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UNIVERSE WITHOUT SINGULARITIES 
A GROUP APPROACH TO  DE SITTER COSMOLOGY 
Ignazio Licata 
Isem, Institute for Scientific Methodology, Pa, Italy 
IxtuCyber for Complex Systems, 
via Favorita 9, 91025, Marsala (TP), Italy 
Ignazio.licata@ejtp.org  
Abstract: In the last years the traditional scenario of “Big Bang” has been deeply modified by 
the study of the quantum features of the Universe evolution, proposing again the problem of using 
“local” physical laws on cosmic scale, with particular regard to the cosmological constant role. The 
“group extention” method shows that the De Sitter group univocally generalizes the Poincarè group, 
formally  justifies the cosmological constant use and suggests a new interpretation for Hartle-
Hawking boundary conditions in Quantum Cosmology. 
Key-words: Group Methods in Theoretical Physics; Projective Relativity; De Sitter 
Universe;Quantum Cosmology. 
1. Introduction 
There are strong theoretical coherence reasons which impose to critically reconsider the 
approach to cosmological problem on the whole. The Quantum Cosmology’s main problem is to 
individuate the proper boundary conditions for the Universe’s wave function in the Wheeler-DeWitt 
equation. These conditions have to be such to allow the confrontation between a  probability 
distribution of states and the observed Universe. In particular, it is expected to select a path in the 
configuration space able to solve the still open problems of the Big-Bang traditional scenario: flat 
space, global homogeneity (horizon problem) and the “ruggedness” necessary to explain the tiny 
initial dishomogeneities which have led to the formation of the galactic structures. 
The inflationary cosmology ideas has partly supplied with a solution to the standard model 
wants by introducing the symmetry breaking and phase transition notions which are at the core of 
Quantum Cosmology. The last one also finds its motivation in the necessity to provide with a 
satisfactory physical meaning to the initial singularity problem, unavoidable in GR under the 
condition of the Hawking-Penrose theorem (Hawking & Ellis, 1973). 
The Hartle-Hawking “no-boundary” condition seems to provide a very powerful constraint for 
the Quantum Cosmology main requirements, but appears as an “ad hoc” solution which could be 
deduced by a fundamental approach. Particularly, the mix of topologies  used to conciliate the 
without boundary Universe symmetry with the Big-Bang evolutionary scenario is unsatisfactory. 
We realize that  most part of the Quantum Cosmology problems inherit the uncertainties of the 
Fridman model in GR, so they derive from the euristic use of the local laws on cosmic scale. 
A possible way-out is the Fantappié-Arcidiacono group approach which allows to individuate a 
Universe model without recourse to arbitrary extrapolations of the symmetry groups valid in 
physics. 
The group extension theory naturally finds again the Hartle-Hawking condition on the Universe 
wave function and allows to firmly founding theoretically the Quantum Cosmology. The price to 
pay is a subtle methodological  question on using the GR in cosmology. In fact, in 1952 Fantappié 
pointed out that the problem of the use of local laws to define the cosmological boundary 
conditions is due to the fact that  GR describes matter  in terms of local curvature, but leaves the 
question of space-time global structure indeterminate. It happens because, differently from RR, GR 
has not be built on group base, which thing should be central in building any theory up, especially 
when it aims to express  universally valid statements on physical world, the class of the superb 
theories, how Roger Penrose called them. 
We are going to examine here the foundations of the group extension method (par. 2) and the 
relativity in the De Sitter Universe (par. 3, 4), we introduce the conditions to define matter-fields 
(par.5).In (par.6) we analyze the physical significance of the observers in an istantonic Universe at 
imaginary time, and in (par.7) investigate the physical meaning of an Hartle-Hawking condition in 
an hyper-spherical  universe. 
2. An Erlangen Program for Cosmology  
In 1872 Felix Klein (1849-1925) presented the so-called Erlangen program for geometry, 
centred upon the symmetry transformations group. From 1952, Fantappié, basing on a similar idea 
and in perfect consonance with Relativity spirit, proposed an Erlangen program for physics, where 
a Universe is univocally individuated by a symmetry group which let its physical laws invariant 
(Fantappié,1954, 1959). It has to be underlined that in the theory Universe means any physical 
system characterized by a symmetry group.  
The space-time isotropy and homogeneity principle  with respect to physical laws tells us that 
the physical law concept itself is based upon symmetry.  So the essential idea is to individuate 
physical laws starting from the transformations group which let them invariant. We observe here 
that there are infinite possible transformations group which individuate an isotropic and 
homogeneous space-time. In order to build the next improvements in physics using the group 
extension method, we can follow the path indicated by the two groups we know to be two valid 
description levels of the physical world: the Galilei group and the Lorentz-Poincarè one. It is useful 
to remember that the Galilei group is a particular case of the Lorentz one when ∞→c ,i.e. when  it 
is not made use of the field notion and the interactions velocity is considered to be infinite. Staying 
within a quadrimensional space-time and consequently considering only groups at 10 parameters 
and continuous transformations, Fantappié showed that the Poincaré group can be considered a limit 
case of a broader group depending with continuity on c and another parameter r: the Fantappié 
group; moreover this group cannot be further extended under the condition to stay within a group at 
10 parameters. 
So we have the sequence: 
                              
                                        1031
31 +++ →→ FLG  
Where G is the Galilei group, L the Lorentz one and F the Fantappié final one, from which with 
∞→R , we get the L group. It is shown that such sequence of universes is univocal. 
The Lorentz group can be mathematically interpreted as the group of roto-translations such to 
let that particular object that is the Minkowski space-time invariant. Similarly, the Fantappié group 
is the one of the pentadimensional rotations of a new space-time: the hyper-spherical and at 
constant curvature  De Sitter universe (maximally symmetric). We point out we have obtained the 
De Sitter model without referring to the gravitational interaction, differently from the GR where the 
De Sitter universe is one of the possible solutions of the Einstein equations with cosmological 
constant. From a formal viewpoint we make recourse to pentadimensional rotations because in the 
De Sitter universe there appears a new constant r, which can be interpreted as the Universe radius. 
The group extension mechanism individuates an univocal sequence of symmetry groups; for 
each symmetry group we have a corresponding level of physical world description and a new 
universal constant, so providing the most general boundary conditions and constraining the form of 
the possibile physical laws. The Fantappié group fixes the c  and r constants and defines a new 
relativity for the inertial observers in De Sitter Universe. In this sense, the Theory of Universes- 
based on group extension method- is actually a version of what is sought for in the Holographic 
Principle: the possibility to describe laws and boundaries in a compact and unitary way. 
In 1956 G. Arcidiacono proposed to study the De Sitter S4 absolute universe by means of the 
tangent relative spaces where observers localize and describe the physical events by using the 
Beltrami-Castelnuovo 4P  projective representation in the Projective Special Relativity, PSR 
(Arcidiacono,1956; 1976; 1984).  
We note that we pass from hyper-spherical S4 to its real representation as hyperboloid by means 
of an inverse Wick rotation, rotating τ→it  and associating the great circles on the hyper-sphere 
with a family of geodesics on the hyperboloid. In this way, we get a realization of the Weyl 
principle for defining a Universe model, because it fixes a set of privileged observers (Ellis & 
Williams, 1988). So, the choice of 4P Beltrami-Castelnuovo is equivalent to study a relativity in 
4S . 
3. The Fantappié Group Transformations 
To study the De Sitter 4S  universe according to Beltrami-Castelnuovo representation we have 
to set the projectivities which let the Cayley-Klein interval invariant: 
(1.3)                                 0222222 =+−++ rtczyx . 
