A Research on the a Priori of Physics 9788797246900, 9788797246917

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Regular tetraon basis supporting an active spherical symmetry.

A Research on the

a priori of Physics Critique of Pure Mathematical Reasoning. Founding Ideas for Understanding Physics as an Ethical Epistemology for all Entities in one Universal Nature.

Jens Erfurt Andresen

The Cartesian perpendicular orthonormal basis {σ1 , σ2 , σ3 } need a fourth 1-vector, e.g. u0 = −√⅓ (σ1 + σ2 + σ3 ) to support an active spherical central symmetry around an origo forming an irregular tetraon. Every multiple dependent unit 1-vectors u can do the active spherical walk.

This book is in full copyrights © 2020 to Jens Erfurt Andresen, M.Sc. Physics

© This publication is in copyright. All rights are reserved by Jens Erfurt Andresen, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

A research on the a priori of Physics Volume I, – Edition 1, – Revision 3, December 2020 ISBN-13: 978-87-972469-0-0 (paperback) ISBN-13: 978-87-972469-1-7 (PDF) Legal disclaimer: While the advice and information in this book are believed to be true and accurate at the date of publication, neither the author nor the publisher can accept any legal responsibility for any errors or omissions that may be made. They makes no warranty, express or implied, with respect to the material contained herein. General Disclaimer and Purpose: This book is written for discussion of the foundation of a science for physics It is presumed and strongly recommended that the reader is a graduated physicist. There can be passage in this book where new theories, views and postulates are promoted. The reader shall be prepared to take own ethical judgments at each issue. My endeavour has been not to shorten deductive calculations to make the concept ideas precise, so the brilliant reader must bear with reading all these details. Please let me know of printing errors, mistakes and of course disagreements to what is Nature. Any suggestions to improving of text formulations and explanations are welcome. Anyway, I wish you a good reading and in places an in-depth detailed study. In best regards Jens Erfurt Andresen The author of this book is a graduate physicist M.Sc. (Cand. Scient.) 1983 from NBI-UCPH. Niels Bohr Institute (NBI), University of Copenhagen (UCPH) e-mail: © Jens Erfurt Andresen, M.Sc. Physics, UCPH, Denmark

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A research on the

a priori of Physics

.

A comment to the spherical display figure on the front cover The geometric prerequisite for space is: Two points form a line. Tre points form a plan triangle circumscribed circular 𝑆 1 -symmetry with a center point. Four points form a solid structure in space: The Platonic tetrahedron ideal, circumscribed by spherical surface 𝑆 2 -symmetry, supported by a unit radial basis {uD , uA , uB , uC }, called a tetraon, regular when

uD + uA + uB + uC = 0.

Active outwards directions making a 𝑆 3 -symmetry of information development as a process for any locality center. Essential the fourth direction e.g. uD is linear dependent on the three others.

Hundred years ago, we have an attempt by Niels Bohr to draw an intuition of the spatial structure symmetries of a Methane molecule.

For the last 100 years we have seen the symmetry of a four valent carbon atom as the foundation of all living spatial structure. Here it is essential that it is circumscribable by a 𝑆 2 sphere that has an inside and an outside. As we know every living cell is dependent on a cellular membrane forming a closed cave with an internal active communicating outwards and give an reaction on the inwards information from the external. This outwards principia we presume valid for all active particle entities in physical space of Nature. The analyse of this book treats the question: Does 𝑆3 -symmetry of space have four grades of directions with orientations, and dimensions with sixteen 24 linear independencies including real scalar quantities? © Jens Erfurt Andresen, M.Sc. Physics, UCPH, Denmark

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Main content volume I

Content

vii

Preface

xvii

Prologue

xix

I. The Time in the Natural Space

23

1.

The Idea of Time

23

2.

The Parameter Dependent Mechanics

48

3.

The Quantum Harmonic Oscillator

61

II. Geometry of Physics

123

4.

The Linear Natural Space in Physics

123

5.

The Plan Concept

153

6.

The Natural Space of Physics

231

III. Space-Time Relations in Physics 7.

324

Relation Space of Physics

325

Epilogue 8.

337

Problematisation of the Philosophical approach

337

References

339

List of figures

341

Lexical Index

345

Detailed content list at pages vii–vii

© Jens Erfurt Andresen, M.Sc. Physics, UCPH, Denmark

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a priori of physic

In a Tribute to Baruch Spinoza I exclaim my attitude: Nature itself is the master, in the name* of Physics. This radical ethical idea for one universal substance concept of Nature per se is essential for our epistemology for making a fundamental science we call* physics. By this we can create enlightenment to understand this universal Nature where we live as members. Mathematic is a tool made by humans founded on fundamental geometrical objects we invent for our intuition as we perceive extensive things in nature. The perception of things we can count in sequential order from which we invent the number systems. Our process of counting times of occurrences is often called time. Some people incline in a paraphysical way to call a continuous division of this counting for the river of time. Anyway, promoting ideal laws of mathematic to be a master instance for Nature results in a paraphysical religion of pure mathematics. It is urgent to prevent this aesthetics in using human made mathematical tools when we do our ethical science work building structures that model the way Nature behave. Therefor a subtitle of this bookwork is:

Critique of Pure Mathematical Reasoning The principia is, that founding concepts of mathematic tools has to obey ideas we substantial relate to qualities we exemplify by objects as measurable reference quantities for all categories we can extract from Nature.

This ethics is a Categorical Imperative. In honour of Immanuel Kant.

© Jens Erfurt Andresen, M.Sc. Physics, UCPH, Denmark

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a priori of physics

What we learned from semiotics:1 First, we take as example the novel title by Umberto Eco: The Name of the Rose I again exclaim my attitude: The beauty of a rose is not a quality of the Rose, it is a quality of the observer interpreting process. When you see a rose you interpret, there is a rose. You presume there is something, a conceptual object, that gives you the perception. It is obvious that there are some geometrical primary qualities in Nature that courses you see a rose. You see the example object as an icon for the concept category roses, that you then name the Rose. You categorise you experience with this rose by a name Rose, as a symbol of the physical object. Your perception of a rose is a complicated process. When you in that process experience beauty, it is that interpretation that emerge the quality of beauty. This we categorise as a secondary quality. I learned from a Danish biochemist Jesper Hoffmeyer who wrote some semiotic philosophical works e.g. [1], that there is a fundamental principle, short formulated as: Every living cell has an inside and an outside. This principle I extent to every entity in Nature including what we call elementary particles and is formulated as: The energy captured internal in any entity structure obey the same primary qualities as all the external structure of the universal Nature. An outstanding question is a black hole an entity or just a primary quality structure?

Problem of the classical physics: The concept ideas of mass and force was by Newton presumed as primary qualities. In 20th century’s quantum mechanics and relativity space-time theory they seems to be secondary. In this book volume I have tried to avoid or eliminate the effect of these concepts of mass and force and only let them emerge as secondary qualities. Only the geometrical aspects and counting the times in cyclic frequency energy oscillations is presumed fundamental primary qualities as an a priori of physics. --The analytical result of this book seems to be that the a priori law of physics is Kepler’s Second Law, now fundamental saying:

Any plane angular development is a chronometric constant unit ℏ =1. 1

I was introduced to semiotics by physicist Peter Voetmann Christansen by first following graduate course in Eco-Physics for my orientation followed by a course in Response Theory [34] as a part of my majoring exam 1981 for graduation from UCPH 1983.

© Jens Erfurt Andresen, M.Sc. Physics, UCPH, Denmark

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a priori of physics

Content

Content Content

vii

Preface

xvii

Prologue

xix

I. The Time in the Natural Space

23

1.

The Idea of Time

23

1.1. Primary Quality 1.2. Quantity 1.3. The Causal Action 1.3.1. Logic and Numbers

23 24 24 24

1.3.1.2. The Number Sequence

24

1.3.2. Time, Action and Sequence

25

1.3.2.1. Extension of Time

26

1.3.3. Quantity in Time

26

1.3.3.1. The Passage of Time

26

1.3.4. Speed of Times, the Quantum of Time, and the Frequency 1.3.4.2. The Frequency in Action 1.3.4.3. Associations with the Known Physics

28 28

1.3.5. Continuous Time and Action

29

1.3.5.1. Continuous Timing

1.4.

29

The Cyclic Time

31

1.4.1.1. The Period 1.4.1.2. The Circle Plan

31 33

1.5. The Complex Numbers 1.5.2. The Complex Exponential Function 1.5.3. The Imaginary Approach to the Cyclic Circle of Rotation 1.6. The Complex Oscillation - the Circular Movement 1.6.2. The Cyclic Circle Clock 1.6.2.2. The Cyclic Rotation Oscillation 1.6.2.3. The Time Concept as a Running Wheel 1.6.2.4. Euler Circle as the A Priori Clock

37 38

1.7.1.1. A Entity in Physics and its Quantitative Functions

1.7.2. 1.7.3. 1.7.4. 1.7.5.

38

The Derivative Function The Parameter Derived Quantity The Circular Rotation and the Unitary Group 𝑈(1) The Circular Rotating Oscillator Synchronometry

38 39 39 40

1.7.5.2. The Real Rotation 1.7.5.3. The Internal Oscillation

42 42

1.7.6. The Oscillator Rotation in Physics 1.7.7. Fourier Transformation 1.7.8. The Local Internal Time

43 45 46

1.7.8.2. The Orthogonal Frequencies 1.7.8.3. The local Homogeneous Parameter and the Constant Oscillator Frequencies Jens Erfurt Andresen, M.Sc. Physics,

Denmark

33 33 34 35 36 36 36 37

1.6.3. The Continuous Measure for the Concept of Time 1.7. The Cyclic Rotation

©

27

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2.

Content

1.7.8.4. Orthogonality and Dependency in the Information Problem

47

The Parameter Dependent Mechanics

48

2.1. The Lagrange Formalism 2.1.1. The Lagrange Function 2.1.2. Action 2.1.3. The Conservative Energy 2.2. The Hamilton Function

48 48 49 49 50

2.2.1.2. Generalised Canonical Quantities 2.2.1.3. The Poisson Bracket

50 51

2.2.2. The Operator Quantised Circle Oscillator

52

2.2.2.2. The Spectrum of Oscillators

52

2.2.3. Quantised Probability

53

2.2.3.2. Heisenberg Picture 2.2.3.3. Schrödinger Picture

53 54

2.2.4. 2.2.5. 2.2.6. 2.2.7.

Stationary Eigenstates and Eigenvalues Commutator Relations The Energy Frequency Quantised Evolution Parameter Momentum

2.2.7.1. The Canonical Quantum Operators 2.2.7.2. The Measurable Expectation Values of Quantum Mechanics

57 58

2.3. A classical Formulation of the Cyclic Rotation Oscillation 2.3.1. Hamilton Formulation for the Harmonic Oscillator

58 58

3.

2.3.1.2. The Energy of an Oscillator 2.3.1.3. The Lagrange Function for the Cyclical Rotating Oscillator

59 59

The Quantum Harmonic Oscillator

61

3.1.2. 3.1.3. 3.1.4. 3.1.5. 3.1.6. 3.1.7.

The Quantum Real Scalar Field for the Linear Harmonic Oscillator Ladder Operators of the Quantum Harmonic Oscillator Eigenstates in the Real Field Linear Quantum Harmonic Oscillator The Quantum Number Operator The Ground State The Traditional Rotational Movement Seen as a Cyclical Object, a Circle Oscillator

3.1.7.2. The Transversal Plane 3.1.7.3. The Rotating Direction with Orientation 3.1.7.4. An Idea of a Primary Quantum Operator

61 62 62 63 64 65 66 67 67

3.1.8. Classical Angular Momentum 3.1.9. Quantising the Angular Momentum

68 68

3.1.9.2. Differential Operator in 3 Dimensions

69

3.2. The Two-Dimensional Quantum Harmonic Oscillator 3.2.1. The Plane Super-positioned Hamilton Operator 3.2.2. The Angular Momentum Operator 3.2.3. Ladder Operators of the Plane Quantum Mechanical Harmonic Circle Oscillator 3.2.4. The Circular Rotating Oscillator Eigenvalues

©

54 55 56 57

70 70 70 71 72

3.2.4.2. The Rotating Circle Oscillator in Polar Coordinates 3.2.4.3. Annihilation and Creation Operators in a Polar Plane

73 74

3.2.5. Ground State of the Circle Oscillator 3.3. Excitation of the Plane Harmonic Circle Oscillator 3.3.1. The First Excited States of the Circle Oscillator 3.3.2. Higher Excited States of the Circle Oscillator

75 78 78 80

Jens Erfurt Andresen, M.Sc. Physics, Denmark

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Content

3.3.2.1. The Possibility of the Second Excited States of the Circle Oscillator 3.3.2.2. The Possibility of a Third and Higher Excited States of the Circle Oscillator

3.3.3. The Plane Excited Circle Oscillator 3.3.4. The Possible Excitation of a Circular Oscillator with ± Signed Orientation. 3.3.4.2. 3.3.4.3. 3.3.4.4. 3.3.4.5. 3.3.4.6.

The Oscillation Freedom from Portable Energy as the concept of Rest Mass The Qualitative Unit of the Circle Oscillator Entity Polar Radial Distribution of the Angular Momentum Over the Circular Oscillator Plan The Area Distribution of the Action Over the Transversal Plane The Energy Intensity Momentum

3.3.5. Frequency Scaling of the Circle Oscillator

81 82 83 84 86 86 87

88

3.3.5.2. Examples of Commonly Used Reference Clocks Seen as One Circle Oscillator 3.3.5.3. The Relative Reference for the Circle Oscillator and the Autonomous Norm 3.3.5.4. Scaling of the Frequency Energy in The Propagation

88 88 89

3.4. The Quantum Excited Direction 3.4.1. The Direction of the First Excitation Described in Cylindrical Coordinates.

90 90

3.4.1.2. 3.4.1.3. 3.4.1.4. 3.4.1.5.

