Gravity goes quantum: really?

Just for fun, I had a session about rather hypeish “How gravity goes quantum” of NASA (see this) with Google to revive what I remember of the model of Partanen and Tulkki from previous discussions with Tulkki and about the review of their model that I wrote (see blog post). I also proposed the role of the octonions then and they are indeed introduced in their recent article.

The core fields are the space-time dimension field and 8-D spinor field defined in empty Minkowski space.

1. The space-time dimension field is essentially the tetrad field of GR and allows us to construct the space-time metric formally. Tetrad components would transform in GR by local Lorentz transformations acting as non-compact gauge symmetry, which however is not used.
   1. The space-time is the flat Minkowski space M⁴ globally. Non-trivial topologies are not possible. This is an extremely strong limitation and one loses most of GR. One has just gauge theory in M⁴.
   2. Metric as gravitational field is defined purely algebraically as an analog of vierein field by standard rules. The dimension field. as tetrad is a quantum field in M⁴ and the metric is constructed in terms of it. The products of vielbein components involve singularities and normal ordering is required. The construction of Christoffel symbols and curvature leads to horrible non-linearities. Poincare symmetries are obtained. Equivalence Principle and general coordinate invariance are claimed but it is difficult to take this claim seriously. The reason is that the action of the general coordinate coordinate transformations is very different from the action of gauge symmetries although the physical content of these symmetries is the same.

One should should show that general coordinate invariance emerges from their theory but the definition of the space-time metric and curvature tensor, Ricci tensor and Ricci tensor as its companions is extremely difficult since quantum fields are in question. The same problems are encountered as in general relativity.

8-spinor field would naturally correspond to the spinors of the standard model having besides spin degrees of freedom naturally electroweak spin and em charge. Color quantum numbers missing. This would however lead to problems since M⁴ spinors have non-compact gauge group SL(2,C) if one wants the gravitational gauge symmetries of GR. Color quantum numbers are missing.

Symmetry group is assumed to be SU(8) assigned with 8-spinors and it is compact. Gravitation is assigned with 4-D Cartan group U(1)⁴. The remaining 3-D Cartan algebra U(1)³ should represent standard model Cartan algebra which is however 4-D. It is assumed that electromagnetic U(1) is shared by the gravitational Cartan group and standard model gauge group.

The identification of the symmetries as la arger symmetry group SU(8) is not consistent with the notion of internal symmetries in Minkowski space allowing SL(2,c)× SU(2)_R×SU(2)_L at most as symmetries. Color symmetries remain missing in standard interpretation.

The authors have clearly picked several ideas from TGD (congratulations for a good taste!) and try to fuse them to their own theory.

1. Also in TGD empty Minkowski space plays a key role but space-times are surfaces in H=M⁴×CP₂ and the dynamics is purely geometric. Poincare symmetry is not lost as in GR. The space-time surfaces representable as graphs M⁴→ CP₂ represent only special solutions important in the long length scale limit.

CP₂ type extremals, cosmic strings are not surfaces of this type and are essential for the description of particles and monopole flux tubes are central for physics in all scales. Without non-trivial space-time topologies TGD would predict only one fermion generation. This is actually the situation also in the model of Partanen and Tulkki.

In TGD, the induced metric is a genuine metric and Also spinor connection and spinor structure of H is induced and gives rise to electroweak gauge potentials. Also spinors are induced to the space-time surface.

In TGD one does not quantize the induced geometry. Instead, one introduces the notion of “world of classical worlds” (WCW) consisting of space-time surfaces as analogs of 4-D Bohr orbit so that one has wave mechanics in WCW with fermions included. Fermion fields are free fields in H identifiable as leptonic and quark-like fields and there are no divergence problems. The geometry of WCW is unique from its existence and has maximal symmetries: this is true already for the loop spaces.

The color degrees of freedom correspond to isometries of CP₂ and here comes the most dramatic prediction: an entire hierarchy of standard model physics is predicted since color is now associated with the CP₂ isometries and corresponds to color multiplets for the partial waves. Electroweak symmetries correspond to holonomomies of CP₂ and one can identify electroweak interactions in fermion spin degrees of freedom as color interactions.

Infinities cancel since holography = holomorphy principle solving the field equations implies that there is no path integral. Radiative corrections are obtained but are manifestly finite. No renormalization as elimination of infinities is needed.

Number theoretic vision defines the second half of TGD, complementary to the physics as geometry view and all basic number theory, including octonions and quaternions, is involved. Octonions are mentioned also in the recent article. I suggested a possible role of octonions in their theory in a discussion with Mikko Partanen (see the earlier blog post).

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/2026/05/gravity-goes-quantum-really.html