The (1.3) meets the time axis in the two 0tt ±=  “singularities”, where crt =0  is the time it 
takes light to run the Universe r radius. In this case the singularities’ meaning is purely geometrical, 
not physical, and they represent the hyperboloid rims (1.3), since the De Sitter universe is lacking in 
“structural” singularities. The 4S  invariant transformations are the 5-dimensional space rotations 
which lead on the 4P  observer’s space the projectivities that let the (1.3) unchanged. 
Let’s introduce the five homogeneous projective coordinates (Weierstrass condition): 
(2.3)                                  2rxx aa = , with   .4,3,2,1,0=a  
The ix  space-time coordinates, with i = 1,2,3,4 are: 
(3.3)                          xx =1 ,    yx =2  , zx =3 , ictx =4 . 
The connection between the (2.3) and (3.3) is given by the relation: 
(4.3)                         0xxrx ii =  
from which, owing to (2.3), we get the inverse relation: 
(5.3)                        arx =0 , axx ii =  , 
where 2222 11 γα −+=+= rxxa ii   , with rx
=α  and 0tt=γ . 
The searched transformation between the two 'O and O  observers consequently has the form: 
(6.3)            
ax = bab xα   with abα  orthogonal matrix. 
Limiting ourselves, just for simplicity reasons, to the 410 ,, xxx variables and following the 
standard method, also used in RR, we get 3 families of transformations: 
A) the space translations along the x axis, given by the  ( 10 , xx ) rotation: 
(7.3)       ϑϑ sincos 01
1 xxx +=  
                 
                +−= ϑsin1
0 xx ϑcos0x  
                 4
4 xx = . 
Using the (4.3) and putting αϑ == r
Ttg , we get the space-time transformations with T 
parameter: 
(8.3)        
'     ,  
1 2' . 
The (8.3) for r  indeterminate, i.e. ∞→r , are reduced to the well-known space translations of 
the classical and relativistic cases, connected by the T parameter. 
B) the T0 parameter time translation, given by the ( 40 , xx ) rotation: 
(9.3)       0004
4 sincos ϑϑ xxx +=  
               +−= 04
0 sinϑxx 00 cosϑx  
               1
1 xx = . 
          
                
Putting γϑ itTitg == 000   we obtain: 
(10.3)      
=     ,     
Also the (10.3), when ∞→r  are reduced to the known cases of classical and relativistic 
physics. 
C) the V parameter inertial transformations, given by the ( 41 , xx ) rotation: 
(11.3)           04011
sincos ϕϕ xxx +=  
                      +−= 01
4 sinϕxx 04 cosϕx  
                       0
0 xx = . 
Putting βϕ icVitg == , here we find again the Lorentz transformations: 
(12.3)             
x  , 
The (A), (B) and (C) transformations form the Fantappié projective group which for two 
variables (x,t) and three parameters (T,T0,V), with T translations and V velocity along x, can be  
written: 
(13.3)          
( )[ ]
( ) ( ) 0
ttrxab
bTctax
αβγβγα
γβγαβ
( )[ ]
( ) ( ) 0
ttrxab
bTtcxa
αβγβγα
αβγαβ
where we have put 221 γα −+=a  and 22 )(1 βγαβ −+−=b , with rx=α , cV=β  and 
0tt=γ . 
For ∞→r  we get a = 1 and 21 β−=b , and from (13.3) we obtain the Poincaré group with 
three parameters (T, T0,V). 
The Fantappié group can be synthesized by a very clear geometrical viewpoint, saying that the 
De Sitter universe at 21 r constant curvature shows an elliptic geometry in its hyper spatial global 
aspect (Gauss-Riemann) and an hyperbolic geometry in its space-time sections (Lobacevskij). 
Making the “natural” r unit of this two geometries tend towards infinity we obtain the parabolic 
geometry of Minkowski flat space. 
4. The Projective Relativity in De Sitter Universe 
The Projective Special Relativity (PSR) widens and contextualizes the relativistic results in De 
Sitter geometry.Just like in any physics there exists a wll-defined connection between mechanics 
and geometry. Therefore the PSR makes use of the notion of observer’s private space, redifining it 
on the basis of a constant curvature.  
In PSR it is introduced a space temporal double scale which connects a ( τχ , ) point of S4 with a 
(x,t) one of P4 by means the (1.3) projective invariant. Given a AB straight line and put as R and S 
the intersections with (1.3), the projective distance is given by the logarithm of the (ABRS) bi-ratio: 
(1.4)             ( ) ( ) ( ) ( ) ( )ASBRBSARtABRStAB ⋅⋅== log2log2 00 . 
From the (1.4) we obtain: 
    (2.4)     
rarctg=χ    and   
00 log
From the (2.4) second one, similar to the Milne’s formula, we can see that the “formal” 
singularities are related to the projective description which depicts a universe with infinite space 
and finite time, whereas the De Sitter one is with finite space and infinite time. It is important to 
underline that such equivalence between an “evolutionary” model and a “stationary” one, 
differently from what is often stated, is purely geometrical and has nothing to do with the physical 
processes, but it deals with the cosmological observer definition.We will speak again about such 
fundamental point further. 
The addition of durations’ new law: 
(3.4)                            
1 tdd
it is obtained by the (10.3) formulae and finds its physical meaning in the appearing of the new 
crt =0 , interpretable as the “universe age” for any 
4P observer family. 
Let us consider a uniform motion with U velocity, given by '' Utx = , by means of Fantappié 
transformations we have a uniform motion with W  velocity given by: 
(4.4)               
( ) ( ) ( )
cUVcVU
−+−++
For the visible universe of the O  observer, inside the light-cone, it is valid the condition 
γα ±=  and a=1 , and the (4.4) can be simplified as: 
(5.4)                
( )( )
cVcUcVU
For V = c then W=c, according to RR, while for U=c we have: 
(6.4)               ( ) ( ) ccVcVccW ≠+−±= 112 2α . 
The  (6.4) expresses the possibility of observing hyper-c velocity in PSR. The outcome is less 
strange than it can seem at first sight, because now the space-time of an observer is defined not only 
by the c constant but also by r, and the light-cone is at variable aperture. In straighter physical terms 
it means that when we observe a far universe region of the crt =0  order,  the cosmic objects’ 
velocity appears to be superior to c value, even if the region belongs to the light-cone of the 
observer’s past. For b=0 we obtain the angular coefficients of the tangents to the (1.3) Cayley-Klein 
invariant starting from a P point of the Beltrami-Castelnuovo projection, which represent the two 
light-cone’s straight lines. Differently from RR, here the light-cone’s angle is not constant and 
depends on the P point according to the formula: 
(7.4)             ( )222 γαϑ += atg . 
From the (7.4) derives the C variation of the light velocity with time: 
(8.4)          
21 γ−
C , with 
from which follows that ∞→C in the  two 0t±  singularities  which fix the limit duration according 
to the addition of durations’ new law (3.4).  
Another remarkable consequence of the projective group is the expansion-collapse law, that is 
the connection between the two singularities. Differentiating the (10.3) and dividing them we obtain 
the  velocities’ variation law for a translation in time: 
 (9.4)         ( ) 002' 11 txttVV γγγ −+=− . 
For 1=γ  and 00 tT =  we have the law of projective expansion valid for 00 <<− tt : 
(10.4)           
= , or also 
If ( )00 == tγ , we can write 
(11.4)              HxtxV == 0 , ( )αβ = , 
where 01 trcH == is the well-known Hubble constant. 
The analogous procedure will be followed for the law of projective collapse valid for 00 tt << , 
with 1−=γ  and 00 tT −= : 
(12.4)               
= , or 
We note that in singularities the expansion-collapse velocity becomes infinite. In PSR such 
process, differently from GR, is not connected to gravitation, but derives from Beltrami-
Castelnuovo geometry. 
