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Physical Science International Journal
15(3): 1-19, 2017; Article no.PSIJ.34613
ISSN: 2348-0130
On a Plausible Triple Electro-gravito-informational
Significance of the Fine Structure Constant
Andrei-Lucian Drăgoi1*
1Independent Researcher in Theoretical Physics and Fundamental Biology, Bucharest, Romania.
Author’s contribution
The sole author designed, analyzed and interpreted and prepared the manuscript.
Article Information
DOI: 10.9734/PSIJ/2017/34613
Editor(s):
(1) Aleksey Anatolievich Zakharenko, The International Institute of Zakharenko Waves (IIZWs), Krasnoyarsk, Siberia, Russia.
(2) Abbas Mohammed, Blekinge Institute of Technology, Sweden.
Reviewers:
(1) Stanislav Fisenko, Moscow State Linguistic University, Russia.
(2) Auffray Jean-Paul, New York University, USA.
(3) El-Nabulsi Ahmad Rami, Cheju National University, South Korea.
(4) Luis Acedo Rodríguez, Instituto Universitario de Matemática Multidisciplinar, Polytechnic University of Valencia, Spain.
(5) Francisco Bulnes, Technological Institute of High Studies of Chalco, Mexico.
(6) Lung-Chien Chen, National Taipei University of Technology, Taiwan.
Complete Peer review History: http://www.sciencedomain.org/review-history/20204
Received 1st June 2017
Accepted 18th July 2017
Published 27th July 2017
ABSTRACT
In the last century, a small minority of physicists considered a hypothetical binary logarithmic
connection between the large and the small constants of physics, which also implies a base-2
power law (Fürth, 1929; Eddington, 1938; Teller, 1948; Salam, 1970; Bastin, 1971; Sirag, 1980,
1983; Sanchez, Kotov and Bizouard, 2009, 2011, 2012; Kritov, 2013). This paper brings to
attention a plausible triple electro-gravito-informational significance of the fine structure constant,
with its implications in a plausible four fields unification pattern at Planck scales: this triple
significance is based on the existence of a unifying global scaling factor of nature which appears in
a hypothetical fine tuning of all the non-zero rest masses of the all the elementary particles in the
Standard Model. Furthermore, this paper also proposes the dimensional relativity hypothesis (DRH)
stating that the 3D appearance of space (or the 4D nature of spacetime) may be actually explained
by the relative magnitude of the photon (angular) quantum momentum (and the hypothetical
graviton quantum momentum respectively) and this global scaling factor (GSF): DRH also includes
a generalized electrograviton model (EGM) for any hypothetical graviton. This paper also proposes
a set of strong (and very strong) gravity constants and a gravitational field varying with the energy
(and length) scale, all with potential importance in the unification of the four fundamental fields.
Each of these hypotheses is a potential update for the Standard Model of particle physics.
Short Research Article
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
2
Keywords: Fine structure constant with triple electro-gravito-informational significance; unifying
global scaling factor of nature; four fields unification; the standard model of particle
physics; dimensional relativity hypothesis; electrograviton model.
1. INTRODUCTION [1]
In 1929, the German physicist R. Fürth proposed
the adimensional constant
32 128
16 2
as a
possible “connector” between gravitational and
quantum mechanics constants [2].
Arthur Eddington (1937) and Dirac (1937) have
remarked the coincidence of the large
adimensional numbers in physics which can be
reformulated as:
1/2 40
/ / 10
v H e
G
a a R r N
(
1/a
137
is the inverse of the fine structure
constant [FSC] at rest
21/137/
ee ckq
;
41
1/ 3.1 10
vv
GG
a
is the inverse of a variant of
the gravitational coupling constant [GCC]
41
/ 1/ 3.1 10/
v p e
GGm m c
;
0
/
H
RHc
9
14.5 10 light years
is the Hubble radius of
the observable universe, which is a function of
the Hubble constant
071.9 / /H km s Mpc
;
22
/
e e e e
r k q m c
15
2.8 10 m
is the
classical radius of the electron at rest;
80
10N
is the approximate number of nucleons in our
observable universe, a number which can be
estimated by astrophysical methods).
In 1938, Arthur Eddington proposed that the
number of protons in the entire Universe should
be exactly equal to:
256 79
136 2 1.57 10N
(
N
was later called the Eddington’s number
Edd
N
) and Eddington hypothesized that square
root of
Edd
N
should be close to Dirac’s big
number (which he invoked in his large number
hypothesis) such as
256
136 2
Edd
N
128
136 2
39
3.97 10
. Later on, Eddington
changed
136
to
137
(using the new
experimental values of
[re]determined in his
life time) and (re)insisted that
had to be
precisely
1/137
, a fact which attracted irony at
that time [3]. However, Eddington’s statement
also implied the adimensional constant
128
2
,
which wasn’t given proper attention for the next
10 years (Kritov [4]).
In 1948, Edward Teller proposed a possible
logarithmic connection between
and
2 39
/ 10
N
Gm hc
of the form
12
ln /
N
Gm hc
, with
N
m
being the
standard rest mass of a nucleon (proton or
neutron) [5].
In 1970, Abdul Salam also brought in attention a
possible logarithmic connection between
v
G
and
[6].
In 1971, Edward Bastin invoked the observation
2
//
vv p
G
a Gm c
38
1.7 10
99% 127
2
and
proposed the derivation of
1/ 137a
from
the exponent 127 by summing 127 with its series
of digits, such as 127+(1+2+7)=137 [7].
In 1980, Saul-Paul Sirag also proposed an
alternative interpretation of the binary logarithmic
relation between
1/a
and
1/
vv
GG
a
,
such as
2
100.6%
log 137.84
v
G
aa
.
(Sirag [8]).
John D. Barrow and Frank Tipler probably didn’t
know about Salam’s (1970), Bastin’s (1971) and
Sirag’s (1980, 1983) works on this subject, when
they wrote in 1986 that: „Edward Teller appears
to have been the first who speculated that there
may exist a logarithmic relation between the fine
structure constant
and the parameter
2 39
/ 10
N
Gm hc
of the form
12
ln /
N
Gm hc
[equation 4.23] (in fact
1 59
ln 3.27 10
[corrected estimation] and
the formula is too insensitive to be of very much
use in predicting exact relations)“ [9,10]. (
N
m
also stands for the nucleon [proton/neutron] rest
mass).
Regrettably, Barrow and Tipler also ignored
Eddington’s works on the subject which could
have inspired them to analyze the much more
“sensitive” binary logarithm variant
2
2
log /
N
Gm hc
instead of the natural
logarithm variant
2
ln /
N
Gm hc
. This paper
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
3
proposes additional arguments against Barrow
and Tipler superficial analysis of this subject and
continues the works of all the authors previously
cited who “advocated” in the favor of this binary
logarithm connection.
