Complex Numbers Quantum Mechanics in Euclidean Spaces

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COMPLEX NUMBERS AND QUANTUM MECHANICS

IN EUCLIDEAN UNIVERSE

This article describes the Physical reality Markos Georgallides : Tel-00357 -99 634628

in terms of Impulse and Unit AB,( matter) Civil Engineer(NATUA) : Fax-00357-24 653551( not as with Big Bang) . It is shown what 15 , N . Mylona St , 6010 , Larnaca Cyprusis matter and how is connected across the Expelled from Famagusta town occupieduniverse . Euclidean Geometry is proved by the Barbaric Turks .to be the dual Model of Space since this isParticle and Wave as Complex numbers. Email < [email protected] >

In previous articles has been explained :

1. The first dimensional Unit AB is the geometrical Shape that has as Position the

( direction ÂB , BÂ ) and as Magnitude ( the length AB = 0 → N → ∞ ) . ( F1-3)ds = An Infinitely small increment of length AB in the direction AB , BA .∞ = An Infinitely great magnitude AB in the direction AB , BA .Any point C is on Straight line AB , when then exists equation CA + CB = AB ,i.e. the Whole AB is equal to the Parts CA and CB . ( equation ).

2. Spaces of Unit AB are ( in Plane ) the Infinite Regular Polygons inscribed in thecircle with AB as Side , (repetition of Unit AB ) , the Nth Space , the Nth Unit Tensorof the N equal finite Elements ds , and the ∞ Spaces line AB ↔ . ( F.1-6 , F.2 )

3. Anti-Spaces of Unit AB are ( in the three dimentional space ) the Symmetrically

Infinite Regular Solids inscribed in the Sphere with AB as side of the Solid ,( Harmonic Repetition of Unit BA , symmetrical to AB ) , the Nth Anti-Space ,the Nth Unit Tensor of the N equal finite Anti-Elements and the ∞ Spaces ,line BA ↔ . ( F1-6 , F2 )

4. Sub- Spaces of Unit AB are ( in Plane ) the Infinite Regular Polygons inscribedin the circle with AB as diameter , ( Harmonic Repetition of the Roots in UnitAB ) and in Nth Sub-Space , the Nth Unit Tensor of the N finite Roots and incase of ∞ Elements are the points on the circle ) . ( F1-6 ) , ( F.3 )

The Superposition of Spaces , Anti - Spaces and Sub-Space Layers of Unit AB

is shown in F4 . Remark : (+) Spaces , (-) Anti-Spaces , ( ± ) Sub-Spaces , of aunit AB are between magnitude ( Point = 0 =Nothing ) and the Infinite magnitude( ↔ = ± ∞ = Infinite ) which means that all Spaces are in one Space .Because in Spaces and Anti-Spaces , the ∞ Spaces of Unit AB is line AB ↔ , and in Sub-Spaces , the ∞ Sub-Spaces of Unit AB are the points on the circlewith AB as diameter , then this ordered continuum for points on the circle of Unit AB and on line AB shows the correlation of Spaces in Unit AB . ( Monas)

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5. For N = ± ∞ , Point B of Unit AB is in Infinity , N →↔← ,

The ∞ Spaces , Anti - Spaces and Sub-Spaces are created from ∞ Points only,without Position and Dimension , and are within these infinite enclosed points.

Since Spaces , Anti – Spaces and Sub-Spaces are created from Unit AB , and are Property of this Unit only , therefore these are a Restrained System ( S )

Presupposition for Unit AB = ds (displacement ds ) , is Point A to move at the Newposition B (A= B) , which means an IMPULSE ( P ) removes point A to B. Since in each Restrained System ( S ) the Work done ( W) by Impulse ( P ) on a

Virtual displacements ( ds > 0 ) is zero , or W = ∫ P.ds = 0 A-B

Therefore , Each Unit AB = ds > 0 exists , by this Inner Impulse ( P ) .i.e. The Position and Dimension of all Points which are connected across Universe

and that of Spaces exists , because of this Static Inner Impulse , on the contrary

should be one point only ( Primary Point = Black Hole → ds = 0 ) . Impulse is ∞ and may be Vacuum , Momentum or Potential or Induced Potential.

