Hidden Variable links the Relativity
and the Quantum theory
Shogo Niimi
Far-east Institute of Technology, 89 Nishigo Kashima Gamagori Aichi 443-0037 JPN.

Albert Einstein. He discovered great achievements such as the photoelectric effect that leads to the quantum theory. Even if he did not
approved the quantum theory. His idea is summarized in the word that “God does not play dice”. He thought that the quantum theory
should have hidden variable. Generally, this idea has been denied. However, I discovered this variable that links relativity and quantum
theory. I could find the light velocity, the Planck's constant also the function that can calculate the gravitational constant by use thisvariable. In addition to it, I got the function that expresses electron. In here, we will get new way of forward to the Grand Unified Theory.
Generally, most of people have been believe in the space wasborn with the Big Bang. However still now, that is theimportant issue how was the space born and is growing up.
For example, there has the idea of multiple universes that ourworld is only one of many of the spaces. This idea came fromthe suggestion that the space-time has 11 dimensions. It
ascends to the Kaluza-Klein theory. The string theory wascreated in this theory. As for this theory, using a string forsolve the problem that impossible to express the matter fromthe material particle. After that, the superstring theory wascreated for solve the problems of the string theory. The
space-time began to be discussed that has 10, 11 dimensions
or so, at the point. Also, a superstring got to have a lot of
faculties. It connects the other end, snaps, vibrates andwinds. Then, D-brane was created for a superstring solves
many problems, be fluttering and connecting with themembrane.
Now, the membrane became superior to asuperstring by the M-theory exploited that the membrane has
the inner structure. I do not find that will this story continue
forever. The new hero is born every time the new problem
happens. However, always you must pay attention to the
new hero. They have the key that connects with the Grand
Unified Theory also solve the space theory. Above-mentioned
new hero “the membrane that has structure” appears in mysuggestion. He plays the lead character in here. I call him“gap space”. I suggest that our space – the whole space – hasbeen dividing the real space, imaginary space also the gap
space, exists between both, kinds of multiple universes. This
whole space has some curvature. I got the light velocity with
the Planck’s constant in this whole space. In addition, got
the electric charge and the electronic mass. Finally, The
gravitational constant that has important issue was gotten.
Actually, the solution of the electric charge has width. It
seems to be the Heisenberg uncertainty relation in thequantum theory. This relation is that impossible to fix theother quantity in case of fix some quantity. I consider all ofabove, hidden variable seems to be the curvature of the whole
space, as Einstein needed it in the quantum theory.
Ensuring all of them by the gap space. Will he win the
Oscar? – Refer to 1) up to the superstring theory –The space phase transition
In the latest observation result is that our space is13.7-milliard years old2). However, I seem that she is active
and not the elderly. She has the born moment as I can counthere age. You call the moment the Big Bang. The substanceof the Big Bang is the space phase transition. I suggest this-１-

[Fig.1]////I suppose that the base of our space is empty world H0 [1]. H0
has the special inner degree of freedom that “the fluctuation” and has
zero dimension. This fluctuation originates in “mathematical zero has
the solution”. H0 separates real space R3 and imaginary space I3 by thefluctuation [2]. Each R3 and I3 has 3 dimensions. In this case, whole
space H6 has 6 dimensions. Ordinary, R3 recombines with I3
immediately and revert to empty world H0 [3]. However, supposing thatcomplex space G3 is composed by R3 and I3 [2]. As a result, R3, G3 andI3 compose whole space H9 that has 9 dimensions and it is not possible
to revert to H0 [4]. This H9 is our present space. However, we live in isreal space R3