The recurrence of
128
2
and
1/
22
a
factors in
these (probably just apparent) numerical
coincidences suggests that base-2 power law
may have a significant role in numerical relations
of these physical constants, predicting the
existence of a universal (large) scaling factor of
nature (indissolubly related to the fine structure
constant) with important implications in a
possible fine-tuning of all non-zero rest masses
of all known elementary particles in the standard
model, with implications for the existence of life
forms in our universe.
2. THE EXISTENCE OF UNIFYING
GRAVITATIONAL SCALING FACTOR
BASED ON A HYPOTHETICAL
ELECTRO-GRAVITATIONAL SIGNIFI-
CANCE OF THE FINE STRUCTURE
CONSTANT
There is a high probability to exist a profound
(direct and/or indirect) connection between the
(very) large and the small adimensional
constants (invariants or quasi-invariants) in
physics. If there is a connective function between
these two types of adimensional constants, the
most plausible and natural candidate for this is a
logarithmic function: also note that all the running
coupling constants of all non-gravitational
fundamental forces are logarithmic functions of
the energy/length scales.
Let us consider a generalized gravitational
coupling constant
,/
Gx y x y
m m Gm m c
and its inverse
, , /1/
GGx y x y x y
a m m a m m c Gm m
for any two identical or distinct non-zero rest
masses
x
m
and
y
m
of any pair of elementary
particles (EPs) in the standard model (SM). Let
us consider a base-2 logarithmic function
,
xy
f m m
which compares
,
Gxy
a m m
with
the inverse of the fine structure constant (FSC)
2
1/ / 137
ee
a c k q
such as:
2
log ,,/
Gx y x y
f m m a m m a
(2-1)
Interestingly, the function
,
xy
f m m
has its
values relatively close to 1, in the double closed
interval
0.817, 1.085
with an arithmetic
average of
0.93
which is also respected on
the “diagonal” values (as detailed in the Table 1
and Fig. 1: obviously, the values of
,
xy
f m m
tend to be super-unitary for the combinations of
the lightest EPs and sub-unitary for the rest of
EPs.
As seen from the previous table and figure, the
sub-unitary values seem to predominate, as the
three leptonic flavors of neutrino (which are
almost surely lighter than the electron) weren’t
yet graphed, given the difficulty in determining
their exact non-zero rest masses.
The values of the function
f
are also plotted in a
graph, sorted in ascending order, revealing a
stair-like shape with a quasi-linear trend line: see
the Fig. 2a.
The electron neutrino
0
e
rest mass is
estimated to be in the interval
2
0.2, 2 /eV c
[11]. For
.2
1.85 /
e
hyp
m eV c
(which is the last
experimental estimation) and
y
m
m , m , m , m , m , m ,
m , m , m , m m m,,
u c s t
db
e W Z H
,
.
,
e
hyp
y
f m m
1.201, 1.193, 1.134,
1.161, 1.083, 1.122,
, 1.161, 1.131,
1.091, 1.09, 1.086
1.217
, which
extends the interval of values of
f
to
0.817, 1.217
(by adding more super-unitary
values to it and equilibrating the previous “mix”
which was dominated by sub-unitary values): this
last interval is approximately centered in value 1
and is relatively symmetrical and “equilibrated”
around this value with two sub-unitary and super-
unitary “wings”
1 0.2
; the arithmetic average
of this extended interval of
f
values increased
to
0.97
and preserves the stair-like shape
(with a quasi-linear trend line) when graphed
values are graphed in ascending order: see the
Fig. 2b.
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
4
Table 1. The values of the function
2
log /,/
x y x y
f m m c Gm m a
in the double closed
interval
0.817, 1.085
Elementary
particles
Quarks*
Leptons**
Bosons***
2/3
u
1/3
d
2/3
c
1/3
s
2/3
t
1/3
b
e
/
W
0
Z
0
H
2/3
u
1.053
1.046
0.987
1.014
0.935
0.974
1.069
1.013
0.983
0.943
0.942
0.938
1/3
d
1.046
1.038
0.979
1.006
0.927
0.966
1.061
1.005
0.976
0.935
0.934
0.931
2/3
c
0.987
0.979
0.92
0.947
0.868
0.908
1.002
0.946
0.917
0.877
0.875
0.872
1/3
s
1.014
1.006
0.947
0.974
0.895
0.935
1.029
0.973
0.944
0.903
0.902
0.899
2/3
t
0.935
0.927
0.868
0.895
0.817
0.856
0.951
0.895
0.865
0.825
0.824
0.82
1/3
b
0.974
0.966
0.908
0.935
0.856
0.895
0.99
0.934
0.904
0.864
0.863
0.859
e
1.069
1.061
1.002
1.029
0.951
0.99
1.085
1.029
0.999
0.959
0.958
0.954
1.013
1.005
0.946
0.973
0.895
0.934
1.029
0.973
0.943
0.903
0.902
0.898
0.983
0.976
0.917
0.944
0.865
0.904
0.999
0.943
0.913
0.873
0.872
0.868
/
W
0.943
0.935
0.877
0.903
0.825
0.864
0.959
0.903
0.873
0.833
0.832
0.828
0
Z
0.942
0.934
0.875
0.902
0.824
0.863
0.958
0.902
0.872
0.832
0.83
0.827
0
H
0.938
0.931
0.872
0.899
0.82
0.859
0.954
0.898
0.868
0.828
0.827
0.824
Note: The super-unitary values (corresponding to the combinations between the lightest elementary particles)
were marked in italics.
*only the average estimated rest mass of quarks was considered (without their individual rest mass determination
uncertainties);
** only the leptons with known rest masses were tabled (such as the three leptonic flavors of neutrinos were
excluded for the moment);
*** only the bosons with non-zero rest masses were tabled;
Fig. 1. The values of the function
2
log /,/
x y x y
f m m c Gm m a
in the double closed
interval
0.817, 1.085
.1.349,
ee
hyp
f m m
also adds to this interval
1 0.2
as an apparently “isolated” value, but
which pushes the arithmetic average of
f
values to
0.98
and
1.01
(for the diagonal
values only) (values even closer to 1) and which
may suggest the existence of EPs (with non-zero
rest masses and probably zero-charged) even
lighter than the neutrinos to fill the gap between
1.2 and 1.349 and even to extend this interval.
0
0.2
0.4
0.6
0.8
1
1.2
1-1.2
0.8-1
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
5
The most plausible candidates to fill that gap (at
least partially) are the sterile neutrinos (neutrinos
with right-handed chirality which are well-
motivated theoretically, as all other known
fermions have been observed with left and right
chirality), in the case they shall be confirmed to
have non-zero rest energies <1eV. Other
plausible candidates are the lightest
supersymmetric particles (hypothetical particles
proposed by supersymmetric models), probably
neutralino (which is the most plausible candidate
for the main constituent of the hypothetical dark
matter), the gravitino and the lightest sneutrino.
