Does quantum criticality quite generally imply Einstein's equations in TGD?

Does quantum criticality quite generally imply Einstein's equations in TGD?

Jacobsen has argued that Einstein's equations follow from thermodynamic arguments (see this. Empty space Einstein's equations would follow from the vanishing of the second variation of volume. I have discussed this from the TGD point of view in (see this).

This is very interesting from the point of view of TGD for the following reasons.

1. Quantum criticality is the basic guiding principle of TGD. In the same way that holomorphy determines the dynamics of 2-D quantum critical systems, its 4-D generalization would determine the dynamics of the space-time surfaces. Holography in turn follows from the condition that the definition of the geometry of the "world of classical worlds" (WCW) consisting of 3-surfaces realizes 4-D general coordinate invariance. Quantum classical correspondence is realized if one can assign to a given 3-surface almost unique space-time surface as its "Bohr orbit" identifiable as a preferred extremal of a classical action. It indeed turns out that there is a slight failure of determinism.

   This gives rise to holography =holomorphy vision (see this and this) predicting that space-time surfaces are holomorphic surfaces and therefore minimal surfaces. This is actually true for any classical action which is general coordinate invariant and constructible in terms of the induced geometry.

2. The details of the classical action action would be seen in the vacuum functional since the classical action defines the Kähler function of the "world of classical worlds" (WCW)(see this. They could also be seen via the boundary conditions at partonic surfaces serving as interfaces of regions with Minkowskian and Euclidean signature of the induced metric (see this and this) and stating that conserved classical charges are indeed conserved. The dream is that also the boundary conditions following holography= holomorphy vision are universally satisfied.
3. Note that the number theoretical vision suggests another way to fix the classical action. Number theoretical vision strongly suggests that the exponential of the action defining the vacuum functional is a number theoretic invariant and I have considered some natural candidates for it (see this).

In Jacobsen's case the vanishing of the second variation of volume in special space-time coordinates extremizing it gives rise to vacuum Einstein's equations. Now space-time is a 4-surface and the variation could be with respect to a subset of its embedding space coordinates. By holography= holomorphy principle, the space-time surface is a minimal surface. Hamilton Jacobi structure (see this) defines natural holomorphic coordinates but it is natural to require manifest general coordinate invariance.

One can consider the variation of 4-volume with respect to the local H-J coordinates of the space-time surface as an analogy for what Jacobson did.

1. General coordinate invariance implies that only the variation of the boundaries or interfaces between Minkowskian and Euclidean regions defining the holographic data matter. This conforms with the universality of H-H vision. The variation would be in the direction of the normal of the partonic orbits as interfaces of Minkowskian and Euclidean regions. This is like varying the length of a string requiring extremum.
2. A possible interpretation would be as a procedure finding extremum of WCW Kähler function with respect to the zero modes which do not appear in its line element. These maxima are of special importance in the functional integral (see this and this) and indeed critical.
3. This would give a single field equation for the H coordinate in the normal variation. Since the field equations depend on the classical action only at these boundaries, this option could in very general sense give rise to the analogs of Einstein's equations: gravitational field would have as a source the matter assigned with the partonic orbits. This would conform with the physical picture: matter at the partonic orbits affects the fields in the interior by boundary conditions.
4. The second variation would be for the normal components of the canonical momentum currents (or isometry currents) defined by the classical action. They would depend only on the first derivatives of H coordinates only and would give rise to second order partial differential equations.

   If the action involves a boundary term (such as Chern-Simons term allowed by light-likeness) (see this and this), one can obtain field equations coupling the boundary dynamics to the normal currents. The equations are very different from Einstein's equations but the physical idea is the same: vacuum Einstein's equations outside the matter are replaced with analogs of massless field equations solved by H-H vision.