6 . Any two Points A , B on Primary Space or Anti-Space , consist the firstDimentional Unit AB , so itself , using the same Principles of Equality( AB = BA ) Inequality and Stability , creates all Spaces , Anti-Spaces and Sub-Spaces of Unit AB and Since are Property of this Unit only , therefore eachof these bounded Spaces is a Restrained System of Unit AB . Fig 4

Impulse PA , P B at points A , B is Property of these points only and since also Spaces , Anti-Spaces and Sub-Spaces of Unit AB is a Restrained System of this

Unit AB , therefore , Impulses P A , P B at points A ,B , and the Spaces , Anti –Spaces and Sub-Spaces of Unit AB is a New Restrained System .A New Unit with a quite different Quality . THE FIRST MATERIAL UNIT [FMU] Η Πρώτη Υλική Μονάδα , i.e.

The [FMD] is a DIPOLE → the First Material Dipole [ FMD = AB – PA , P B ].Δηλαδή : < Η Πρώτη Υλική Μονάδα τού Σύμπαντος είναι το Δίπολο ΑΒ >

Fig . 8 PÂ = PB A → . . . . . . . ← O → . . . . . . . ← B = [ F.M.D ]

This Dipole AB may be created , by the opposite displacement ds ↔ of onecentral point O , or by the Rotation of Dipole AB through this point O .Central point O may be the Prime Point or all infinite points of Spaces .The position of points A , B in Spaces and Anti-Spaces fixes the type of Dipole.[FMD] can be created and is a Monad , ( that which is one ) , and it is a part of the whole AB . Impulses PA , P B cannot be created because exist at pointsA , B and are of the same essence as that of Impulse P .



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7. The infinite combinations of these two In-depended variables ( P , ds→Δs) containin itself , all infinite Static Primary Spaces . On point B exist infinite common Impulses ( P = 0 → ÂB → ∞ ) , due to the Stability of the Primary Spaces and Anti - Spaces [ W = P . Δs = 0 , and P,AB = (ΣLi.Pi) / Δs) i = 1 → ∞ , where Δs , Liis any dimensional unit distance ] , and connects the many points within infinity .

Applying Principle of Virtual Work on any point B with P = 0 , then Arises at this point B Impulse P from infinite points of Primary Space ( Δs = ∂ W / ∂ Ps ).

Any Unit AB = 0 → AB → ∞ of the Primary Space create infinite SpacesAnti-Spaces and Subspaces which are < a bounded State for each Space> .The beginning of the Change ( Impulse P ) is enclosed ( bounded ) in Position Band is continual in each ds , and it is the Essence and Cause of Impulse P .i.e. exists an influence flow ( perpetual flux = change = activity ) ↔ or

on ↓ Communicator = Medium ↓

Impulse P → [ Bounded Primary Space- Anti-Space ] → Bounded Impulse PA

PÂ = PB

A → . . . . . . . ← O → . . . . . . . ← B

8 . Between points A , B of [PNS] and Anti-Space , i.e. [ FMD = AB – PA , P B ] , andon line AB exist infinite points Ai , Bi with Impulses Pi A , Pi B that attract each

other , and so forming the first Motion of the bounded Impulses Pi A , Pi B .Between points A , B of [PNS] and Anti-Space , i.e. [ FMD = AB – PA , P B ] , andon line AB exist infinite points Ai , Bi which form ∞ [ FMDi = AiBi – Pi A , Pi B ]that collide each other , and so forming the first motion of Dipole [ AiBi ] . All movein the first Unit AB ↔ ∞ , with PA = P B = ∞ . [ FMD = ∞ – ∞ , ∞ ] . Fig.7

Motion occurs into these bounded States of [PNS] and Anti-Space , and because of the different Impulses P A , P B of points A , B and that of Impulses Pi A , Pi B ,

either on straight lines AB or on tracks of Spaces ,Anti-Spaces ,Sub-Spaces of AB.