Fig. 2a. The values of the function
f
(without considering neutrino rest mass
combinations) sorted in ascending order
Fig. 2b. The values of the function
f
(also
considering the combinations of the three
neutrino rest masses of ~1eV/c2) sorted in
ascending order
Furthermore, even if the proton and the neutron
aren’t elementary particles,
,
py
f m m
and
,
ny
f m m
also have values that “fit” in the
same approximate interval
1 0.2
, as those
nucleons have rest masses relatively close (with
the same order of magnitude) to the tauon rest
mass, for which
0.999,e
f m m
:
1.006,e
N
f m m
.
A special case (disserving a separate
discussion) is
, 0.9991
e
f m m
, with
2
log / e
c Gm m
136.914
, which is
strikingly close to
2
/137.036
ee
a c k q
.
The mass “ambitus“ defined by the double
closed interval
,
e
mm
(with
,
e
m m m
)
has a special significance in particle physics, as
it is defines the minimum and the maximum
elementary masses that can be “stored” on a
negative/positive elementary charge in the case
of charged leptons/antileptons, accordingly to the
Standard Model (SM). The closeness
99.91%
2
log / e
c Gm m
2
/ee
c k q
suggests that the gravitational energy of an
electron(/positron)-tauon(/antitauon) pair is
centered (base-2) logarithmically with
unexpected high accuracy around the
electromagnetic dimensionless
2
/ee
a c k q
:
this is an indirect proof that gravity may partially
break the charge symmetry of the
electromagnetic field and allow two or more
distinct masses for the same charged EP, but
preserving the base-2 logarithmic law defined by
,1
xy
f m m
in relation to the electromagnetic
field energy quanta
/
ph
Ec
, so that
2
/
/22
ee
c k q a
e
c Gm m
and
2
log / e
a c Gm m
.
The secondary “diagonal” function
2
2
log /
xx
a c Gm
, with
x
m
m , m , m , m , m , m ,m
m , m , m , m m m,,
u c s t
db
e W Z H
has values in
the approximate interval
110,190
with two
relatively symmetrical “halves” below and above
1/ 137a
and a stair-like graph shape, as
represented in the Fig. 2c.
0.8
0.85
0.9
0.95
1
1.05
1.1
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
6
Fig. 2c. The values of the function
x
a
sorted
in ascending order, also considering three
neutrino rest masses of ~1eV/c2
One may notice from the previous figure that all
the step-10 subintervals of
x
a
values
110,120
,
120,130
,
130,140
,
140,150
and
180,190
is occupied by a stair-like group of
four, three or at least two EPs with non-zero rest
masses. The “gap” consisting of subintervals
150,160
,
160,170
and
170,180
may be
also occupied by other light EP undiscovered
yet, as already discussed previously.
This paper considers that it is very unlikely for
the relatively large diversity of EPs non-zero rest
masses to be strongly centered logarithmically
around
1/ 137a
only due to a “simple”
coincidence: by contrast, it is very plausible that
the existence of this “unity in diversity” to be the
consequence of a more profound law of nature,
as
is also the expression of a charge-
anticharge symmetry (which was verified to be
exact with very high accuracy) with no clear and
definitive explanation yet.
,
xy
f m m
may prove not only a fine tuning of
all the elementary rest masses alone (by their
combinational pair products), but more
importantly and especially a fine tuning of big G
magnitude, which has an essential role in
significantly “assuring” the “centering” of the
f
values of around 1. The gravitational coupling
function
,
Gxy
mm
of any pair of elementary
rest masses appears to be (base-2)
logarithmically fine-tuned with the fine structure
constant at rest
. It is also very plausible that
both big G and the set of elementary masses to
be actually inter-correlated and both determined
by a still unknown (more profound) law of nature,
which may be in fact a general/universal
propriety of spacetime itself.
In conclusion,
,
xy
f m m
appears to help in
predicting the products of any pair of elementary
non-zero rest masses
xy
mm
as a function of
Planck mass
/
Pl
mcG
, so that:
2
1 0.2 1 0.2
/
22
Pl
xy aa
m
cG
mm
(2-2a)
The factor
1/ 41
1.8 1022
a
a
n
which
emerges in the previous equation can be
proposed as a unifying gravitational/mass
scaling factor of the Standard model, so that:
2
1 0.2
Pl
xy a
m
mm n
(2-2b)
0.5 0.1
Pl
xy a
m
mm n
(2-2c)
The existence of
1/
22
a
a
n
in the function
,1
xy
f m m
justifies the hypothesis that the
fine structure constant may have a dual/hybrid
electromagnetic and gravitational (electro-
gravitational) significance, so that
a
n
may be
regarded as a unifying electro-gravitational
scaling factor with the propriety that
41
2 1.8 10/a
x y a
Pl
m m m n
and
2
log a
na
.
A similar base-2 logarithmic function
,
xy
rrff
can be conceived to compare the most important
length scales of our universe in pairs
,
xy
rr
such as:
2
, log //
x y x y
ff r r r r a
(2-3a)
, log /
x y a x yn
ff r r r r
(2-3b)
Given the estimated radius of our observable
universe (ou)
26
4.4 10
ou
Rm
, the classical
100
110
120
130
140
150
160
170
180
190
200
0 5 10 15
a_x
a~137
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
7
electron radius
22
/
ec e e e
r k q m c
15
2.8 10 m
,
the radius of the proton (as determined by
scattering using electrons)
15
0.87 10
p
rm
and the Planck length
3 35
/ 1.62 10
Pl
l G c m
, for
,
xy
rr
, , ,
ou ec p Pl
R r r l
,
,
xy
rrff
has the following
values:
1.49,
ou Pl
Rlff
,
0.99,
ou e
Rrff
,
1.01,
ou p
Rrff
,
0.48,
pPl
rlff
,
0.49,
ePl
rlff
. Interestingly, all these values
tend to concentrate around values 1/2, 1 and 3/2
which are multiple integers of 1/2. The length
26
5 10
a ec
n r m
is relatively close to
26
4.4 10
ou
Rm
, so that
1.14
a ec ou
n r R
and
99.9%
2
log ( / ) 136.85
ou ec
R r a
. The length
ap
nr
26
1.6 10 m
is also relatively close to
ou
R
so that
0.35
a p ou
n r R
and
101.1%
2
log ( / ) 138.54
ou p
R r a
. In other words,
when expressed in
p
r
and
e
r
units, the
observable universe has ~137 length “octaves”
which also suggests the gravitational
significance of FSC. The next “natural” ff-value in
the series
/ 2 *nwith n N
is
2,
xPl
Rlff
,
which predicts a radius
2 21
10
x a ou
Pl
R l n R
which is a potential candidate for the real radius
of our present universe (which is already
predicted by superstring theory to be with at
least 3 orders of magnitude larger than
ou
R
).