process below.
First, I take the empty world that impossible to call it
space. Somewhere possible to twitches caprice even if it isthe empty space. For example, I give you a vector that has
size and the direction with arrow. Some product of the sametwo vectors becomes zero. My reverse explanation, it musthave the vector as solution even if zero. There is possibility
that the event happens in the empty world in this way.
Generally, you call it “fluctuation”3). Notice, this phenomenon
is a little different from pair creation of particle andantiparticle.
The solutions of the empty world always appear anddisappear in caprice. I replace it in the phenomenon of thespace that they disappear as soon as encounter even if the real
space and the imaginary space appear in the whole space. As
for this process, you could take the relation between the real
number and the imaginary number with the same size. For
example, you give the real number a and give the imaginary
number ia. i is the symbol of the imaginary number. Thesummation of their that a2 is the square of a and -a2 is thesquare of ia become zero. Pair creation and annihilation ofvirtual particle with anti-particle by the fluctuation that issimulated this calculation. However, I do not have therestrictions that the same solutions always appear in thewhole space. Also, do not ensure that they disappear without
doing anything. Sometime, I suppose that they start theirrecoverable action as impossible to recover empty space.
Actually, this action is that the real-imaginary space appears
between both spaces before both spaces meets after the real
space and the imaginary space appeared in the whole space.
This real-imaginary space is the complex space, named gap
space. As a result, the whole space has 9 dimensions by each
3 spaces has 3 dimensions [Fig.1]. Incidentally, I do not takethe time dimension in here. I3 represents imaginary space, G3
represents gap space and R3 represents real space on the
figure. Also, H9 represents whole space. I consider ofcompactification of the extra dimensions now. This issue
appears from that our world has 3 dimensions. In otherwords, compactification is the operation that the concealmentof the extra dimensions in case of our world was created inover 3 dimensions space. By the way, you live in the worldthat is real space R3. Therefore, do not necessary to considerof the compactification even if the whole space has 9
dimensions. Changing a viewpoint, I am possible to express
that already made compactification of the extra dimensions
when the space was born.
Unfortunately, unsaved the serious matter. The whole
space had the general solution – call it elementary function –by the gap space appeared in there. I understand that he has
the radius of the special space, call it Sub Bromwich=Wagner
sphere4) [Fig.2], when investigate these elementary functions indetail. It is a unique radius that has a unit of angle. It has
more important thing, this radius is also the radius of thewhole space. The curvature of the whole space is restrictedby it. This curvature has the reciprocal of the unit of angle
that are advantages for me. It depends on this, I cancalculate of the elementary functions.
By the way, how should I take gap space in modern science?
It will be the membrane that has structure also partitions
space tiny, named holographic membrane5) on the M-theory6).
For example, you can imagine a soap bubble [Fig.3]. The
membrane that the gap space also has the solution ofoscillation. The real part of oscillation has not zero on theboundary between the gap space and the real space, also
appears in the real space. On the other side, the imaginary
part of oscillation has zero on the boundary, never have an[Fig.2]////I suppose that c1
on [r2, ir3], c2 on [r1, ir2]
and c3 is Bromwich=
Wagner circle on complex
plane [r3, ir1]. Bromwich=
Wagner circle is a virtual
circle on the complex plane
and agrees the Laplacetransform. Incidentally, i is
the symbol of an imaginary
number that in front of r.
The virtual sphere can becomposed in the real space by the combination of these three circles. I
call it Sub Bromwich=Wagner sphere. Vector P can be placed in thissphere. P is a function with angle ωτ, and ωτ is limited byanother angle θ. It is written as P(ωτ= wτsinθ). It is interesting,w that appeared here is the radius of Sub Bromwich=Wagner sphere
and whole space H9, and has a unit of angle. The reciprocal of w is thecurvature of the space.

