To Develop a Virtual Model of Microscopic Quantum Gravity

We would like to suggest that, by considering three virtual gravitational constants assumed to be associated with gravitational, electromagnetic and strong interactions along with a strongly interacting virtual nuclear elementary charge, a workable model of final unification can be developed. In a verifiable approach, Newtonian gravitational constant and Fermi’s weak coupling constant can be interrelated via nuclear and electromagnetic gravitational constants.


Introduction
The most desirable cases of any unified description are: a) To implement gravity in microscopic physics and to estimate the magnitude of Newtonian gravitational constant.b) To develop a model of microscopic quantum gravity.c) To simplify the complicated issues of known physics.d) To predict new effects, arising from a combination of the fields inherent in the unified description.
Proceeding further, we also suggested the role of a new elementary charge associated with nuclear interactions and strong coupling constant [13,14].In this paper, by considering the word 'virtual' and by refining and re-arranging the old semi empirical relations, we make a bold attempt to fit the Fermi's weak coupling constant and Newtonian gravitational constant [15,16].Estimated magnitudes are It may be noted that, success of any unified model either depends on its physical/mathematical back ground or depends on its wide range of applications.It is clear from the below proposed applications (1 to 53) that we could satisfactorily fit the nuclear, electroweak and Newtonian gravitational constants through unified semi-empirical relations.This sincere attempt is to be reviewed and ascertained by the scientific community.We would like to appeal that, with respect to currently believed String theory and Quantum gravity models, proposed semi empirical relations and proposed assumptions, can be given some consideration in developing a 'workable model' of TOE.

Three basic assumptions of final unification
In our earlier publications, we proposed the following three assumptions.In this paper we review them with the word, 'virtual'.
Assumption-1: Magnitude of the virtual gravitational constant associated with the electromagnetic interaction is, where e e is the currently believed electromagnetic elementary charge and s α is the currently believed strong coupling constant.Like quarks, the strong interaction elementary charge is experimentally undetectable and can also be called as 'invisible elementary nuclear charge'.
Note-1: e G can be estimated with, π πε Note-2: s G can be estimated with,

Important and interesting results
Considering the following semi empirical results one can understand and validate the role of the proposed three assumptions.

A) Ratio of rest mass of proton to electron:
With reference to : With reference to ( ) and : With reference to ( ) and : e N G G  With reference to ( ) , and : B) Electron and proton rest masses with respect to Planck mass : This can be compared with a broad range of quoted values, 0.113 to 0.12 pertaining to various measurements.

D) Magnetic moment of proton [14]:
This can be compared with the recommended value of This can be compared with the recommended value of ( ) Note: Considering K) Planck's constant: Note: ( ) ( )

M) Atomic radii [21]:
Let, By considering the periodic arrangement of atoms, further research can be carried out.

N) Photoelectric work functions [22]:
Let, Z W = Photoelectric work function of Z.
where A R is the radius of atom, A is the atomic mass number and factor ( ) For the preparation of table, we consider, From the table, it is possible to say that, 1) As atomic number increases, factor f starts from ~3.5 and gradually reaches to ~2.5.
2) Actual atomic radii for light atoms seem to be less than the reference values and for heavy atoms, actual atomic radii seem to be higher than the reference values.

O) Earth's magnetic dipole moment
Planet's earth's magnetic dipole moment can be understood with: ( ) This can be compared with the estimated order of magnitude of Earth's magnetic dipole moment .Based on this relation, other solar planets, exo-planets and neutron star's "mass dependent" order of magnitude of 'magnetic dipole moments' can be estimated [23].See table-3.It may be noted that, for 30 hot Jupiters, on an average, estimated value is roughly 0.21 times the reference values.See table -2.