2 82
3.2 10
a
n
is also close to the gravity-
related ratios between the rest-mass of our
observable universe (ou)
54
3.1 10
ou
M kg
and the non-zero rest masses of the proton
p
m
and electron
e
m
, such as:
81
/ (1.8 10 )
ou p Edd
M m N
,
84
/ 3.4 10
ou e
Mm
and
82
/ 7.9 10
ou p e
M m m
.
Additionally,
/4 75.75 / /
pa
ckm s Mpc
rn
105%
0
H
,
with
071.9 / /H km s Mpc
being the
Hubble constant as determined by the latest
measurements from 2016 with the Hubble
telescope [12]. This
a
n
-based predicted value is
very close to the first good estimate
075 / /H km s Mpc
proposed in 1958 by
the influent American astronomer Allan Sandage
[13].
Additionally,
3 23
/ 4 6.01 10
a
an
is very close
to the numerical value of the Avogadro constant
23
6.023 10 /
A
N molec mole
, so that
99.8%
3
/4 aA
a n N
.
Furthermore,
a
n
is a plausible candidate for the
warp factor
W
proposed in Randall–Sundrum
type 1 (RS1) universe model which proposes a
5D anti-de Sitter bulk space, with all EPs (except
for the graviton) being localized on a (3 + 1)D
brane or branes. This 5D space is extremely
warped, a warp caused by the energy gradient
between two branes with positive and negative
energy respectively: a positive-energy
“Planckbrane” (where gravity is a relatively
strong force) and a negative-energy “Tevbrane”
(our home with the Standard Model particles): in
RS1 these two branes are separated in the 5th
dimension (which is not necessarily large) by
approximately 16 energy/length units (as based
on the brane and bulk energies). The warping (or
red-shifting) of the 5th extra spatial dimension is
analogous to the warping of spacetime in the
vicinity of any massive object as proposed by
Einstein in his General Relativity Theory (GRT):
this warping generates a large ratio of energy
scales between the Planckbrane and the
Tevbrane, so that the natural energy scale at
one end of the extra dimension is much larger
than at the other end [14,15].
3. THE HYPOTHETICAL ELECTRO-
GRAVITON AS BASED ON THE
ELECTRO-GRAVITATIONAL SIGNIFI-
CANCE THE FINE STRUCTURE
CONSTANT
,1
xy
f m m
can be also written as:
2/
log /
xy
c
Gm m
a
(3-1a)
/2
/aa
xy
cn
Gm m
(3-1b)
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
8
/xy
aGm m
nc
(3-1c)
Based on previous equations, this paper
proposes a model for the hypothetical spin-2
graviton analogous to the spin-1 photon (with
wavelength
and energy
/
ph
Ec
),
with zero-rest mass and moving with maximum
speed
c
(or close to
c
, as the speed of
gravitational waves [16]). As it is modeled
analogous to the electromagnetic field quanta
(the photons), this hypothetical graviton may be
called “electrograviton” (eg), as it is also based
on a (reduced) gravitational Planck-like constant
76
/ 5.1 10
eg a
n Js
, so that an eg with
wavelength
is predicted to have an energy
defined by:
/
eg eg
Ec
(3-2)
Measuring the value of FSC at rest (with very
high accuracy attained by using experiments
based on the quantum Hall effect) may be
considered an indirect method to essentially
determine the electro-gravitational scaling factor
1/
22
a
a
n
, which can further be used to
redefine FSC at rest, such as:
.
2
log
redef
a
na
(3-3a)
.
2
1/ 1/ log
redef
a
an
(3-3b)
The existence of the hypothetical graviton
(modeled in this paper as an electrograviton)
may also imply the existence of subtle (at least
theoretically distinguishable) subquantum states
for any physical system (PS) (composed of one
or more EPs) generated by the absorption of one
or more egs [17,18,19]. The total number of
possible (at least theoretically) distinguishable
states of a PS
S
N
can be calculated as the
product between the total number of quantum
states
qS
N
and the total number of gravitonic
(subquantum) states
gS
N
, such as [20]:
S qS gS
N N N
(3-4a)
2 2 2
log log log
S qS gS
N N N
(3-4b)
The absorption of one photon by a PS increases
qS
N
with one unit (one additional possible
quantum state). Analogously, if hypothesized
that the absorption of each individual eg adds a
distinct supplementary possible subquantum
(gravitonic) state to a receiver PS then (so that
gS
N
increases with one unit, one additional
possible gravitonic/subquantum state), the
absorption of (a number of)
a
n
egs means
receiving
2
log 137
a
ngbits
([subquantum]
gravitonic bits): on the other hand, the energy-
absorption of
a
n
egs with
/
eg eg
Ec
is
equivalent to the absorption of one photon with
the same wavelength
and energy
/a eg
ph
E c n E
; if the absorption of
each individual photon adds a distinct
supplementary possible quantum state to a
receiver PS then, the absorption of a photon
means receiving
1137qbit gbits
. Each gbit
represents a “octave”-like group of gravitonic
states similar to the “octave” interval used in
music to define any (double closed) interval of
frequencies (f)
,2ffHz
.
FSC expresses an informational equivalence
between electromagnetism and gravity and
137a
may be used as an interconversion
factor between qbits and gbits, so that FSC
1/a
can be regarded as the probability of
targeting a specific subquantum (gravitonic)
octave of states (defined by each gbit) of any
PS. In this view, a photon can be defined as an
EP containing
1137qbit gbits
so that the
probability of a real electron/positron (at rest) to
emit a real photon (Feynman’s interpretation of
FSC) may measure in fact the base-2 logarithmic
probability of an electron/positron to emit a
photon in a specific octave of subquantum
(gravitonic) states.
The author of this paper had also demonstrated
that
a
n
and the redefined
.
2
log
redef
a
na
can
both help predicting a quantum gravitational
coupling constant
Gq
a
for an electron/positron
pair which approximates the empirical
2/
e
GcGm
45
1.75 10
with very high
accuracy[1], such as:
[1] Discovered in 2014 and included in a document registered
at the Romanian Copyright Office (ORDA) with the
registration number 2546 / 26.03.2015. URL
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
9
99.6%
45
3/2 1.74 10
2
1
Gq G
aa
an
(3-5a)
This quantum
Gq
a
can be interpreted as a
linearithmic probability and can also be used to
predict a quantum big G scalar
q
G
with the
same high accuracy, such as:
2
99.6%
11 3 1 2
,
6.648 10
/
q Gq e
q
with
G m kg s G
G c m
(3-5b)
In conclusion, FSC may actually have a triple
electro-gravito-informational significance, so that
a
n
can be considered a unifying
(quantum/subquantum) electro-gravito-
informational scaling factor, such as
41
1/ 1/10
a
n
can be interpreted as the
probability to target a specific subquantum
(gravitonic) state of any EP and
2
1/ log 1 /137
a
n
can be interpreted as
the base-2 logarithmic variant of that (same)
probability, which is the probability to target a
specific octave of subquantum (gravitonic) states
of any EP.
