22 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
symmetry. [Being in a Lorentzian noncompact framework, we did not ad-
dress here questions such as the resolvent of D when £ is a Hilbert space,
which we did not need at this stage; eventually one may however have
to deal with the reasons that motivated the additional requirements on
triples in the Riemannian compact context; note that here the restriction
to an open orbit was needed in order to have bounded commutators [D,
in the Hilbertian context, but a good choice of£ could lift the restriction.]
That framework should be naturally extendible to the supersymmetric
context, which is the one considered in
with modified spectral
triples and is natural also for the problems considered here since e.g.
DiEBRac and D(HO) are UIR of the corresponding supersymmetries.
(vii) If we want to incorporate "everything," the (external) symmetry
be the Poincare group SO(l,
·lR.4 in the ambient Minkowski space (pos-
sibly modified by the presence of matter) and
3)q in the qAdS4
black holes, or possibly some supersymmetric extension. The unified (ex-
ternal) symmetry could therefore be something like a groupoid. The lat-
ter should be combined in a subtle way (as hinted e.g. in
"internal" symmetry associated with the various generations, colors and
flavors of (composite) "elementary" particles in a generalized Standard
Model, possibly in a noncommutative geometry framework analogous to
what is done in
[Co06, CCM, Ba06].
There would of course remain
the formidable task to develop quantized field theories on that back-
ground, incorporating composite QED for photons on AdS as in
and some analog construction for the electroweak model (touched in part
and for QCD, possibly making use of some formalism coming
from string theory.
(viii) The Gelfand isomorphism theorem permits to realize commutative invo-
lutive algebras as algebras of functions on their "spectrum." Finding a
noncommutative analog of it has certainly been in the back of the mind
of many, since quite some time (see e.g.
We now have theories
and many examples of deformed algebras, quantum groups and noncom-
mutative manifolds. The above mentioned quadruples could provide a
better understanding of that situation.
[AFFS] E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Massless particles, con-
formal group, and de Sitter universe, Phys. Rev. D (3) 23 (1981), no. 6, 1278-1289.
[AC90] D. Arnal and J.-C. Cortet, Representations* des groupes exponentiels, J. Funct. Anal.
92 (1990), no. 1, 103-135.
[BHTZ] M. Baii.ados, M. Henneaux, C. Teitelboim, and J. Zanelli, Geometry of the 2
hole, Phys. Rev. D (3) 48 (1993), no. 4, 1506-1525
[BTZ] M. Baii.ados, C. Teitelboim, and J. Zanelli, Black hole in three-dimensional space-time,
Phys. Rev. Lett. 69 (1992), no. 13, 1849-1851
[Ba06] J.W. Barrett, A Lorentzian version of the non-commutative geometry of the standard
model of particle physics, J. Math. Phys. 48 (2007) 012303
[Bie95] P. Bieliavsky, Espaces symetriques symplectiques, Thesis, Universite Libre de Bruxelles
[Bie98] __ , Four-dimensional simply connected symplectic symmetric spaces, Geom. Ded-
icata 69 (1998), no. 3, 291-316.
[Bie02] __ , Strict quantization of solvable symmetric spaces, J. Symplectic Geom. 1
(2002), no. 2, 269-320