Since in Primary bounded Space AB is always ds = 0 → AB → ∞ and since alsois ↓ Communicator = Medium ↓

Impulse P→ on [ Bounded Primary Space - Anti-Space ] is→ Bounded Impulse PA , PB then

Points Ai ,Bi with Pi A , Pi B ( of Spaces , Anti-Spaces , Subspaces of AB ) move inSpaces of AB (Primary Space is not moving ) and in case of ∞ Space ,circular motion.

Subspaces of AB are the very small moving bounded Spaces with ds > = 0 .The Infinite Impulse ( P = ∞ ) on Neutral [ B.N.S ] moves these Bounded Spaceswith Infinite velocity ( v = ∞ m/s ) and zero frequency f = 0 . → [ M . ray ]while The Infinite Impulse ( P = ∞ ) on Neutral [ B.N.S ] moves these Bounded Sub-Spaces with Finite velocity ( v = u m/s ) and finite frequency f ≠ 0 . Fig.6 Impulse P didn’t appear in Space , because Space began inside of Impulse P .



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9. Points A , B on Primary Space is the Positive Dipole which creates matter .Points A , B on Primary Anti-Space is the Negative Dipole which creates theopposite of matter , or Anti - matter ( the negative matter ) .Points A , B either be on one Space and Anti-Space alternatively ,then is created the Neutral matter and Neutral Anti-matter.

Bounded Primary Spaces- Anti-Spaces can be created , so Dipole AB on Spacesare the Units of matter , and Impulse P ↔ [ FMD = AB – PA , P B ] → PA , P B i.e. [ Dipole AB = Matter ] is the communicator of Impulse [ P ] , with the BoundedImpulses ( PA , P B ) or [ P ] ↔ [ FMD = AB – PA , P B ] → PA , P B . Fig.7- 8

Every material Dipole is consisted of AB = [ ds > 0 ] and as , AB is the first dimentional Unit which create Spaces , Anti-Spaces (F2) and Sub-Spaces (F3)then Impulses PA , P B . exist on their superposition ( F4 ) i.e.

Impulse into the bounded Spaces of AB , exist as a deficite of Impulse at points

A and B . For AB = 0 then Impulse exists in every point as Angular Momentum and For AB =∞ Impulse exist in all Spaces of AB .

Bounded Impulses PA , PB cannot be created because exist as the Dipole of Spaces.Since Neutral matter is consisted of these two opposite elements ( + , - ) , thereforeall derivatives in this bounded Neutral Space can alternatively exist itself , as comprise both elements .

Since Matter is made up of Electrons , Protons and Neutrons i.e. [ (-) (+) (± = 0 )]and Anti - Matter is made up of Anti- Electrons , Anti-Protons and Anti-Neutrons i.e. - [ (-) (+) ( ± = 0 ) ] = [ (+) (-) ( ∓ = 0 ) ] , therefore any Combination per two

for Matter and Anti-Matter is equal to 15 and for Neutral Matter C ⁴₂

= 6 . F.5 The Combinations of the Infinite < Dipole Points > create all qualities of matter .

Considering mass as the Inertial property of matter , a measure of its energy content , and since matter is any Dipole AB with its properties , then what ismeasured is the material Dipole in a different Space .(Electricity , Magnetism , LightTemperature , etc ) , while Bounded Impulses ( PA , PB ) are not . Matter deduces theLaws of Nature at ( PA ,PB ) by the interconnection of Spaces in Primary Neutral Space.

10. Algebraic Numbers :

According to F4 Monas AB = 0 ↔ AB ↔ ± ∞ , Spaces , Anti-Spaces , Sub-Spaces of

AB are the Infinite Regular Polygons , on circle with AB as Side , and on circlewith AB as diameter . According to De Moivre’s formula , the n-th roots on the unitcircle AB are represented by the vertices of these Regular n-sided Polygon inscribedin the circle which are Complex numbers in the general form w = a + b. i wherew = a + b.i = r. e(iφ) = r [cosφ + i.sin φ] , a and b = Real Numbers , r = √ a²+b²

(±) i = Imaginary Unit .