More ambitiously, this paper proposes
41
10
a
n
to be a first rank adimensional constant that may
vary with the energy scale of measurement (as
detailed later in this paper), but with its value at
rest remaining constant on the entire history
(including future) of our universe. This paper also
proposes a contraction in the set of adimensional
constants of the Standard Model, so that FSC to
be redefined as
.
2
1/ 1/ log
redef
a
na
,
independently of the combination of dimensional
constants
2/ 1/137
ee
k q c
, constants which
may be considered just (dimensional) second
rank parameters with that combinational value
fine-tuned near
2
1/ 1/ log 1/ 137
a
na
.
4. THE DIMENSIONAL RELATIVITY
HYPOTHESIS (DRH) BASED ON THE
UNIFYING SCALING FACTOR
a
n
[1]
An interesting (probably just apparent)
coincidence emerges when comparing
h
and
/
eg a
n
with a global (angular)
momentum parameter of the observable
universe (ou) at rest
89
1.37 10
ou ou ou
L E t Js
:
71
3.14 10
ou
EJ
is the approximate resting
energy of ou determined from the experimental
measurements of the average energy density of
ou
ou
which is estimated to be very close to
the critical energy density established by the
Friedmann model as
22
0
3 8 //
cH G c
so
that
310
8.73 10 /
ou c Jm
and the
volume of ou
33 80
4 / 3 3.6 10
ou ou
V R m
(derived from the radius of ou
26
4.4 10
ou
Rm
);
9
13.8 10
ou
t years
is the age of the present
ou (as determined by specific astrophysical
methods).
The closeness of the positive reals
ph
d
and
eg
d
to the positive integers 3 and 4 respectively
(denoting the number of apparent/perceptual
dimensions of the 3D space alone and the 4D
spacetime respectively, as gravity is modeled by
General Relativity in a 4D Minkowski space) may
suggest that the number of dimensions of ou
may not be absolute, so that it may not be
correct to (a priori) predefine the number of
dimensions of space/spacetime as pure
observational arbitrary parameters without also
considering the type of gauge boson (photon,
electrograviton, etc. and its specific (angular)
momentum quantum ,
eg
, etc.) used to
observe/measure that space/spacetime and its
number of dimensions
x
d
. An arbitrary
x
d
may be extracted from an arbitrary triad
,,
x x x
nhL
as
log /
x x x xn
d L h
: this fact
suggests that it may not be correct to define
x
d
a priori, based only on empirical/experimental
observation, without also defining the triad
,,
x x x
nhL
from which this
x
d
was extracted:
this is because a fixed
x
d
(as we
associate space with a 3D reference
frame) also implies a fixed ratio
22
log / log / / log
x x x x x x xn
d L h L h n
, which
3
log / 2.98
aou ou a
ph n
d L L n
(4-1a)
4
log / 3.98
aeg ou eg ou eg an
d L L n
(4-1b)
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
10
also implies a strict correlation in the variation of
all the elements of the triad
,,
x x x
NhL
.
Given the relativity of
log /
x x x xn
d L h
from
the triad
,,
x x x
NhL
, this paper launches the
dimensional relativity hypothesis (DRH) which
states that: “the dimensions (D) of the
observable universe (ou) may not be Euclidean
but fractal and the number of dimensions (d) of
ou may not be absolute (a priori defined) and
integer but relative and fractionary, depending on
the electro-gravito-informational unifying scaling
factor
a
n
and the (angular) momentum “key”-
quantum we use to study the global angular
momentum
ou
L
(using our mind, senses and
their extensions as observational/measuring
tools)”. As we generally use light (photons) to
perceive and study space (together with virtual
photons which were demonstrated to permeate
all space, as proved by the Casimir effect), the
fact that
3
ou a
Ln
may generate the “3D
space” appearance: there are studies which also
show that time may not exist as a “4th dimension”
in the relativistic microcosm (the quantum level
of reality). The fact that we perceive time at the
macroscopically level (as part of an apparent “4D
spacetime”, with a 4th dimension modeled and
measured using a classical linear time function)
may be also an appearance generated by gravity
(mediated by [electro]gravitons with the very
small
eg
momentum-quantum) and to the
relation
4
ou eg a
Ln
.
The human brain uses photons (light) to observe
an apparent “empty” space, so that it may be the
“victim” of the illusion governed by
3
ph
d
,
which generates the appearance of a “3D
spacetime”, in which time is not an additional 4th
dimension, but only an abstract/artificial function
which records a sequence of changes/events in
that 3D space. The human brain also uses a
combination of photons (light) and (quantized)
gravity to observe the movements of objects in
space, so that it may be also the “victim” of the
illusion governed by
4
eg
d
, which generates
the appearance of a “4D spacetime”, with an
additional spatial 4th dimension attached to a
perceptual “3D space”.
This hypothesis can also offer an escape from a
potential tautology, as when we measure
different parameters of a quantum particle (QP),
we use algorithms and equations based on the a
priori assumption that space has three
(Euclidean or non-Euclidean) dimensions
3
x
d
, which may be essentially an illusion
created by
3
ph
d
: it is also the case in this
paper, when
ou
V
was calculated using the same
3D space a priori assumption.
