What is missing in the Bronstein cube encoding traditional perspective on quantum gravity is ...

... a fourth dimension for the N-direction!

//this post has been re-edited on Saturday the 10th March 2018

Almost exactly three years ago I wrote in a post : in the Bronstein's diagram ... it is hard to miss the lack of a block labeled "Thermodynamics / Statistical physics" on the first line but this is another story for another post with the following question : is there any universal constant one should associate to the missing block, like a large integer N ...? 

I am delighted was excited to find in arxiv today two days ago an article doing trying to do the job better than I would ever have dreamed of could. I must confess I have not read it yet thoroughly but what I did makes sense to me and is clearly exposed informative (even if I would like to add some explicit comments when my professional duties will give me more leisure time).

... the proper setting for thinking about quantum gravity, and for exploring the many issues it raises (mathematical, physical, conceptual), is broader than the traditional one of ‘quantizing General Relativity’, well captured by the Bronstein cube. It is best pictured as a Bronstein hypercube, in which the non-spatiotemporal nature of the fundamental building blocks suggested by most quantum gravity formalisms (and even by semi-classical physics), and the need to control their collective dynamics, are manifest. This allows the proper focus on the problem of the emergence of continuum spacetime and geometry from such non-spatiotemporal entities. We ... argue... that modern quantum gravity approaches are well embedded into this conceptual scheme, and have already started producing many results on the issues that are put to the forefront by it. The quantum gravity world is therefore even richer, more complex but also more exciting than traditionally thought, and we are already actively exploring it. More surprises should be expected.

The Bronstein hypercube of quantum gravity
Daniele Oriti (Submitted on 7 Mar 2018)

To have a better pictorial representation of what quantum gravity is about, then, the Bronstein cube should be extended to an object with four (a priori) independent directions, to a ‘Bronstein hypercube’, as in the picture {above}. The fourth direction is labeled N, to indicate the number of quantum gravity degrees of freedom that need to be controlled to progressively pass from an entirely non-geometric and nonspatiotemporal description of the theory to one in which spacetime can be used as the basis of our physics. A complete theory of quantum gravity will sit at the same corner in which it was sitting in the Bronstein cube (which is obviously a subspace of this hypercube), but the same theory admits a partial, approximate formulation at any point along the N-direction ending at that corner. Only, the more one moves away from it, the less the notions of continuum spacetime and geometry will fit the corresponding physics. One could say that the definition of a theory of quantum gravity will be provided in the opposite corner (looking only at the two ends of the N direction, while keeping both G, h and 1/c finite), because it is at this point that the definition of the fundamental degrees of freedom of the theory and of their basic quantum dynamics will be put on the table. This is sensible, but it is also true that providing a complete definition of the same theory amounts to making sure it is well-defined up to the opposite end, even though the same theory will always be used in some approximation or truncation. (The picture is adapted from the one at en.wikipedia.org/wiki/Tesseract).

//addendum Saturday the 18th of September thanks to a welcome comment!

For a mathematically consistent description of nature, in which all constants, k, h, c⁻¹, and lₚ,  in natural Planck units have the value 1, we lack a secure conceptual basis. The relativistic quantum field theory understands itself as a local, causal quantum theory. The precise formulation of locality and causality requires the notion of a bundle of light cones. A relativistic theory of gravitation deprives the relativistic quantum field theory this essential basis!  One thing seems to be certain:  A relativistic quantum theory of matter, radiation and gravitation will cause revolutionary changes in the space-time concept will result. As repeatedly did, lastly by Doplicher, Fredenhagen and Roberts, it can be argued that in such a theory, events can at best only be localized in space-time regions of a "mean of an "average extension" lₚ, which must have the consequence that in space-time cells of finite extent only a finite number of degrees of freedom can be excited (and thus all ultraviolet divergences should be eliminated), and that the space-time even on very small scales must have quantum mechanical properties" . The superstring theory comes closest to such ideas at present. Its relation to a relativistic quantum theory of matter, radiation and gravitation could plausibly be compared with that of the quantum theory of atomic structure of Bohr and Sommerfeld to the quantum mechanics of atoms, as it was discovered by Born, Heisenberg, Jordan, Schrödinger and Pauli. I summarize these remarks in a graphical representation (see fig. 1 [below]).

Jürg Frölich, Max-Planck-Medaille Lecture

Ein Blick zurück, ein Blick nach vorn

Tentative English translation: 

"Planck's Hypercube": 

0 Non-relativistic continuum mechanics. 

A Hamiltonian mechanics, symplectic geometry; Newtonian Space-time. 

B Classical relativistic field theory, infinite-dimensional symplectic geometry; Minkowski space-time. C Non-relativistic quantum mechanics, non-commutative Geometry of quantum mechanical phase spaces; Functional integration . 

D Special-relativistic QFT, non-commutative geometry infinite-dimensional quantum phase spaces; infinitely dimensional Symmetries, BRST and BV cohomology. 

E Newtonian gravity and its formulation as geometric Theory within the framework of Riemannian geometry. 

F General relativity, Lorentzian geometry. 

G “Non-relativistic quantum gravity”, non-commutative Riemannian geometry. 

M quantum gravity, "M-theory", "quantum or string Geometry ". 

k, h↔ deformations of the "observables" and "state spaces" 

c ⁻¹, lₚ ↔ deformations of the “symmetries”