-２-

influence in the real space. I need this result.
We can scarcely catch antiparticle in the reality world.
The experimenter only has the privilege that catch
antiparticles. Actually, antiparticles should still be exists
stably in the imaginary space. However, antiparticles cannot
come out from the imaginary space. I appreciate that the gap
space exists for the stability of the world.
The light velocity also the Planck’s constantI cannot win your confidence even if the explanations that are
so many words. The reality should prove philosophy. The
most excellent proof in here is to get gap space. However, I
have never met a trusted experiment result, for now. The
next way, I should take the experiment value on the physicalphenomenon by the computation. No one trusts me if I
cannot get the light velocity about 300 million meter per
second at least. However, I have a limitation to get theequipments. Do not take huge experimental facilities. Do
not have the big researcher group. No budget and funds. I
only have myself – none secretary – and a cheap personal
computer with the computation software.
However, you do
not despise it. I needed 100 thousand￡ to buy it that has
the same ability, to got the elementary function before 20
years ago. Unfortunately, I was not able to buy it. In themeantime, I solved a lot of issues even if in such situation.
Examined each variable in detail and cleared their relations.
It was the preparation to do correct computation. Then, I got
the materials of civilization, now. Be able to compute up to
15th decimal place use this software. Actually, I need 20thdecimal place but tolerate it. Also use some special functions.
However, I must compute the elementary function in the
logical limitation and have not intention even if want the light
velocity. Do not give the elementary function value in my[Fig.3]////The membrane of a soap
bubble not only divides the space
but also has structure.
That
fantastic ray is generated by thisstructure. In other words, the ray
is created by the phase difference
that is generated by the reflection of
the light from the obverse and thereverse of the membrane.
The gap space is equivalent to this membrane. At first sight, theinformation that cannot be seen but clarifies by this space participates in
it. The gap space vibrates same as the membrane of the soap bubble
is wavered by the wind. Sighting from the real space, the gap space isthe barrier that confines the imaginary space. Sighting converse fromthe imaginary space, the real space is confined by the gap space. By
the way, I do not know whether the gap space burst same as a soap
bubble, at last.

discretion. The computer does value calculation. I only givethe computation table and the numerical formula. Then,
made some computation tables; computed them in verticallyand horizontally every time. I found interesting value in the
48.9-radian vicinity of the elementary function that has theangle at some day, which took about one year and a half forthe computation [Column-1]. It was [3.51767E-43] that divided
the Planck’s constant by the light velocity. E-43 has ameaning that divide 3.51767 by 43 zeros line with 1 as the
symbol of Microsoft Excel7). Generally, it is expressed as 10-43.
Is this result may just an accident? It is a result that I
computed the elementary function in vertically. Therefore,
computed it in horizontally.
Then, certainly got value[299792458] with light velocity. Yes! Got not only it but also
value [1.05457E-34] with Planck’s constant that appears at theside of the light velocity [Column-2, 3]. Their wonderfulresults were gotten in the real space.

[Fig.4] ////There are velocity v -blue line-, angular momentum  -redline-, real part of magnetic flux Qr -green line-, imaginary part ofmagnetic flux Qi -yellow line- in the figure. The x-axis sets angle θfrom zero to π/2. The y-axis sets each physical quantity. Except foranother frame of v, scale agrees of all physical quantities. Primarily, I
should represent that divide the graph, v and  are set in the real
space, Qr and Qi are set in the gap space but get together them in one
graph that visible. There are light velocity [299792458(m1s-1)], Planck’s
constant[1.05457E-34(V1A1s2)]
andabsolute
value[1.77807E-13(V1s1)] of the magnetic flux at about 48.9 in ωτ. The
dotted line of the position ofθ=π/2 is the boundary that divides real
space and gap space. The imaginary articles do not appear on the real
space same as Qi has zero on the boundary. I use exponent notation in
Microsoft Excel as a [E-34] and so on, also the unit notation by Isao
Imai -Tokyo University-.

