Hyperanalyticity of space-time

Aleksandr Rybnikov

August 21, 2018

Abstract

New mathematical object - hyperanalytic function is introduced. The

convergence of hyperanalytic functions is substantially above than the con-

vergence of analytic functions. A speci�c sample of hyperanalytic function

is the reticulum function (RF). This function describes the reticulum space-

time. RF can't be decomposed into the Fourier series and, therefore, RF

does not provide the conservation of parity as the analytic functions do.

Thanks to this, the RF can be decomposed in an endless series of two primi-

tive hyperanalytic functions by sequential attempts of decomposition in the

even and odd functions. The unique parameter of such series is the �ne

structure constant α. It allows combine all fundamental interactions intothe Naturally-Uni�ed Quantum Theory of Interactions. The price of such

quantum uni�cation is the reticulum space-time.

Keywords: �ne structure constant α, theory of everything, parity non-conservation.

1

CONTENTS CONTENTS

Contents

1 Introduction 3

2 Decomposition of RF 5

2.1 The mean value of RF . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 First di�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Even di�erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Odd di�erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Three-dimensional RF . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Quantum derivative with respect to time 10

4 Uni�ed Theory of Interactions 12

4.1 Interaction 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Interaction 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Interaction 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Interaction 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 Interaction 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.6 Interaction 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Appendix 18

2

1 INTRODUCTION

1 Introduction

It is known that there is a fundamental connection between analyticity of thefunction and the convergence of its Fourier coe�cients. The better the function,the faster its coe�cients tend to zero, and vice versa. The power decrease of Fouriercoe�cients is inherent in functions of the Ck class while exponential to analyticalfunctions. Here there is a possibility of existence of the hyperanalytic functions,for which the decrease of the Fourier coe�cients corresponds to tetration[1].

Examples of such functions were �rst obtained with conventional arithmetic[2].Therefore for this article calculations have been made with a program that providesthe necessary accuracy in each case1.

Natural hyperanalytic function occurs when considering reticulum with a stepL, in which nodes there are not de�ned yet objects. The distribution of center'sobjects can be described using the reticulum functions (RF). The de�nition of aone-dimensional RF is based on the following identity2 .

1

σ√2π

∫ ∞−∞

e−12( xσ)2dx =

1

σ√2π

∫ L2

−L2

∞∑n=−∞

e−12(x−nL

σ)2dx = 1. (1)

From here RF3 is

R(x) =1

σ√2π

∞∑n=−∞

e−12(x−nL

σ)2 . (2)

Graph 1. Hyperanalytic function R (x)4.

1The long arithmetic programs in the application were set up for calculations using more than

100 signi�cant digits.2For the �rst time the value of this one-dimensional integral was calculated in 1729 year by

Leonard Euler during his work at the St. Petersburg Academy Of Sciences. Karl Friedrich Gauss,

whose name is called the subintegral function of the �rst integral is born on April 30, 1977.3The MATHEMATICAL DOUBLE-STRUCK CAPITAL font is used to highlight all hyper-

analytic functions and constants produced from them.4All calculations were performed at the values L = 1 and σ=0.4992619105929628

3

1 INTRODUCTION

It is obvious that the RF can not be laid out in the Fourier series because itdoes not have antiderivative that can be expressed as elementary functions. Byvirtue of this RF cannot be decomposed into even and odd functions5, while anarbitrary analytic function f can be only presented in the form of sum of odd andeven functions in the interval [a, b]:

f (x) = g (x) + h (x) ,

where

g (x) =f (x− a)− f (b− x)

2,

h (x) =f (x− a) + f (b− x)

2.

Due to this the RF can be laid out in an endless row of two primitive hyperan-alytic functions by sequential attempts to decompose on even and odd functions.Thus, the RF can be decomposed by the simplest way, but such a series is not onelike the orthonormal basis of Fourier series.

5The absence of a certain parity is the non-preservation of parity.

