Kaluza-Klein compact extra dimensions detection
Eduardo Valencia
Advanced Gravity A.C.
7305 San Dario
Ave. STE G 48-226
LAREDO TX 78045
Criteria for compact dimension detection are
derived from classical and quantum motion of mass probes in 5 dimensions,
confirming the higher dimensional nature of space-time within experimental
accuracy.
PACS numbers: 11.10.Kk, 04.50.+h
The recent search
for extra spatial dimensions has been performed with experiments focused on a
measurement of the universal gravitational constant G, currently measuring the
force of gravity at the sub-millimeter range. Such a deviation from Newton's
inverse-square law, in accordance with higher dimensional gravity models, can
be taken as proof of the existence of large spatial dimensions at low energy,
physical interactions detected in 4-dimensional space-time, are induced by
dynamic behavior of fields and physical observable quantities defined over such
extra dimensions, offering the possibility for indirect detection of such
dimensions within actual experimental energy range. Although string theory
postulates 26 compact extra dimensions at the Planck scale, and large compact
and extended dimensions may exist, the scope of the present work focuses on the
5-dimensional nature of space-time and realistic matter states derived
from the dynamics on such 5-dimensional manifold, that can be detected
experimentally at current energies, despite the energy scale magnitude of the
Planck sized dimensions.
The criteria for
the detection of spatial compact extra dimensions, as originally postulated by
O. Klein [1] at the Planck scale, are given by the
analysis of the motion of a mass probe in 5 dimensions. As the motion occurs in
5-dimensional space-time, physical characteristics will be adquired by the mass
probe, such characteristics manifested in 4-dimensional space-time must match
the experimental physical measured values in laboratory, revealing the true
physical nature of the higher dimensional space-time and therefore indicating
the without any sign of ambiguity the presence of an extra compact dimension.
The first verifiable prediction derived from this higher dimensional classical
motion, indicates the existence and origin of the electric quantised charge 
, in 4-dimensions, and then the corresponding quantum law of
motion, the Dirac equation for massless fermions propagating in 5-dimensional
space-time will yield the energy spectrum, revealing the origin and magnitude
of 4-dimensional rest mass for spin ½ fermions within experimental laboratory
accuracy. The fundamental constant of Quantum Electrodynamics 
is required to
contribute to the Kaluza-Klein gravitational coupling constant 
.
Realistic unification through the
Kaluza-Klein approach requires d=5 manifold topology to be: 
, the coordinate points are described by 
and the spatial extra
dimension radius 
is of Planck length
order. Finally the extra spatial dimension is required to be periodic 
. The d=5
metric of the space-time and action are:
(1)
so far the gravitational coupling constant usually given by:
(2)
For the mass probe 5-dimensional motion, consider the Lagrangian[2] :
(3)
the geodesic law of motion derived:
(4)
corresponds to the 4-dimensional Lorentz force motion law, if the electric charge ansatz is:
(5)
The Quantum Electrodynamics fundamental constant rule definition, consistent with the 5-dimensional geometrical background and the 4-dimensional experimentally measurements obtained for alpha, requires the definition of Kaluza-Klein 5-dimensional alpha probability density distribution:
(6)
in order to calculate the probability associated to alpha in four dimensional space-time:
(7)
The accuracy of
the probability value derived from the 5-d probability density distribution, is
in agreement with the corresponding probability value given by the standard
definition of alpha, as well as with the measurements obtained from experiment.
The v vev energy scale, characterizing the 4-dimensional Standard Model
spontaneous symmetry breakdown phenomenology and the Higgs mechanism,
parameterizes the probability distribution and fixes the probability measure.
The v vev input is set at 246 Gev, and the observed electromagnetic
strength, becomes defined when the spontaneous symmetry breakdown occurs at 246
Gev. The Higgs particle estimated radius r_H magnitude is assumed to be 
The 4-dimensional
alpha probability definition and value 
obtained from the parameterized probability density
distribution definition (6), will be retained for all motion criteria and
operator definitions.
The parametric probability density definition can be validated further, if one assumes an energy scale symmetry breakdown value v of electron rest energy order, and the radius magnitude of the hypothetical associated boson to be of classical electron radius order i.e.:
(8)
In order to identify the electric charge measured in 4-dimensional space-time, from the 5-dimensional probe motion described by (4), the alpha contribution to the gravitational coupling is:
(9)
Taking the mass
probe fifth dimensional momentum component as 
, the definition of the electric charge ansatz yields the
quantised electron charge:
(10)
The calculated charge corresponds to the observed experimental value,
Therefore the Lorentz force charge ansatz, now can be recast in operator form, given the following definition for the charge operator:
(11)
where the charge amplitude 
satisfies:
(12)
While the mass probe 5-d classical motion manifests the electric charge in 4 dimensions, the quantum motion will require a masless charged probe spin ½ fermion satisfying the Dirac law in 5 dimensions [3] :
(13)
the Dirac
operator defined over the fifth dimension acting on the massless charged
fermion probe wavefunction 
leads to the
corresponding 4-dimensional energy spectrum eigenvalues:
(14)
The 5-dimensional quantum motion of a massless charged fermion, manifests massive spin ½ electrons in 4-dimensional space-time for the following definition for the 5-d Dirac Operator:
. (15)
The electron mass gap
The mass operator establishing an equivalence relationship between the electric charge and the electron rest mass is given by:
(16)
where r_e is the classical electron radius.
Such operator is
defined over the compact dimension,
acting upon the charge amplitude
previously defined, defines the electron mass gap for the fifth dimension. The spectrum values derived from
the 5-dimensional quantum motion law, clearly shows that zero modes are
massless and do not have electric
charge, as well as the fact that the first mode
equates the experimentally measured value of the electron
mass 
The classical and quantum motion of probes over the 5-dimensional space-time, predict the manifestation of quantised electric charges and the electron mass spectrum values corresponding to the experimental accuracy available in 4-dimensional space-time, electron states are a manifestation of the 5-dimensional geometry unifying gravity and electromagnetic forces. Therefore, such classical and quantum motion criteria indicate the presence of the fifth dimension as a compact extra dimension of Planck radius at each point of 4-dimensional space-time, confirming beyond any doubt the physical existence of the fifth dimension as postulated by O. Klein. The physical structure of the 5-dimensional space-time proposed by the Kaluza-Klein unification is not anymore a mere theoretical assumption, and the criteria set the first step towards string and brane theory experimental verifications.
I want to acknowledge useful conversations with G. Torres.