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On Monas AB = 0 ↔ AB ↔ ± ∞ exists < a bounded State for each of InfiniteSpaces > and the [ Dipole AB = Matter ] is the communicator of Impulse [ P ] of Primary Space , with the Bounded Impulses ( PA , P B ) of Dipole or

[ P ] ↔ [ FMD = AB – PA , P B ] → PA , P B .

We will show that Monas AB and Complex Numbers is the only manifold forPhysical reality .

a. Exists ²√ 1 = ± 1 or [ -1 ↔ +1 ] , therefore xx coordinate system representsthe one-dimensional Space and Anti-Space . ( the Straight line) , 1.1 = 1 , (-1).(-1) = 1

b. Exists ³√ 1 = [ 1 , -½ + (√ 3.i ) /2 , -½ - (√ 3.i ) /2 ] therefore xx-yy coordinatesystem represents two - dimensional ± Spaces and ± Complex numbers .( the Plane )1.1.1 = 1 , [-½ +(√3.i ) /2 ]³ = 1 , [-½ -(√3.i ) /2] ³ = 1

+ ic. Exists ∜1 = ²√ ²√1 = ²√ ±1 = [ +1 , -1 ] , [ ²√ -1 = + i , - i ] or -1 ↔ +1 , ↨

therefore coordinate systems xx - yy represents all these Spaces . - i( ± Real and ± Complex numbers ) , where Monas = 1 = ( that which is one ) .represents the three-dimensional Space and Anti-Space. (the Sphere) . [±1] =[± i] =1

Remarks :

1. Spaces and Anti-Spaces are continuous and represent Real numbers . (a)2. The Model of nature is not built on Complex numbers because Complex numbersspring out of Spaces , Anti-Spaces and Sub-Spaces of the FDU ( ds = 0 → AB → ∞ ) and represent reality . The roots of Monads are the same Monads in Space andAnti-Space as well as Imaginary Monads in Sub-Space i.e.The Harmonic repetition of the roots ( Principle of Equality ) composes units with no

need to be Image or real dimensions .Image or Real dimensions exist in Euclidean Geometry as the vertices of the RegularPolygons ( and Anti-Polygons ) on any First dimensional unit AB . The geometricalVisualization of Complex numbers , springs from formula √ 1 = ± 1 , ± i (c)and since ± 1 is the one dimentional real Space ( the straight line ) the vertical axisis on ( Harmonic repetition of Spaces ) the other one dimentional Imaginary Spacewhich is conveyed . Since dimension needs ( N-1) points then (c) is representing theSpace with three dimensions (dx ,dy ,dz ) which is Natural , Real and Complex numbers3. Natural numbers with their discrete nature symbolize discontinuity of Spaces .4. Monads ( Entities) are the Harmonic repetition of their roots , and since roots arethe combinations of purely real and purely Imaginary numbers , which is a similarity

with Complex numbers ( Real and Image ) then , Monads are composed of Real and Imaginary parts as Complex Numbers are . i.e.Objective reality contains both aspects ( Real and Imaginary , discrete , AB , andContinuous , PA , P B , etc ) .This Duality exists in Euclidean Geometry to all Elements .Following Principle of Equality and In-equality ( Quality = Quantity) then :



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1a Point is nothing , Everything and it is Anywhere .2b Straight line is 0 and ±∞ ( the first dimensional unit AB is the first material Dipole)3c Plane is Positive , Negative , ± Neutral and ± Complex .4d Space is Positive , Negative , ± Neutral and ± Complex .

5. Considering Infinite ( Primary Points ,FMD ,Spaces, Anti-Spaces ,Sub-Spaces of AB ) as a Parameter, then are Quantized and Engulfed in Primary Neutral Space .Euclidean Geometry corresponds to the Physical Universe and laws of Spaceswithout inapplicable intuitions and incomprehensible .

6. Laws of Physics are located in Geometrical Formation of Universe and utilizedby Impulse P , through the bounded Impulses PA , P B , of Points .