4

2 DECOMPOSITION OF RF

2 Decomposition of RF

2.1 The mean value of RF

As it follows from (1.1) the mean value of RT is 1. However as will be seen from thefurther, it is expedient to choose the greater value of the decomposition's constantmember. Introduce the following de�nitions:

R (0) = Rmax =1

σ√2π

∞∑n=−∞

e−12(

−nσ )

2

,

R (1/2) = Rmin =1

σ√2π

∞∑n=−∞

e−12(

1/2−nσ )

2

.

Then A0 is the mean value of RF:

A0 =Rmax + Rmin

2.

Graph 2. First di�erence R (x)− A0

2.2 First di�erence

One can approximate �rst di�erence by the following way:

A1 (x) =Rmax − Rmin

2cos (2πx) .

Let introduce parameter of the �ne structure α as function of σ:

α (σ) =1

2

Rmax − RminRmax + Rmin

. (3)

5

2.3 Even di�erences 2 DECOMPOSITION OF RF

Now A1 (x) can be expressed:

A1 (x) =Rmax + Rmin

2(2α (σ) cos (2πx)) .

The choice of the name and symbol of this parameter is due to the fact that

α (0.4992619105929628) = α =e2

4π�0~c

is the value known in physics as a �ne structure constant[3].

2.3 Even di�erences

Even di�erences are a primitive hyperanalytic function V(2i× 2πx), which is qua-sisymmetric relative to the point x=0.25.

Graph 3. Second di�erence R (x)− A0 − A1 (x) = V(2× 2πx)

Its symmetrical part approximated in the following way:

A2i (x) = c2i (cos (2i× 2πx)− 1) .

Using the value

R (1/4) = R1/4 =1

σ√2π

∞∑n=−∞

e−12(

1/4−nσ )

2

6

2.4 Odd di�erences 2 DECOMPOSITION OF RF

de�ne the amplitude for c2:12

(Rmax+Rmin2

− R1/4)= 2α4. This de�nition allows to

select approximation A (x) in the form:

A (x) =Rmax + Rmin

2(1 + 2α (σ) cos (2πx)) + 2α4 (cos (2× 2πx)− 1) .

2.4 Odd di�erences

Odd di�erences are a primitive hyperanalytic function W ((2i− 1)× 2πx), whichis quasiantisymmetric relative to the point x=0.25.

Graph 4. Third di�erence R (x)− A (x) = W (2πx)

Quasiantisymmetry of W (2πx) follows from the fact that the integral of A (x)di�ers from 1:∫ 1/2

−1/2A (x) dx− 1 = 1

4(Rmax + Rmin) +

1

2R1/4 − 1 ' 1.02E − 34.

Thus functionW ((2i− 1)× 2πx) should be decomposed in the even and odd func-tion. Its even part is:

Wqs ((2i− 1)× 2πx) = W ((2i− 1)× 2πx) +W ((2i− 1)× 2π (0.5− x))2

= V(2(i+1)×2πx).

This is seen from the following picture. However, as shown above, V(2i× 2πx) isnot an even function.

7

2.4 Odd di�erences 2 DECOMPOSITION OF RF

Graph 5. Hyperanalytic function V(4× 2πx)

Graph 6. Hyperanalytic function W (3× 2πx)

The odd part of W ((2i− 1)× 2πx) is no longer a hyperanalytic function andis equal to:

W qa ((2i− 1)× 2πx) = W ((2i− 1)× 2πx)−W ((2i− 1)× 2π (0.5− x))2

.

It can be approximated with any degree of accuracy following way:

A(W qa ((2i− 1)× 2πx)) = β(cos (3(2i− 1)× 2πx)− cos ((2i− 1)× 2πx)),

where β is a normalizing multiplier.

8

2.5 Three-dimensional RF 2 DECOMPOSITION OF RF

Thus, the approximation of R(x) is:

A (x) =Rmax + Rmin

2(1 + 2αcos (2πx))

+ 2∞∑i=1

α4i

(cos (2i× 2πx)− 1)

+2

Wmax

∞∑i=1

α9i2

(cos (3× (2i− 1)× 2πx)− cos ((2i− 1)× 2πx)) , (4)

where Wmax is a normalizing multiplier.