Annihilation of an Excited Circle Oscillator Change of Direction of an Excited Circle Oscillator The Fundamental Substance for an Entity and the Extensive Difference The Substance of the Concept of a Photon

3.4.2. The Linear Movement 3.4.2.1. 3.4.2.2. 3.4.2.3. 3.4.2.4. 3.4.2.5.

92 92 93 93

94

The Concept of the Straight Line, through what? The Unitary Direction as an Abstract Interpretation An Interpretation of the Angular Excited Quantum An Interpretation of the Excited Direction The Past Versus the Depth

94 95 96 96 97

3.4.3. The Phase Angle and Parameter Dependent States 3.4.4. The Principle of Superposition

99 100

3.4.4.1. 3.4.4.2. 3.4.4.3. 3.4.4.4.

The Simple Superposition of the Two Degenerated subtons The General Authentic Superposition in one Direction The Monochromatic Transversal Plane Wave The Amplitude for a Monochromatic Transversal Plane Wave

3.4.5. Linearly Polarization

3.5. Modulation of a Quantum Mechanical Field 3.5.1. The Macroscopic Modulation 3.5.2. The Carrier 3.5.2.2. 3.5.2.3. 3.5.2.4. 3.5.2.5. 3.5.2.6.

100 101 101 102

103

3.4.5.1. Is the Idea of a Linearly Polarized ‘Photon’ an elementary particle? 3.4.5.2. One double±subton Interpreted as a Progressive Wave 3.4.5.3. Superposition of Linear Polarized double±subtons

103 104 105

107 107 107

Macroscopic Coherent Amplitude Modulation Phase Modulation Amplitude Modulation at Mutual Frequencies QAM Modulation The Two Helicities

110 110 112 112 113

3.5.3. Elliptical Polarisation

113

3.5.3.2. The Elliptical Polarized Monochromatic Transversal Plane Wave

113

3.5.4. One double±subton as a Real Qubit 3.6. The Cyclic Quantum Oscillator Idea

115 116

3.6.1.2. The Direction 3.6.1.3. The Spectrum in One Direction

117 118

3.6.2. The Development of Entities in Physics 3.6.3. Conclusion in Traditional Terms for the Concept of Time and Energy 3.6.3.1. The Fundamental Quality of the Fundamental Quantum

©

80 81

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Content

3.6.3.2. The Concept of Time as Complementarity to Frequency-Energy. 3.6.3.3. The Causal Action of Light Gives the Extension 3.6.3.4. Space

II. Geometry of Physics 4.

123

The Linear Natural Space in Physics

123

4.1. The Linear Algebraic Space 4.1.1. The Abstract Linear Space, a Vector Space 4.1.1.1. 4.1.1.2. 4.1.1.3. 4.1.1.4. 4.1.1.5. 4.1.1.6. 4.1.1.7. 4.1.1.8. 4.1.1.9.

124 124

Algebra of a Linear Spaces The Dimensions of a Linear Algebra of a Linear Vector Space Sum of Subspaces The Simplest Linear Space for a Quality of Physics Definition of One Dimension of First Grade The Simplest Multidimensional Space The Real Numbers as a Vector Space The Reality of Time as a Vector Space The Abstract 1-Dimensional Vector Space for Real Numbers

124 124 125 125 126 126 126 127 128

4.1.2. The Real Spatial Linear Vector Space

129

4.1.2.2. Multiple Linear Spatial Dimensions

129

4.1.3. The Vector Space of Complex Numbers

130

4.1.3.2. The Complex Scalar 4.1.3.3. A Multi-dimensional Complex Vector Space

131 131

4.1.4. Vector Spaces of Infinite Dimensions

132

4.1.4.2. The Vector Space of Fourier Integrals

133

4.1.5. Qualitative Substance of a Vector Space

135

4.1.5.1. Linear Relationship of Geometry 4.1.5.2. The Geometry

135 135

4.2. The Geometric Space 𝔊 4.2.2. The Dimensions and Qualitative Grades of Geometric Space 4.3. The Idea of a Point - in the Geometric Space 𝔊 4.3.1.1. 4.3.1.2. 4.3.1.3. 4.3.1.4. 4.3.1.5.

4.4.

137 138 139

No Extension. The Euclidean Elements: The Quality of the Concept of Points No Quantity The Concept of the Simplest Primary Quality Geometric Points are not objects

139 139 140 140 140

The Straight Line 𝔏 in Geometry of Physical Space

141

4.4.1.1. Euclidean Elements for the Concept of a Straight Line 4.4.1.2. Additional Features –

4.4.2. The Concept of Geometric Vectors 4.4.2.2. 4.4.2.3. 4.4.2.4. 4.4.2.5. 4.4.2.6. 4.4.2.7. 4.4.2.8. 4.4.2.9. 4.4.2.10. 4.4.2.11. 4.4.2.12. 4.4.2.13.

©

121 121 121

144

Geometric Translation The 1-vector Concept Addition of Co-linear 1-vectors Multiplication of 1-vector with a Real Scalar The unit object for a linear direction The Linear Extension from a 1-vector The Parametric Development of a Straight Linear Ray The Parametric Span of a Straight Line Co-linear 1-vectors The Spatial Line as a Real Linear Vector Space ℝe𝟏 A 1-vector Intuited as a Translation The Translation of a Geometric 1-vector

Jens Erfurt Andresen, M.Sc. Physics, Denmark

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Content

4.4.3. The Straight Line Idealism 4.4.3.1. 4.4.3.2. 4.4.3.3. 4.4.3.4. 4.4.3.5. 4.4.3.6.

149

The Objective Reality of a Difference The Concept of Simple Memory The Simplest Subject The Simplest Object The 1-vector Concept as the Primary Quality of First Grade (pqg-1) The Scalar as a Simple Pure Quantity

4.4.4. Relationship Between the Concepts the 0-vector Scalar and the 1-vector 4.4.4.1. 4.4.4.2. 4.4.4.3. 4.4.4.4. 4.4.4.5.

5.

The scalar product between the co-linear 1-vectors The Unitary Co-Linear Direction Vector and the Inverse Geometric 1-vector The Zero-vector Representing All Points The First Grade Object, a Geometric 1-vector → a Subject in a Substance as Idealism The linear paper folding

The Plan Concept

5.1.

Additional A Priori Judgments to the Euclidean Plane Geometry The Concept of an Angle The Concept of Different Angles The Concept Circular Arc of Angle The Concept of the Primary Quality of Second Grade (pqg-2) The Quantity of an Angle Addition of angular quantities The Angular Quantity as a Sector Area The perpendicular tangent to the circle in the plan

The Additive Algebra for Vector Spaces of Geometric Substance The Linear Span of the Geometrical Plane from 1-vectors Bilinear Forms Clifford Algebra The Combined Linear Space

163 163

5.2.3. The Inner Product Algebra 5.2.4. The Geometric Product 5.2.5. The Outer Product of Geometric Vectors

165 165 166

The Bivector Orientation as a Sequential Operation Bivector Quantity and Form Structure Quality Direction A Bivector Multiplied by a 1-vector The Category a Bivector

5.2.6. The Orthonormal Bivector Object as a Unit for the Circular Rotation in a Plane The Hodge Coordinate for the Pseudoscalar Span in the 𝔓 plane Concept Operations with the Unit Bivector Pseudoscalar for a Plane The Form Structure of the Plane Subject 𝒊 has Arbitrary Shaped Objects The Unit-bivector 𝒊 Multiplied by a 1-vector

5.2.7. The Unitary Rotor Operator as a Concept 5.2.7.1. 5.2.7.2. 5.2.7.3. 5.2.7.4.

©

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166 167 168 168

169 169 169 170 171

171

The Geometric Rotor in the Euclidean Plane Intuition of the Bivector and the Scalar for the Interpretation of a Rotation The Plane-segment Unit Rotor Independence of any pqg-1 Direction in a Plane Quality

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

160 161 161 162 162

162

5.2.2.2. The Inner Symmetric Product of Geometric Vectors 5.2.2.3. The Scalar Product

5.2.6.2. 5.2.6.3. 5.2.6.4. 5.2.6.5.

155 155 156 156 156 157 158 158 159

160 160

5.2.2. The Geometric Algebra with Direct Product

5.2.5.2. 5.2.5.3. 5.2.5.4. 5.2.5.5.

150 151 151 151 152

153

5.2. The Plane Geometric Algebra 5.2.1. Addition of 1-vectors in the Plane 5.2.1.2. 5.2.1.3. 5.2.1.4. 5.2.1.5. 5.2.1.6.

150

153

The Geometric plane 𝔓 5.1.1.2. 5.1.1.3. 5.1.1.4. 5.1.1.5. 5.1.1.6. 5.1.1.7. 5.1.1.8. 5.1.1.9. 5.1.1.10.

149 149 149 149 150 150

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Content

5.2.8. The Exponential Function with one plane direction Bivector as Argument 5.2.8.2. 5.2.8.3. 5.2.8.4. 5.2.8.5.

The Product of Rotors The Rotor Product With a 1-vector A Bivector is Self-identic by a Rotation in its Own Plan The Simple 1-rotor Algebra

5.2.9. A Complex Quantity in Space Called a Plane Spinor 5.2.9.1. The Complex Quantity as a Geometric Product of Two 1-vectors 5.2.9.2. The 1-Spinor as a Generator Radius-vector Multiplied to a Basis 1-vector

5.2.10. The Circle Oscillating 1-rotor in Development Action as One Plane Substance 5.3. The Rotor Concept as the Primary Quality of Even Grades (pqg-0-2). 5.3.1.2. Two Points Define the Primary Quality of First Grade (pqg-1) 5.3.1.3. The Orthonormal Basis for Circular Plane Symmetry

5.3.2. The Geometric Algebraic Complex Plane

175 176

176 177 178 179

183

5.3.3.2. The Cartesian Coordinate System and the Plane Pseudoscalar Concept 5.3.3.3. The Parity Inversion of the 2-dimensional Descartes Extension Coordinate 5.3.3.4. The Extension Grade One Parity Inversion of Scalars and Bivectors

5.3.4. The Qualities of the Geometric Algebra of the Plane 5.3.4.1. The Rotation Symmetrical Plane Concept 5.3.4.2. The 2-dimensional Plane 5.3.4.3. The Orthonormal Reference for the 2-dimensional Plane

5.3.5. A Grade-2 Object, 2-blade Bivector → a Subject in a Substance as Idealism 5.3.6. The Inadequate Cartesian 𝒙, 𝒚 Coordinate System 5.3.7. The 1-vector Product Complex Quantity and Polar Coordinates of Plane Concept 5.3.7.2. A plane Rotation and Dilation added to a Translation

5.3.8. The Magnitude of a Multivector 5.4. Transformation of Geometric 1-vectors in the Euclidean plan 5.4.1. Parallel Translation of a Vector 5.4.2. Reflections 5.4.2.1. Reflection in a Geometric 1-vector 5.4.2.2. Reflection Along a Geometric 1-vector 5.4.2.3. Reflection Through a Non-normalized 1-vector

184 184 185

186 186 186 187

188 189 190 191

192 193 193 193 193 194 194

5.4.3. The Projection Operator From one 1-vector to Another 1-vector 5.4.4. Reflection in a Plane Surface as a Physical Process 5.4.5. Rotation Inside one and the Same Plane Direction The Half Angle Rotor of a Euler Rotation The Idea of an Active Rotation The Invariant Direction of a Rotor The Duality of Direction

194 195 196 197 197 197 197

5.5. Inherit Quantities of the Algebra for the Euclidean Geometric Plane Concept 5.5.2. The Auto Product Square in the Euclidean plane 5.5.3. The Nilpotent Operation 5.5.3.2. The Spanned Spaces of Nilpotence Zero Divisors 5.5.3.3. The Spanned Spaces of Mutual Annihilation Zero Divisors

5.5.4. The Idempotent Operation

199 199 199 201 201

202

5.5.4.2. The Paravector Concept 5.5.4.3. The Projection of a Paravector on Its Idempotent Basis 5.5.4.4. Non Measurable Fictive Magnitude of Paravectors Jens Erfurt Andresen, M.Sc. Physics, Denmark

175

181 181 181

5.3.3. Cartesian Coordinates for the Plan idea

©

174 174 174 174

179

5.3.2.2. The Polar Coordinates of a Plane Idea 5.3.2.3. Primary Qualities of the plane Concept 5.3.2.4. The Area Concept as Generator for Circle Rotation.

5.4.5.2. 5.4.5.3. 5.4.5.4. 5.4.5.5.

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Content

5.6. The Real Matrix Representation for the Plane Concept 5.6.1. The Fundamentals of Matrices in a Plane Algebra 𝒢2 5.6.1.1. 5.6.1.2. 5.6.1.3. 5.6.1.4.

Matrices for a Cartesian 1-vector Concept for an Euclidean plane ℝ12 Examples of Matrices of Geometric Multivectors The Matrices of the Geometric Algebra 𝒢2(ℝ) An Example of a Matrix in 𝒢2(ℝ)

205 205 205 205 206 208

5.7. Plane Concept Idea of an Non-Euclidean Clifford Algebra 209 5.7.1. Plane Geometric Clifford Algebra with Minkowski Signature for Measure Information 209 5.7.1.2. An Entity Seen from the External Far Distant as a Null Signal 5.7.1.3. The -bivector as an Information Signal

5.7.2. The Traditional Display of the Minkowski -plane

212

5.7.2.2. The Traditional Display of the Minkowski space 5.7.2.3. Minkowski Space with Display of Three Extension Dimensions

5.7.3. The Paravector Space and the Minkowski 1-vector in STA 5.7.4. Lorentz Rotation in the Minkowski -plane 5.7.4.2. 5.7.4.3. 5.7.4.4. 5.7.4.5.