From the Fantappié group it also follows a new formula for the Doppler effect: 
(13.4)              ( ) ( ) 2' 11 αββωω ++−= , 
where ω is the frequency. For 1=β , which is V=c, we get nothing but the traditional 
proportionality between distance and frequency, αωω =' . For V=0 there follows a Doppler effect 
depending on distance: 
(14.4)               2' 1 αωω += . 
The z red-shift is defined by '1 ωω=+ z  and the (13.4) becomes: 
(15.4)                ( ) ( ) ( ) 21111 αββ ++−=+ z , 
which was historically introduced- in a 1930 Accademia dei Lincei famous memoir- by 
Castelnuovo to explain the “new” Hubble observations on galactic red-shift. If  we are placed on the 
observer’s light-cone where the (12.4) becomes )1( ααβ −= , the (15.4) will be: 
(16.4)                    ( )α−=+ 111 z . 
The red-shift tends towards infinity for x = r, and hyper- c  velocities are possible if z > 1. 
As everybody would naturally expect, modifying geometry implies, as well as in RR, a deep 
redefinition of mechanics. In PSR, the m mass of a body varies with velocity and distance according 
(17.4)                   bamm 20= .    
From the (17.4) it follows that for a = 0, in singularities, the mass is null, while on the light-
cone, for b = 0, ∞→m . The mass of a body at rest varies with t according to: 
(18.4)                     ( )20 1 γ−= mm , 
from which we deduce that at the initial and final instant, 1±=γ , the mass vanishes. 
Another greatly important outcome (Arcidiacono, 1977) is the relation between m mass and the    
J polar inertia momentum of a body: 
(19.4)                       2mrJ =  
A remarkable consequence is that the universe M mass varies with t: 
(20.4)                         ( ) ( )
MtM +−= γ , 
where M0 is the mass for 0=t , and J  the polar momentum with respect to the observer. 
So the overall picture for an inertial observer in a De Sitter Universe is that of a universe 
coming into existence in a singularity at –t0 time, expanding and collapsing at t0 time and where c 
light velocity is only locally constant. In the initial and final instants the light velocity is infinite and 
the global mass is zero while in the expansion-collapse time it varies according to (20.4). In the 
projective scenario the space flatness is linked to the observer geometry in a universe at constant 
curvature. All this is linked to the fact that in PSR the translations and rotations are indivisible. In 
the singularities there is no “breakdown” of the physical laws because the global space-time 
structure is univocally individuated by the group which is independent of the matter-energy 
distribution. In this case, the singularities in 4P  are – more properly- an horizon of events with a 
natural “cosmic censure” fixed by observers’ geometry. 
5. The Projective Gravitation 
The connection between the metric approach to Einstein gravitation and Fantappié-Arcidiacono 
group one is the aim of Projective General Relativity(PGR), which describes a universe globally at 
constant curvature and locally at variable curvature. It can be done by following the Cartan idea, 
where any  4V  Riemann manifold is associated with an infinite family of Euclidean, pseudo-
Euclidean, non-Euclidean spaces tangent to it in each of its P points. Those spaces’ geometry is 
individuated by a holonomy group. The Cartan connection law links the tangent spaces so as to 
obtain both the 4V  local characteristics (curvature and torsion) and the global ones (holonomy 
group). The GR holonomy group is the one at four dimension rotations, i.e. the Lorentz group. So 
we get a general method which builds a bridgeway up between differential geometry and group 
theory (Pessa, 1973; Arcidiacono, 1986) 
To make a PGR it is introduced the 5V  Riemann manifold which allows as holonomy group the 
De Sitter-Fantappié one, isomorphic to the 5S five-dimensional rotations’ group. The 5V  geometry 
is successively written in terms of Beltrami projective inducted metric for a  anholomonous 4V  
manifold at variable curvature. The Veblen projective connection: 
(1.5)              { }ABCABC =π = ( )BCSCSBBSCAS gggg ∂−∂+∂2
defines a projective translation law which let the field of the Q quadrics invariant in the tangent 
spaces, in each 4V  point, 0== BAAB xxgQ ,where ABg  are the coefficients of the five-dimensional 
metric, the Kx are the homogeneous projective coordinates, and (ABC)=0,1,..,4.From the (1.5) we 
build the projective torsion-curvature tensor: 
(2.5)             SBC
BCDR ππππππ −+∂−∂= . 
So the gravitation equations of Projective General Relativity are: 
(3.5)              ABABAB TRgR χ=− 2
with ABT energy-momentum tensor, and χ Einstein gravitational constant. The (2.5) tensor is 
projectively flat, i.e. when it vanishes we get the De Sitter space at constant curvature. The deep 
link between rotations and translations in 4S  naturally leads  the (3.5) to include the torsion, 
showing an interesting formal analogy with Einstein-Cartan- Sciama-Kibble spin-fluids  theory. The 
construction is analogous to the GR one, but in lieu of the relation between Riemann curvature and 
Minkowski s-t, we get here a curvature-torsion connected to the De Sitter-Fantappié holonomy  
group. It has to be noted that, in concordance with the equivalence principle, the PGR gives a metric 
description of the local gravity, valid for single( i.e., non cosmological) systems. 
It is here proposed again the problem of the relations between local physics and its extension on 
cosmic scale. In fact, if we take the starting expression of standard cosmology based upon GR, i.e. 
let us consider the whole matter of Universe, and transfer it within the ambit of PGR, we can ask 
ourselves if the torsion role, associated to the rotation one, could get a feed-back on the background 
metric, modifying it deeply. Generally, the syntax of a purely group-based theory does not get the 
tools to give an answer, because it is independent from gravity and the hypotheses on ABT . For 
example, Snyder (Snyder, 1947) showed that in a De Sitter space it is introduced an uncertainty 
relation linked to a curvature of the kind: 21 rxx ki ≈∆∆ . Only a third quantization formalism, able 
to take into account the dynamical two-way  inter-relations between local and global, will succeed 
in giving an answer. 
The essential point we have to underline here is that the introduction of a cosmological constant, 
both as additional hypothesis on Einstein equations or via group, is a radical alternative to the 
“machian philosophy” of the GR. 
So, for a Universe without metter-fields we assume the constant curvature as a sort of “pre-
matter” which describes in topological terms the most general conditions for the quantum vacuum. 
Therefore the Einstein equations in the following form are valid: 
(4.5)           ABAB gG Λ=  and ( ) ABAB gRR Λ−= 2 , 
with their essentially physical content, i.e. the deep connection among curvature, radius and matter-
energy’s density vacρ  by means of the cosmological constant: 
       (5.5)            
vac π
6. De Sitter Observers, Singularities and Wick Rotations 
From a quantum viewpoint the 4S  interesting aspect is that it is at imaginary cyclic time and 
without singularities. It means that it is impossible to define on De Sitter a global temporal 
coordinate. So it has an istanton feature, individuated by its Euler topological number which is 2 
(Rajaraman,1982). This leads to a series of formal analogies both with black holes’ quantum 
physics and the theoretical proposals for the “cure” for singularities. 
Let us consider the De Sitter-Castelnuovo metric in real time: 
(1.6)             222
2 11 Ω+��
drdrr
ds , 
where 2222 sin ϕϑϑ ddd +=Ω in polar coordinates. 
