### Transition from flat to hyperbolic geometry and q-deformation

In Thinking Allowed Original there was a link to a very interesting article with title "Designing Curved Blocks of Quantum Space-Time...Or how to build quantum geometry from curved tetrahedra in loop quantum gravity" telling about the work of Etera Livine working at LPENSL (I let the reader to learn what this means;-).

**The idea of the article**

The popular article mentions a highly interesting mathematical result relevant for TGD. The idea is to build 3-geometry - not by putting together flat tetrahedra or more general polyhedra along their boundaries - but by using curved hyperbolic tetrahedra or more generally polygons) defined in 3-D hyperbolic space - negative constant curvature space with Lorentz group acting as isometries - cosmic time=constant section of standard cosmology.

As a special case one obtains tesselation of 3-D hyperbolic space H^{3}. This is somewhat trivial outcome so that one performs a "twisting". Some words about tesselations/lattices/crystals are in order first.

- In 2-D case you would glue triangles (say) together to get curved surface. For instance, at the surface of sphere you would get finite number of lattice like structures: the five Platonic solids tetrahdron, cube, octahedron, icosahedron, and dodecahedron, which are finite geometries assignable to finite fields corresponding to p=2, 3, and 5 and defining lowest approximaton of p-adic numbers for these primes.

- In 2-D hyperbolic plane H
^{2}one obtains hyperbolic tilings used by Escher (see this).

- One can also consider decomposition of hyperbolic 3-space H
^{3}to lattice like structure. Essentially a generalization of the ordinary crystallography from flat 3-space E^{3}to H^{3}. There are indications for the quantization of cosmic redshifts completely analogous to the quantization of positions of lattice cells, and my proposal is that they reflect the existing of hyperbolic crystal lattice in which astrophysical objects replace atoms. Macroscopic gravitational quantum coherence due to huge value of gravitational Planck constant could make them possible.

Back to the article and its message. The condition for tetrahedron property stating in flat case that the sum of the 4 normal vectors vanishes generalizes, and is formulated in group SU(2) rather than in group E^{3} (Euclidian 3-space). The popular article states that deformation of sum to product of SU(2) elements is equivalent with a condition defining classical q-deformation for gauge group. If this is true, a connection between "quantum quantum mechanics" and hyperbolic geometries therefore might exist and would correspond to a transition from flat E^{3} to hyperbolic H^{3}.

** Let loop gravity skeptic talk first**

This looks amazing but it is better to remain skeptic since the work relates to loop quantum gravity and involves specific assumptions and different motivations.

- For instance, the hyperbolic geometry is motivated by the attempts to quantum geometry producing non-vanishing and negative cosmological constant by introducing it through fundamental quantization rules rather than as a physical prediction and using only algebraic conditions, which allow representation as a tetrahedron of hyperbolic space. This is alarming to me.

- In loop quantum gravity one tries to quantize discrete geometry. Braids are essential for quantum groups unless one wants to introduce them independently. In loop gravity one considers strings defining 1-D structures and the ordinary of points representing particles at string like entity might be imagined in this framework. I do not know enough loop gravity to decide whether this condition is realized in the framework motivating the article.

- In zero energy ontology hyperbolic geometry emerges in totally different manner. One wants only a discretization of geometry to represent classically finite measurement resolution and Lorentz invariance fixes it at the level of moduli space of CDs. At space-time level discretization would occur for the parameters charactering strings world sheets and partonic 2-surfaces defining "space-time genes" in strong form of holography.

- One possible reason to worry is that H
^{3}allows infinite number of different lattice like structures (tesselations) with the analog of lattice cell defining hyperbolic manifold. Thus the decomposition would be highly non-unique and this poses practical problems if one wants to construct 3-geometries using polyhedron like objects as building bricks. The authors mention twisting: probably this is what would allow to get also other 3-geometries than 3-D hyperbolic space. Could this resolve the non-uniqueness problem?

I understand (on basis of this) that hyperbolic tetrahedron can be regarded as a hyperbolic 3-manifold and gives rise to a tesselation of hyperbolic space. Note that in flat case tetrahedral crystal is not possible. In any case, there is an infinite number of this kind of decompositions defined by discrete subgroups G of Lorentz group and completely analogous to the decompositions of flat 3-space to lattice cells: now G replaces the discrete group of translations leaving lattice unaffected. An additional complication from the point of view of loop quantum gravity in the hyperbolic case is that the topology of the hyperbolic manifold defining lattice cell varies rather than being that of ball as in flat case (all Platonic solids are topologically balls).