The Lorentz Transformation of a paravector The Lorentz boost The Doppler Effect of the Lorentz Boost The Space-Time Algebra STA from a 4-dimensional 1-vector basis

5.7.5. The planes of Space-Time Algebra and the Euclidean Cartesian plane 5.7.5.1. Founding Summary of Minkowski Space with Euler and Lorentz Rotations 5.7.5.2. Mapping Operation Between STA planes and the Euclidean Cartesian plane 5.7.5.3. Exponential Function in the Plane Concepts

5.8.

The Exponential Function of Arbitrary Multivectors

217 218 219 220 221 222

223 223 223 225

5.8.1.2. The Hyperbolic Functions of Multivectors 5.8.1.3. The cosine and sine Functions of Arbitrary 𝒢𝑛 Multivectors

226 226

5.8.2. Exponential and Hyperbolic Functions in the Plane Concept

227

5.8.2.2. The 1-Spinor in the Euclidean Cartesian plane 5.8.2.3. The Lorentz 1-Spinor in the Line Direction Paravector Space 5.8.2.4. The Lorentz 1-Spinor in the Minkowski -plane

227 227 228

The Natural Space of Physics

228 229 229 229 229

231

6.1. The Classic Geometric Extension Space ℨ 6.1.2. Additional A Priori Judgments to the Euclidean Stereo Space Geometry 6.1.2.1. The concept of Spatial Angular Structure

231 233 234

6.1.3. The Euclidean 1-vector Space for Natural Space.

234

6.1.3.2. Covariant Cartesian Coordinates 6.1.3.3. Contravariant Coordinates 6.1.3.4. The Classical Cartesian Coordinate System for Position Points in ℨ-Space

234 234 235

6.1.4. A Curiosum, the Concept of a Tetraon in a Tetrahedron

235

6.1.4.2. The Six Bivector Angular Planes of the Regular Tetraon

236

6.2. The Geometric Algebra of Natural Space 6.2.1. Addition of Bivectors 6.2.2. The Trivector concept

237 237 237

6.2.2.2. The Magnitude of a Trivector

©

214 216

226

5.8.3. A Philosophical Conclusion on All This Exercise 5.9. Concluding Summary on the Algebra for the Geometric Plane Concept 5.9.1. The Euclidean Plane Concept 5.9.2. The Non Euclidean Plane Concept 5.9.3. General Exponential Series

6.

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6.2.3. The Trivector and the ℨ-space Chiral Pseudoscalar

238

6.2.3.2. The Cartesian Orthonormal Basis 1-vector Set of Primary Quality of Third Grade (pqg-3) 6.2.3.3. The Hodge Coordinate for the Pseudoscalar Span in ℨ Space

6.2.4. The Geometric Algebraic Basis of a ℨ-space

240

6.2.4.1. The 1-vector Basis 6.2.4.2. The Transversal Bivector Basis as a Dual Basis of a ℨ Space

6.2.5. The Geometric Algebra for Euclidean ℨ-space 6.2.5.2. 6.2.5.3. 6.2.5.4. 6.2.5.5. 6.2.5.6. 6.2.5.7.

The Unitary group 𝑈(1) for Plane Combination of Rotors Multiplication Combination of Direction Different 1-rotors in ℨ-space Comment on the Ontology of Directions and Possibility of Location The Abstract Generalised Rotor Form

6.3.6. Rotation of Multivectors 6.3.7. Framing a Field for a Geometric Algebra in ℨ-space 6.4. The Geometric Clifford Algebra 6.4.1.1. 6.4.1.2. 6.4.1.3. 6.4.1.4.

The Quadratic Form in general The Clifford Algebra for Complex Numbers The simple Euclidean Plane Geometric Clifford Algebra 𝒢2,0 Plane Subjects in Euclidean ℨ Space Clifford Algebra 𝒢3,0

6.4.2. The Pauli Basis for combined direction structures of ℨ-space 6.4.2.2. The Pauli Matrices as generator operators 6.4.2.3. The Pauli Basis Generated from 1-vectors

241 241 242 242 242 243

245 245 246 247 248 248 248 249 249

250 250 251 251 251 251 252

253 253 253

6.4.3. The Quaternion Picture

254

6.4.3.1. An Anti-Euclidean Geometric Algebra 𝒢0,2 6.4.3.2. Quaternions ℍ

254 255

6.4.4. The Versor as a Direction 2-Rotor

256

6.4.4.2. The Traditional View on the Spatial Rotation Problem 6.4.4.3. The Four Real Scalar Coordinates for the Versor Quaternion Direction

6.4.5. The Two Parameter Quaternion 2-Spinor

256 257

260

6.4.5.2. The Two State Observable of a Fundamental Entity in ℨ Space

261

6.4.6. Euler Angles for a Rotor in ℨ-space

263

6.4.6.2. The Other Euler Angle Sequence

266

6.4.7. The Transversal Bivector Idea Dual to a 1-vecetor make Foundation for Rotations 6.4.7.1. The two Orthogonal Rotors as Generators for a Local Entity

6.4.8. The Rotated Direction in ℨ-space 6.4.8.2. Rotation of a Chosen Direction in ℨ Space 6.4.9.1. 6.4.9.2. 6.4.9.3. 6.4.9.4. 6.4.9.5.

267 268

270

Review of the Quantum Mechanical Circle Oscillator Multi Excitations of Angular Momentum Internal in One Entity Intuition of Two Perpendicular Exited Circle Oscillators Inside one Entity Breaking the Spherical Symmetry in to One Direction External and Internal Directions of an Entity in ℨ space

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6.4.9. Oscillations in ℨ-space

©

240 240

241

The Commuting Pseudoscalar for ℨ-Space The Mixed Product Between a 1-vector and a Bivector The Simple Product of Three 1-vectors Even and Odd Multivector in general Product of Two Bivectors Commutator Product of Multivectors in Geometric Algebra

6.3. The ℨ-space Structure Quality Described by Multivectors 6.3.2. The Even and the Odd Geometric Algebra 6.3.3. Operational Structure of the Trivector Chiral Volume Pseudoscalar 𝓲 of ℨ Space 6.3.4. Rotation in ℨ-space 6.3.5. Multiplication Combination of Rotors 6.3.5.1. 6.3.5.2. 6.3.5.3. 6.3.5.4.

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6.4.9.6. Complementarity

274

6.5. The Angular Momentum in ℨ Space 6.5.1. One Quantum of Angular Momentum 6.5.1.1. 6.5.1.2. 6.5.1.3. 6.5.1.4. 6.5.1.5.

275 275

1-vector Angular Momentum The Bivector Angular Momentum The Angular Momentum Operator Excitation of Angular Momentum in three directions of ℨ Space Commutator Products of Angular Momentum Bivectors and their Dual 1-vectors

275 275 276 276 277

6.5.2. The Commutator Relations in Geometric Algebra for Angular Momenta

278

6.5.2.2. Orthogonal Chirality of Angular Momenta in the Even 𝒢0,2 Geometric Algebra

279

6.5.3. Thoughts About Symmetry Braking and Quantisation of Direction in ℨ-space 6.5.4. Chiral Orientation of Combined Angular Momentum Bivectors and Dual 1-vectors

280 282

6.5.4.2. The Total Angular Momentum of a local entity in ℨ space

282

6.5.5. The Quantum Stats of the Locally Combined Angular Momentum

284

6.5.5.2. The Quantum Ladder Step Operations in ℨ space 6.5.5.3. Excitation of Angular Momentum in ℨ space

285 286

6.5.6. The Spin½ of a Directional Entity of Locality in ℨ-space

287

6.5.6.1. The fundamental first excitation of ℨ space 6.5.6.2. Symmetry Braking of the Half Spin Ψ½ Entity,

287 287

6.5.7. Synthesis of the Locality of Entities in ℨ-space 6.5.8. The Idea of One Spin½ Entity in Physical ℨ-space 6.5.8.1. 6.5.8.2. 6.5.8.3. 6.5.8.4. 6.5.8.5. 6.5.8.6. 6.5.8.7.

290 291

The Extension Distribution of the Wavefunctions The Internal Oscillating Wavefunction Components The Oscillator Fluctuating Versor Wavefunction for the Entity Ψ½ in ℨ space Versor Eigenwave-Function as for the Stationary State Existence of an Entity Ψ½ One Eigen-Versor Separated in 1-Spinors Angular Momentum Wavefunction Components The Versor Quaternion Spin½ entity Ψ½ Eight Qualitative States of a Spin½ Entity Ψ½ in ℨ Space

6.5.9. The Internal Auto Synchronisation of an Indivisible-Atomic-Elementary Entity 6.5.10. A Hypothetic Thought Intuition of One Four-Angular-Momenta Function 6.5.10.2. 6.5.10.3. 6.5.10.4. 6.5.10.5. 6.5.10.6. 6.5.10.7.

The Autonomous Regular Tetraon Basis for Four-Angular-Momenta Function An Autonomous Four-Angular-Momenta Function Information from a Tetraon Symmetric entity Ψ½ The Regular Tetraon Symmetry Quantity Cargo of One Indivisible Spin½ entity quality The Freedom of the Tetraon Angular Composition in one Fermion Multiple Circular Oscillators as Structure Form Qualities for Spin½ Fermions

6.5.11. The Full Geometric Algebra 𝒢3(ℝ) for Spin½ Fermions 6.5.11.2. The Non Quaternion Grades ≤3 for of Indivisible Entities Ψ½ in ℨ Space

6.5.12. The strange Intuition of Locality in ℨ-space 6.5.13. The Fundamental Concept of Direction Locality in Space 6.6. Identical Entities in ℨ-Space

291 291 292 294 294 295 297

298 300 300 301 301 303 304 304

305 305

307 308 309

6.6.1.1. Classification of Fermions in the Structure of ℨ-space 309 6.6.1.2. The Simple Versor Form and the 𝑆𝑈(2) Algebra of the Complex 2×2 Matrix Group 309 6.6.1.3. Three Linear Independent Internal Components of Angular Momenta of Fermion Structure 310

6.6.2. Various Fermions given by Tetraon Symmetry from the Platonic Tetrahedron Idea 6.6.3. The Categorical Quality of Spin½ Fermions in ℨ-space 6.6.3.2. The Categorical Classification of Identical Spin½ Fermions in ℨ Space

6.6.4. The Idea of One Interaction Direction

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Multiple Numbers of Spin½ Fermions

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312

6.6.4.1. The One Whole Quantity Charged Fermion 6.6.4.2. The Field of Information About one Charge

6.7.

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6.7.1. External Qualities of Charged Fermions

314

6.7.1.1. Identical Charged Fermions 6.7.1.2. Opposite Charge Fermions 6.7.1.3. The Bohr-Rutherford Atomic Model

314 314 314

6.7.2. Mutual Exclusive Extension of Fermions

314

6.7.2.2. The Pauli Exclusion Principle

314

6.7.3. Orbital Angular Momentum 6.7.3.1. 6.7.3.2. 6.7.3.3. 6.7.3.4. 6.7.3.5.

315

The Squared Perpendicular Part of Orbital Integer Quantum Number Excitation The Orbital Angular Momentum of Multiple Spin entities Ψ½ Atomic Shells and Subshells Categories of Atomic Quantum Numbers Atoms in Practise

315 315 316 316 317

6.7.4. The Spatial Wavefunction Probability Distribution Structure of atoms 6.8. Conclusion on Topological Structure of ℨ-space Founded in Physics 6.8.1. Conclusion on the Local Situated Topological Structure of Natural ℨ-space

318 319 319

6.8.1.2. The Rest Mass Problem of Fermions

320

6.9. External Relations Between Fermions in an Extended Space of Information 6.9.1. A New Break Through for Physics Foundation in Human Knowledge of Nature

321 321

6.9.1.1. Extension of Space by Grassmann Exterior Products

323

III. Space-Time Relations in Physics 7.

324

Relation Space of Physics

325

7.1. Vision of Relations by the Development Concept of Physics in Nature 325 7.1.2. The Space-Time as Development Information of Extension Relation is Called 𝔇-space 326 7.1.3. The Full Geometric Space-Time Algebra 𝒢1,3(ℝ) for Physical Relations in 𝔇-space 326 7.1.3.2. 7.1.3.3. 7.1.3.4. 7.1.3.5. 7.1.3.6. 7.1.3.7.

The Multivector Decomposition in a Sum of Grades for 𝔇-space of Physics Conjugation in Geometric Algebra Reversion of the Odd Chirality Volume Pseudoscalar 𝓲 in the 𝒢3(ℝ) Algebra for ℨ-space Reversion of the Even Helicity Pseudoscalar 𝓲 in the Algebra 𝒢1,3(ℝ) for 𝔇-space The STA Bivector Field 𝑭 of STA in the Information Development 𝔇-space of Physics Difference Between the Pseudoscalar Concepts for 𝔇-space and ℨ-space

7.1.4. The Odd and Even Part of the Geometric Space-Time Algebra 𝒢1,3(ℝ) 7.1.4.2. 7.1.4.3. 7.1.4.4. 7.1.4.5.

The Transcendental Ignorance of the Odd part of the Geometric Algebra for 𝔇-space The Even Closed Geometric Algebra of 𝔇-space The Even Geometric Spinor Quality in the Algebra 𝒢1,3+(ℝ) of 𝔇-space The Composite Rotor Structure in 𝔇-space

7.1.5. Stop of this volume 7.2. For inspiration to further work 7.3. Further Work on These Issues is Postponed 7.3.1. The Foundation of Physics is Nature Itself

331 331 331 333

335

Epilogue

©

331

333 334 335 335

7.3.1.1. Nature per se as a priori of Physics

8.