As
ou ou ou
L E t
is a function of both the
energy
ou
E
and age
ou
t
of ou, a generalized
function
,,
x x x x
d E t h
can be defined next:
, log /,
x x x x a x x xntd E t h E h
(4-2a)
,,
x x x x
d E t h
predicts that if we could
(theoretically) existed and could have used the
same photons with the same
x
h
(as the
present photons have) in a very early historical
epoch of ou (defined by
x ou
tt
, the same
x ou
EE
and the same
a
n
) our space would
have looked more like a ~2.5D space (like in the
first second after the hypothetical Big Bang) or
even ~1.5D (like in the first Planck time interval
5 44
/ 5.39 10
Pl
t G c s
after the
hypothetical Big Bang):
1 , 2.56,
x ou
d E s
(4-2b)
, , 1.51
x ou Pl
d E t
(4-2c)
,,
x x x x
d E t h
also predicts that if we would (at
least theoretically) exist and use the same
photons with the same
x
h
(as the present
photons have) in a very distant future of ou
(defined by
x ou
tt
, the same
x ou
EE
and
the same
a
n
) our space would have looked
more like a ~4D space, for example in the future
moment corresponding to the age of
50
10
x
t years
measured after the hypothetical
Big Bang:
50
,10 , 3.95
x ou
d E years
(4-2d)
More interestingly, all known elementary
particles (EPs) (including those EPs with
extreme low/high non-zero rest masses like the
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
11
neutrinos [nn] and the Higgs [H] boson,
except the photon and gluon) have non-zero
rest energies in the interval
,1.85 125
nn H
EEeV GeV
which is
relatively “centered” in
0.5
nn H MeVEE
(more or less ~5-6 orders of magnitude):
log /
EP a ou EPn
d E E
has values in the
interval
1.9, 2.2 D
which is relatively
centered around 2(D). This may explain why
QPs (first treated as superstrings by the string
theories [STs]) can also be generalized and
modeled as 2D surfaces (supermembranes or 2-
branes) that may exist in an 11D spacetime as
proposed by M-Theory (MT) and supergravity
theory (which combines the principles of
supersymmetry and general relativity). A
a
n
-
based 11D universe may have a total angular
momentum
11
tot eg a
Ln
. In this view, bosons
may be modeled as open 2-branes and fermions
may be modeled as closed 2-branes [21,22]. The
same QPs may be regarded as 2-branes in a 4D
spacetime or as 1-branes (strings) in a 2D
(holographic) universe: this sustains the
holographic principle (HP) proposed by Gerard't
Hooft’s but also the AdS/CFT correspondence
(aka Maldacena duality or gauge/gravity
duality).
The definition
log /
x a ou xn
d L h
, can be used
to inversely define a specific (angular)
momentum quantum associated to any d-frame
of reference (with a number of
d
dimensions)
such as:
/d
ou a
hf d L n
(4-3a)
35
3 2.4 10hf Js
is relatively close (with
approximately the same order of magnitude) to
the reduced Planck constant
, so that
30.23hf
. The rest energies of W/Z
bosons
80.4
WGeVE
,
91
ZGeVE
and their
mean (measured) lifetimes
25
3 10
WZ
tt s
may help defining two reduced (angular)
momentum-like quanta
W
and
Z
(with
2
being associated with a full mean lifetime
/WZ
t
),
such as:
6 26 (3)
2
WW
WEt hf
(4-3b)
7 29 (3)
Z Z Z
h E t hf
(4-3c)
W
h
and
Z
h
are relatively close but larger than
30.23hf
(with ~1 order of magnitude),
and that is why the W/Z bosons may be
considered “heavy” photons, or high-momentum
photons (unstable excited states of the photon
with non-zero rest energies/masses and
tendency to decay asymmetrically into pairs of
distinct leptons) which is also the essential part
in the successful unification of electromagnetic
field (EMF) and the weak nuclear field (WNF) as
the electroweak field (EWF). As
( ) ( )
log / 2.97 3
a ou
W Z W Z
n
dL
,
observing our space by using W/Z bosons also
generates the same 3D space appearance.
Analogous to
W
and
Z
, one can also
calculate a reduced (angular) momentum-like
quantum for the Higgs boson (HB)
H
by using
the HB non-zero rest energy
125
H
E GeV
and
its mean lifetime
22
1.56 10
H
ts
(with
2
being associated with a full mean lifetime
H
t
),
such as:
4718 20770 (3)
2
HH
HEt hf
(4-3d)
H
L
is with ~3-4 orders of magnitude larger than
30.23hf
and that is why HB may be
considered a “very heavy” photon, or very-high-
momentum photon (a very high and unstable
excited state of the photon with a non-zero rest
energy/mass and tendency to decay
symmetrically into pairs of identical/opposite-
charge W/Z bosons, photons, leptons). As
log / 2.9 3
H a ou Hn
dL
, observing our
space by (at least theoretically) using HBs may
also generate the same 3D space appearance.
Based on
hf d
function, DRH also predicts
and defines a quantum
G
function
q
Gf
associated to any integer/fractional dimensional
d-frame (with
d
dimensions), such as:
2
/
qe
Gq
Gf d c m hf d
,
(4-4a)
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
12
with
4
qq
G Gf G
(4-4b)
Based on the
q
Gf d
general definition, DRH
predicts a hypothetical (very plausible) strong
gravity constant (SGC) associated with a 3D
frame and generated by a strong gravity field
(SGF) measured by a quantum momentum close
to
3hf h
, such as:
31 3 1 2
3 10
q
Gf m kg s
and
(4-5a)
41
1.5 10 G
(4-5b)
The majority of authors have calculated a value
for this hypothetical SGC
from
25 3 1 2
inf 10 m kg s
(corresponding to
inf 2.84d
) up to
37 3 1 2
sup 10 m kg s
(corresponding to
sup 3.14d
), with most of
estimations between
28 3 1 2
10 m kg s
and
32 3 1 2
10 m kg s
, with a average
3
avr
d
.
(Seshavatharam and Lakshminarayana S.
[23,24,25]; Perng [26]; Fisenko et al. [27,28,29];
Recami et al. [30,31,32]; Fedosin [33,34,35];
Tennakone [36]; Stone [37]; Oldershaw [38,39];
Mongan [40]; Sivaram and Sinha [41]; Dufour
[42]).
SGF may act as a confinement force between
the 2-branes contained in the same 3-brane (like
our 3D space) stabilizing that 3-brane.
Furthermore, DRH also predicts that there may
exist a set of very strong gravity fields (VSGF)
associated to the 2D (the frame of 2-branes) and
1D (the frame of strings/1-branes) which may
manifest at scales progressively smaller and
even smaller the Planck length scale, such as:
72 3 1 2 82
2 10 10
q
Gf m kg s G
(4-6a)
113 3 1 2 123
1 10 10
q
Gf m kg s G
(4-6b)
VSGF (2) is associated with
2
q
Gf
and may
act as a confinement force between the strings
contained in the same 2-brane stabilizing that 2-
brane. VSGF (1) is associated with
1
q
Gf
and
may act as a confinement force between the
points contained in the same 1-brane (string),
stabilizing that 1-brane.
1
q
Gf
is a potential candidate for the upper
bound of a plausible finite G that limits the
growth to infinity of the strength of gravity when
approaching infinitesimal length scales possibly
inferior to the Planck length scale (as possibly in
the black holes): the predicted hypothetical
asymptotical freedom of gravity.