P) Neutron star mass and radius
M m represent the masses of neutron star [24] and neutron, then, 3.175 Note: By considering , 2  mass of neutron star can be estimated to be 1.5875M  .This is just greater than the famous Chandrasekhar mass limit of 1.4M  .
2) If NS R represents the neutron star radius, then, ( )

To understand proton's melting point
With reference to Hawking black hole temperature formula [25], melting point of proton [26] can be understood with: Based on this relation and with reference to up quark, other quark melting points can be expressed with the following kind of relation. where represents the ratio of mass of any quark to mass of up quark.Based on this relation, for up quark of rest energy 2 MeV, its corresponding

Fitting medium, heavy and super heavy nuclear charge radii
For medium, heavy and super heavy atomic nuclei, nuclear charge radii [27,28,29] can be fitted with the following simple relation.( )

Fitting and Understanding Fermi's weak coupling constant
Quantitatively the famous Fermi's weak coupling constant [13,14] can be fitted with the following relation.
When distance between two protons is close to 0 , gravitational force of attraction between them can be expressed as, Based on this idea, W F can be expressed in the following ways.
It may be noted that, this relation is free from all numerical factors and accuracy mainly depends Clearly speaking, s is the ratio of specific charge of proton associated with s e to specific charge of electron associated with .e e Using this ratio, proton-neutron stability relation can be fitted directly in the following way [30].
where s A is the estimated stable mass number of .

B) Nuclear binding energy:
Nuclear binding energy potential can be expressed with the following relation.
Based on the new integrated model proposed by N. Ghahramany et al [31,32] and with reference to relation (31), it is possible to show that, ( ) where, .
Based on this strange and simple relation and with reference to our recent publication [37] and first four terms of the semi empirical mass formula (SEMF), close to the beta stability line,

( )
for Z 2 to 100 , = it is possible to show that, ( ) ( ) This can be compared with first four terms of the semi empirical mass formula where : , estimated binding energy range is (857 to 1140) MeV and can be compared with reference binding energy [38] range of (806 to 1105) MeV.It is for further study.With reference to SEMF, close to the beta stability line, it is also possible to show that, ( ) ( ) Let, ( ) In comparison with SEMF, by replacing s A with A and s N with ( ) A Z − in relation (36) and by considering a multiplication factor of the kind ( ) ( )

Z A s A A
− associated with each term, binding energy of A can be estimated approximately.

Neutron life time and Avogadro number
Neutron life time [39] can be fitted in the following way, ( ) where, By considering the unified atomic mass unit as well as neutron proton rest masses, ( ) 3 888.5 sec In this way, results of bottle experiments and beam experiments both can be fitted [13].

Strange results connected with Planck scale Schwarzschild radius
A) Conceptual thought: Schwarzschild radius of Planck mass plays a vital role in electroweak and strong interactions.
Schwarzschild radius of Planck mass

B) Strange result connected with ( )
and Based on this relation, If, recommended , from relation (43), it is also possible to show that, where,

C) Strange result connected with Proton-electron mass ratio
From above relations, ( )

'System of units' independent Avogadro number and Molar mass unit
If, atoms as a whole believed to exhibit electromagnetic interaction, then molar mass constant and Avogadro number, both can be understood with the following simple relation.
Where atom m is the unified atomic mass unit and mole M is the molar mass unit or gram mole.
Thus it is very clear to say that, directly and indirectly 'gravity' plays a key role in understanding the molar mass unit.Based on these relations, "independent of system of units" and "independent of ad-hoc selection of exactly one gram or one kilogram", it may be possible to explore the correct physical meaning of the famous 'Molar mass unit' and 'Avogadro number' in a unified approach.It may be noted that, Avogadro number and 'gram mole' are having many applications in solid state physics, gas dynamics/thermodynamics and basic chemistry/electrochemistry.By considering the following relation, it is possible to couple the Avogadro number with the observed fundamental interactions.
12. Relation in between ( ) , , and : It may be noted that, fitting the gravitational constant with elementary physical constants is a very challenging issue.G. Rosi et al say [15]: "There is no definitive relationship between N G and the other fundamental constants, and there is no theoretical prediction for its value, against which to test experimental results.Improving the precision with which we know N G has not only a pure metrological interest, but is also important because of the key role that N G has in theories of gravitation, cosmology, particle physics and astrophysics and in geophysical models".
In this context, we would like to stress the fact, with currently available standard theoretical models, it may not be possible to fit and verify the Newtonian gravitational constant with elementary physical constants.With the following semi empirical relations and with further research, in a verifiable approach, it is certainly possible to explore the back ground physics of the role of the Newtonian gravitational constant in microscopic physics.
In a semi empirical approach it is also noticed that, 4 With reference to the recommended and experimental values of the Newtonian gravitational constant, with trial-error and by introducing an ad hoc factor of ( ) The ad-hoc fitting factor ( ) 3 2 needs a detailed explanation and is for further study.