328 328 329 329 330 330

337

Problematisation of the Philosophical approach

337

References

339

List of figures

341

Lexical Index

345

Jens Erfurt Andresen, M.Sc. Physics, Denmark

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a priori of physics

Preface

Preface Motivation of this Book After my graduation as a Candidate in the Science of Physics from University of Copenhagen (UCPH) in 1983, I earned an academic position in the Royal Danish Post & Telegraph as a specialist in radio wave propagation for network planning of cellular phone systems. Through this work and my participation in COST (European Cooperation in Science and Technology) projects 207, and 231, I learned that the encoding of information signal in the propagating radio waves was possible. An encoding with a preserved phase alignment direction of the signal waves even in a multireflection delay spread with Rayleigh fading probability of the received added wave front that certainly not are coherent. A philosophical question arose for me: Why is this phase information preserved in a radio signal that’s not local receiver time-phase-coherent in its wave front due to the multi delay spread in its propagation. The encode method that came possible at that time was Orthogonal Frequency-Division Multiplexing (OFDM).2 We physicist immediately know that the preservation of frequency as a quantum is embedded in each photon ℎ𝜈 = ℏ𝜔 of he radio signal. Is the quantum phase information of the radio signal also embedded in each photon? The Philosophical Change After the big political change in Europa around 1990 and my personal fiasco making a commercial business out of inventing and improving new electronics for audio amplification I was so mature that I began reading Immanuel Kant’s philosophy etc. From this I found that; to make a persistent impact to our world I had for my ambition to write a book concerning the problem of how we humans can know anything about physics. This project was first started around 2003. I started my writing in my mother tongue Danish to keep all philosophical concepts clear in my mind. I was a novice as an author. My only experience in writing was from my candidate thesis and technical rapports from my work. By studying I. Kant, I found that space and time is a priori transcendental to the intuition. To experience something in space a change of space has to happen. New Studies in Modern Physics What I not learned at university was the later new Standard Model with its zoo of elementary particles. It was exceedingly difficult to find literature that told what it really was, how and why. Someone told that reading Theodor Frankel, The Geometry of Physics [2], could get you some of way to the understanding. I studied this book over some years in my spare time. I also got hold of Peskin & Schroeder, An Introduction to Quantum Field Theory [3]; followed by Mark Srednicki, Quantum Field Theory [4], which i read more carefully. The problem with all these is, that nobody seems to care how all this mathematical theory connect to a physical nature. My problem is, I am keen to know how nature works, not how fantastic humans is able to construct mathematics. Meanwhile all this frustration in braking through the mythological surface narrative of modern physics I proceeded in my endeavour to find a foundation epistemology for understanding physics.

The Philosophical Concept of Time and Space We can only experience the nature when change happens. Therefor the first hundred pages of this book criticise the traditional time concept perspective. The space aspect here is restricted to the cyclic circular movement, as ancient time concept was cyclic. The timing is dual reciprocal to the frequency energy ℎ𝜈 = ℎ1⁄𝑇. For the fundamental timing concept only two conceptual direction qualities are essential, the one into the future and the other the information phase angle direction. Due to the principal idea of causality, analytically these has no opposite orientation. 2

The technical explanation of the OFDM have very much in common with the wave packet idea of Schrödinger quantum mechanics. This encoding principia is nowadays used all over e.g. for Wi-Fi and smart devices.

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Preface

The Space Extension Our approach to space has from ancient time been geometric. Descartes: space has extension. When you look at a wall tabula in front of you, you can draw an abstract circle. One turn in the circle you measure symmetric with six straight radius segments. Imagine a Mercedes star support from center by three radius vectors. You divide in three areas left, right and down. Extend this to the regular platonic tetrahedron with four vertex corners supported by four radius vectors from center of the circumscribed sphere. Then you have a figure I call a regular tetraon shown at the frontpage as a carbon-methane molecule symmetry. I promoted this tetraon icon idea to my mascot3 for understanding a fundamental symmetry of what we traditional call a three dimensional 3D space for our classical space for our universal Nature. The modern mathematic approach to geometry and all its symmetries and interconnections is often so complicated, that it is unaffordable to understand for one person. We have to go back to the natural foundation of mathematics and build it step by step so that each step has a understandable physical foundation in nature. First time I found this done seriously systematic was when I in 2011 found the concept of Geometric Algebra on Wikipedia in a search after knowledge about quaternions. It led me to David Hestenes’s Oersted Medal Lecture 2002 [5]. Studying this and other of Hestenes research papers I was able to start building a new founding epistemology of a space concept of physics for the enlightenment of our universal Nature. The space concept as the geometry of physics starts at page 123. The geometric elements p.137-139. The line p.141. The geometric plane p.153. The idea of the geometric algebra starts at p.160, first detailed for the Euclidean plane concept. Inherit qualities p.199 leading to matrix representation p.205. Then the geometric algebra for the anti-Euclidean plane concept p.209. The idea of a natural Euclidean space founded on three dimensions start at page 231, and its geometric algebra p.237. A Space of Relations When Nature establish relations across space extension the information signal takes time. Therefor the traditional Minkowski metric of four dimensional non Euclidean founding vector space is used. To establish an epistemological enlightenment of the fundamental relations that can measure extensions by one isotropic unit of development we will use ideas from a geometric algebra approach introduced by David Hestenes 1966 in a book named Space-Time Algebra [6]. The goal is to continue this work using a substantial fourth dimension of one quantum count of timing-development unit direction quality 𝛾0 orientated into the future. This development concept make spatial information relation possible. I believe all this is necessary a priori knowledge for us to understand Nature by intuition in a new enlightenment etic. Practical Issues of Writing this Book Starting this research work of developing an epistemology I was not much of an writer. Therefor the first part page 23-197 was written in Danish. I realised my effort needed a wider audience, therefor I took the text into google translate to get a version in English. Obvious it was necessary to fit the text to a hopeful understandable level. In 2017 I started writing direct in English about ℨ-space in chapter 6 from page 231-308. After this in primo 2019 page 199-229 the plane concept was expanded with extra abstract properties Chapter 5.5; Plane matrix representation 5.6; Non-Euclidean plane 5.7; and Generalised geometric algebraic exponential functions 5.8; to Chapter 5.9. From June 2019 continued at Chapter 6.6 onto 7.3, page 309-337. Jens Erfurt Andresen the 9th November 2019. 3

This icon idea tetraon I found and named in 2003. When I later, at the one hundred anniversary of Niels Bohr’s atomic model, saw a video made by ‘Danmarks Radio’ about his work, I found Bohr’s drawing of that Methane molecule, that’s shown at the frontpage comment of this book. This picture is graphed direct from an appropriate frame cut from this anniversary video.

©

Jens Erfurt Andresen, M.Sc. Physics, Denmark

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Prologue

Prologue (i)

An Introduction to Natural Space of Physics I live in the middle of a big sphere. When I look out into the night, furthest away I see the starry sky. All I see in the world is across. The paper, the screen, the wall, yes even the celestial sky is across. All that is crossed has two dimensions 2D. It has a length (width) and a breadth (height), said otherwise x and y coordinate or when it comes to the celestial (starry) sky the angular coordinates 𝜑 and 𝜃. When I look along a straight line in any direction, I call a plane perpendicular to that sight a transversal plane, transversal to that aiming line. When I point my right arm and index finger, I can take my left index finger and move it around the right in a circular cylinder motion. There is something outside my right arm and index finger, that is the physical space. The natural space of physics is, of course, also inside the arm. The questions then are: What qualities does space really possesses? What impact has space on everything?

(ii)

How Do You Experience the Physical Space Around You? When you look at the plane surface the screen, the paper, a wall, or a stone in front of you, you can determine how it rotates. It is not rotating you properly would say. Now you have just made a measurement of a physical rotation and found that the relative rotation is zero in this case. When we see the celestial sky crossing over us, we would see that it is rotating relative to us. We know that the ancient Greek philosophers were able to see and understand that it rotates (1 turn per star night). To view the light from the paper or screen in front of you, keep it placed transversely (across) in front of you. Today we know about the light that the photons propagate with an invariant (isotropic) speed. The light can only go forward, thus it possess causality. The light with which we see a thing comes from that thing → into our eyes never opposite inverted. The space we can look through, and thus perceive the existence of, depends on the propagation of light. This travel of the light creates the passage of time. The passage of time comes therefore from this cause. As we though know analytically, time can only go forward. Time with negative sign corresponds to expectations of something in the future. We can now understand, that the time and space around us depends on the passage of light. The term "right now and here" we use for our own location in space. The theory of relativity has taught us that the term "right now and there" has no meaning. When we predict something, we can say "right there and then it will happen". We must wait and see if it is a true or a false prediction. We can meaningful say "right there and then it happened". When we look at the stars, we know that what we see in the sky happened a long time ago, but to us it happens now. When we look at other people and objects, we know that is already the past that we see. Even when we see ourselves in the mirror, it is the past we see. In addition, we see ourselves from the outside in the mirror. I am inside myself and can see myself from the outside using the mirror. The light changes direction in the mirror and the depth of the image changes sign at the mirror. The mirror image shows right hand on right side, just as with left hand to the left side. We cannot have empathy with our mirror image because we cannot give hand with the right hand. When we give the right hand to another person, our hands meet across. The inner spirit of the other is an outer to me, while my inner is the outer of the other. The understanding of this concept is called empathy. The word empathy is usually only used when it comes to human emotional understanding of the other (or the others). Similarly, when dealing with a physical entity (e.g. an elementary particle), we must assume that each entity has its own internal, which relates to the outside.

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Prologue

(iii)

An Image of the World A local event consists of 4 dimensions. In 3D computer graphics you always count on 4 dimensions. As is known, there are two dimensions on the screen (2D for width and height) plus one dimension for the depth of the image plus one dimension for the timing. The last two dimensions overlap each other as they relate to the causality of the light as the cause of the temporal development. The reason we must divide the depth and time is computationally due to the duality between the transmission of light from the object and the reception of the light in the subject as the receiving interpreter. (the eye and the brain or a measuring instrument, that in both cases is an object to the programmer.) Historically, the term "right now and here" is called for a point. But in the mathematical philosophy the point has no extension, and therefore no existence over time in space-time. A physical entity, somewhere in space, will always be extended to have a local existence. We compare and refer here to the idea of indefiniteness of quantum mechanics.

(iv)

Extension and Propagation of Space The way we can see the shape and extension of space is that the light is propagating. We know that the stars had to be there, because the light comes to us from them. The only way we can measure a distance in space is by means of light. You might say, that at least locally you can just measure the distance with a stick, ruler, or tape measure. Here you must remember that light is needed to see the scale, and what holds the atoms in place in a material stick is light (electromagnetism). Thus, the light is the only thing that expands the space for us and gives it the form (structure) and extension. We just must keep in mind that in itself light is invisible, as we need light to see light, which is a causal contradiction.

(v)

The structure of light The traditional interpretation of light is, that it can be perceived as both waves and particles. Let me try to describe an updated image of the light structure. Here I will start with the Euclidean geometry. Here the concept of the line and the plane is very central. Considered as a particle, the light moves along straight lines. As a wave, the light propagates as a transversal plane wave. How shall we understand this? The direction of light is perpendicular to a transversal plane. Basically, the newest view of wave oscillation of light is a rotation in the transversal plane. The light always rotates transversely to the direction of movement. In physics, we represent the light direction by a vector k, which is perpendicular to the oscillation plane. Popularly said, a vector is a line-segment with a direction and a length, pictorially an arrow. When we consider light as one particle (one photon) the photon has one specific energy 𝐸. The energy of the photons is linked to angular frequency 𝜔, 𝐸 = ℏ⋅𝜔 = ℎ⋅𝜈 , where ℏ is Planck's constant (ℎ = 2𝜋ℏ) and the light cyclic frequency is 𝜈 = 2𝜋𝜔. An example of this quantity is the colour of visible light, that is determined by the photon energy. The particle momentum of the photos is ℏ⋅k, given by the wave vector k for the direction of light perpendicular to the transversal oscillation plane. k has the length magnitude |k| = |𝜔|⁄𝑐 = 𝐸 ⁄ℏ𝑐 . We imagine an imaginary photon, whose oscillation is a rotation in a circle, as it propagates FORWARD in the direction of light perpendicular to the plane of the circle. The entity that rotates is the electromagnetic field. A local field value in the free space rotates as it moves forward. This makes the rotation a spiralling helix movement. The line arrow, the vector k represents the progressive helix spiral movement. The imaginary photons can occur in two different states, left and right modes. We will say the imaginary photons have the helicity +1 or − 1. We can now interpret the light consisting of an electromagnetic field wave whose local value rotates and moves forward at the speed of light.

©

Jens Erfurt Andresen, M.Sc. Physics, Denmark

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a priori of physics

Prologue

The photon energy is given by the rate of rotation as the frequency of the rotation 𝐸 = ℏ⋅𝜔 . Since the imaginary4 photon is massless, its angular momentum is the same as its line momentum ℏ⋅k. In four dimensions we can express a photon by a four component vector 𝑘= 𝜔⁄𝑐 +k. _______ In summary, all visual space depends on light, and the propagation of light cause the idea of time. In all we have the traditional concept of space-time. Now we will proceed to develop an examination of an epistemology as an ethics foundation for understanding the a priori of physics.

First chapters (pages 23-197) is original written in Danish and translated to English primo 2017:

4

Imaginary just as the imaginary complex number plane, see later throughout this book.

