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

Tuesday, July 22, 2025 0:57

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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.

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.

The details of the classical action action would be seen in the vacuum functional since the classical action defines the K”ahler 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. Could one understand the finding of Jacobsen in this framework?

The vanishing second variation for volume means classical criticality required in the TGD framework by quantum criticality (in TGD, the classical theory is an exact part of quantum theory). In GRT, the vanishing variation would give rise to Einstein’s equations in empty space.

But how to get Einstein’s equations coupled to matter? These equations are proposed at the long length scale limit of TGD. The gravitational field of GRT and gauge potentials are proposed to correspond to the sums of the corresponding induced quantities for the space-time sheets of the many-sheeted space-time defining these fields in a metrically deformed M⁴ defining the QFT limit of TGD.

The difference between GRT and TGD is that space-time surfaces in TGD are generalized holomorphic minimal surfaces in H=M⁴× CP₂. The representation as a 4-surface means a geometric constraint. This is like a mechanical problem for point-like particles subjected to geometric constraints. Actually the particles are replaced with 3-surfaces, or rather with their almost-deterministic Bohr orbits. The constraints give rise to constraint forces and their natural interpretation would be as an addition of the energy momentum tensor of matter serving as a source of the gravitational field.

This suggests that Einsteins’ equations with matter would be natural, not only at the QFT limit but quite generally, and reflect the quantum criticality in the TGD Universe assumed quite generally and fixing various coupling parameters so that coupling constant evolution is discrete and coded by the number theoretic vision in terms of algebraic extensions of rationals. Indeed, one can ask whether quantum criticality could quite generally correspond to Einstein’s equations for a suitable counterpart of energy momentum determined by classical action. If so, quantum criticality would determine that classical action and give Einstein’s equations.

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). For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/07/does-quantum-criticality-quite.html

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