7. A Dipole acquires mass by moving through Spaces . Time also is then derivedand doesn’t exist since Motion is related to other objects .Since Dipole [ FMD = AB – PA , P B ] may have AB = 0 → ds → ∞ and also , for

Immense amount of PA

= 0 → PA

→ ∞ , PB

= 0 → PB

→ ∞ , the infinitecombinations of infinite Dipole reveal discrete and Continuity in every point of Primary Neutral Space . Because of this periodic nature ( Property , not Time ) of points , appearance (+) and disappearance (-) of manifold occurs .This is why particles loosing mass [ FMD = AB – PA , P B ] disappear and also loosing[ PA , P B ] reappear .8. Dipole [ FMD = AB – PA , P B ] does not mix points and Impulses, because ds > 0and points come out of Impulse P ( two separate domains ) as Impulse P connectspoints in PNS and influences [ PA , P B ] on Dipole AB . Since Dipole AB may bein any Space with different [ PA , P B ] then in case PA = P B = ∞ and ds = 0 we havea Black Hole i.e.a Translation ( violation) of a Space Dipole ( Gravity , Electromagnetic spectrum ,etcEnergy and Physical laws ) into Primary Neutral Space .Since mass = [ FMD =AB – PA , P B ] and ds = 0 then an enormous amount of Impulsesis created as well as emission of X-ray photons , with very small frequencies .The transformation from one Space to another one , doesn’t delete the laws of conservation of Energy . Dipole in [PNS] create Spaces , Anti-Spaces, Sub-Spaceswhich Vibrate and are the origin of the other Spaces .

9. The Stability of Dipole itself appears to depend on the continually effect ion of Impulse P on the bounded Impulses [ PA , P B ] ( zero point energy ) i.e.In generation by the ( quantum-fluctuation ) motion of charged particles that constitutematter ( ds , PA , P B ) = Dipole or [FMD = AB – PA , P B] , Impulse P , still follows theconservation of Energy law , and this because Impulse P with the enormous amountof energy effects on the bounded Impulses PA , P B of Dipole [ FMD = AB – PA , P B ].Since frequency of P is near zero [ M.ray ] Electromagnetic radiation throughoutSpace is not observable unless a new Media is found for this .This property had not leaked into Universe once , but it is a continuous affection onthe bounded Impulses on Points of Primary Neutral Space .



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FUNDAMENTAL PRINCIPLES .1. The Elements of Euclidean geometry are , Points , straight Lines , circle .etc,

F.12. The Space Layers ( Regular Solids ) with sides equal to AB = 0 → ∞

The Increasing Plane Spaces by Repetition of the same Unit AB . [ 9 ]

F.2

3. The Sub-Space Layers ( Regular Solids on AB ) as Roots of AB = 0 → ∞The Decreasing Plane Spaces by Repetition of the same Unit AB . [ 9 ]

F.3

4.The Superposition of Plane Space , Anti-Space Layers and Sub-Space Layers : The simultaneously co-existence of Spaces , Anti-Spaces and Sub-Spaces of anyUnit AB = 0 → ∞ , i.e. , Euclidean , Elliptic , Spherical , Parabolic , Hyperbolic ,Geodesics , Metric and Non-metric geometries , exists in Euclidean Model as anSub case within .The Interconnection of Homogeneous and Heterogeneous SpacesAnti-Spaces and Subspaces of the Universe . The Unity of Opposites .[9]

F.4



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5. Matter , Anti-matter , Positive –Negative Neutral Matter .

Fig 5

6. Velocities of Matter , Anti-matter , ± Neutral Matter in [ PNS ]

Fig . 6

Fig . 7

7. The First Material Dipole [ FMD = AB – PA , P B ] = [ FMD ]

Fig . 8



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THE AXIAL – CENTRIFUGAL MOTION OF SPACES

a. B΄←A→ B c One degree of Freedom (linear). AB = AB ́

b. c (bʳ ) a ┼ (b) Two degrees (Plane) ab=abʳ =ac=acʳ , ab ┴ ac , abʳ ┴ acʳ

c. ª人 ʙ cʳ Three degree (Globe) ª ʙ = ª c = ª ⁿ , ª ʙ ┴ ªc ┴ ª ⁿ

ⁿ

1. Since Primary point O may be at point A and in all directions ( all degrees of freedom ) , only when exists another one point A΄ such that OA + OA΄ = 0 andor OA = - OA΄ and OÂ = - O Â΄ = - OÂ , then this process is equivalent , inthe two dimentional space , to the two equal and perpendicular First dimentionalUnits OA ┴ OB and OA΄ = - OA , OB΄ = - OB . ( b ).This is the motion in Dipole AB , AB´, [ matter] ┴ [ Anti-matter] = MotionPeriodic motion of ← C→ , transformed to Axial-Centrifugal [ A , ADEH ] .