DRH also proposes a generalized
electrograviton model (EGM) in which there is a
distinct electrograviton (0-spin, 1-spin or 2-spin)
associated with each dD frame (with d being a
positive integer number of dimensions) with its
own specific angular quantum momentum, such
as:
/d
eg ou a
hf d L n
(4-7a)
and
2
/
q e eg
Gq
Gf d c m hf d
(4-7b)
In this view, the Newtonian/relativistic gravity is
mediated by the 4D-frame electrograviton (4-eg),
with an angular quantum momentum measured
by
4
eg eg
hf
which generates a
gravitational field with strength measured by
4
q
G Gf
. In the same view, SGF is predicted
to be mediated by a 3D-frame electrograviton (3-
eg) with
3
eg
hf
, which has a strength also
measured by
41
3 1.5 10
q
Gf G
. The
photon (which is its own antiparticle), the W/Z
bosons and HB may all be considered different
types of 3-egs because
() 3
HW Z ph
d d d
.
In this way, the DRH-based SGF may co-predict
(retrodict) the existence of the Higgs field (HF),
as the 3D-frame eg (3-eg) has some striking
scalar similarities with HB, which is a scalar QP
(the only known scalar QP in nature, first
predicted to exist in 1960s) with 0-spin and even
parity. HB is defined as the quantum excitation of
one [of the four] components of HF: HB is a very
plausible candidate for the 3-eg (predicted by
DRH) and vice versa.
This DRH sub-hypothesis also implies that
3
q H q
Gf d Gf
. However, the
mainstream considers that more studies are
needed to firmly confirm if the ~125GeV boson
discovered in CERN's Large Hadron Collider
(LHC) has properties matching those predicted
by Standard Model (SM) for HB, or whether,
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
13
more than one type of HB exist (as predicted by
some theories). The 100% confirmation of HF
existence depends on the final confirmation of
HB existence, as HF is detected through its
excitations (the HBs, which are difficult to obtain
and detect).
HF is predicted to be tachyonic (as the
symmetry-breaking of HB [through condensation]
only occurs under certain conditions), and has a
"Mexican hat" shaped potential with non-zero
strength at any distance (also manifesting in
empty space and permeating the entire
observable universe and possibly all our
universe, similar to both electromagnetic field
(EMF) and the predicted SGF.
In its vacuum state, HF breaks the weak isospin
symmetry of the electroweak field (EWF) and
generates the W and Z bosons of WNF, which
have very large non-zero rest masses of about
(80-90)GeV. HF may also explain the non-zero
rest masses of other elementary QPs like quarks
and leptons (that are predicted to be normally
massless when considering the symmetries
controlling their interactions), by using other HF-
based mechanisms alternative to the Higgs
mechanism.
,
xy
f m m
can be generalized/extended
for any (reduced) Planck-like constant
, , ,
xW Z H
, such as:
2
2
log /
/
,, x x y
x y x x e e
c Gm m
f m m c k q
(4-8)
All fermionic EP non-zero rest masses can be
considered the result of symmetry breaking of
high-momentum bosons (like the W/Z and HB)
but keeping
,,
x y x
f m m
close and centered
on value 1. Not only “injecting” energy in a
photon (by frequency increase of that photon)
may generate fermionic particle-antiparticle pairs
(with non-zero rest masses), but also injecting
momentum in a photon may generate “(very)
heavy photons” (like W/Z bosons and HB) which
further decay in fermionic pairs.
DRH considers very plausible the possibility that
the symmetry-breaking condensation of HB to
also generate not-only the W/Z-bosons (also 3-
egs), but also the 4-egs which mediate the
gravitational field (GF): this implies GF to be a
residual SGF (SGF may also be a residual
VSG[2], as VSG[2] may be a residual VSG[1]),
and may contribute to the
a
n
-based explanation
of the hierarchy problem, as
43/
q H a
Gf d Gf n
(with an approximate
same order of magnitude).
DRH also predicts that VSGFs are probably
mediated by the 1/2D-frame egs (1-egs and 2-
egs) quantized by
1
eg
hf
and
2
eg
hf
, which
generates
1
q
Gf
and
2
q
Gf
and may also
have 0-spin and even parity (like HB and the 3-
eg).
In checkpoint conclusion, DRH (as based on the
universal scaling factor
a
n
) offers important
explanations and predictions (mainly the
generalization of the electrograviton model for
any relative frame with d dimensions).
We can also define a (generalized) reduced
(angular) momentum-like quantum
x
for any
bosonic or fermionic EP or non-EP with non-zero
rest energy
x
E
(with
2
being associated with
a full mean lifetime
x
t
of that particle) so that:
,2
xx
x x x Et
Et
(4-9)
As
,
x x x
Et
can be applied to fermions also,
the values of
,
x x x
Et
and
,
x ou x
dL
log /
a ou xnLh
for the main
elementary/composite fermions (the electron[e],
the muon[µ], the tauon[τ], the up-quark [u] from
the free proton/hydrogen atom, the down-quarks
[d1 and d2] from the free proton and from the
free neutron respectively, the charm-quark [c],
the strange-quark [s], the top-quark [t], the
bottom-quark [b], the proton[p] and the unstable
free neutron[n]) based on their rest energies
(
0.5
e
E MeV
,
106E MeV
,
1777E MeV
,
2
u
E MeV
,
(1,2) 5
d
E MeV
,
1290
c
E MeV
,
100
s
E MeV
,
173
t
E GeV
,
4
b
E GeV
,
938
p
E MeV
and
940
n
E MeV
) and their
estimated mean lifetimes (
26
10
e
t yrs
,
6
10ts
,
13
10ts
,
31
110
up
d
t t t yrs
,
2886
n
d
t t s
,
12
10
c
ts
,
8
10
s
ts
,
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
14
25
10
t
ts
and
12
10
b
ts
) are represented in
the Table 2.
Table 2. The value of
,
x x x
Et
and
x
d
for
the main known particles of our universe
,
x x x
Et
x
d
54
, 1.8 10
e x e e
Et
1.7
e
dD
16
, 5.6 10
xEt
2.6dD
11
, 1.2 10
xEt
2.7dD
59
, 1.8 10
u x u u
Et
1.5
u
dD
59
11
, 3.7 10
x
d d d
Et
11.5
d
dD
24
22
, 1 10
x
d d d
Et
22.4
d
dD
11
, 3.4 10
c x c c
Et
2.7
c
dD
14
, 3 10
s x s s
Et
2.6
s
dD
*
,21
t x t t
Et
*
3
t
dD
12
, 1.6 10
x
b b b
Et
2.7
b
dD
61
, 7.2 10
p x p p
Et
1.5
p
dD
26
, 2 10
n x n n
Et
2.3
p
dD
It is noticeable that
,
x x x
Et
corresponding to
the main known fermionic particles except the
top-quark (t) (marked * in the previous table) has
values with many (11 to 61) orders of magnitude
larger than
, , ,
xW Z H
and
dimensional d-frames with
x
d
values
condensing around ~1.5D and ~2.5D: this is an
important distinction between fermions and
bosons (which have
x
d
values around
3
ph
dD
and
4
eg
dD
if considering the
existence of the hypothetical electrograviton).