ElectronPrimarily, the sufficiency to the light velocity and the Planck’s
constant were only gotten in the identical condition in theworld for me. However, you know that it is not described how
material is born.
Therefore, this discussion is aninsufficiency for you. At least, you need electron that I havesearched it before 30 years ago. I needed more than 1 year
for this great challenge.
First, I examined the elementary function in the gap
space. The elementary function in here, it becomes thecomplex function that was combined the imaginary and thereal number. In the gap space, the solution is different fromthe real space completely. The whole picture of this solution
is vibrating beautifully [Fig.4]. Now, pay attention to thevicinity of the 48.9 radian. Two pieces of value that I havenever seen them lined the side of the light velocity and the
Planck’s constant. The combination of two pieces of value isthe solution of the complex function that has the real number
and the imaginary number. However, you do not take natureof this function by appeared values. I described that a vector
has size, just now. In addition to it, the complex function has
size. At first, I did not understand this size that has themeaning. When examining well, I found that it was thesquare of the product between the light velocity and the
Planck’s constant. The value has about [1.77807E-13]. By
the way, the light velocity can be replaced the combination of
the dielectric constant in the vacuum and magnetic
permeability by James C. Maxwell (Edinburgh U. K.). This
mean is that the light velocity same as the combination of theelectric field and the magnetic field. However, the relation
between the electric field and the magnetic field does not havethe unit of speed. Therefore, I accept that they have thesame unit as they have the same meaning. In other words,
regarding that the light velocity has the unit of the resistance.
Then, the product between the light velocity and the Planck’s
constant has the unit that is the square of the magnetic flux.
After all, this complex function has the unit that is the sameas the magnetic flux [Column-4].
The unit of theelectromagnetism is very complicate. Actually, this result
was gotten by the idea of I. Imai8) – Tokyo University –.
All the actors gathered in here. However, the story is not
so easy.
Actually, I deal with the angle here is the
combination of time and the frequency. Unfortunately, I
cannot operate isolated time in the logical limitation in case ofcomputing all functions here. Planck time is not an exception.
I cannot give the time any value and do not have to give it.
To give time is intention for the value adjusting only.
Therefore, I always cannot give only combination of time andthe frequency the value. Also, cannot operate most of the-３-

physical quantity expressions in this strong limitation. Of
course, they have the distance and acceleration. Actually, theactors gathered only three the light velocity, the Planck’s
constant and the magnetic flux. It will be the outside of the
expectation of Carlo Collodi (Author/1826-1890/Italy).
However, I have an idea that inspires a life in Pinocchio. It isthe cancellation of time at the time. For example, the timefunction is tied in the time, but it has the possibility that isreleased by the integral. The integral is the mathematicaltechnique for the evaluation in the function by gathering
objects of some area. Here, the electric current works a partof Pinocchio. The electric current can be made with the
combination of the energy and the magnetic flux. The energy
is a kind of Pinocchio. Pinocchio runs here by the operationof the cancellation of the time in the logical limitation. Thatis the expression of the electric charge.
Correctly, the electric charge is composed by the magnetic
flux and the angular momentum that has the Planck’s
constant also another material [Column-5]. This material isthe angle that is made by the radius of the Sub
Bromwich=Wagner sphere and the time. Of course, it isreleased from the time. The light velocity and the Planck’s
constant have this angle. In the computation result, theelectric charge has a width of 0.05% around [1.60217E-19].
Also, the angle has a width of about 0.05% in 48.9 vicinity.
These cause that the space has the curvature, no less. In
other words, the light velocity, the Planck’s constant and theelectric charge are appeared in our world by the space has thecurvature. This width should cause of the uncertainty in thequantum theory. All expressions have the angle that haswidth. Therefore, this angle should hidden variable thatlinks relativity and quantum theory.
Electron is composed by electric charge and electronicmass. The mass is created by the Higgs mechanism in the
The Higgs
standard theory – the G-W-S theory9) –.
mechanism expects that Higgs boson as a seed of mass has
greater mass and exists in our world. However, I do not use
this idea in here; only use the information about the light
velocity, the Planck’s constant, the electric charge and theangle. Like the case of the electric charge, I must exclude of
the concept with time and length. However, there are thenecessary and sufficient even if a little information. First, I
want to get information by the electric charge. Got the resultthat the electric charge has width a short while ago. Of
course, this width is limited at the width of some angle.
Mathematically, the width of the electric charge is gotten by
integrate – the definite integral – the electric current. Thiswidth has [8.2E-23]. Next, I want to get information from theangle. There are two kinds of angle; one of angle decides thecurvature of the space, be discussed here already. The widthof this angle has [0.025235]. Another one exists in Sub
Bromwich=Wagner sphere, it has [0.032126471]. Actually,
this one is limited by the angle that decides the curvature of
the space, also is used to integrate the electric current. We
must discussed width by the mass is not a material particle.
In fact, the string theory progressed on this understanding.
Next, I need the value that is the base of the fine structure
constant. The fine structure constant does not have the unitthat is used to explain the fine structure of the atomic
spectrum. It is gotten from the squared of electric charge andthe ratio of Planck’s constant to light velocity. Both of them
have same unit. Therefore, the solution of the ratio does nothave a unit. Actually, this value has width of 0.05% around[1.37036E-05]. Now, I get the expression of electronic mass
and value [Column-6]. The mass value has width, also that
central value agrees with experimental value [9.1093897E-31].
It is obvious with above discussion; the mass is created by the
space that has the curvature.