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– xxi –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 1. The Idea of Time

I. The Time in the Natural Space 1. The Idea of Time The widely common opinion is that a phenomenon in physics takes place in space and time. This view, I will not question. However, I ask; what is space and what is time? Immanuel Kant pointed out that space and time are conditions for human cognition; concepts of space and time cannot be experienced empirically. Immanuel Kant categorised: SPACE as a necessary external condition for the observation of the physical world, and TIME is required as an internal condition of keeping track of phenomena in physics. • The passage of time cannot be measured as the time has passed while. • The form (structure) of space cannot be known by measurement as the structure is a condition included in the measurement concept. This paradox, that space and time cannot be perceived empirically, Immanuel Kant called a transcendental property. The property with which we intuit the world. Space and time are thus necessary a priori concepts of human cognition. General cognition – a priori – the TIME and the SPACE – is itself transcendental concepts. Since René Descartes, it has been commonly accepted to consider the world as divided into the inner subjective and outer objective. Later, I will define the difference between the inner local and the outer world. I would just here anticipate that not only ‘the one’ but also ‘the other’ is local and has its own interior, and all the rest of the exterior is in the global world. All this participate in an universal Nature. First, some thoughts on the concept of time and its apparent subjective inner character, then some about the objective nature of space as an external condition for cognition in an epistemology.

1.1. Primary Quality5 • •

Primary quality of time comes from the causal action. (action is the cause) The memory can register change and distinguish ‘the one’ from ‘the other’ and then ‘the third’. The memory is an internal local capability. • Primary quality of space is the chiral extension (hand turned extent6). • The conscious awareness can distinguish the left from the right side, like the exterior from the interior (also called the front to back relation7).8 These definitions of primary quality for time and space, that involve memory and awareness, can be deepen. • The memory records the historical events in the local.9 Time has a direction: The memory can remember from past events, the memory cannot remember the future. • A conscious awareness places the events locally in space. Our consciousness distinguishes between own inner location and a second outer local event. 5

Primary quality is by Galileo, Descartes, Newton, Locke, etc. an attribute of the thing in itself, while a secondary quality is the perceived attribute. 6 Hand turned extent requires multiple grades and dimension of space volumes. This is described further in later chapters below. 7 When you see yourself in the mirror, your right hand is seen in the right side and your left hand in the left side of the picture, but the interior is hidden behind the exterior masks of your face. Your back is hidden, as well as your interior. 8 One could also mention here a difference between up and down, but it is already implicitly given the difference between right and left, as it also is a transversal property (circling around the way between the inner and outer). Up and down concept is usually associated to gravity. Here in the first place we will keep the gravity out of the pure concepts of space and time. 9 In addition to our own memory, the content text in a book or other dataset is also memory. The memory is internal and therefore inside a locality.

©

Jens Erfurt Andresen, M.Sc. Physics, Denmark

– 23 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 1. The Idea of Time – 1.3. The Causal Action –

Together, time and space have a primary quality that we call ‘local’ as opposed to ‘global’. The outer local event can be learned and may be remembered in the inner local memory. The global is valid for all times and all over space and can be categorised as a secondary quality.10

1.2. Quantity • The quantity of time is the result of counting 1, 2, 3, ... how far have we counted. • The quantity of space has different qualities as volume, area, distance, and action. All these with directions. The properties of space will be explored later below in chapters II. p.123 →.

1.3. The Causal Action 1.3.1. Logic and Numbers

All human logic is based on syllogisms. An important example used very often is Nr. A All humans are fallible. 1 B All philosophers are humans. 2 C All philosophers are fallible. 3

(1.1)

This logical inference is a syllogism based on the causal action. From statement A and B to the conclusion C. If we first have phrase A and then another sentence B, we can form a third phrase C, as a causal consequence of A and B. We write A, B ⇒ C Sometimes in logic, the interchangeability of the two terms are allowed, we say that the commutative law applies the premises. From the example: First sentence B, so another phrase A, which follows the third sentence C. "All philosophers are humans. and All humans are fallible." We remove only the underlined between the concept of 'humans' and gets the conclusion C. In each sentence the causal consequence, is not commutative, since e.g. the sentence "All fallible are humans” is not the same as sentence A. In the syllogism, the sequence applies the causal consequence, one cannot from C and A conclude B, or from C and B one cannot conclude A. A causal sequence is in principle not commutative. Normally, the A B C is not the same as C B A or B A C. This applies for example to the alphabetical order. a, b, c, d, e, f, g, h, … A priori: To make a syllogism, one must be able to count to three in memory.

1.3.1.2. The Number Sequence

(1.2)

10

The count is a causal action. Do not swap order of the number figures. 1 2 3 is not the same as 2 1 3. The count is not commutative. In contrast, by an ordinary addition of numbers the commutative law applies 2+1= 3 ⇔ 1+2=3. When we count the causal action applies, so the order of the numbers is essential. We shall always add the number 1 as follows: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, … This process is not commutative, the sequence order as a direction is necessary as reading direction in this book is a direction. When we count time, a quantity appears. (see more below 1.3.3). – How many times do you count one more?

The perception of a global property is a secondary quality. The unlimited world cannot be measured, therefor not an object.

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 24 –

A Research on the a priori of Physics –

December 2020

– 1.3.2. Time, Action and Sequence – 1.3.1.2 The Number Sequence –

The count process is adding the number one, +1, to a previous number 𝑛, whereby we get (1.3) (1.4)

(1.5)

(1.6) (1.7)

+1

𝑛+1 ← 𝑛. I call +1 the counting operator. We call the count figures for the natural numbers ℕ ≡ 0,1,2,3,4, … Zero is taken into the natural numbers, since the count to 1 requires that we distinguish any {1} from non {0}. The natural counts ℕ represent a sequential order 0 0,

and

these according to (3.128)-(3.129) and (3.130).

The active principle requires that the rotation phase angle 𝜙 is a monotone variable 𝜙 ≡ 𝜔𝑡 ∈ℝ ⃗⃗ . The rotation intuit as object is created from the by the information development parameter 𝑡 ∈ℝ ±𝑖𝜙 noumenon idea of an active rotation 𝑒 = 𝑒 ±𝑖𝜔𝑡 , as they occur in formulas (3.145) and † (3.146) for the creation operators 𝑎⊙± . ©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 78 –

A Research on the a priori of Physics –

December 2020

– 3.3.1. The First Excited States of the Circle Oscillator – 3.2.4.3 Annihilation and Creation Operators in a Polar Plane –

These two operators (3.145), (3.146) commute with 𝐿̂3 = (𝑎+† 𝑎+ − 𝑎−† 𝑎− )ℏ, hence by (3.103) (3.147)

† † [𝑎⊙+ , 𝐿̂3 ] = [𝑎⊙− , 𝐿̂3 ] = 0.

By this they have the same eigenstate as 𝐿̂3 in the eigenvalue equation 𝐿̂3 |𝑛, 𝑚⟩ ≐ ℏ𝑚|𝑛, 𝑚⟩. Therefore, the two excited states for 𝜌 ≥0; the progressive with 𝑛 = 1, 𝑚 = +1 , (3.148)

1

1 2

† |0,0⟩ = 2 ( 4 𝜌𝑒 −2𝜌 ) ⊙𝑒 +𝑖𝜙 = 2𝑟̃ (𝜌)⊙𝑒 +𝑖𝜔𝑡 , 𝜓+⊙ = |1, +1⟩ = |1,0⟩ = 𝑎⊙+ √𝜋

and the retrograde with 𝑛 = 1, 𝑚 = −1, (3.149)

↕ 1

† |0,0⟩ = 2 ( 4 𝜌𝑒 𝜓−⊙ = |1, −1⟩ = |0,1⟩ = 𝑎⊙− √𝜋

1 − 𝜌2 2

where 𝜌 ≥0

) ⊙𝑒 −𝑖𝜙 = 2𝑟̃ (𝜌)⊙𝑒 −𝑖𝜔𝑡 .

It is noteworthy here, that these first excitations are direct autonomous normalised (3.150)

∞

2

∞

∞ 4

⟨1, ±1|1, ±1⟩= ∫0 𝜓±∗ (𝜌)𝜓± (𝜌) 𝑑𝜌 = ∫0 (2| 𝑟̃ (𝜌)) 𝑒 ±𝑖𝜙 𝑒 ∓𝑖𝜙 𝑑𝜌 = ∫0 when the ground state is prescribed ⟨0,0|0,0⟩ = 1 ⇒ 𝐴0 = I.e. that the idea 2𝑟̃ (𝜌) is confirmed for 𝜌 ≥0.

1 4

√𝜋

√𝜋

2

𝜌2 𝑒 −𝜌 𝑑𝜌 = 1,

.

When we write 𝑟̃ (𝜌) = +½ pd1 (𝜌) and 𝑟̃ (−𝜌) = − 𝑟̃ (𝜌) = −½ pd1 (𝜌) , the parity inversion contradiction can be written for 𝜌 ≥0 (3.151)

2𝑟̃ (𝜌)= 𝑟̃ (𝜌) − 𝑟̃ (−𝜌) = +½ pd1 (𝜌) − ½ pd1 (−𝜌) = +½ pd1 (𝜌)+½ pd1 (𝜌) = pd1 (𝜌).

The excitation energy of the first state is here; according to (3.113) and (3.84), 1 (3.152) 𝐸𝜔 = 𝐸𝜔,1 − 𝐸𝜔,0 = ℏ𝜔 . We will by intuition look at the circular rotation described by the polar coordinates, the radial magnitude 𝜌 ≥0 and the phase angle 𝜙 ≡ 𝜔𝑡 ∈ℝ, that develops with the parameter ⃗⃗ . 𝑡 ∈ℝ So, we compare the classical view with the cyclical oscillator (3.117) or (3.48) ∗ (𝑡, 𝑞𝜔 𝑟) = 𝑟𝑒 𝑖𝜔𝑡 , and writes for 𝜃=0 ⇒ 𝑒 𝑖0 =1, for the progressive orientation (3.153)

† ∗ (𝑡, 𝑞𝜔 𝑟) = 𝑟𝑒 +𝑖𝜔𝑡 ∼ |1, +1⟩𝜔,0 = |1,0⟩𝜔,0 = 𝑎⊙ 0 |0,0⟩𝜔,0 = +

with 𝑛 = 1, 𝑚 = +1 ; (3.154)

and the retrograde with

1

1 2

(2 4 𝜌𝑒 −2𝜌 ) 𝑒 +𝑖𝜔𝑡 , √𝜋

𝑛 = 1, 𝑚 = −1

† 𝑞𝜔 (𝑡, 𝑟) = 𝑟𝑒 −𝑖𝜔𝑡 ∼ |1, −1⟩𝜔,0 = |0,1⟩𝜔,0 = 𝑎⊙ 0 |0,0⟩𝜔,0 = −

1

1 2

(2 4 𝜌𝑒 −2𝜌 ) 𝑒 −𝑖𝜔𝑡 . √𝜋

Looking at the absolute squares we can in principle set 𝑟 = 1, as (3.155)

∞ 4

⟨|𝑞𝜔 (𝑡, 𝑟)|2 ⟩ = 𝑟 2 ∼ ⟨1, ±1|1, ±1⟩ = ⟨ |2𝑟̃ (𝜌)|2 ⟩ = ∫0

√𝜋

2

𝜌2 𝑒 −𝜌 𝑑𝜌 = 1

The requirement for an entity A𝛹±𝜔 in physics, is when created in the event A by excitation † |0,0⟩, it establishes a geometric plane ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈ 𝑈(1) | ∀𝜃 ∈ℝ } for the circle 𝑎± oscillator and a start phase angular direction 𝜙0 = 𝜔𝑡0 in this plane through this event A. The problem here is phase direction relative to what? More on this later below in section 3.5.4 and chapter 6.

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 79 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.3. Excitation of the Plane Harmonic Circle Oscillator –

3.3.2. Higher Excited States of the Circle Oscillator ̂𝜔 |𝑛, 𝑚⟩ = ℏ𝜔(𝑛+1)|𝑛, 𝑚⟩. Now excitation 𝑛 >1 for the energy eigenvalue equation (3.113) 𝐻 3.3.2.1. The Possibility of the Second Excited States of the Circle Oscillator

For the double-excited state 𝑛 = 𝑛+ + 𝑛− = 2 , there are three cases (3.156) 𝑚 = 𝑛+ 𝑛− = −2, 0, +2. First, we from (3.128) and (3.129) write (3.157)

† |1, ±1⟩0 = 𝑎±

𝑒 ±𝑖𝜙 2

𝜕

𝑖 𝜕

1

1 2

(𝜌 − 𝜕𝜌 ∓ 𝜌 𝜕𝜙) 4 𝜌𝑒 −2𝜌 𝑒 ±𝑖𝜙 = √𝜋

1 1 4

2 √𝜋

1 2

(2𝜌2 1 ± 1)𝑒 −2𝜌 𝑒 ±𝑖𝜙 𝑒 ±𝑖𝜙 ,

then from this 1

2

(𝑎+† ) |0,0⟩ =

P±⋅(

1 2

1

𝜌2 𝑒 −2𝜌 ) ⊙𝑒 +𝑖2𝜙 for 𝑛=2, 𝑚 = +2,

(3.158)

|2, +2⟩ = |2,0⟩ =

(3.159)

|2,0⟩ = |1,1⟩ = 𝑎+† 𝑎−† |0,0⟩ = 𝑎−† 𝑎+† |0,0⟩ = P±⋅( 4 (𝜌2 − 1) 𝑒 −2𝜌 ) ⊙

(3.160)

|2, −2⟩ = |0,2⟩ =

√2

4

√2 √𝜋

1 2

1

√𝜋

1 √2

(𝑎−† )2 |0,0⟩ =

1 2

1

P±⋅(

4

for 𝑛=2, 𝑚 = 0,

√2 √𝜋

𝜌2 𝑒 −2𝜌 ) ⊙𝑒 −𝑖2𝜙 for 𝑛=2, 𝑚 = 2.