Discussion
It is true that, unless stringent requirements are met, in general, speculative alternatives to currently accepted theories cannot be accepted or published.Scientific papers having content that lie outside the mainstream of current research must justify by including a clear, detailed discussion of the motivation for the new speculation, with reasons for introducing any new concepts.If the new formulation results are in contradiction with the accepted theory, then there must both be a discussion of which experiments could be done to verify that the conventional theory needs improvement, and also an analysis showing the consistency of the new theory with the existing experiments.In this context, we appeal that, 1) Subject of final unification is having a long unsuccessful history.Clearly speaking, so far, no model succeeded in implementing the Newtonian gravitational constant or Planck scale in nuclear and electroweak interactions.
2) Even though, the basic idea of String theory [42,43] is very simple, very interesting and highly intuitive, there are no concrete new predictions on low energy scales and high energy scale predictions are beyond the reach of current technology.
3) It may be noted that, since 1992, J. E Brandenburg is working on 'GEM unification theory' [44] and proposed an interesting and unified relation, Compared to J. E Brandenburg and other available models of current unification theories, in this paper, with reference to Planck scale, we presented a variety of multipurpose arithmetic relations pertaining to nuclear, electroweak and astrophysical applications.

4)
As the current unification paradigm is failing in developing a 'practical unification procedure', the point that we wish to emphasize here is that, by considering three new assumptions, we presented a number of applications connecting micro-macro physical systems and finally developed arithmetic relations for understanding the role of the Newtonian gravitational constant in microscopic physics.We appeal the mainstream physicists to see the possibility of considering the proposed relations for further investigation with respect to strong, electroweak and gravitational interactions collectively.

5)
Following this kind of computational approach, it is certainly possible to reproduce another set of arithmetic relations by using which, in near future, it may be possible to find a set of absolute relations having sound physical reasoning and strong mathematical back up.

Conclusion
As it is inevitable to unite gravity and other three atomic interactions, if one is willing to explore the possibility of incorporating the proposed assumptions either in String theory models or in Quantum gravity models, certainly, background physics assumed to be connected with proposed semi empirical relations can be understood and in near future, a 'workable' or 'practical' model of "everything" can be developed.Based on relations ( 28) and ( 29), Fermi's weak coupling constant and the three gravitational constants can be fitted in a unified approach and finally, Newtonian gravitational constant can be estimated accurately with microscopic physical constants.

=
Estimated root mean square radius of proton of finding electron in any orbit labeled with 1, 2, 3,.. n = further research can be carried out.Bohr radius can be addressed with, be considered as characteristic constant connected with revolving electron.L) Magnetic flux quantum in super conductivity [20]: unit.A = Atomic mass number.

≅
This energy can be compared with currently believed QCD energy scale of 270 MeV.
is for further study.Based on these relations,

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 November 2017 doi:10.20944/preprints201711.0119.v1
It may be noted that, this relation is free from arbitrary numbers and can be compared with the following relation available in recent literature.See table-3.
c a ≅