The arc in the real space, the imaginary space expresses thediffracted wave, I can suppose that they are theelectromagnetic wave.
The issue is that do I replaceresistance block G to something of the gap space theory. I
consider about this. Generally, seem to be that the mass does
not born in resistance-less. The Light velocity was decisionby that the vacuum has the dielectric constant and magnetic
permeability. Therefore, seem to be that the vacuum has
viscosity. I think that this viscosity becomes resistance andgenerates mass. Therefore, I accept that this resistance was
generated by the whole space has the curvature. Resistanceblock G has this meaning in the figure. Also, I can explainthat G relates to the gravitational constant by accept that thecurvature generates gravitation. By the way, G is expressed
the reverse of the resistance on the electromagnetism. As
described a little while ago, the unit of the resistance is equal
to the unit with velocity. Therefore, the key point is velocity
or the product in the time to the acceleration. The reversal ofthem is resistance block G. The gravitational constant that isgotten lined the side of the light velocity and the Planck’s
constant after computed them.
However, there aredifferences about 2% between this value and experimental
value. Why do not get the experimental value? Yes! This
computation value must be exists in the ideal field. For
example, the nonlinear equation like that Einstein equation or
Schrödinger equation11) gets the effect of gravitational field or
particle itself. Actually, the nonlinear equation expresses thereal world very well by takes the effect of itself. Therefore,
the difference between computation value and experimental
value can be described by put the correction as follows.
The approximate value of the one particle field potentialthat is gotten by Einstein equation has a unit that is thesquare of the speed12). This one particle field potential can begiven to resistance block G. The self-correction by the

The gravitational constant
As you know, the gravitational constant is the most importantvalue same as the light velocity and the Planck’s constant.
Out of the assumption that I could explain it at first.
Therefore, postponed it than research of electron. I describe
them here in my order of the research process.
Generally, the gravitational constant has deep relation
with mass. In this case, the electronic mass must be hint of it.
Therefore, I must get electronic mass by all means. There ispossibility that Planck mass exists in here. However, I foundthat it is not possible to reach the solution even if approach it
contrary to my expectations. Therefore, take another way.
All physical quantity is limited by the curvature of the space isa fact. Then, the gravitational constant must be limited by
the curvature of the space, and precede electron same as thelight velocity. In this way, I suppose that one of elementaryfunction express the gravitational constant. However, I could
not find such a expression at first sight. Therefore, I will
restructure the process that approach for the gravitational
constant.
First, I take Kármán Vortex Street10) on thehydrodynamics as you know. Why Kármán whirlpool? I
think that situation of electron is born in the gap spaceresembles situation of Kármán whirlpool is born. In this way,electron can be supposed to Kármán whirlpool itself [Fig.5]. q
is Kármán whirlpool electron and resistance block G isequivalent to the isolate isle in the natural world. The
vertical line in the gap space expresses the transverse wave.

-４-[Fig.5]////Whole space H9 is composed by real space R3, gap space G3
and imaginary space I3. For example, I suppose that there is a small
hole on the boundary of G3 and R3, also set resistance G. The verticalline in G3 is the vibration of magnetic flux Q. Real part Qr of Q leaks out
R3 as I explained in [Fig.4]. Then, Qr collides with G and has an
impact by the angular momentum that exists in R3. In this result,electron q same as Kármán whirlpool appears in their. This electron q
takes the base as the minimum value of Planck’s constant . By theway, the unit of resistance G [V1A-1] agrees with the unit of speed[m1s-1]. G, i.e. v is the base of the gravitation, supposing that thegravitation is a kind of resistance. Certainly, gravitational constant GN isgotten by the combination of v, 1/w that is the curvature of the spaceand ο(omicron) that is the self-correction by the gravitation field. οisgotten by the infinite product of the curvature of the space, but replace it
to an approximate expression [‒sinwτ]. Incidentally, I took an isolate
isle as resistance G figuratively, not a reality. Correctly, the space has
resistance. It is clear by the relation between the velocity and theresistance. Incidentally, the relation between G3 and I3 is same as theabove.