**The notion of finite measurement resolution**

The notion of finite measurement resolution emerged first in TGD through the realization that von Neumann algebras known as hyper-finite factors of type I_{1} (perhaps also of type III_{1}) emerge naturally in TGD framework. The spinors of "world of classical worlds" (WCW) identifiable in terms of fermionic Fock space provide a canonical realization for them.

The inclusions of hyperfinite factors provide a natural description of finite measurement resolution with included factor defining the sub-algebra, whose action generates states not distinguishable from the original ones. The inclusions are labelled by quantum phases coming as roots of unity and labelling also quantum groups. Hence the idea that quantum groups could allow to describe the quantal aspects of finite measurement resolution whereas discretization would define its classical aspects.

p-Adic sectors of TGD define a correlate for cognition in TGD Universe and cognitive resolution is forced by number theory. Indeed, one cannot define the notion of angle in p-adic context but one can define phases in terms of algebraic extensions of p-adic numbers defined by roots of unity: hence a finite cognitive resolution is unavoidable and might have a correlate also at the level of real physics.

The discrete algebraic extensions of rationals forming a cognitive and evolutionary hierarchy induce extensions of p-adic numbers appearing in corresponding adeles and for them quantum groups should be a necessary ingredient of description. The following arguments support this view and make it more concrete.

** Quantum groups and discretization as two manners to describe finite measurement resolution in TGD framework**

What about quantum groups in TGD framework? I have also proposed that q-deformations could represent finite measurement resolution. There might be a connection between discretizing and quantum groups as different aspects of finite measurement resolution. For instance, quantum group SU(2)_{q} allows only a finite number of representations (maximum value for angular momentum): this conforms with finite angular resolution implying a discretization in angle variable. At the level of p-adic number fields the discretization of phases exp(iφ) as roots U_{n}=exp(i2π/n) of unity is unavoidable for number theoretical reasons and makes possible discrete Fourier analysis for algebraic extension.

There exist actually a much stronger hint that discretization and quantum groups related to each other. This hint leads actually to a concrete proposal how discretization is described in terms of quantum group concept.

- In TGD discretization for space-time surface is not by a discrete set of points but by a complex of 2-D surfaces consisting of strings world sheets and partonic 2-surface. By their 2-dimensionality these 2-surfaces make possible braid statistics. This leads to what I have called "quantum quantum physics" as the permutation group defining the statistics is replaced with braid group defining its infinite covering. Already fermion statistics replaces this group with its double covering. If braids are present there is no need for "quantum quantum". If one forgets the humble braidy origins of the notion begins to talk about quantum groups as independent concept the attribute "quantum quantum" becomes natural. Personally I am skeptic about this approach: it has not yielded anything hitherto.

- Braiding means that the R-matrix characterizing what happens in the permutation of nearby particles is not anymore multiplication by +1 or -1 but more complex operation realized as a gauge group action (no real change to change by gauge invariance). The gauge group action could in electroweak gauge group for instance.

What is so nice that something very closely resembling the action of quantum variant of gauge group (say electroweak gauge group) emerges. If the discretization is by the orbit of discrete subgroup H of SL(2,C) defining hyperbolic manifold SL(2,C)/H as the analog of lattice cell, the action of the discrete subgroup H is leaves

"lattice cell" invariant but could induce gauge action on state. R-matrix defining quantum group representation would define the action of braiding as a discrete group element in H. Yang-Baxter equations would give a constraint on the representation.

This description looks especially natural in the p-adic sectors of TGD. Discretization of both ordinary and hyperbolic angles is unavoidable in p-adic sectors since only the phases, which are roots of unity exist (p-adically angle is a non-existing notion): there is always a cutoff involved: only phases U

_{m}=exp(i2π/m), m<r exist and r should be a factor of the integer defining the value of Planck constant h_{eff}/h=n defining the dimension of the algebraic extension of rational numbers used. In the same manner hyperbolic "phases" defined by roots e^{1/mp}of e (the very deep number theoretical fact is that e is algebraic number (p:th root) p-adically since e^{p}is ordinary p-adic number!). The test for this conjecture is easy: check whether the reduction of representations of groups yields direct sums of representations of corresponding quantum groups.