These state expressions of radial density magnitude are shown in Figure 3.6. As they are even functions, this contradicts Newton’s 3rd law for the parity inversion balance: P±⋅(𝑟̃2 (𝜌)) = 𝑟̃2 (𝜌) − 𝑟̃2 ( 𝜌) = 0 for 𝑚 = ±2,

or = 𝑟̃𝑚=0 (𝜌) − 𝑟̃𝑚=0 ( 𝜌) = 0 for 𝑚=0.

Figure 3.6 The double excitation of a single ground state provides an even density functions for 𝑛=2; 𝑚= 2,0,2. That by parity difference is resulting in pdm2 (𝜌) = 𝑟̃2 (𝜌) − 𝑟̃2 (−𝜌) = 0 , for 𝜌 ≥0 as illustrated by a black line. This is the situation for the double excitation in one and the same plane through the event.

The even functions have self-symmetric distribution around an imagined center. They therefore do not give a contribute as a difference over the ⊙ plane. Therefore, the judgment is, that the double excitations in the plane of the rotation in the circle oscillation do not contribute to proper elementary entities in reality. In this way, we shall claim † † that two simultaneous successive operating excitations 𝑎± 𝑎± |0,0⟩ in same ⊙ transversal plane has no existence as one and the same entity. This has its cause in the parity inversion balance 𝑒 𝑖𝜋 = 1 along a straight line in the plane according to (3.120), which then results in a lack of contradiction and therefore disappears. The parity inversion factor P± =0 just displays these cancelations is characteristic for the action of the unitary 𝑈(1) group of elements ⊙ ~ 𝑒 𝑖𝜃 from the plane ground state |0,0⟩. † † This issue of auto cancelling of double excitation 𝑎± 𝑎± |0,0⟩ is limited to one and the same event location A of circulation ⊙ in one and the same plane direction. † |0,0⟩ = |1, ±1⟩ from the same ground However, the creation of two or more simultaneous 𝑎± ⊙ ⊙ state plane ⊙ at A, are creations of multiple indistinguishable entities A𝛹±𝜔 ~ A|1, ±1⟩𝜔 .

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 80 –

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December 2020

– 3.3.3. The Plane Excited Circle Oscillator – 3.3.2.2 The Possibility of a Third and Higher Excited States of the Circle Oscillator – 3.3.2.2. The Possibility of a Third and Higher Excited States of the Circle Oscillator

The third excitations in the ⊙ plane attempted written (3.161)

|3, ±3⟩0 = =

(3.162)

|3, ±1⟩0 = =

1 √6

3

† (𝑎± ) |0,0⟩0 =

1 𝑒 ±𝑖𝜙

𝜕

1 √3

† |2, ±2⟩0 𝑎±

𝑖 𝜕

√6 2 1 † 𝑎 |2,0⟩0 √2 ± 1 𝑒 ±𝑖𝜙 √2

2

1

1 2

1

(𝜌 − 𝜕𝜌 ∓ 𝜌 𝜕𝜙)P± 4 𝜌2 𝑒 −2𝜌 𝑒 ±𝑖2𝜙 = P± √𝜋

𝜕

𝑖 𝜕

1 4

√6 √𝜋

1 2

1

(𝜌 − 𝜕𝜌 ∓ 𝜌 𝜕𝜙)P± 4 (𝜌2 − 1) 𝑒 −2𝜌 = P± √𝜋

1

1 4

√2 √𝜋

1 2

𝜌3 𝑒 −2𝜌 𝑒 ±𝑖3𝜙 ,

1 2

𝜌3 𝑒 −2𝜌 𝑒 ±𝑖𝜙 .

Figure 3.7 A thought tripled excitation of the ground state |0,0⟩ to |3, 𝑚⟩ is an odd function in one and the same plane. (In this case without normalization factor)

Third excitations in the same plane is admittedly odd functions but builds on the lack of reality of the double excitations, therefore, we assume that it will not appear in the same ⊙ plane. Since the involuting parity operation for double excitations in (3.158)-(3.160) for the same plane gives P± =0 the even 𝑛=2 excitation is missing as a starting condition for additional excitations in this plane, e.g. (3.161)-(3.162) inherits P±=0, etc. The higher excitations can be obtained by the inclusion of several independent transversal planes (multiple dimensions) and thereby get physical reality, but we will not elaborate more on this here.97 The multiple frequency 𝑛𝜔 does not occur in the circle oscillator based on multiple use of † † † creation operators 𝑎± … 𝑎± 𝑎± |0,0⟩ in the same ⊙ plane. We therefore only get first excitations of a plane circle oscillator where 𝑛=1. Here all the angular frequencies ∀𝜔 ∈ℝ are permitted, as all higher multiplications 𝜔 = 𝑛? 𝜔? of some basis frequency † |0,0⟩ = A|1, ±1⟩𝜔 is of course also permitted, in an idea of a circle oscillator excitation at A; 𝑎±𝜔 ⊙ ⊙ for our intuition of entities A𝛹±𝜔 ~ A|1, ±1⟩𝜔 . The only way we can distinguish these entities is by the given quantities 𝜔 ∈ℝ and by the distinguishable transversal planes with ⊙ symmetries. We remember that all 𝜔′ ≠ 𝜔 are orthogonal in this planes. 3.3.3. The Plane Excited Circle Oscillator

In the case of light and electromagnetic waves, the concept of a transversal plane wave is already well established in the epistemology of physics. My contention is that the circle oscillator and its symmetry 𝑈(1) is an a priori intuition of the transversal plane for a propagating wave. This review hereh in section 3.3 claim that we are forced only to look at the case of a first circular excitation 𝑛=1 of a plane for a transversal propagating wave entity. The Eigenvalue equation (3.113) demands that the angular frequency energy 𝜔 is a given quantity of the quality of excited circle oscillator creation in a plane symmetry 𝑈(1) of ⊙. 97

The 20th century quantum mechanics books dealing fully with the Angular Momenta in three dimensions 𝐿𝑥 , 𝐿𝑦 , 𝐿𝑧 . E.g. look in [8], [7], [9], and emeritus [30], [32], [38], alternativity [31] .

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 81 –

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December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.3. Excitation of the Plane Harmonic Circle Oscillator –

3.3.4. The Possible Excitation of a Circular Oscillator with ± Signed Orientation.

The plane concept allows a circle rotation through the plane around a locus situs event point in the plane with a 𝑈(1) symmetry, which requires an angular dualism 𝜃\𝜙 (3.139), from an eternal ∂ ∂ ∂ constancy ∂𝜙 𝜃 = ∂𝜙 ⊙ = ∂𝜙 |0,0⟩ = 0, for all angles ∀𝜃 ∈ [0,2𝜋[ for the formulation ⊙𝜃 = 𝑒 𝑖𝜃 , bearing in mind the symmetry as (3.137) ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈ 𝑈(1) | ∀𝜃 ∈ℝ }. † One active creation at a given angular frequency energy 𝜔 with the creation operator 𝑎⊙±𝜔 from the ground state, where we presuppose 𝜌 >0, gives as in (3.148)-(3.149) (3.163)

⊙ † ⊙ |0,0⟩⊙ 𝜓±𝜔 = |1, ±1⟩𝜔 = 𝑎⊙±𝜔

𝜕

𝑖 𝜕

𝜕

1

1 2

= 𝑒 ±𝑖𝜙 (𝜌 − 𝜕𝜌 ∓ 𝜌 𝜕𝜙) |0,0⟩⊙ = 𝑒 ±𝑖𝜙 (𝜌 − 𝜕𝜌) |0,0⟩⊙ = 2 4 𝜌𝑒 −2𝜌 ⊙ 𝑒 ±𝑖𝜔𝑡 √𝜋

Here the angular phase coordinate 𝜙 of the state is translated into the energy angular frequency ⃗⃗ , so that 𝜙 = 𝜔𝑡. The development goes from 𝜔 multiplied by the development parameter 𝑡 ∈ℝ the past to the future as an direction dimension different from the circle oscillators plane. I say and claim, that the plane of the circle oscillator is transversal to the development, and that this past to future direction will be governed by the angular momentum. The excited ground state at quantum 𝜔 ∈ℝ is an active revolving plane circular symmetric distribution of frequency energy (3.164)

† |1, ±1⟩⊙𝜔 = 𝑎±𝜔 |0,0⟩⊙ = ⊙2𝑟̃ (𝜌)𝑒 ±𝑖𝜔𝑡 , for 𝜌 > 0.

We saw above that the resulting transversal radial distribution (3.144) is composed of parity inversion contradiction (3.151) 2𝑟̃ (𝜌) = 𝑟̃ (𝜌) − 𝑟̃ (−𝜌). These are shown for intuition in Figure 3.8, where the normal vector 𝑛⃗ to the transversal plane pointing into the future.

Figure 3.8 The intuition that the ground state is excited to a harmonic circle oscillator with the angular momentum 𝐿⃗3 in which the density magnitude is shown as a parity inversion contradiction (red-blue). The transversal plane is represented by the polar coordinates (𝜌, 𝜃) is equivalent to the complex number 𝜌𝑒 𝑖𝜃 . The excitation is an active rotation by the factor 𝑒 ±𝑖𝜔𝑡 producing the past |𝜙| = |𝜔 ̂|𝑡 , rearward from the rotation in the transversal plane, consistent with its momentum along the normal vector 𝑛⃗, |𝑛⃗|=1. The rotation has an angular momentum 𝐿⃗3 = ±1ℏ𝑛⃗ corresponding to the two possible state orientations of the rotation. Note here; the rotation 𝑒 +𝑖𝜔𝑡 is shown by 𝐿⃗+3 = +ℏ𝑛⃗. The development parameter 𝑡 is drawn backward along with the angular development phase 𝜙 in the rotation, |𝜙| = 𝑡⁄|𝜔 ̂|. Since 𝜔 ̂ is the measure norm |𝜔 ̂| ≡ 1 (helix pitch 1). The drawing of the density magnitudes 𝑟̃ (𝜌) and 𝑟̃ (−𝜌) represents its own dimension (and has to be interpreted totally different from the development parameter dimension) and is displayed as rotation symmetric disc (red & blue circles) generated by the 𝑈(1) rotational symmetry ⊙~ 𝑒 𝑖𝜃 through the plane 𝜌𝑒 𝑖𝜃 . The density is intuit as the thickness of the disc pd1 (𝜌) = +½ pd1 (𝜌) − ½ pd1 (−𝜌) = 𝑟̃ (𝜌) − 𝑟̃ (−𝜌) = 2𝑟̃ (𝜌).

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 82 –

A Research on the a priori of Physics –

December 2020

– 3.3.4. The Possible Excitation of a Circular Oscillator with ± Signed Orientation. – 3.3.4.2 The Oscillation Freedom from Portable Energy as the concept of Rest Mass –

The circular symmetry plane as a background is here expressed by a complex number 𝜌𝑒 𝑖𝜃 ∈ℂ from the polar coordinates (𝜌, 𝜃) , which are in correspondence with the symmetry of the unitary rotation group 𝑈(1) of unitary operators (3.165) 𝑈𝜃 : 𝜃~𝜃 → ⊙𝜃 = 𝑒 𝑖𝜃 ~ 𝑒 𝑖𝜃 ∈ 𝑈(1). 𝜌 is a dilation factor from a unitary number 𝑒 𝑖𝜃 , that define the plane by coordinates (𝜌, 𝜃). The two opposite distribution functions are shown as red opposite blue. These distributions are not only rotation 𝑈(1) symmetric in the plane, but information development rotates also by the factor 𝑒 ±𝑖𝜔𝑡 . In this intuition, we see them as a distribution of angular momentum in the plane. This is shown in Figure 3.8 as circles of the rotation over the transversal plane. The development dimension produces a direction into the past (along the green line to the left in the Figure 3.8), wherein the circle oscillator produces a cylindrical helical structure, of which helicity has pitch ±1, relative to the development norm |𝜙| = |𝜔|𝑡. (Figure 3.8 only display helicity +1) We remember the eigenvalue equations for the rotating circle oscillator excitation for the Hamilton operator (3.113) and the angular momentum operator (3.114) is written ̂𝜔 |1, ±1⟩ ≐ (1ℏ𝜔 + ℏ𝜔)|1, ±1⟩ (3.166) 𝐻 𝜔 (3.167) 𝐿̂3 |1, ±1⟩ ≐ ±1ℏ|1, ±1⟩ We note that when energy is measured in the same unit as the angular frequency, ℏ=1. In (3.167) the eigenvalue of the angular momentum is ±1ℏ. This means that the angular momentum98 has a direction represented by the two oppositely oriented vectors ⃗ −3 = −𝐿 ⃗ +3 , ⃗ ±3 | = ℏ = 1 (3.168) 𝐿 where |𝐿 ⃗ +3 = ℏ𝑛⃗, as shown in Figure 3.8, Here, the progressive rotation is represented by, 𝐿 ⃗ −3 = − ℏ𝑛⃗, where 𝑛⃗ is the normal vector to the rotation plane. and the retrograde by 𝐿 3.3.4.2. The Oscillation Freedom from Portable Energy as the concept of Rest Mass

̂𝜔 that is The circle oscillator excitation gives an eigenvalue for the Hamilton function 𝐻 (3.169) 𝐸𝜔,1 = 2ℏ𝜔 = (ℏ𝜔 + ℏ𝜔) = 1𝐸𝜔 + 𝐸𝜔,0 = 𝑇𝜔 + 𝑉𝜔 Here we compare with the classic formulation (2.92) 𝐻𝜔 = 𝑇𝜔 + 𝑉𝜔 , where the kinetic energy then is 𝑇𝜔 = 1𝐸𝜔 = ℏ𝜔 and the potential energy is 𝑉𝜔 = 𝐸𝜔,0 = ℏ𝜔 , which is the binding to the ubiquitous ground state that possess the 'highest' symmetry of a plane. This potential energy is what I call the internal binding energy of the circle oscillator to its own parity inversion contradiction through a locus situs circular centrum in its plane. This internal coupling to the ground state is an intrinsic property of an excitation to an entity 𝛹±𝜔 . Just as (2.96) we have the formulation 𝐻𝜔 = 𝑇𝜔 + 𝑉𝜔 = 2𝑇𝜔 and further the function 𝐿𝜔 = 𝑇𝜔 − 𝑉𝜔 = 0 , which represents the portable energy. A physical entity 𝛹±𝜔 of the excited plane harmonic circle oscillator therefore has no interaction with the surroundings and moves freely through its surroundings (without any external forces). But moves with a kinetic energy (3.170) 𝑇𝜔 = ℏ𝜔 ~ 𝑚𝜔 𝑐 2, where an abstract internal relativistic quantity 𝑚𝜔 as an inertia factor (internal ’mass’)99, with speed 𝑐, and proportional with 𝜔. We see, that 𝜔 ∈ℝ is the real quantity, that defines a quantum excitation of the circle oscillator, which just is a scalar eigenvalue in (3.166). The fact that the kinetic and potential energies balance 𝐿𝜔 = 𝑇𝜔 −𝑉𝜔 = 0 means that the excited plane quantum harmonic circle oscillator is free from portable energy and thereby do not carry any rest mass along a path produced as a development parameter.