gravitational field can be seen by reconstruct the expression inthis way. The self-correction by the gravitational field is thatthe gravitational field gets the effect of the mass by the mass
was created on the gravitational field, and the mass gets theeffect of the gravitational field, and then _ _ _ the way of thecorrection that the effect continues just like infinitely of itself.
Generally, it is expressed by the infinite series as thesummation infinitely or the infinite product as the productinfinitely.
They often diverge infinity, the known
gravitational theory are suffered from them. The operationof renormalization13) was developed for solve the problem.
However, it is not the universal. Therefore, I did not use it
for the electric charge in this paper. Also, I do not need it for
the gravitational constant. I can make the self-correctionthat does not diverge the infinite series or the infinite productby use the curvature of the space. Therefore, I take a term of
the self-correction gotten by the infinite product, here. The
value is about 0.9778. This result has width, the average
agrees with experiment value [6.67259E-11] [Column-7].
I describe another idea. The maximum velocity may befaster than c about 2% as the correction of the gravitational
constant is unnecessary by the fact of the photon is dominatedin the gravitation. In this case, the maximum velocity has
the relation in the expending universe14) 15). Actually, it ispossible to say that the self-correction by the gravitational
field is the correction with the velocity by the one particle fieldpotential has a unit of the speed.
Discussion
I present the light velocity, the Planck’s constant, thegravitational constant, the electric charge and electronic mass
for you in this paper. Surely, we are close the Grand Unified
Theory now. However, I have one of dissatisfactory point. I
do not make that they will be able to answer a question “doesthe mass exist in the world” clear. Here, I will show the one
direction.

The mass exists around us very much. However, it isvery difficult to demonstrate it logically. In addition to it,
Higgs boson is not confirmed. Therefore, I will take up
Planck mass as the polestar. Of course, I do not take it withthe gravitational constant. If the other method were here, I
should use it. I got the interesting result that take off theingredient of the electric charge from the expression ofelectronic mass after considering variously. This value has
about half Planck mass.
In other words, this value
multiplied by twice of the self-correction is the approximationof Planck mass. It has more accuracy. They are not perfect
but electronic mass and Planck mass are represented by theexpressions that are very akin [Column-8]. In this result, I
hope that all mass must be represented by unified form.
Then, the substantiality with mass will be proved. The
perfect Grand Unified Theory will be built with sooner orlater.
I want to discuss one point supplementary. I hope thatthey are connected with the experiment and the application.
I described that the electric charge is composed by the angularmomentum, the magnetic flux and the curvature of the spacein the ex-section. However, the composition way of theelectric charge is not only it. For example, it is possible tocompose by the moment of force and the electric potential or
the electric moment and the moment of inertia. Also, theelectric charge is gotten by the cancellation of time with the
combination of the angular momentum and the electricmoment. Of course, all cases need the curvature of the space.
There are a variety of cases but every event does not happen
on the high-energy physics. At least one of expression of theelectric charge is a cheap and convenient instrument thatcreates electron, I hope.////////////////////////////////////////////////////////////□//[Column]//A part of the expressions list.
//1, Angle////ωτ= wτsinθ
//2, Velocity include the light velocity////v = ω2 exp (+στ) cosωτ/(σ2+ω2)//3, Angular momentum include the Planckʼs constant//// = ω2 exp (-στ) cosωτ/(σ2+ω2)//4, Magnetic flux////Q = ω2 exp (-iστ) cosωτ/(σ2+ω2)//5, Electric charge//q = wτ/Q/////= τsinθ/U///////[U : Electric moment]
6, Electronic mass//mq =⊿qβsin⊿ωτ////[β: Base of the fine structure constant]
7, Resistance of the space include the gravitational constant//GN =ο/(wτv)///////////////[ο(omicron): Self-correction of the///////////////////////////////////////////////gravitational field]
8, Approximation of Planck mass//mPL≒-2sinwτβsin⊿ωτ///[ο≒-sinwτ]Reference
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