2.5 Three-dimensional RF

Threed-imensional RF R (x, y, z) can be obtained from the de�nition (1.2):

R (x, y, z) = R2maxR (x) .

Thus, the approximation of the three-dimensional RF is also the series of the �nestructure constant α along any axis of the reticulum three-dimensional space, andthe constant itself is a function of the dimensionless parameter σ, which is equalto quotient of the "diameter" of some physical object, located in each cell, to thegrid step L.

9

3 QUANTUM DERIVATIVE WITH RESPECT TO TIME

3 Quantum derivative with respect to time

To quantize the time the direct use of the lattice idea is too formal. It is thereforeappropriate to use a de�nition of derivative with respect to time but withoutmoving to the limit. Let R (t) is RF on a unit interval [−T/2,T/2] and τ = σ èT = 1:

R (t) =1

τ√2π

∞∑i=−∞

[exp

(−12

(t+ T/4− i

τ

)2)− exp

(−12

(t− T/4− i

τ

)2)].

(5)

Graph 7. Hyperanalytic function R (t)

By consistently subtracting sinuses, one can show that the approximation of theR (t) has the following form:

A (t) =∞∑k=0

(−1)k+1 aksin (2π (2k + 1) t) . (6)

Let use k + 1 equations with di�erent values of l to determine the coe�cient'svalues ak:

k∑i=0

(−1)i aisin(2i+ 1

2l + 1

2π

4

)= R

(1

4 (2l + 1)

).

Given that A (1/4) is numerically equal to 2 (Rmax (τ) + Rmin (τ))α (τ), equa-tion (3.2) can be written as follows:

αeff (t, τ) =1

2 (Rmax (τ) + Rmin (τ))

∞∑k=0

(−1)k+1 aksin (2π (2k + 1) t) .

10

3 QUANTUM DERIVATIVE WITH RESPECT TO TIME

R (t) is also a hyperanalytic function, as the next approximation takes place:

αeff (t, τ) =∞∑k=0

(−1)k+1 α(2k+1)2sin (2π (2k + 1) t) . (7)

Graph 8. Second harmonic.

Graph 9. Third harmonic.

11

4 UNIFIED THEORY OF INTERACTIONS

4 Uni�ed Theory of Interactions

The resulting decomposition of hyperanalytic functions by the degrees of α allowsto assert that a Naturally-Uni�ed Quantum Theory of Interactions exists, becausethe information that can be extracted from the function e−

12( xσ)2 (or more precisely

from formulas (1.2) and (3.1)) is at least equivalent to that contained in givenbelow �gure.

Graph 10. Force dependence on distance[4]

The key to decoding this information is the reticulum model of Space-Time.Independence of the obtained results from the size of L and T is therefore funda-

12

4.1 Interaction 1 4 UNIFIED THEORY OF INTERACTIONS

mentally important6. This means that when considering the each interaction isdue to a reticulum with speci�c values7. The reticulum Space-Time model allowsto highlight four interactions from the decomposition of the RF:

strong magnetic interaction (1),

electromagnetic interaction (2),

interference electro-weak interaction (3),

weak interaction (4),

and three interactions from the decomposition of the quantum derivative withrespect to time:

electromagnetic interaction (2),

gravitational interaction (5),

unknown interaction (6).

4.1 Interaction 1

The mean value of RT is equal 1 eventually. It is therefore advisable consider itsvalue relative to the coe�cient of the second member. In this case, it will have aknown physical value[6]:

qSqe

=qNqp

=1

2α,

where qS and qN - are the Dirac monopole's magnetic charges, qe is the charge ofelectron, qp is the charge of positron. It means that the reticulum space is formedby the Dirac monopole's. Such a model was described in article[2].

4.2 Interaction 2

Coe�cient at cos (2πx) is equal 2α. Assuming that the deuce re�ects the existenceof two photon states, it can't be taken into account when estimating intensity. Inthis way, as it should be to expect the intensity of electromagnetic interaction isα.