As we have seen in PSR, the singularity in Hcr = becomes an horizon of events for any 
observer when it passes to the Euclidean metric with it−→τ : 
(2.6)           ( )222
22 sincos
Ω++= rddrH
dds ττ , 
with a close analogy with the Schwarzschild solution’s case. The τ period is Hπβ 2= ; for the 
observers in De Sitter it implies the possibility to define a temperature, an entropy and an area of 
the horizon, respectively given by: 
(3.6)     1
−== β
Tb ; π
From the (3.6) we get the following fundamental outcome: 
(4.6)        AS
which is the well-known expression of the t’Hooft-Susskind-Bekenstein Holographic 
Principle(Susskind,1995). The (4.6) connects the non-existence of a global temporal coordinate 
with the information accessible to any observer in the De Sitter model. In this way we obtain a deep 
physical explanation for applying  the Weyl Principle in the De Sitter Universe, and sum up that in 
cosmology, as well as in QM, a physical system cannot be fully specified without defining an 
observer. G. Arcidiacono stated that the hyper-spherical Universe is like a book written with seven 
seals ( Apocalypse, 6-11), and consequently two operations are necessary to investigate its physics: 
1) inverse Wick rotation and 2) Beltrami-Castelnuovo representation. That’s the way we can 
completely define a relativity in  De Sitter. 
The association of imaginary time with temperature gets a remarkable physical significance 
which implies some  considerations on the statistical partition function (Hawking, 1975). For our 
aims it will be sufficient to say that such temperature is linked to the (4.6) relation, i.e. to the 
information that an observer spent within his area of events. Which thing has patent implications 
from the dynamical viewpoint, because it is the same as to state that, as well as in Schwarzschild 
black hole’ s case, the De Sitter space and the quantum field defined on it behave as if they were 
immersed in background fluctuations. The transition amplitude from a configuration of a φ  generic 
field  in dttt =− 12  time will be given by the 
iHdte−  matrix element which acts as a ( )1U  group 
transformation of the ( ) ( )timespace UU 11 ⇔ . It means that a transition amplitude on 4S  will appear to 
an observer as the ( )tR scale factor’s variation with H variation rate. 
It makes possible to link the hyper-spherical description with the Big-Bang evolutionary 
scenario and to get rid of the thermodinamic ambiguities which characterize its “beginning” and 
“ending” notions. The last ones have to be re-interpretated as purely quantum dynamics of the 
matter-fields on the hyper-sphere free of singularities.  
7. Physical Considerations for Further Developments 
Such considerations suggest a research program we are going here to shortly delineate ; it 
furthermore develops the analogy between black holes, istantons and De Sitter Universes (see – for 
example – Frolov, Markov, Mukhanov,  1989;Strominger, 1992). It is known that the Hartle-
Hawking proposal of “no-boundary” condition removes the initial singularity and allows to 
calculate the Universe wave function (Hartle-Hawking, 1989). In fact, it is possible – as in the usual 
QFT- to calculate the path integrals by using a Wick rotation as “Euclidization” procedure. In such 
way also the essential characteristics of the inflationary hypotheses are englobed (A. Borde, A. 
Guth and A. Vilenkin, 2003). The derived formalism is similar to that used in the ordinary QM for 
the tunnel effect, an analogy which should explain the physics at its bottom (Vilenkin, 1982; S.W. 
Hawking and I.G. Moss, 1982). 
The group extension method provides this procedure with a solid foundation, because the De 
Sitter space, maximally symmetric and simply connected, is univocally individuated by the group 
structure, and consequently is directly linked to the space-time homogeneity and isotropy principle 
with respect to physical laws. The original Hartle-Hawking formulation operates a mix of 
topologies hardly justified both on the formal level and the conceptual one. The “no-boundary” 
condition is only valid if we works with imaginary time, and the theory does not contain a strict 
logical procedure to explain the passage to real time. This corresponds to a quite vague attempt to 
conciliate an hyper-spherical description at imaginary time with an evolutive one at real time 
according to the traditional Big-Bang scenario.In fact, it has been observed that the Hartle-Hawking 
condition is the same as to substitute a singularity with a “nebulosity”.  
The spontaneous proposal, at this point, is considering the Hartle-Hawking conditions on 
primordial space-time as a consequence of a global charaterization of the hyper-sphere and directly 
developing quantum physics on 4S .Which thing does not contradict the quantum mechanics 
formulation and its fundamental spirit, which is to say the Feynman path integrals. In other words, 
quantum mechanics has not to be applied to cosmology for the Universe smallness at its beginning, 
but because each physical system – without exception- gets quantum histories with amplitude 
interferences. We point out that such view is in perfect consonance with the so-called quantum 
mechanics Many Worlds Interpretation ( Halliwell, 1994). The “by nothing creation” means that we 
cannot “look inside” an istanton (hyper-spherical space), but we have to recourse to an 
“evolutionary” description which separates space from time. The projective methods tell us how to 
do it. 
An analogous problem– to some extent – is that of the Weyl Tensor Hypothesis. Recently, 
Roger Penrose has suggested a condition on the initial singularity that, within the GR, ties entropy 
and gravity and makes a time arrow emerge (Penrose,1989). It is known that the ABCDW  Weyl 
conformal  tensor describes the freedom degrees of the gravitational field. The Penrose Hypothesis 
is that 0→ABCDW in the Big-Bang, while ∞→ABCDW  in the Big-Crunch. The physical reason is 
that in the Universe’s initial state we have an highly uniform matter distribution at low entropy 
( entalpic order), while in Big-Crunch, just like a black hole, we have an high entropy situation. 
This differentiates the two singularities and provides a time arrow. In an hyper-spherical Universe 
there is no “beginning” and “ending”, but only quantum transitions.Consequently, the Penrose 
Hypothesis can only be implemented in terms of projective representation within the ambit of PGR. 
Finally, we can take into consideration the possibility to build a Quantum Field Theory on 4S . 
A QFT, for T tending towards zero, is a limit case of a theory describing some physical fields 
interacting with an external environment at T temperature. Without this external environment we 
could not speak of dechoerence , could not introduce concepts such as like dissipation, chaos, noise 
and, obviously, the possibility to describe phase transitions would vanish too. Therefore, it is of 
paramount importance to write a QFT on De Sitter background metric and then studying it in 
projective representation. If we admit decoherence processes on 4S , it is possible to interpret the 
Weyl Principle as a form of Anthropic Principle: the “classical”  and observable Universes are the 
ones where it can be operated a description at real time. 
In conclusion, it is possible to delineate an alternative, but not incompatible with traditional 
cosmology scenario.The Universe is the quantum configuration of the quantum fields on 4S .Thus 
developing a Quantum Cosmology coincides with developing a Quantum Field Theory on a space 
free of singularities.The Big-Bang is a by vacuum nucleation in an hyper-spherical background at 
imaginary time, and so the concepts of “beginning”, “expansion” and “ending” belong to the space-
time foreground and gain their meaning only by means of a suitable representation which defines a 
family of cosmological observers. 
Acknowledgements: 
I owe my knowledge  of the group extension method to the regretted Prof. G. Arcidiacono 
(1927 – 1998), during our intense discussions while strolling throughout Rome.  
Special thanks to my friends E. Pessa and L. Chiatti for the rich exchange of viewpoints and e-
mails. 