- In TGD framework H
^{3}is identified as light-cone proper time=constant surface, which is 3-D hyperboloid in 4-D Minkowski space (necessary in zero energy ontology). Under some additional conditions a discrete subgroup G of SL(2,C) defines the tesselation of H^{3}representing finite measurement resolution. Tesselation consists of a discrete set of cosets gSL(2,C). The right action of SL(2,C) on cosets would define the analog of gauge action and appear in the definition of R-matrix.

The original belief was that discretization would have continuous representation and powerful quantum analog of Lie algebra would become available. It is not however clear whether this is really possible or whether this is needed since the R-matrix would be defined by a map of braid group to the subgroup of Lorentz group or gauge group. The parameters defining the q-deformation are determined by the algebraic extension and it is quite possible that there are more than one parameters.

- The relation to integrable quantum field theories in M
^{2}is interesting. Particles are characterized by Lorentz boosts in SO(1,1) defining their 2-momenta besides discrete quantum numbers. The scattering reduces to a permutation of quantum numbers plus phase shifts. By 2-particle irreducibility defining the integrability the scattering matrix reduces to 2-particle S-matrix depending on the boost parameters of particles, and clearly generalizes the R-matrix as a physical permutation of particles having no momentum. Could this generalize to 4-D context? Could one speak of the analog of this 2-particle S-matrix as having discrete Lorentz boosts h_{i}in sub-group H as arguments and representable as element h( h_{1}, h_{2}) of H: is the ad hoc guess h= h_{1}h_{2}^{-1}trivial?

- The popular article says that one has q>1 in loop gravity. As found, in TGD quantum deformation has at least two parameters are needed in the case of SL(2,C). The first corresponds to the n:th root of unity (U
_{n}= exp(i2π/n)) and second one to n×p:th root of e^{p}. One could do without quantum group but it would provide an elegant representation of discrete coset spaces. It could be also powerful tool as one considers algebraic extensions of rationals and the extensions of p-adic numbers induced by them.

One can even consider a concrete prediction follows for the unit of quantized cosmic redhifts if astrophysical objects form tesselations of H

^{3}in cosmic scales. The basic unit appearing in the exponent defining the Lorentz boost would depend on the algebraic extension invlved and of p-adic prime defining effective p-adcity and would be e^{1/np}.

For a summary of earlier postings see Links to the latest progress in TGD.

## 2 Comments:

Matti,

In the case of the 4D polytope called the 24 cell it is more than a spherical top (that is topological ball) and is an exception in four space to all dimensions above and below it. This trend of thinking so too narrow to reach new understanding if we at least know the n-dimensional Euclidean geometry. This is nothing else that the generation problem where iterative looping (not sure it applies to the deepest idea of gravity) is an isolation. But even if this building of a real space is the case and case only as a hierarchy fractal it oscillates in duality tripling the partial fractal. Have we not understood this from Riemann when we cross an Euclidean boundary - or of Feynman as you also mentioned when we rotate his diagrams 90 degrees to describe particles? I would say that in a sense the hyperbolic and the elliptical are primordially equivalent in the description and that to build such an elaborate theory as they put forth is equally only half the picture. In the bigger picture such infinite lattices would then be remote and not hold up, even with the know consideration of complex hypernumbers. The age of such physics is over and in my opinnion your general take on things still stands among a handful of other future mainstream theories.

The main new observation of the posting came clear only one day after writing it as it often happens. The TGD inspired conjecture that finite measurement as a discretisation and described using quantum groups are aspects of one and same thing. The connection is very concrete and testable: the reduction of representations

of groups to representations of discrete subgroups should

give representations of corresponding quantum groups.

*Discretization defined by a coset space of Lie group with discrete subgroup can be described in terms of gauge group action defined by the right action of group leaving the coset invariant.

*Braid statistics makes sense by strong form of holography implying that 2-surfaces (string world sheets and partonic 2-surfaces are basic objects as far as scattering amplitudes are considered - space-time sheets.

* R-matrix defining braiding and quantum group is representable as this gauge action. This is very beautiful result and in p-adic context quantum groups are unavoidable since phases and exponents of hyperbolic angles must be discretized.

*Also the observation that e^p is p-adic number and e thus an algebraic number (p:th) root is essential and makes possible p-adic discretisation of hyperbolic "phases" and and also implies discretisation of Lorentz

boosts and thus cosmic redshifts if astrophysical objects from hyperbolic tesselations. A testable prediction.

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