6This is due to the fact that the normal distribution is in�nitely divisible distribution.7The experimental con�rmation of this demonstrated the result of work[5], that the optical

transparency of the monoatomic 2M-layer of graphene depends only on the dimensionless values:

constant �ne structure α and number π.

13

4.3 Interaction 3 4 UNIFIED THEORY OF INTERACTIONS

4.3 Interaction 3

As seen in Fig. 10 dependence of the magnitude of the weak forces on the distancehas signi�cantly di�erent areas. At the starting site this dependence parallel to thedependence of electromagnetic forces. Odd di�erences contain cos (2i× 2πx) witha double argument that corresponds to the description of interference interaction ofphotons and neutrino. Coe�cient at cos (2× 2πx) is equal 2α4. Assuming that thedeuce re�ects the existence of two photon and neutrino states, it can't be taken intoaccount when estimating intensity. Because frequency increase (compared to theprevious cosine), the value of the intensity of interference of the weak interactionhas value

√2α4 = 4.01×10−9. This value corresponds to end of a straight section.

Odd di�erence with other values coe�cients and other frequencies are repeatedregularly for subsequent generations of leptons.

The importance of identifying this interaction is unique property of the primi-tive hyperanalytic function V(2i× 2πx) - not saving parity.

4.4 Interaction 4

The actual weak interaction corresponds to the cos (3× 2πx) and cos (2πx) withcoe�cient 2α9. Assuming that the deuce re�ects the existence of two neutrinostates, it can't be taken into account when estimating intensity. In view of theincreasing frequency in three times the value of the intensity of the interferenceinteraction should be multiplied by

√3. Due to the function W ((2i− 1)× 2πx)

is not normalizied the coe�cient should be divided by its maximum value Wmaxequal to ∼= 1.5396. As a result get the value

√3α9/Wmax = 6.60 × 10−20. This

value corresponds to the end of the curved section. Odd di�erences with othervalues of coe�cients and other frequencies are repeated regularly for subsequentgenerations of leptons, overlapping the entire range of together with the interactionof 3.

The importance of identifying this interaction is a unique property of a primi-tive hyperanalytic function W ((2i− 1)× 2πx) - the nonconservation of parity.

Interactions of 3 and 4 are related to each other by a set of two unorthogo-nal hyperanalytic functions. This means that interacting leptons described initiallyas mixed states, regardless of whether the neutrino are massless or not. Thus, thetransformations of neutrino one generations in the neutrino of another generationare natural quantum phenomenon.

As seen in Fig. 10 dependence of the magnitude of the weak forces on thedistance has a plot at which the speed of their reduction is described precisely bycoe�cients of decomposition of the RF. The values of the lower bounds given inthe following table show that the leptons of the fourth generations cannot existbecause interactions 3 and 4 overlap each other eventually in this range due

14

4.5 Interaction 5 4 UNIFIED THEORY OF INTERACTIONS

to the di�erent speeds of reducing the lower bounds of each interaction.

Generation Interaction 3 Interaction 4

1√2α4

√3α9/Wmax

2√4α16

√6α36/Wmax

3√8α64

√9α81/Wmax

4.5 Interaction 5

Since the �rst coe�cient of decomposition of the quantum derivative with respectto time has already identi�ed as the intensity of electromagnetic interaction, itcan be expected that the second factor is related to the only remaining interaction- gravitational. To get the intensity of gravitational interaction[4] it is enoughto square second coe�cient α9 and multiply it by sqrt3 (to account for anotherfrequency).

The resulting value is less than a percentage exceeds the constant of gravita-tional interaction:

Gm2p}c

= 5.906× 10−38.

This discrepancy gives an upper estimate of the quantum amendment, which canbe added to the law of gravity.

First we show how the constant G will look like if instead of the mass protonmp enter a new associated mass of proton mpa. In this case, the value of G will bethe following type8:

G =√3α18

~cm2pa

.9 (8)

Based on data10 in the following table, we obtain:

mpa = 1.68082× 10−27.8The obtained formula reveals the latent quantum-relativistic status of the law of gravity itself.