 References 
Arcidiacono, G.(1956),  Rend. Accad. Lincei,20, 4 
Arcidiacono, G. (1976), Gen. Rel. And Grav., 7, 885 
Arcidiacono, G.(1977), Gen. Rel. And Grav.,7, 865 
Arcidiacono, G.(1984), in De Sabbata,V. & T.M.Karade (eds), Relativistic Astrophysics and      
Cosmology, World Scientific, Singapore 
Arcidiacono, G. (1986), Projective Relativity,Cosmology and Gravitation, 
Hadr.Press,Cambridge,USA 
Borde, A. , Guth,A., Vilenkin, A. (2003),Phys. Rev. Lett. 90 
Ellis, G.F.R. & Wiliams,R. (1988), Flat and Curved Space-Times, Clarendon Press 
Fantappié, L. (1954), Rend. Accad. Lincei,17,5 
Fantappié, L. (1959), Collectanea Mathematica,XI, 77 
Frolov, V.P., Markov,M.A., Mukhanov,V.F.(1989), Phys.Lett.,B216,272 
Halliwell,J.J. (1994), in Greenberger,D. & Zeilinger,A.(eds), Fundamental Problems in 
Quantum Theory, New York Academy of Sciences,NY 
Hartle, J.B. & Hawking,S.W. (1983), Phys.Rev.D,28,12 
Hawking,S.W. & Ellis, G.F.R.,(1973) The Large Scale Structure of Space-Time,Cambridge 
Univ.Press 
Hawking,S.W. (1975), Commun.Math.Phys.,43 
Hawking, S.W. & Moss, I.G. (1982), Phys.Lett.,B110,35 
Pessa,E. (1973), Collectanea Mathematica,XXIV,2 
Rajaraman,R.(1982), Solitons and Istantons,North-Holland Publ.,NY 
Snyder, H.S. (1947), Phys.Rev., 51,38 
Strominger, A. (1992), Phys.Rev.D,46,10 
Susskind, L. (1995), Jour.Math.Phys.,36 
Vilenkin, A. (1982), Phys.Lett.,117B,1.

  In the last years the traditional scenario of Big Bang has been deeply
modified by the study of the quantum features of the Universe evolution,
proposing again the problem of using local physical laws on cosmic scale, with
particular regard to the cosmological constant role. The group extention method
shows that the De Sitter group univocally generalizes the Poincare group,
formally justifies the cosmological constant use and suggests a new
interpretation for Hartle-Hawking boundary conditions in Quantum Cosmology.

Introduction 
There are strong theoretical coherence reasons which impose to critically reconsider the 
approach to cosmological problem on the whole. The Quantum Cosmology’s main problem is to 
individuate the proper boundary conditions for the Universe’s wave function in the Wheeler-DeWitt 
equation. These conditions have to be such to allow the confrontation between a  probability 
distribution of states and the observed Universe. In particular, it is expected to select a path in the 
configuration space able to solve the still open problems of the Big-Bang traditional scenario: flat 
space, global homogeneity (horizon problem) and the “ruggedness” necessary to explain the tiny 
initial dishomogeneities which have led to the formation of the galactic structures. 
The inflationary cosmology ideas has partly supplied with a solution to the standard model 
wants by introducing the symmetry breaking and phase transition notions which are at the core of 
Quantum Cosmology. The last one also finds its motivation in the necessity to provide with a 
satisfactory physical meaning to the initial singularity problem, unavoidable in GR under the 
condition of the Hawking-Penrose theorem (Hawking & Ellis, 1973). 
The Hartle-Hawking “no-boundary” condition seems to provide a very powerful constraint for 
the Quantum Cosmology main requirements, but appears as an “ad hoc” solution which could be 
deduced by a fundamental approach. Particularly, the mix of topologies  used to conciliate the 
without boundary Universe symmetry with the Big-Bang evolutionary scenario is unsatisfactory. 
We realize that  most part of the Quantum Cosmology problems inherit the uncertainties of the 
Fridman model in GR, so they derive from the euristic use of the local laws on cosmic scale. 
A possible way-out is the Fantappié-Arcidiacono group approach which allows to individuate a 
Universe model without recourse to arbitrary extrapolations of the symmetry groups valid in 
physics. 
The group extension theory naturally finds again the Hartle-Hawking condition on the Universe 
wave function and allows to firmly founding theoretically the Quantum Cosmology. The price to 
pay is a subtle methodological  question on using the GR in cosmology. In fact, in 1952 Fantappié 
pointed out that the problem of the use of local laws to define the cosmological boundary 
conditions is due to the fact that  GR describes matter  in terms of local curvature, but leaves the 
question of space-time global structure indeterminate. It happens because, differently from RR, GR 
has not be built on group base, which thing should be central in building any theory up, especially 
when it aims to express  universally valid statements on physical world, the class of the superb 
theories, how Roger Penrose called them. 
We are going to examine here the foundations of the group extension method (par. 2) and the 
relativity in the De Sitter Universe (par. 3, 4), we introduce the conditions to define matter-fields 
(par.5).In (par.6) we analyze the physical significance of the observers in an istantonic Universe at 
imaginary time, and in (par.7) investigate the physical meaning of an Hartle-Hawking condition in 
an hyper-spherical  universe. 
2. An Erlangen Program for Cosmology  
In 1872 Felix Klein (1849-1925) presented the so-called Erlangen program for geometry, 
centred upon the symmetry transformations group. From 1952, Fantappié, basing on a similar idea 
and in perfect consonance with Relativity spirit, proposed an Erlangen program for physics, where 
a Universe is univocally individuated by a symmetry group which let its physical laws invariant 
(Fantappié,1954, 1959). It has to be underlined that in the theory Universe means any physical 
system characterized by a symmetry group.  
The space-time isotropy and homogeneity principle  with respect to physical laws tells us that 
the physical law concept itself is based upon symmetry.  So the essential idea is to individuate 
physical laws starting from the transformations group which let them invariant. We observe here 
that there are infinite possible transformations group which individuate an isotropic and 
homogeneous space-time. In order to build the next improvements in physics using the group 
extension method, we can follow the path indicated by the two groups we know to be two valid 
description levels of the physical world: the Galilei group and the Lorentz-Poincarè one. It is useful 
to remember that the Galilei group is a particular case of the Lorentz one when ∞→c ,i.e. when  it 
is not made use of the field notion and the interactions velocity is considered to be infinite. Staying 
within a quadrimensional space-time and consequently considering only groups at 10 parameters 
and continuous transformations, Fantappié showed that the Poincaré group can be considered a limit 
case of a broader group depending with continuity on c and another parameter r: the Fantappié 
group; moreover this group cannot be further extended under the condition to stay within a group at 
10 parameters. 
So we have the sequence: 
                              
                                        1031
31 +++ →→ FLG  
Where G is the Galilei group, L the Lorentz one and F the Fantappié final one, from which with 
∞→R , we get the L group. It is shown that such sequence of universes is univocal. 
The Lorentz group can be mathematically interpreted as the group of roto-translations such to 
let that particular object that is the Minkowski space-time invariant. Similarly, the Fantappié group 
is the one of the pentadimensional rotations of a new space-time: the hyper-spherical and at 
constant curvature  De Sitter universe (maximally symmetric). We point out we have obtained the 
De Sitter model without referring to the gravitational interaction, differently from the GR where the 
De Sitter universe is one of the possible solutions of the Einstein equations with cosmological 
constant. From a formal viewpoint we make recourse to pentadimensional rotations because in the 
De Sitter universe there appears a new constant r, which can be interpreted as the Universe radius. 
The group extension mechanism individuates an univocal sequence of symmetry groups; for 
each symmetry group we have a corresponding level of physical world description and a new 
universal constant, so providing the most general boundary conditions and constraining the form of 
the possibile physical laws. The Fantappié group fixes the c  and r constants and defines a new 
relativity for the inertial observers in De Sitter Universe. In this sense, the Theory of Universes- 
based on group extension method- is actually a version of what is sought for in the Holographic 
Principle: the possibility to describe laws and boundaries in a compact and unitary way. 
In 1956 G. Arcidiacono proposed to study the De Sitter S4 absolute universe by means of the 
tangent relative spaces where observers localize and describe the physical events by using the 
Beltrami-Castelnuovo 4P  projective representation in the Projective Special Relativity, PSR 
(Arcidiacono,1956; 1976; 1984).  
We note that we pass from hyper-spherical S4 to its real representation as hyperboloid by means 
of an inverse Wick rotation, rotating τ→it  and associating the great circles on the hyper-sphere 
with a family of geodesics on the hyperboloid. In this way, we get a realization of the Weyl 
principle for defining a Universe model, because it fixes a set of privileged observers (Ellis & 
Williams, 1988). So, the choice of 4P Beltrami-Castelnuovo is equivalent to study a relativity in 
4S . 
3. The Fantappié Group Transformations 
To study the De Sitter 4S  universe according to Beltrami-Castelnuovo representation we have 
to set the projectivities which let the Cayley-Klein interval invariant: 
(1.3)                                 0222222 =+−++ rtczyx . 
The (1.3) meets the time axis in the two 0tt ±=  “singularities”, where crt =0  is the time it 
takes light to run the Universe r radius. In this case the singularities’ meaning is purely geometrical, 
not physical, and they represent the hyperboloid rims (1.3), since the De Sitter universe is lacking in 
“structural” singularities. The 4S  invariant transformations are the 5-dimensional space rotations 
which lead on the 4P  observer’s space the projectivities that let the (1.3) unchanged. 
Let’s introduce the five homogeneous projective coordinates (Weierstrass condition): 
(2.3)                                  2rxx aa = , with   .4,3,2,1,0=a  
The ix  space-time coordinates, with i = 1,2,3,4 are: 
(3.3)                          xx =1 ,    yx =2  , zx =3 , ictx =4 . 
The connection between the (2.3) and (3.3) is given by the relation: 
(4.3)                         0xxrx ii =  
from which, owing to (2.3), we get the inverse relation: 
(5.3)                        arx =0 , axx ii =  , 
where 2222 11 γα −+=+= rxxa ii   , with rx
=α  and 0tt=γ . 
The searched transformation between the two 'O and O  observers consequently has the form: 
(6.3)            
ax = bab xα   with abα  orthogonal matrix. 
Limiting ourselves, just for simplicity reasons, to the 410 ,, xxx variables and following the 
standard method, also used in RR, we get 3 families of transformations: 
A) the space translations along the x axis, given by the  ( 10 , xx ) rotation: 
(7.3)       ϑϑ sincos 01
1 xxx +=  
                 
                +−= ϑsin1
0 xx ϑcos0x  
                 4
4 xx = . 
Using the (4.3) and putting αϑ == r
Ttg , we get the space-time transformations with T 
parameter: 
(8.3)        
'     ,  
1 2' . 
The (8.3) for r  indeterminate, i.e. ∞→r , are reduced to the well-known space translations of 
the classical and relativistic cases, connected by the T parameter. 
B) the T0 parameter time translation, given by the ( 40 , xx ) rotation: 
(9.3)       0004
4 sincos ϑϑ xxx +=  
               +−= 04
0 sinϑxx 00 cosϑx  
               1
1 xx = . 
          
                
Putting γϑ itTitg == 000   we obtain: 
(10.3)      
=     ,     
Also the (10.3), when ∞→r  are reduced to the known cases of classical and relativistic 
physics. 
C) the V parameter inertial transformations, given by the ( 41 , xx ) rotation: 
(11.3)           04011
sincos ϕϕ xxx +=  
                      +−= 01
4 sinϕxx 04 cosϕx  
                       0
0 xx = . 
Putting βϕ icVitg == , here we find again the Lorentz transformations: 
(12.3)             
x  , 
The (A), (B) and (C) transformations form the Fantappié projective group which for two 
variables (x,t) and three parameters (T,T0,V), with T translations and V velocity along x, can be  
written: 
(13.3)          
( )[ ]
( ) ( ) 0
ttrxab
bTctax
αβγβγα
γβγαβ
( )[ ]
( ) ( ) 0
ttrxab
bTtcxa
αβγβγα
αβγαβ
where we have put 221 γα −+=a  and 22 )(1 βγαβ −+−=b , with rx=α , cV=β  and 
0tt=γ . 
For ∞→r  we get a = 1 and 21 β−=b , and from (13.3) we obtain the Poincaré group with 
three parameters (T, T0,V). 
The Fantappié group can be synthesized by a very clear geometrical viewpoint, saying that the 
De Sitter universe at 21 r constant curvature shows an elliptic geometry in its hyper spatial global 
aspect (Gauss-Riemann) and an hyperbolic geometry in its space-time sections (Lobacevskij). 
Making the “natural” r unit of this two geometries tend towards infinity we obtain the parabolic 
geometry of Minkowski flat space. 
4. The Projective Relativity in De Sitter Universe 
The Projective Special Relativity (PSR) widens and contextualizes the relativistic results in De 
Sitter geometry.Just like in any physics there exists a wll-defined connection between mechanics 
and geometry. Therefore the PSR makes use of the notion of observer’s private space, redifining it 
on the basis of a constant curvature.  
In PSR it is introduced a space temporal double scale which connects a ( τχ , ) point of S4 with a 
(x,t) one of P4 by means the (1.3) projective invariant. Given a AB straight line and put as R and S 
the intersections with (1.3), the projective distance is given by the logarithm of the (ABRS) bi-ratio: 
(1.4)             ( ) ( ) ( ) ( ) ( )ASBRBSARtABRStAB ⋅⋅== log2log2 00 . 
From the (1.4) we obtain: 
    (2.4)     
rarctg=χ    and   
00 log
From the (2.4) second one, similar to the Milne’s formula, we can see that the “formal” 
singularities are related to the projective description which depicts a universe with infinite space 
and finite time, whereas the De Sitter one is with finite space and infinite time. It is important to 
underline that such equivalence between an “evolutionary” model and a “stationary” one, 
differently from what is often stated, is purely geometrical and has nothing to do with the physical 
processes, but it deals with the cosmological observer definition.We will speak again about such 
fundamental point further. 
The addition of durations’ new law: 
(3.4)                            
1 tdd
it is obtained by the (10.3) formulae and finds its physical meaning in the appearing of the new 
crt =0 , interpretable as the “universe age” for any 
4P observer family. 
Let us consider a uniform motion with U velocity, given by '' Utx = , by means of Fantappié 
transformations we have a uniform motion with W  velocity given by: 
(4.4)               
( ) ( ) ( )
cUVcVU
−+−++
For the visible universe of the O  observer, inside the light-cone, it is valid the condition 
γα ±=  and a=1 , and the (4.4) can be simplified as: 
(5.4)                
( )( )
cVcUcVU
For V = c then W=c, according to RR, while for U=c we have: 
(6.4)               ( ) ( ) ccVcVccW ≠+−±= 112 2α . 
The  (6.4) expresses the possibility of observing hyper-c velocity in PSR. The outcome is less 
strange than it can seem at first sight, because now the space-time of an observer is defined not only 
by the c constant but also by r, and the light-cone is at variable aperture. In straighter physical terms 
it means that when we observe a far universe region of the crt =0  order,  the cosmic objects’ 
velocity appears to be superior to c value, even if the region belongs to the light-cone of the 
observer’s past. For b=0 we obtain the angular coefficients of the tangents to the (1.3) Cayley-Klein 
invariant starting from a P point of the Beltrami-Castelnuovo projection, which represent the two 
light-cone’s straight lines. Differently from RR, here the light-cone’s angle is not constant and 
depends on the P point according to the formula: 
(7.4)             ( )222 γαϑ += atg . 
From the (7.4) derives the C variation of the light velocity with time: 
(8.4)          
21 γ−
C , with 
from which follows that ∞→C in the  two 0t±  singularities  which fix the limit duration according 
to the addition of durations’ new law (3.4).  
