conditions that CP2 represents and extremal of EYM action requires cosmological constant in Euclidian regions determined by the constant curvature of CP2 and one can ask whether the average value of cosmological constant over 3-space could correspond to the cosmological constant explaining accelerating cosmic expansion.
Entropic gravity is generalized in TGD framework so that all interactions are entropic: the reason is that in zero energy ontology (ZEO) the S-matrix is replaced with M-matrix defining a square root of thermodynamics in a well defined sense.
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