98 99

In the classical world, we describe the angular momentum by a vector 𝐿⃗ = 𝑟 × 𝑝 with a direction in space (or a bivector 𝐋=r∧p). We know from 20th century physics that a classical external rest mass has no meaning in this, 𝐿𝜔 = 𝑇𝜔 − 𝑉𝜔 = 0 .

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 83 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.3. Excitation of the Plane Harmonic Circle Oscillator –

To determine the quality of excitations we examine their eigenvalue equation for the angular momentum of the circle oscillator (3.167), that give rise to the two angular momentum vectors of (3.168), which sets two opposite orientations of a direction in something we call space. ⃗ +3 = +ℏ𝑛⃗ into the future, the other opposite 𝐿 ⃗ −3 = −ℏ𝑛⃗ back into the past, which The one 𝐿 constitutes each side of the circle oscillator plane. – See Figure 3.8 . ⃗ ±3 | = ℏ = 1. We remember from (3.168) that |𝐿 3.3.4.3. The Qualitative Unit of the Circle Oscillator Entity

To understand the quality of a circle oscillator we are looking for the unit of its fundamental quantity. In addition, we introduce an autonomous normalization of the angular frequency, that is, we put 𝜔 = |𝜔 ̂| = 1, where 𝜔 ̂ stands for a unit of angular frequency as a norm for 𝜔 ̂ itself, an 2 autonomous norm. Furthermore, we set ℏ=1, 𝑐 = 1 in our intuition of the unit idea to concern it ⃗ +3 | = |𝐿 ⃗ −3 | = 1, and 𝐿 ⃗ +3 = −𝐿 ⃗ −3 . all, in the way that |𝐿 We introduce a unit vector as the direction in some space outwards from the active circular rotation ̂ ⃗ +3 = 𝑛⃗ = ⃗𝟏 ~ ⃗𝟏 (3.171) 𝜔 ⃗̂ = 𝐿 ~ 𝐿̂3 ~ 𝜔 ⃗̂ , where the unit norm for the angular frequency vector is written |𝜔 ⃗̂ |=|𝜔 ̂| ≡1. Thus, alleged: ̂ The direction vector 𝜔 ⃗ for the rotation represent the quality for a circle oscillator. Hence the quality direction is substantial to the concept of a transversal circular rotation.

Figure 3.9 An intuition of the density distribution of the angular momentum for a retrograde excited circle oscillation. The negative direction 𝐿⃗−3 = − 𝑛⃗ for the retrograde rotation just points into the past from the transversal plane. The transversal plane is represented by the polar coordinates (𝜌, 𝜃), which is equivalent to the complex number 𝜌𝑒 𝑖𝜃 . The magenta circle-ring represent the unitary rotational group 𝑈(1), where ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈𝑈(1) | ∀𝜃 ∈ℝ}. 1

1 2

This dictates that the radial distribution 2 4√𝜋 𝜌𝑒 −2𝜌 of the excitation of the angular momentum must be rotation symmetric in the ⊙ plane. We let the autonomous normed vector ⃗𝜔 ⃗⃗̂ for the angular frequency determine the excitation with the creation 1

1 2

† † − 𝜌 ±𝑖𝜔𝑡 2 |0, 0⟩ = |1, ±1⟩𝜔 operator 𝑎±𝜔 . The excited states (3.163) is 𝑎±𝜔 ⊙⃗𝜔 . ⃗̂ ⃗⃗ = 2 4 𝜌𝑒 ⃗⃗̂ 𝑒 ⃗⃗⃗̂ ⃗⃗⃗̂ √𝜋

The direction of the unitary rotational symmetry ⊙𝜔 ⃗⃗̂ = 𝐿⃗+3 . ⃗⃗⃗̂ is given perpendicular out from the direction ⃗𝜔 ̂ ̂ ̂ Hence the transversal plane for ⊙𝜔 ⃗⃗ as normal vector ⃗𝜔 ⃗⃗ ⊥⊙. 𝜔 ⃗ is the direction FORWARD into the future. ⃗⃗⃗̂ has ⃗𝜔 Here, in this figure, the retrograde excitation 𝑚 = −1, is illustrated as a unitary (𝜌=1) spiral cylinder −𝑖𝜔𝑡 |1, −1⟩⃗𝜔 = 𝑒 𝑖𝜃 𝑒 −𝑖𝜔𝑡 = 𝑒 −𝑖𝜔𝑡+𝑖𝜃 , stretching out in the past, with a positive development parameter 𝑡, or ⃗⃗̂ ~ ⊙𝑒 phase |𝜙|=|𝜔 ̂|𝑡. This retrograde excitation is in line with eigenvalue equation (3.114) 𝐿⃗−3 |1, −1⟩𝜔 ⃗⃗⃗̂ ≐ −1|1, −1⟩𝜔 ⃗̂ ⃗⃗ . ± 2 ̂ ⃗ The intuition in this figure assumes that all conditions are normalized (1 ≡ |𝜔 ̂| = |𝜔 ⃗ | = |𝐿3 | = ℏ = 𝑐 = 1).

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 84 –

A Research on the a priori of Physics –

December 2020

– 3.3.4. The Possible Excitation of a Circular Oscillator with ± Signed Orientation. – 3.3.4.3 The Qualitative Unit of the Circle Oscillator Entity –

The plane of circle oscillators is transversal and those points out a direction 𝜔 ⃗̂ for information development (a course to sail in space-time). The sign of the direction orientation associated to angular momentum depends on the sign of the active rotation in the transversal plane in accordance with (3.167). The positive direction for information development of the transversal plane in a circle oscillator ⃗ +3 . The positive FORWARD orientation of the 𝑛⃗ is given in accordance with the direction of 𝐿 direction into the future we use as an arrow vector direction for the angular frequency oscillation ⃗ +3 = 𝑛⃗. (3.172) 𝜔 ⃗̂ = 𝐿 This is the primary quality of any development through a plane as given by an active creation of a circle rotation with a necessary energy quantity 𝜔 (the angular frequency). The active flow direction 𝜔 ⃗̂ by the transversal plane into the future gives the positive orientation of rotation in the transversal plane. – (You may use the right-hand rule.) The two possible orientations of rotation 𝑚 = ±1, is given by (3.167). ⃗ −3 = −𝜔 The retrograde angular momentum vector 𝐿 ⃗̂ pointing into the past, and appoint thus the coordinate direction that represent the past, 'As a tail pulled by the circle oscillators momentum' shown in Figure 3.9. ⃗ ±3 = ±1⋅𝜔 We have the two cases of rotation orientations 𝐿 ⃗̂ ± ⃗ 3 represents the active angular rotation in the transversal plane. • The angular momentum 𝐿 • The rotation axis direction 𝜔 ⃗̂ represents the active progress through the transversal plane.

Figure 3.10 The intuition of expectation for the radial distribution of the active angular moment over the transversal plane for a retrograde 𝐿⃗−3 = −𝜔 ⃗̂ excited circle oscillation. First the integrated contribution of the excited angular state throughout ⃗⃗ − ⃗− ⟨1, − 1|1, − 1⟩⃗𝜔 the transversal plane, we expect ⟨1, − 1|𝐿⃗−3 |1, − 1⟩𝜔 ̂ ⃗⃗ 𝐿3 = 𝐿3 . Then a detailed action contribution from a ⃗̂ ⃗⃗ = tiny circle ring 𝑑𝜌 at radius 𝜌 for the expectation of the angular momentum distribution 1 2 2

1 |2 4 𝜌𝑒 −2𝜌 | 𝐿⃗−3 𝑑𝜌 = √𝜋

©

4 √𝜋

2 𝜌2 𝑒 −𝜌 𝐿⃗−3 𝑑𝜌,

and by radial integration

∞ 4

∫0

√𝜋

2 ∞ 𝜌2 𝑒 −𝜌 𝐿⃗−3 𝑑𝜌 = 𝐿⃗−3 ∫0

4 √𝜋

2

𝜌2 𝑒 −𝜌 𝑑𝜌 = 1𝐿⃗−3

Since the intuition in this figure assumes that all conditions are normalized (1= |𝐿⃗±3 | = |𝜔 ⃗̂ | = |𝜔 ̂| ≡ 1= ℏ = 𝑐 2 ) then we equivalent in the classical picture conclude that 𝑚𝜔 =1 as a angular initial ‘mass’ in a circling ring at medium radius ⃗ ±3 |). 𝑟 = 𝜌̅ = 1 from (3.66) corresponds to a moment of inertia 𝐼3,𝜔 = 𝑚𝜔 𝑟 2 = |𝐿⃗−3 |⁄𝜔 = 1. (1= 𝑟 = 𝑚𝜔 = 𝐼3,𝜔 = |𝐿 The illustrative idea with the classic image of the angular momentum viewed as a plane rotating disc or ring with an equivalent mass, giving a moment of inertia and kinetic energy (normed as = 1), around in the transversal plane. In the classic view there is no action out from the plane. A contradiction to this, the created quantum circle oscillation produce a helix field through the passed displayed by a phase-parameter |𝜙| = |𝜔 ̂|𝑡. I would try to express the picture that the past ̂ ⃗ press the transversal plane FORWARD in the direction 𝜔 ⃗ = 𝟏 and that the kinetic-energy~momentum~power~1; with the ̂ ⃗ ±3 |=1=ℏ=𝑐 2 ). – This has the external quantity ⃗⃗ | =|𝐿 angular momentum 𝐿⃗±3 as a ‘motor’, internal autonomous norm (|⃗𝜔 ℏ𝜔. December 2020 Volume I, – Edition 1, – Revision 3,

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 85 –

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.3. Excitation of the Plane Harmonic Circle Oscillator – 3.3.4.4. Polar Radial Distribution of the Angular Momentum Over the Circular Oscillator Plan

We note that the excited state has radial density distribution from (3.143) (3.144) as (3.173)

pd1 (𝜌) =

1 2

1

2 4 𝜌𝑒 −2𝜌 , for ∀𝜌 ≥ 0

2𝑟̃ (𝜌) =

√𝜋

∞

∞ 4

2

The excited state normalized as (3.150) ∫0 𝜓±∗ (𝜌)𝜓± (𝜌)𝑑𝜌 = ∫0 𝜋 𝜌2 𝑒 −𝜌 𝑑𝜌 = 1 √ From here we infer that the action of the angular momentum is radial distributed as (3.174)

2

4

|pd1 (𝜌)|2 =

|𝜓± (𝜌)| =

√𝜋

𝜌2 𝑒 −𝜌

2

integrated along the rings with radius 𝜌 in the transversal plane as indicated in Figure 3.10. 3.3.4.5. The Area Distribution of the Action Over the Transversal Plane

The activity is spread across the transversal plane, radially, per area element we have (3.175)

2𝜋

𝑑𝐴 = 𝜌 𝑑𝜃 𝑑𝜌 = 2𝜋𝜌𝑑𝜌 ,

because ∫0 𝑑𝜃 = 2𝜋

from the angular symmetry.

A forward-flowing volume element at the radial coordinate 𝜌 is then (3.176) 𝑑𝑉 = 𝜌 𝑑𝜃 𝑑𝜌 𝑑|𝜙| = 𝜌 𝑑𝜃 𝑑𝜌 |𝜔 ⃗̂ |𝑑𝑡 = 2𝜋𝜌 𝑑𝜌 |𝜔 ⃗̂ |𝑑𝑡. Thus, the radial factor 1⁄2𝜋𝜌, that multiplied at the radial distribution pf1 (𝜌) to get the area probability density function (3.177)

𝑝𝑑1 (𝜌)

pdA (𝜌, 𝜃) =

2𝜋𝜌

2𝑟̃ (𝜌)

=

2𝜋𝜌

1

=

and alternatively, the radial action |pf1 the intensity (the activity per unit area) (3.178)

|𝑝𝑓1 (𝜌)|2

(𝜌, 𝜃) =

2𝜋𝜌

1

=

4

2𝜋𝜌 √𝜋

1 2

1

( 𝜋𝜌 4

√𝜋 2 (𝜌)|

1 1

𝜌 𝑒 −2 𝜌 ) =

4

𝜋 √𝜋

1 2

𝑒 −2 𝜌

for ∀𝜌 > 0

multiplied by 1⁄2𝜋𝜌 to provide

2

𝜌2 𝑒 −𝜌 =

2 1 𝜋 √𝜋

𝜌𝑒 −𝜌

2

for ∀𝜌 > 0

The radially dependent transversal distribution of the intensity is illustrated in Figure 3.11.