Another remarkable consequence of the projective group is the expansion-collapse law, that is 
the connection between the two singularities. Differentiating the (10.3) and dividing them we obtain 
the  velocities’ variation law for a translation in time: 
 (9.4)         ( ) 002' 11 txttVV γγγ −+=− . 
For 1=γ  and 00 tT =  we have the law of projective expansion valid for 00 <<− tt : 
(10.4)           
= , or also 
If ( )00 == tγ , we can write 
(11.4)              HxtxV == 0 , ( )αβ = , 
where 01 trcH == is the well-known Hubble constant. 
The analogous procedure will be followed for the law of projective collapse valid for 00 tt << , 
with 1−=γ  and 00 tT −= : 
(12.4)               
= , or 
We note that in singularities the expansion-collapse velocity becomes infinite. In PSR such 
process, differently from GR, is not connected to gravitation, but derives from Beltrami-
Castelnuovo geometry. 
From the Fantappié group it also follows a new formula for the Doppler effect: 
(13.4)              ( ) ( ) 2' 11 αββωω ++−= , 
where ω is the frequency. For 1=β , which is V=c, we get nothing but the traditional 
proportionality between distance and frequency, αωω =' . For V=0 there follows a Doppler effect 
depending on distance: 
(14.4)               2' 1 αωω += . 
The z red-shift is defined by '1 ωω=+ z  and the (13.4) becomes: 
(15.4)                ( ) ( ) ( ) 21111 αββ ++−=+ z , 
which was historically introduced- in a 1930 Accademia dei Lincei famous memoir- by 
Castelnuovo to explain the “new” Hubble observations on galactic red-shift. If  we are placed on the 
observer’s light-cone where the (12.4) becomes )1( ααβ −= , the (15.4) will be: 
(16.4)                    ( )α−=+ 111 z . 
The red-shift tends towards infinity for x = r, and hyper- c  velocities are possible if z > 1. 
As everybody would naturally expect, modifying geometry implies, as well as in RR, a deep 
redefinition of mechanics. In PSR, the m mass of a body varies with velocity and distance according 
(17.4)                   bamm 20= .    
From the (17.4) it follows that for a = 0, in singularities, the mass is null, while on the light-
cone, for b = 0, ∞→m . The mass of a body at rest varies with t according to: 
(18.4)                     ( )20 1 γ−= mm , 
from which we deduce that at the initial and final instant, 1±=γ , the mass vanishes. 
Another greatly important outcome (Arcidiacono, 1977) is the relation between m mass and the    
J polar inertia momentum of a body: 
(19.4)                       2mrJ =  
A remarkable consequence is that the universe M mass varies with t: 
(20.4)                         ( ) ( )
MtM +−= γ , 
where M0 is the mass for 0=t , and J  the polar momentum with respect to the observer. 
So the overall picture for an inertial observer in a De Sitter Universe is that of a universe 
coming into existence in a singularity at –t0 time, expanding and collapsing at t0 time and where c 
light velocity is only locally constant. In the initial and final instants the light velocity is infinite and 
the global mass is zero while in the expansion-collapse time it varies according to (20.4). In the 
projective scenario the space flatness is linked to the observer geometry in a universe at constant 
curvature. All this is linked to the fact that in PSR the translations and rotations are indivisible. In 
the singularities there is no “breakdown” of the physical laws because the global space-time 
structure is univocally individuated by the group which is independent of the matter-energy 
distribution. In this case, the singularities in 4P  are – more properly- an horizon of events with a 
natural “cosmic censure” fixed by observers’ geometry. 
5. The Projective Gravitation 
The connection between the metric approach to Einstein gravitation and Fantappié-Arcidiacono 
group one is the aim of Projective General Relativity(PGR), which describes a universe globally at 
constant curvature and locally at variable curvature. It can be done by following the Cartan idea, 
where any  4V  Riemann manifold is associated with an infinite family of Euclidean, pseudo-
Euclidean, non-Euclidean spaces tangent to it in each of its P points. Those spaces’ geometry is 
individuated by a holonomy group. The Cartan connection law links the tangent spaces so as to 
obtain both the 4V  local characteristics (curvature and torsion) and the global ones (holonomy 
group). The GR holonomy group is the one at four dimension rotations, i.e. the Lorentz group. So 
we get a general method which builds a bridgeway up between differential geometry and group 
theory (Pessa, 1973; Arcidiacono, 1986) 
To make a PGR it is introduced the 5V  Riemann manifold which allows as holonomy group the 
De Sitter-Fantappié one, isomorphic to the 5S five-dimensional rotations’ group. The 5V  geometry 
is successively written in terms of Beltrami projective inducted metric for a  anholomonous 4V  
manifold at variable curvature. The Veblen projective connection: 
(1.5)              { }ABCABC =π = ( )BCSCSBBSCAS gggg ∂−∂+∂2
defines a projective translation law which let the field of the Q quadrics invariant in the tangent 
spaces, in each 4V  point, 0== BAAB xxgQ ,where ABg  are the coefficients of the five-dimensional 
metric, the Kx are the homogeneous projective coordinates, and (ABC)=0,1,..,4.From the (1.5) we 
build the projective torsion-curvature tensor: 
(2.5)             SBC
BCDR ππππππ −+∂−∂= . 
So the gravitation equations of Projective General Relativity are: 
(3.5)              ABABAB TRgR χ=− 2
with ABT energy-momentum tensor, and χ Einstein gravitational constant. The (2.5) tensor is 
projectively flat, i.e. when it vanishes we get the De Sitter space at constant curvature. The deep 
link between rotations and translations in 4S  naturally leads  the (3.5) to include the torsion, 
showing an interesting formal analogy with Einstein-Cartan- Sciama-Kibble spin-fluids  theory. The 
construction is analogous to the GR one, but in lieu of the relation between Riemann curvature and 
Minkowski s-t, we get here a curvature-torsion connected to the De Sitter-Fantappié holonomy  
group. It has to be noted that, in concordance with the equivalence principle, the PGR gives a metric 
description of the local gravity, valid for single( i.e., non cosmological) systems. 
It is here proposed again the problem of the relations between local physics and its extension on 
cosmic scale. In fact, if we take the starting expression of standard cosmology based upon GR, i.e. 
let us consider the whole matter of Universe, and transfer it within the ambit of PGR, we can ask 
ourselves if the torsion role, associated to the rotation one, could get a feed-back on the background 
metric, modifying it deeply. Generally, the syntax of a purely group-based theory does not get the 
tools to give an answer, because it is independent from gravity and the hypotheses on ABT . For 
example, Snyder (Snyder, 1947) showed that in a De Sitter space it is introduced an uncertainty 
relation linked to a curvature of the kind: 21 rxx ki ≈∆∆ . Only a third quantization formalism, able 
to take into account the dynamical two-way  inter-relations between local and global, will succeed 
in giving an answer. 
The essential point we have to underline here is that the introduction of a cosmological constant, 
both as additional hypothesis on Einstein equations or via group, is a radical alternative to the 
“machian philosophy” of the GR. 
So, for a Universe without metter-fields we assume the constant curvature as a sort of “pre-
matter” which describes in topological terms the most general conditions for the quantum vacuum. 
Therefore the Einstein equations in the following form are valid: 
(4.5)           ABAB gG Λ=  and ( ) ABAB gRR Λ−= 2 , 
with their essentially physical content, i.e. the deep connection among curvature, radius and matter-
energy’s density vacρ  by means of the cosmological constant: 
       (5.5)            
vac π
6. De Sitter Observers, Singularities and Wick Rotations 
From a quantum viewpoint the 4S  interesting aspect is that it is at imaginary cyclic time and 
without singularities. It means that it is impossible to define on De Sitter a global temporal 
coordinate. So it has an istanton feature, individuated by its Euler topological number which is 2 
(Rajaraman,1982). This leads to a series of formal analogies both with black holes’ quantum 
physics and the theoretical proposals for the “cure” for singularities. 