Interestingly, the top-quark shares the same ~3D
frame with the bosons.
There is also an interesting dimensional
correspondence between quarks and leptons,
such as the electron, the up-quark and the down
quark from the free proton all share the same
~1.5D frame (which is also the frame of the
proton). The other heavier (and short-lived)
leptons share the same ~2.5D frame with the
other heavier (and more unstable) quarks.
Interestingly, the (reduced) (angular) quantum
momentum of the photon
acts as a “cut-off”
quantum momentum of nature, as all particles
with
,
x x x
Et
are predicted to have non-
zero rest masses (as in the majority of fermions
and bosons) and all particles with
,
x x x
Et
are predicted to have zero-rest
masses (only relativistic masses, as in the case
of the photon and the hypothetical
electrograviton).
The existence of the “empty” ~2D and ~3.5D
frames (which aren’t shared by any known
particles) may be in fact an indirect proof for the
existence of other (still unknown) particles that
may “occupy” this apparently “empty” d-frames
with
35 45
, / 10 ,10
x x x
Et
(for the ~2D
frame) and
30 20
, / 10 ,10
x x x
Et
(for
the ~3.5D frame).
5. THE PREDICTION OF A
GRAVITATIONAL FIELD VARYING
WITH THE ENERGY SCALE
This paper also proposes a set of three simple
a
n
-based functions to describe three
hypothetical variations of the gravitational field
(GF) strength as measured by the quantum
gravitational coupling constants
1Gq
a
,
2Gq
a
and
3Gq
a
, with a variable energy scale
,
ePl
E E E
, with
20.51
ee
E MeVmc
and
519
1.22 10/
Pl
E GeVcG
(the Planck
energy at which unification of all the four
fundamental forces is predicted to occur), such
as:
var
13/2
2
1
Pl
E
E
a
Gq a
En an
and
2
1/e
q Gq
G E E c m
(5-1a,b)
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
15
var
2
23/2
2
1
Pl
E
E
a
Gq a
En an
and
2
22
/e
q Gq
G E E c m
(5-1c,d)
var
3
33/2
2
1
Pl
E
E
a
Gq a
En an
and
2
33
/e
q Gq
G E E c m
(5-1e,f)
31 3 1 2
11.2 10
qPl
G E m kg s
reaches
3
q
Gf
,
72 3 1 2
210
qPl
G E m kg s
reaches
2
q
Gf
and
113 3 1 2
310
qPl
G E m kg s
reaches
1
q
Gf
. The base-10 logarithmic
variation of the functions
1
pE
10 1
log Gq E
,
2
pE
10 2
log Gq E
and
3
pE
10 3
log Gq E
for
,
ePl
E E E
are
represented in the Fig. 3:
Fig. 3. The variation of the gravitational field
strength described by the functions
1
pE
,
2
pE
and
3
pE
This approach also offers the possibility of a
vacuum energy density
vac
that varies inverse-
proportionally to the length scale
(and direct-
proportionally to the energy scale
E
), which
may fill the huge “gap” (varying from 40 to more
than 100 orders of magnitude) between the
observed small
vac
used by general relativity
and the very large
vac
predicted by the
quantum field theory.
2
(1,2,3)
8
vac q
c
G
(5-2)
6. A UNIFICATION PATTERN OF ALL
THE FOUR FUNDAMENTAL FIELDS AT
PLANCK ENERGY SCALE
The running coupling constant of the
electromagnetic field (EMF)
determined in
quantum electrodynamics (QED) using the
beta function can be also written as the
function of a variable energy scale
2
( 0.51 )
ee
E E m c MeV
,
Pl
EE
and
1/137
, such as [43,44]:
2
1 ln /
3
f
e
EEE
or
(6-1a)
fE
may be interpreted/explained and
redefined as the consequence of the variation of
a
n
with a variable energy scale
E
, as described
by the function:
ln(4)
3
//
a a e
nf E n E E
,
(6-1b)
with
2
1/ log a
f E nf E
(6-1c)
The running coupling constant of the weak
nuclear field (WNF)
W
includes the rest
energies of the W/Z bosons (which are the
propagators of the WNF) and is also
based on the Fermi coupling constant
exp.
352
/ 1.1663787 10
F
G c GeV
(with
62 3
1.43585 10
F
G Jm
), which can be
indirectly determined by measuring the muon
lifetime experimentally.
W
can be also written
as a function of a variable energy scale
,
ePl
E E E
, the rest mass/energy of the W+/-
boson
W
m
and
2
WW
E m c
such as
[45,46,47,48]:
3
2
/
/
W
WF
WEE
E G c
fE e
(6-2)
The running coupling constant of the strong
nuclear field (SNF)
S
determined in quantum
chromodynamics (QCD) (also) using the beta
function can also be written as a function of a
variable energy scale
SNF
EE
,
Pl
EE
and
Drăgoi; PSIJ, 15(3): 1-19, 2017; Article no.PSIJ.34613
16
210 40
SNF
E MeV
(the QCD energy scale
of quark confinement as determined
experimentally), such as [49]:
2
7ln /
SSNF
fE EE
(6-3)
The approximated running coupling constants
of GF, EMF, SNF and WNF can all be
represented on the same graph using the base-
10 logarithmic functions
1 10 1
log
GF Gq
p E E
,
2 10 2
log
GF Gq
p E E
,
3 10 3
log
GF Gq
p E E
,
10
log
EMF
p E f E
,
10
log
WNF W
p E f E
,
10
log
SNF S
p E f E
: see the Fig. 4.
Fig. 4. A unification pattern of GF(1,2,3), EMF,
SNF and WNF at the Planck energy scale
7. CONCLUSION
The (running) global scaling factor
ln(4)
3
//
a a e
nf E n E E
, the redefinition of
the (running) fine structure constant
2
1/ log a
f E nf E
(which implies the
contraction of both adimensional parameters into
a single one, which is also associated with a
plausible fine-tuning of both big G magnitude
and all elementary rest masses), the dimensional
relativity hypothesis (DRH) (including the
generalized electrograviton model), the set of
strong (and very strong) (running) gravity
constants (measuring a gravitational field which
varies with the energy scale) are all potential
updates for the Standard Model of particle
physics.
ACKNOWLEDGEMENTS
I would like to express all my sincere gratitude
and appreciation to all my mathematics, physics,
chemistry and medicine teachers for their
support and fellowship throughout the years,
which provided substantial and profound inner
motivation for the redaction and completion of
this manuscript.
COMPETING INTERESTS
Author has declared that no competing interests
exist.
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