The thing is that the product ~× c can be saved only under the simultaneous transformations c→∞ and ~→ 0 according to the correspondence principle. Thus, to speak about the unilateralre�nement of the Newton's Law of gravitation turns out to be wrong in principle.

9In 1922, the Chicago physicist Arthur Lund suggested[7] that the �ne structure constant was

somehow related to the nuclear mass defect, and also considered its possible relationship with

gravity through the ratio:Gm2ee2

=α17

2048π6,

where e è me - charge and mass of electron.10Data taken from Wikipedia 07.03.2018.

15

4.6 Interaction 6 4 UNIFIED THEORY OF INTERACTIONS

Thus, the value of mpa by 9 electronic masses exceeds mass of the proton mp andcan be considered reliable11.

Constant Value

~ 1.054 571 800(13) × 10−34 Äæ · cñ 299 792 458 ì/ñα 7.297 352 566 4(17) × 10−3G 6.674 08(31) × 10−11 ì3 · ñ−2 · êã−1

As an example of an estimate of mpa, one can assume that this value includes themass of the proton mp and the mass of the electron me. Besides, it is necessaryto include mass of neutron mn with coe�cient delta - a fraction of neutrons perproton, which is the tenth for stars and units for planets. It is also necessary tosubtract the energy of the bound nucleons, which is di�erent for stars and planets.Finally, we need to add kinetic energy to nucleon and other possible deposits. Asa result the constant G will become a constant Gik, where, for example, i and k -indices of the sun and planets.

4.6 Interaction 6

The remaining members of the decomposition of the discrete derivative of the RFcan be interpreted only as interactions are essentially weaker than gravitational.

4.7 Conclusion

A clear preference for interactions 3 and 4 in terms of the amount ofinformation can be explained by the fact that the RF can be decomposed inthe an in�nite series of only two primitive hyperanalytic functions.

It is possible to assume that to achieve increase of the information on inter-action 1, for example, when considering the original RF on the complexplane.

There is considerable interest in summarizing the results obtained for thediscrete operator d'Alembert.

The success of a Naturally-Uni�ed Theory of Interactions allows to demandfrom alternative Theories of Everything formulating them in terms of hyper-analytic functions.

11It is more correct to say that now the accuracy of mpa is determined by accuracy G, not viceversa.

16

REFERENCES REFERENCES

References

[1] Tetration � Wikipedia

[2] Aleksandr Rybnikov, Space-time from the Gaussian function's point of view,http://www.gaussianfunction.com

[3] Coupling constant � Wikipedia

[4] 'T H o o f t G. Gauge Theories of the Forces between Elementary Particles.�Scienti�c American, June 1980, v. 242, pp. 90 � 116.

[5] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J.Booth, T. Stauber, N. M. R. Peres, A. K. Geim. Fine Structure Con-stant De�nes Visual Transparency of Graphene. Science 320, 1308 (2008)DOI:10.1126/science.1156965

[6] P.A.M. Dirac,Quantized Singularities in the Electromagnetic Field, Proceedingsof the Royal Society, A133 (1931) pp 60�72.

[7] A. C. Lunn. Atomic Constants and Dimensional Invariants // Physical Review.� 1922. � Vol. 20. � P. 1�14.

17

5 APPENDIX

5 Appendix

The program works with numbers that are represented by strings with �xed length(Mlen). For carrying out the presented in the article computations are enoughnumbers in the range (-99, + 999). Negative the numbers start at 9. The decimalpoint is not included, but it is assumed that it could be between the third and thefourth position. Constants have the following values:

Const c4 As Integer = 4Const c5 As Integer = 5Const c6 As Integer = 6Const c10 As Integer = 10Const c47 As Integer = 47Const c48 As Integer = 48Const c49 As Integer = 49Const c96 As Integer = c48 + c48Const c95 As Integer = c96 - 1Const Zs As String * MLen = "0...0"The following are the programs for four arithmetic operations and operation of

sign's change.

18

5 APPENDIX

19

5 APPENDIX

20