Figure 3.11 The intuition of the first excited state of the circle oscillator is based on the radial distribution function 𝑟̃ (𝜌) =

1 2

1 4

√𝜋

𝜌𝑒 −2𝜌 ∈ ℝ, for ∀𝜌 ∈ ℝ with the odd balance 𝑟̃ (𝜌) = − 𝑟̃ (−𝜌) for ∀𝜌 ∈ ℝ and thus a

parity inversion difference 𝑟̃ (𝜌) − 𝑟̃ (−𝜌) which drives the excitation positive forward direction indicated by ⃗𝜔 ⃗⃗̂ . The area density of this state distributed rotation symmetrically over the transversal plane is then for ∀𝜌 ≥ 0 pdA (𝜌, 𝜃) = (𝜌, 𝜃) =

𝑟̃ (𝜌)

=

𝜋𝜌 |pf1 (𝜌)|2 2𝜋𝜌

1 2 1 1 𝑒 −2𝜌 . But 𝜋 4√𝜋 2 1 −𝜌2

=

𝜋 √𝜋

𝜌𝑒

,

the action of the excitation seen as having an area intensity for ∀𝜌 ≥ 0 (3.178).

The area distribution of the transversal intensity is shown in Figure 3.12. Here the frequency energy is auto-normalized 𝜔 = |𝜔 ⃗̂ | = |𝜔 ̂| ≡ 1 , (its own reference). When we integrate the action of the excited area intensity (𝜌, 𝜃)𝜔 ⃗̂ of a momentum of the ⃗ we get the active power transversal plane with the direction 𝜔 ⃗̂ = ⃗𝟏 (3.179)

©

2 2 2𝜋 ∞ ∞ ∞ 4 2 1 ⃗̂ = 2𝜋𝜔 ⃗̂ ∫0 𝑑𝜌 𝜋 𝜋 𝜌 𝑒 −𝜌 = 𝜔 ⃗̂ ∫0 𝜋 𝜌2 𝑒 −𝜌 𝑑𝜌 = 1𝜔 ⃗̂ = 𝜔 ⃗̂ ∫0 𝑑𝜃 ∫0 𝑑𝜌 (𝜃, 𝜌)𝜔 √ √

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 86 –

A Research on the a priori of Physics –

December 2020

– 3.3.4. The Possible Excitation of a Circular Oscillator with ± Signed Orientation. – 3.3.4.6 The Energy Intensity Momentum –

As just one quantum ⃗𝟏 of ⃗ ±3 = ±1 𝜔 ⃗. • Angular momentum 𝐿 ⃗̂ = ±𝟏 ̂ • Angular frequency energy |𝜔 ⃗ | as one quantum of kinetic energy. • Power as flowing energy 𝜔 ⃗̂ ⁄|𝜔 ̂| from the paste, measured per unit |𝜔 ̂| = |𝜙|⁄𝑡 of the information development dimension, as FORWARD towards the future. • Momentum as we shall see, this can be interpreted as line direction momentum through space.

Figure 3.12 Here the intuition of the transversal area intensity (𝜃, 𝜌) of the progressive circle oscillating rotation induced by the unitary symmetry ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈𝑈(1) | ∀𝜃 ∈ℝ} in the transversal plane is designed as a magenta circle ring ⊙ determining the symmetry of the excitation 𝜔 ⃗̂ ⊥⊙. The excitation is thus a creation from ̂ ⃗⃗⃗ |𝑡 𝑖𝜃 ±𝑖|𝜔 the direction 𝜔 ⃗̂ for the unitary rotational symmetry ⊙𝜔 as active form the transversal plane to 𝜔 ⃗̂. The ⃗̂ ⃗⃗ ≈ 𝑒 𝑒 rotation oscillation is driven by the quantity 𝜔 with the qualitative direction 𝜔 ⃗̂ through the transversal plane 𝜌𝑒 𝑖𝜃 . Here in this figure 𝜔 = |𝜔 ̂| ≡ 1, with a progressive excitation 𝑚 = +1 illustrated as a unitary (𝜌=1) helix cylinder stretching out in the past |𝜙| = |𝜔 ̂|𝑡 , with a positive development parameter 𝑡 = |𝜙|⁄|𝜔 ⃗̂ |. Both the progressive 𝐿⃗+3 = ⃗𝜔 ⃗⃗̂ as here, and the retrograde 𝐿⃗−3 = −⃗𝜔 ⃗⃗̂ excited circle oscillation are intuited as a forward 2 2 1 flowing energy with an area distribution of intensity (𝜃, 𝜌) = 𝜌𝑒 −𝜌 , only with radial dependence and angular 𝜋 √𝜋 symmetry. The excited circle oscillator around 𝜔 ⃗̂ draw an intensity with the transversal plane. – Comment: The reason for the drawing of the relative flatness of the intensity is in the normed scaling (1=|𝜔 ⃗̂ | = |𝜔 ̂|≡1 = ℏ=𝑐 2 =1). ̂ Hereby 𝜔 ⃗ represents the flowing activity as both energy, power as well as momentum ‘motored‘ by the quantum ⃗ for the creation angular momentum 𝐿⃗±3 = ±1 𝜔 ⃗̂ . Overall, this is called the state momentum one quantum ℏ𝜔 ⃗̂ = ℏ⃗𝟏 † ̂ ⃗ ⃗ 𝑎±𝜔 of the circle oscillation direction quality 𝟏 ~ 𝟏 . ⃗⃗⃗̂ 3.3.4.6. The Energy Intensity Momentum

The quality energy eigenvalue 𝜔 in eigenvalue equation (3.166) is interwoven with the angular ⃗ +3 given from eigenvalue equation (3.167) dictates that the energy of a frequency direction 𝜔 ⃗̂ = 𝐿 frequency has a flow direction perpendicular to the angular rotation plane, a direction into future. This current of energy has an intensity flux, spread over the area of the transversal plane as (3.178) above, of which an area element develops into a volume element (3.176). The forward flowing area element through a development unit radian phase angle gives an effective volume element, and the integration of the intensity with (3.179) we get the forward flowing kinetic energy 1 2𝜋 ∞ ∞ ̂ (3.180) ̂|𝑑𝑡 ∫0 𝑑𝜃 ∫0 𝑑𝜌 (𝜌, 𝜃) 𝜔 ⃗̂ = |𝜔 ̂| 2𝜋 ∫0 𝑑𝜌 (𝜌, 𝜃)𝜔 ⃗̂ = |𝜔 ̂| ⃗𝜔 ⃗⃗ = 𝜔 ⃗̂ ∫0 |𝜔 by or through the transversal plane as illustrated in Figure 3.12 ̂ ⃗⃗⃗̂ is We see that a direction ⃗𝜔 ⃗⃗ is intimately linked to the transversal plane whose quality ⊙⊥𝜔 associated with the rotational symmetry of the rotation group 𝑈(1), (3.137) ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈ 𝑈(1) | ∀𝜃 ∈ℝ } . ©

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3.3.5. Frequency Scaling of the Circle Oscillator

In section 3.1.3 we went from the classic quantity 𝑞𝜔 of the oscillator (3.1) to field quantity 𝑞 by multiplying with √𝑚𝜔⁄ℏ defined (3.2), whereby we eliminated 𝑚 in (3.4), but has retained 𝜔 as the characteristic intrinsic quantity. At (3.170) we argued that the kinetic energy flowing FORWARD is equivalent to the kinetic energy rotating in a circle oscillation 𝑇𝜔 = ℏ𝜔~𝑚𝜔 𝑐 2 , as an effective internal angular inertia, which we call 𝑚𝜔 ~ ℏ𝜔⁄𝑐 2 . As we count ℏ = 𝑐 2 = 1, we get the multiplication factor 𝜔 ~√𝑚𝜔⁄ℏ . In this way by intuition we have one classic space-time-dimension as the quality, that possess the quantity 𝑞𝜔 ~ 𝑐 𝑞 ⁄𝜔 . For the radius dimension of the circle oscillator we therefore introduce the scaling transformation 1𝑐 (3.181) 𝜌𝜔 = 𝜔 𝜌. Here it will be necessary to define how we measure the angular frequency 𝜔. We need an angular frequency standard 𝜔0 as measured in a conventional frequency measure in combination with a timing standard, e.g. [Hz] ∼ [s−1 ]periode  1/second, as frequency standard. or alternative the frequency energy in [eV⁄(ℏ=1)] as usual in Particle Physics. 3.3.5.2. Examples of Commonly Used Reference Clocks Seen as One Circle Oscillator

The angular frequency norm for one second is one turn in a circle that lasts 2𝜋 seconds ~ 6.2831853[s] by measuring 𝜔 ̂[s −1 ] = 1[Hz⁄2𝜋] = 1[radian ⋅ s−1 ] for this clock that gives the development information parameter of one radian per second. The radius of this ideal simple reference circle oscillator as a clock is then (3.182) 1⁄𝜔 ̂[s−1 ] = 1[s]𝑐 = 299,792,458[m]. The normal circular 1 secund clock of 1[𝐻𝑧], 𝑓= 2𝜋𝜔 have an classical information radius 1𝑐 (3.183) radius1[𝐻𝑧] = 2𝜋[𝐻𝑧] ≈ 47,713,452 [𝑚]. The most common clock angular frequency unit used in quantum mechanics is electron volts, which has norm (3.184)

eV

𝜔 ̂[eV] = 1 [ ℏ ] = 0.2418 ⋅ 1015 [ Having a radius 𝑟̅𝜔̂⊙[eV] =

1𝑐 ̂ [eV] 𝜔

s−1 ℏ

1𝑐

] = 197.3[nm]ℏ ,

for 𝜔 ̂ = 1[eV] for ℏ = 𝑐 2 = 1 .

= 197.3[nm] and a wavelength 𝜆= 2𝜋𝑟̅𝜔̂⊙[eV] = 1239.7[nm],

which is an infrared wavelength for normal light (visible wavelengths are from 380-750[nm]). 3.3.5.3. The Relative Reference for the Circle Oscillator and the Autonomous Norm

In the relativistic quantum mechanics as above section 3.3.4, we most often set ℏ = 𝑐 2 = 1 . From an intrinsic frequency normal 𝜔 ̂, that by definition of course is normed 𝜔 ̂ ≡ 1 and a direction ⃗1ref , which magnitude is measured as 1 radian in 𝜔 ̂, i.e. ⃗ ref | = 1[radian] = 1[𝜔 (3.185) |1 ̂ −1 ]. From this, any frequency energy 𝜔 >0 conceivably excited as a circle oscillator with the rotation vector (  We hide negative frequency energies in −𝜔 = 𝜔− < 0.) ⃗ ref . (3.186) 𝜔 ⃗ = ±𝜔1 We can for one excitation choose between two different normalizations: ⃗̂ = 𝜔 ⃗̂ |=1, and ⃗ ref , where then |𝜔 • The frequency normal with any direction 𝜔 ̂1 ⃗ auto = 𝜔 ⃗. • Autonomous normalization for the excitation 𝟏 ⃗̂ , which therefore is one quantum 𝟏 This produces a juxtaposition of parallel interpretation 𝜔 ⃗ ∥𝜔 ⃗̂ , namely that ©

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(3.187)

⃗̂ref = 𝜔𝜔 ⃗̂ ↔ 𝟏 ⃗̂ = 𝜔 ⃗̂ref = ⃗1ref . ⃗ ref = 𝜔 𝜔 ⃗ ref = 𝜔 𝜔 ⃗ auto , 𝜔 ⃗̂ ↔ 𝜔 ⃗ = 𝜔1 ̂1 𝜔 We determine the excited entity 𝛹𝜔 to the angular frequency energy 𝜔 with our reference 𝜔 ̂, and get a relative ratio to the autonomous judgment |𝜔 ⃗̂ | ≡ 1 of the quantity 𝜔. The normal use of a frequency standard 𝜔 ̂, 𝜔 ̂ ≡ 1[𝜔 ̂], (e.g. (3.184)) for a excited entity 𝛹±𝜔 in physics with an angular frequency energy 𝜔, which fundamental quantum provides a direction quality 𝜔 ⃗̂ ≡ ⃗𝟏auto , which we scale by 𝜔 to a quantitative direction from the reference ⃗ ref that for us defines the objective direction of 𝛹±𝜔 . The magnitude of vector 𝜔 ⃗ = 𝜔1 ⃗̂ | = 1[𝜔 directional unit vector |𝜔 ̂ −1 ] is measured with the unit for the corresponding information development parameter 𝑡 = |𝜙|⁄𝜔. ̂ ⃗ auto , the polar (𝜌, 𝜃) Just as the direction in intuition can be viewed autonomous normed ⃗𝜔 ⃗⃗ ≡ ⃗𝟏 radius coordinate 𝜌 is viewed as isotropic in the transversal plane with this unit norm. Thereby the 1𝑐 radial coordinate is scaled from the unitary rotation per (3.181) 𝜌𝜔 = 𝜔 𝜌

The unitary circle group ⊙= {𝜃→𝑒 𝑖𝜃 |∀𝜃 ∈ℝ} then has a radius in an everyday measure ⊙ 299792458[𝑚𝑠 −1 ] 1𝑐 𝜆 (3.188) 𝑟̅ (𝜔 ⃗)= 𝜔 = = 2𝜋 , ⊙⊥𝜔 ⃗ , 𝜔 = |𝜔 ⃗ |. 𝜔[𝑠−1 ] This is expressed from the measure of an angular frequency energy quantity 𝜔[𝜔 ̂] . We then have, that the fundamental radius of the excited physical entity 𝛹±𝜔 ⃗⃗⃗ is 1𝑐 𝜆 −1 = 2𝜋 measured with the unit [𝜔 ̂ ] for the information development parameter 𝑡. 𝜔 ⊙ Hence, the speed around the transversal unitary circle ