Astrophysics and particle physics tend to join together through various problems linked to the unification of the fundamental interactions of nature and the specificity of gravity with respect to the other interactions, with the ultimate goal to elaborate a new physics beyond General Relativity and the Standard Model of particle physics. The Standard Model which unifies the weak/electromagnetic/strong interactions in a single frame is based upon the common character of those interactions, a quantum field theory description of gauge symmetries within the 4-dimension space-time of special relativity. Gravitation is independently described by General Relativity, a metric theory connecting the geometry of space-time with the density of impulsion-energy. Nuclei, atoms and molecules are described by quantum theory, whereas the universe at large scale is described by General Relativity. Now in spite of its successes, General Relativity cannot be the ultimate description of gravity as it exhibits an embarrassing singularity at t = 0 and does not take into account the quantum effects, predominant in the primordial universe and at very small scales; moreover the significance of the cosmological constant L is unclear. On the other hand the Standard Model accounts for the bestiary of fields and particles of ordinary matter but cannot be extended so as to include gravitation which cannot be handled through the renormalization procedures of quantum field theory. The minimal coupling between gravitation and particle physics is made by replacing, whenever needed, the pseudo Euclidean Minkowski metrics 
of special relativity by the 
metrics which expresses the curvature of space-time by its material content.
Extra dimensions in addition to the usual 4 dimensions of space-time are a common ingredient of the theoretical attempts aiming at combining gravitation with the other interactions [1-5]. Historically, the Kaluza-Klein approach in the 1920’s assumes a 5th dimension, which is enough to account for the tensor field of gravitation and for the vector field of electromagnetism. In addition, it predicts an extra scalar field and it is a common feature of unification theories to predict the existence of additional fields and particles. However, the later evidence of weak and strong interactions has shown that the original KaluzaKlein model was not sufficient for a comprehensive description of the fundamental interactions. The idea of introducing additional dimensions in order to account for all the known fundamental interactions including gravitation has flourished from the 1980’s, e.g. with the string theories which require at least 10 dimensions to be consistent [
According to certain models, the size of those extra dimensions is very small, of the order of Planck’s length (10−35 m). The common image is that of a plain seen from the top of a hill: if the hill is high enough, the height of the grass is not perceived by the eye and the plain seems to be a 2-dimension domain. The extra dimensions are supposed to be compactified and wrapped so that they are hidden to us. According to other works, they could be much larger, up to 0.1 mm [
The goal of this paper, essentially a phenomenological approach, is to show the possibility, through a relatively simple mathematical apparatus, that a few basic assumptions on the geometrical properties of a 10-dimension space-time allow deriving the main features of the fundamental interactions: types and symmetries of the particles (hadrons and leptons), types and symmetries of the gauge fields. Moreover, this description, combined with gravitation and with the postulate of a minimal Lagrangian, predicts the existence of additional fields resulting from the coupling of gravitation with the other force fields; those hypothetical extra fields do not act the same way upon hadrons and leptons, thus inducing a violation of the equivalence principle.
locally invariant under the Lorentz group
. On another hand a curved surface of dimension d can be embedded inside a flat space with 
dimensions; hence 
can be embedded inside a flat space with 10 dimensions. We will hereafter make the basic assumption that our physical universe is a flat (pseudo) Euclidian space with 10 dimensions
, invariant under some symmetry group
.
be the tangent space to 
in any point M of
, and 
be the orthogonal space to 
in M. 
is locally invariant under
, i.e. 
is invariant under
. 
is a 4- dimension flat space in which one can define a system of 4 (pseudo) orthonormal coordinates 
with µ ranging from 0 to 3, i.e. one time coordinate 
and 3 space coordinates
. Given such a local reference frame for
, one can find an infinity of other equivalent reference frames by applying to 
any combination of rotations and Lorentz transformations while conserving the invariance of the pseudo Euclidian norm 
[
as resulting from a force field which is identified with gravitation.
is a 6-dimension space. A priori we do not know anything about the metrics of this space but we will hereafter assume that 
is a flat space invariant under a symmetry group which conserves the true Euclidian norm 
where 
denotes a set of 6 orthonormal coordinates, i.e. it is the orthogonal group 
or one of its subgroups. We restrict ourselves to the special orthogonal group 
as we will see later on that the transformation 
with i ranging from 1 to 6, equivalent in the “internal” space 
to what parity and time reversal represent in the “orbital space”
, expresses the charge conjugation.
but they have an extension in the other dimensions. Thus in any point M of 
there are 4 “orbital” degrees of freedom along 
and 6 “internal” degrees of freedom in the transverse directions.
. In the base
any infinitesimal transformation of 
can be written as [
is a fully antisymmetric real 6 × 6 matrix, 
and 
are two antisymmetric 3 × 3 matrices, 
is a 3 × 3 matrix and 
denotes the transposed matrix of
; 
has got 15 infinitesimal generators.
can be written as
, where 
is a scalar times the 3-dimension identity
, 
is a 3 × 3 antisymmetric matrix and 
a null-trace 3 × 3 symmetric matrix. 
can then be expressed as the sum of 3 matrices
and 
generate 2 subgroups of 
and 
with respectively 4 and 8 infinitesimal generators. Those 2 subgroups have rather similar structures since
is the single generator of the special orthogonal group
.
is isomorphous to 
and that 
is isomorphous to
.We also note that 
and 
are 2 invariant, or distinguished, subgroups of
. Those 2 groups have in common 
as a maximal subgroup, where 
denotes the 2-dimension identity.
do not generate an invariant subgroup of 
since
are the Pauli matrices (the pair
, 
does not generate a group).
can be symmetry groups for the extension of the physical states in
. In other words, 
shall be of the form
and 
is relevant to characterize the physical states, which accordingly can be classified after the irreducible representations of 
and
, the same as those of 
and 
respectively. We can thus evidence two types of “internal” physical states, each type being characterized by a symmetry group: the “hadronic states” experiencing a 
- symmetry, and the “leptonic states” experiencing a 
-symmetry. Those two local symmetries can be interpreted through the existence of two gauge fields respectively acting upon the hadrons for 
and upon the leptons for
.
. 
like 
has 8 infinitesimal generators which subtend a 8-dimension representation of
. Any given irreducible representation of 
is characterized by the two numbers Y (hypercharge) and I (isospin), each base vectors of the irreducible representation 
is characterized by the 3 numbers Y, I and mI. The strong interaction is mediated through a set of 8 neutral massless vector fields, the so-called “gluons”.
and 
be the two fundamental conjugated 3-dimensionals representations of 
or
from which all the others representations can be built [
and the isospin singlet 
are a set of base states for
, the doublet 
and the singlet 
are a set of base states for
. As 
has 6 dimensions, there are two such distinct triplets, the so-called quarks, plus their images by charge conjugation.
. It has the same symmetry as but it is intrinsically different from the usual electro-weak interaction. 
like 
has 4 infinitesimal generators which subtend a 4-dimension representation of
. Any given irreducible representation of 
is characterized by the two numbers Z (pseudo electro-weak hypercharge) and J (pseudo electro-weak isospin), the representations 
and 
are conjugated, the dimension of the representation is 2J + 1. Each of the base vectors of the irreducible representation 
is characterized by the 3 numbers Z, J, mJ. The “pseudo electro-weak interaction” is mediated through a set of 4 neutral massless vector fields, that we call “gluinos”. Those gluinos, which are bosons, shall not be confused with the so-called gluinos of the super symmetric theories, where they are the fermion partners of the gluons. The existence of this pseudo electro-weak interaction mediated by 4 extra bosons is a major difference with the Standard Model (see 4.3 hereunder).
and 
be the two fundamental conjugated 2-dimension representations of 
or 
from which all the others representations can be built. The two states 
constitute a set of base states for
, the 2 states 
constitute a set of base states for
. As 
has 6 dimensions, there are actually 3 distinct pairs of such doublets, plus their images by charge conjugation.
a 3-dimension hyper surface
. Let 
and 
respectively be the tangent space and the orthogonal space to 
in some point N of
. In 
the particles have 6 degrees of freedom, 3 of them in the tangent space 
and the 3 others in the orthogonal space
. Both 
and 
are 3-dimension flat spaces, invariant under the 3d special orthogonal group
. Actually, the choice of N has no effect on what follows as all the points on 
are equivalent. The symmetry properties above only depend upon the coordinates of M on
, which is consistent with the existence of a gauge field experienced by both hadrons and leptons since we have already noticed that 
and 
have in common 
as a maximal subgroup. This gauge field is associated to the symmetry group
, isomorphous to the special unitary symmetry
, and it is identified with the weak interaction [
and
. Since 
has three infinitesimal generators which subtend a 3-dimensional representation 
of
, the weak interaction is mediated through a triplet of neutral massless vector fields.
to 
in any point N is also a 3-dimension space invariant under
. Since a 2-dimension surface can be embedded within a 3-dimension flat space, let us consider within 
a 2-dimension hyper surface 
invariant under
, i.e. 
is a sphere. Let 
be the tangent plane to 
in any point P of
. 
is invariant under the 2d special orthogonal group 
in any point P of
. Actually, the choice of P has no effect on what follows as all the points on 
are equivalent. The symmetry properties above only depend upon the coordinates of M on
, which is consistent with the existence of agauge field experienced by both hadrons and leptons and associated to the special unitary symmetry group
, isomorphous to the unitary symmetry group
. This interaction is identified with the electromagnetic interaction. It eventually lifts the residual degeneracy of the multiplets associated to the irreducible representations of
.
has only one infinitesimal generator and its irreducible representations are of dimension 1, each of them being characterized by a relative integer number Q, which is identified with the electric charge. The representations 
and 
are conjugated, the totally symmetric representation 
being self-conjugated. The interaction is mediated through a single massless vector field which is identified with the photon.
in P is a line
; in the space 
the particles have 3 degrees of freedom, 2 of them in the plane
, and the third one along
. 
is a 1-dimension space whose only symmetry is 
i.e. the trivial identity
; that can be interpreted as featuring an additional scalar field. As a consequence, both hadrons and leptons can experience a same scalar interaction mediated through a particle which can be identified with the BEH boson.
there is a local extra 6d Euclidian space 
in which is inscribed a 3d hypersphere. In any point of that hypersphere there is a 3d Euclidian subspace in which is inscribed a sphere. That unfolding, coming with a group-to-subgroup declination, reveals the interactions embedded as Russian dolls. The hadronic states are classified according to the symmetry 
isomorphous to 
, and the leptonic states are classified according to the symmetry 
isomorphous to 
.
and 
are orthogonal subgroups of
, the BEH boson has no electric charge and the photon has no mass. Due to the effect of the electromagnetic field on the one hand, and of the BEH field on the other hand, the fundamental representation 
of 
is splitted into a singlet (charge
, mass
) and a doublet (charge
, masse
) so that the 3 mediators of the weak interaction above gain an electric charge and a mass; they are respectively identified with the 
and 
particles. The same mechanism confers an electric charge and a mass to hadrons and leptons.
, 1 photon. The vector character of the gluons, gluinos, 
,
,
and photon is verified in the Annex [
gauge symmetry. It has been shown [
for hadrons and 
for leptons, built from the irreducible representations of 
and 
or
.
the charge conjugation simply appears as the equivalent of what are parity and time reversal in the “orbital space”
, i.e. the transformation 
within ranging from 1 to 6 is analogous to 
with 
ranging from 0 to 3.
, which has been assumed to be globally invariant under a symmetry group
. In any point M of the 4-dimension surface 
is the tensor product of a 4-dimensional “orbital” space invariant under the Lorentz group 
and of a 6-dimension “internal” space 
invariant under
. 
and 
are both invariant, or distinguished, subgroups of
.
.
is changed into
is a tensor quantity featuring the local geometry of
. The impulsion
‘s which allows to impose the 4 conditions
within 
and the conservation of a true Euclidian norm 
within
. We now further postulate the conservation of the full pseudo Euclidian norm
within
. The full 10-dimension space 
is thus assumed to be a pseudo Euclidian space which preserves a pseudo norm 
with 
and 
ranging from 0 to 9, i.e. 
if
, 
otherwise.
, the Lagrangian 
attached to a field 
simply is
attached to the point M of
, we extend to 
what has been done above with the Lorentz invariance in the case of a 4-dimension space-time. The transformation Equation (7) readily becomes
as a function of the 
s it is obvious that
. Applying the Lagrange equations to the 
field, i.e.
‘s so that we can impose the 10 conditions
. With the correspondence
, one equivalently gets
has the following form
and
, 
has 55 distinct components, of which 45 are independent.We can also choose
, with 
and 
ranging from 0 to 3, and
, with 
and 
ranging from 4 to 9, so that
, 
and 
respectively have 10, 21 and 24 distinct components, of which 6, 15 and 24 are independent.
special solutions which can be expressed as a product
,(24)
and Equation (24) for
, the equation of evolution becomes
and integrating over the 6 degrees of freedom of 
yields
, or
.
and a set of 6 additional massless vector fields
can be written as
.
space we have seen that the hadrons experience a symmetry 
isomorphous to 
whereas the leptons experience a symmetry
isomorphous to 
. As a consequence, the interactions 
should be different for hadrons and leptons and thus bodies with different composition would not behave the same way under
. Experimentally that would result in a violation of the equivalence principle [
, whose local symmetry is the Lorentz group
, embedded in a flat 10-dimensional space
. In every point of that “orbital” space
, there is a 6-dimension “internal” space
, orthogonal to the surface
. Simple assumptions about the geometry of
: allow deriving the following: hadrons and leptons, 8 gluons for the strong interaction, 4 gluinos for a pseudo electro-weak interaction, 3 intermediate bosons for the weak interaction, 1 photon, plus an additional scalar field identified with the BEH field.
,
,
and 1 photon, each of them being associated to a symmetry group
. The elementary operations associated to any of them have the form
. More generally, any operation of 
can be written as
in the 4-dimension space-time. Its Lagrangian density is
(A1.4)
and 
ranging from 0 to 3.
is assumed to be a local one, i.e.
, then
times an “internal” state
, the conservation of the Lagrangian imposes inside 
the existence of a vector-type gauge field
since 
exclusively acts upon the variables
.
,
,
and photon.