QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 3

their neutrinos. The Standard Model, of phenomenological origin, encompasses also

the so-called strong interactions, associated with (generally) heavier particles called

baryons (the proton and neutron, and many more), now commonly assumed to be

bound states of "confined" quarks with gluons, in three "colors". It contains 19 free

parameters, plus 10 more in extensions needed to account for the recently observed

neutrino mixing

phenomena, which require nonzero masses for the neutrinos that

are traditionally massless in the Standard Model.

Very recently Connes

[CCM]

developed an effective unified theory based on

noncommutative geometry (space-time being the product of a Riemannian compact

spin 4-manifold and a finite noncommutative geometry) for the Standard Model

with neutrino mixing, minimally coupled to gravity. It has 4 parameters less and

predicts for the yet elusive Higgs particle (responsible for giving mass to initially

massless leptons) a mass that is slightly different (it is at the upper end of the

expected mass range) from what is usually predicted. See also

[Co06],

and

[Ba06]

in a Lorentzian framework.

Previously, in part aiming at a possible description of quantum gravity, but

mainly in order to study nontrivial examples of noncommutative manifolds, Frohlich

(in a supersymmetric context), then Connes and coworkers had studied quantum

spheres in 3 and 4 dimensions

[FGR, CLOl, CDV].

The basic tool there is a

spectral triple introduced by Connes

[Co94].

In the present paper we are devel-

oping a similar approach, but for hyperbolic spheres and using integral universal

deformation formulas in the deformation quantization approach.

The distant hope is that these quantized AdS spaces (at even root of unity)

can be shown to be a kind of "small" black holes at the edge of our Universe

in accelerated expansion, from which matter would emerge, possibly created as

(quantized) 2-singleton states emerging from them and massified by interaction with

ambient dark energy (or dark matter), in a process similar to those of the creation

of photons as 2-Rae states and of leptons from 2-singleton states, mentioned above.

As fringe benefits that might explain the acceleration of expansion of our Uni-

verse, and the problems of baryogenesis and leptogenesis (see e.g.

[Cl06, SS07]).

Physicists love symmetries and even more to break them (at least at our level).

One of the riddles that physics has to face is that, while symmetry considerations

suggest that there should be as much matter as antimatter, one observes a huge

imbalance in our region of the Universe. In a seminal paper published in 1967

that went largely unnoticed for about 13 years but has now well over a thousand

citations (we won't quote it here), Andrei Sakharov addressed that problem, now

called baryogenesis. If and when a mechanism along the lines hinted at above can

be developed for creating baryons and other particles, it could solve that riddle.

Roughly speaking the idea is that there is no reason, except theological, why

everything (whatever that means) would be created "in the beginning", or as con-

ventional wisdom has it now, in a Big Bang. There could very well be "stem cells"

of the primordial singularity that would be spread out, like shrapnel, mostly at the

edge of the Universe. Our proposal is that these could be described mathematically

as quantized AdS black holes. We shall now concentrate our study on them.

1.2. Mathematical introduction. Roughly speaking, a universal deforma-

tion formula (briefly UDF) for a given symmetry

g

is a procedure that, for every,

say, topological algebra A admitting the symmetry

g,

produces a deformation

Ao

of A within the same category of topological algebras. Such a UDF is called

formal

their neutrinos. The Standard Model, of phenomenological origin, encompasses also

the so-called strong interactions, associated with (generally) heavier particles called

baryons (the proton and neutron, and many more), now commonly assumed to be

bound states of "confined" quarks with gluons, in three "colors". It contains 19 free

parameters, plus 10 more in extensions needed to account for the recently observed

neutrino mixing

phenomena, which require nonzero masses for the neutrinos that

are traditionally massless in the Standard Model.

Very recently Connes

[CCM]

developed an effective unified theory based on

noncommutative geometry (space-time being the product of a Riemannian compact

spin 4-manifold and a finite noncommutative geometry) for the Standard Model

with neutrino mixing, minimally coupled to gravity. It has 4 parameters less and

predicts for the yet elusive Higgs particle (responsible for giving mass to initially

massless leptons) a mass that is slightly different (it is at the upper end of the

expected mass range) from what is usually predicted. See also

[Co06],

and

[Ba06]

in a Lorentzian framework.

Previously, in part aiming at a possible description of quantum gravity, but

mainly in order to study nontrivial examples of noncommutative manifolds, Frohlich

(in a supersymmetric context), then Connes and coworkers had studied quantum

spheres in 3 and 4 dimensions

[FGR, CLOl, CDV].

The basic tool there is a

spectral triple introduced by Connes

[Co94].

In the present paper we are devel-

oping a similar approach, but for hyperbolic spheres and using integral universal

deformation formulas in the deformation quantization approach.

The distant hope is that these quantized AdS spaces (at even root of unity)

can be shown to be a kind of "small" black holes at the edge of our Universe

in accelerated expansion, from which matter would emerge, possibly created as

(quantized) 2-singleton states emerging from them and massified by interaction with

ambient dark energy (or dark matter), in a process similar to those of the creation

of photons as 2-Rae states and of leptons from 2-singleton states, mentioned above.

As fringe benefits that might explain the acceleration of expansion of our Uni-

verse, and the problems of baryogenesis and leptogenesis (see e.g.

[Cl06, SS07]).

Physicists love symmetries and even more to break them (at least at our level).

One of the riddles that physics has to face is that, while symmetry considerations

suggest that there should be as much matter as antimatter, one observes a huge

imbalance in our region of the Universe. In a seminal paper published in 1967

that went largely unnoticed for about 13 years but has now well over a thousand

citations (we won't quote it here), Andrei Sakharov addressed that problem, now

called baryogenesis. If and when a mechanism along the lines hinted at above can

be developed for creating baryons and other particles, it could solve that riddle.

Roughly speaking the idea is that there is no reason, except theological, why

everything (whatever that means) would be created "in the beginning", or as con-

ventional wisdom has it now, in a Big Bang. There could very well be "stem cells"

of the primordial singularity that would be spread out, like shrapnel, mostly at the

edge of the Universe. Our proposal is that these could be described mathematically

as quantized AdS black holes. We shall now concentrate our study on them.

1.2. Mathematical introduction. Roughly speaking, a universal deforma-

tion formula (briefly UDF) for a given symmetry

g

is a procedure that, for every,

say, topological algebra A admitting the symmetry

g,

produces a deformation

Ao

of A within the same category of topological algebras. Such a UDF is called

formal