Let us consider the De Sitter-Castelnuovo metric in real time: 
(1.6)             222
2 11 Ω+��
drdrr
ds , 
where 2222 sin ϕϑϑ ddd +=Ω in polar coordinates. 
As we have seen in PSR, the singularity in Hcr = becomes an horizon of events for any 
observer when it passes to the Euclidean metric with it−→τ : 
(2.6)           ( )222
22 sincos
Ω++= rddrH
dds ττ , 
with a close analogy with the Schwarzschild solution’s case. The τ period is Hπβ 2= ; for the 
observers in De Sitter it implies the possibility to define a temperature, an entropy and an area of 
the horizon, respectively given by: 
(3.6)     1
−== β
Tb ; π
From the (3.6) we get the following fundamental outcome: 
(4.6)        AS
which is the well-known expression of the t’Hooft-Susskind-Bekenstein Holographic 
Principle(Susskind,1995). The (4.6) connects the non-existence of a global temporal coordinate 
with the information accessible to any observer in the De Sitter model. In this way we obtain a deep 
physical explanation for applying  the Weyl Principle in the De Sitter Universe, and sum up that in 
cosmology, as well as in QM, a physical system cannot be fully specified without defining an 
observer. G. Arcidiacono stated that the hyper-spherical Universe is like a book written with seven 
seals ( Apocalypse, 6-11), and consequently two operations are necessary to investigate its physics: 
1) inverse Wick rotation and 2) Beltrami-Castelnuovo representation. That’s the way we can 
completely define a relativity in  De Sitter. 
The association of imaginary time with temperature gets a remarkable physical significance 
which implies some  considerations on the statistical partition function (Hawking, 1975). For our 
aims it will be sufficient to say that such temperature is linked to the (4.6) relation, i.e. to the 
information that an observer spent within his area of events. Which thing has patent implications 
from the dynamical viewpoint, because it is the same as to state that, as well as in Schwarzschild 
black hole’ s case, the De Sitter space and the quantum field defined on it behave as if they were 
immersed in background fluctuations. The transition amplitude from a configuration of a φ  generic 
field  in dttt =− 12  time will be given by the 
iHdte−  matrix element which acts as a ( )1U  group 
transformation of the ( ) ( )timespace UU 11 ⇔ . It means that a transition amplitude on 4S  will appear to 
an observer as the ( )tR scale factor’s variation with H variation rate. 
It makes possible to link the hyper-spherical description with the Big-Bang evolutionary 
scenario and to get rid of the thermodinamic ambiguities which characterize its “beginning” and 
“ending” notions. The last ones have to be re-interpretated as purely quantum dynamics of the 
matter-fields on the hyper-sphere free of singularities.  
7. Physical Considerations for Further Developments 
Such considerations suggest a research program we are going here to shortly delineate ; it 
furthermore develops the analogy between black holes, istantons and De Sitter Universes (see – for 
example – Frolov, Markov, Mukhanov,  1989;Strominger, 1992). It is known that the Hartle-
Hawking proposal of “no-boundary” condition removes the initial singularity and allows to 
calculate the Universe wave function (Hartle-Hawking, 1989). In fact, it is possible – as in the usual 
QFT- to calculate the path integrals by using a Wick rotation as “Euclidization” procedure. In such 
way also the essential characteristics of the inflationary hypotheses are englobed (A. Borde, A. 
Guth and A. Vilenkin, 2003). The derived formalism is similar to that used in the ordinary QM for 
the tunnel effect, an analogy which should explain the physics at its bottom (Vilenkin, 1982; S.W. 
Hawking and I.G. Moss, 1982). 
The group extension method provides this procedure with a solid foundation, because the De 
Sitter space, maximally symmetric and simply connected, is univocally individuated by the group 
structure, and consequently is directly linked to the space-time homogeneity and isotropy principle 
with respect to physical laws. The original Hartle-Hawking formulation operates a mix of 
topologies hardly justified both on the formal level and the conceptual one. The “no-boundary” 
condition is only valid if we works with imaginary time, and the theory does not contain a strict 
logical procedure to explain the passage to real time. This corresponds to a quite vague attempt to 
conciliate an hyper-spherical description at imaginary time with an evolutive one at real time 
according to the traditional Big-Bang scenario.In fact, it has been observed that the Hartle-Hawking 
condition is the same as to substitute a singularity with a “nebulosity”.  
The spontaneous proposal, at this point, is considering the Hartle-Hawking conditions on 
primordial space-time as a consequence of a global charaterization of the hyper-sphere and directly 
developing quantum physics on 4S .Which thing does not contradict the quantum mechanics 
formulation and its fundamental spirit, which is to say the Feynman path integrals. In other words, 
quantum mechanics has not to be applied to cosmology for the Universe smallness at its beginning, 
but because each physical system – without exception- gets quantum histories with amplitude 
interferences. We point out that such view is in perfect consonance with the so-called quantum 
mechanics Many Worlds Interpretation ( Halliwell, 1994). The “by nothing creation” means that we 
cannot “look inside” an istanton (hyper-spherical space), but we have to recourse to an 
“evolutionary” description which separates space from time. The projective methods tell us how to 
do it. 
An analogous problem– to some extent – is that of the Weyl Tensor Hypothesis. Recently, 
Roger Penrose has suggested a condition on the initial singularity that, within the GR, ties entropy 
and gravity and makes a time arrow emerge (Penrose,1989). It is known that the ABCDW  Weyl 
conformal  tensor describes the freedom degrees of the gravitational field. The Penrose Hypothesis 
is that 0→ABCDW in the Big-Bang, while ∞→ABCDW  in the Big-Crunch. The physical reason is 
that in the Universe’s initial state we have an highly uniform matter distribution at low entropy 
( entalpic order), while in Big-Crunch, just like a black hole, we have an high entropy situation. 
This differentiates the two singularities and provides a time arrow. In an hyper-spherical Universe 
there is no “beginning” and “ending”, but only quantum transitions.Consequently, the Penrose 
Hypothesis can only be implemented in terms of projective representation within the ambit of PGR. 
Finally, we can take into consideration the possibility to build a Quantum Field Theory on 4S . 
A QFT, for T tending towards zero, is a limit case of a theory describing some physical fields 
interacting with an external environment at T temperature. Without this external environment we 
could not speak of dechoerence , could not introduce concepts such as like dissipation, chaos, noise 
and, obviously, the possibility to describe phase transitions would vanish too. Therefore, it is of 
paramount importance to write a QFT on De Sitter background metric and then studying it in 
projective representation. If we admit decoherence processes on 4S , it is possible to interpret the 
Weyl Principle as a form of Anthropic Principle: the “classical”  and observable Universes are the 
ones where it can be operated a description at real time. 
In conclusion, it is possible to delineate an alternative, but not incompatible with traditional 
cosmology scenario.The Universe is the quantum configuration of the quantum fields on 4S .Thus 
developing a Quantum Cosmology coincides with developing a Quantum Field Theory on a space 
free of singularities.The Big-Bang is a by vacuum nucleation in an hyper-spherical background at 
imaginary time, and so the concepts of “beginning”, “expansion” and “ending” belong to the space-
time foreground and gain their meaning only by means of a suitable representation which defines a 
family of cosmological observers. 
Acknowledgements: 
I owe my knowledge  of the group extension method to the regretted Prof. G. Arcidiacono 
(1927 – 1998), during our intense discussions while strolling throughout Rome.  
Special thanks to my friends E. Pessa and L. Chiatti for the rich exchange of viewpoints and e-
mails. 
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