2. It has been proved [ 8 ] that the two equal and perpendicular Units AB , AB΄ ,( in plane B΄AB ) construct the Isosceles rightangled triangle BAB΄ and the three

circles on the sides as diameters . From any point D on the first circle is conscructedthe square ADEH with vertices on the three circles . This Geometrical Formation isa mooving Machine and is called < Plane Formation of Constructing Squares >

The Plane System of triangle ABB΄ with the three circles on the sides as diametersconsists the Steady Formulation , and square ADEH is the moving Changeable

Formulation of this twin , System - Image ( The Plane System of the Squares –Antisquares ) . On this System of these three circles ( The Plane Procedure which isa Constant System ) is created a continues and also a not continues SymmetricalFormation . The changeable System of the Regular Polygons and the Image ( TheChangeable System of the Regular anti-Polygons ) , as this is in Space and also thisin Time it is proved that , in this Constant System , the Rectilinear motion ( C ↔ )of the Changeable Formation is Transformed into a twin Symmetrically axial -centrifugal rotation ( A , ÂO ) ( the motion of Spaces ) on this Constant System .

3. The conservation of the Total Impulse and Angular Momentum , as well as theconservation of the Total Energy in this Constant System with all properties includedexist in this moving Spaces and Anti-Spaces ( the Images ) of the one - dimensionalpoint units A , A΄ - B , B ΄ , as well as , in Primary ∞ Constant System .

This is the dual nature of Dipole AB = [matter] ┴ [matter] = PA , P B = Particle or Wave. The Property of all Monads is Magnitude (mass) , Potential (Charge ) and ( Position) .Magnitude [ AB or BA ] , Potential [ PA ,P B or P B ,PA ] , Position [ B or B ]

[FMD = B – PA , P B] ┴ [FMD = B – P B , PA]=↺ ↔↻ Axial-Centrifugal motion .Analysis of this Periodic motion ← C→ , and Time Registration follows :



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Periodic motion of ← C→ = of Space( Matter ) and Anti-Space ( Anti-Matter )

Fig ,8

References :

[1] MATRIX STRUCTURAL of ANALYSIS , by J.L.MEEK library of Congress Catalog 1971.[2] DER ZWECK IM RECHT by RUDOLF V. JHERING 1935

[3] EUCLID’S ELEMENTS IN GREEK

[4] The great text of J . L .Heiberg ( 1883-1886 ) and the English translationby Richard Fitzpatrick .

[5] ELEMENTS BOOK 1 .

[6] GREEK MATHEMATICS by, Sir Thomas L .Heath - Dover Publications ,Inc , New York.63-3571.

[7] WIKIPEDIA , THE FREE ENCYCLOPEDIA , Etienne Gilson , 19/8/1978 AUXERRE .

ON SCRIBD & VIXRA

[ 8] A SIMPLIFIED APPROACH OF SQUARING THE CIRCLE . ( MELAN.doc )

[ 9] THE PARALLEL POSTULATE IS DEPENDED ON THE OTHER AXIOMS ( EUCLID.doc )

[ 10] THE MEASURING OF THE REGULAR POLYGONS IN THE CIRCLE ( REGULAR.doc )

[ 11] THE TRISECTION OF ANY ANGLE . ( TRISECTION.doc )

[ 12] THE EUCLIDEAN PHILOSOPHY OF UNIVERSE . ( EUCLID .Spaces.doc ) .

[ 13] UNIVERSE ORIGINATED not WITH BIG BANG ( EUCLID .not.BIG BANG .doc )

by

Markos Georgallides

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Complex Numbers Quantum Mechanics in Euclidean Spaces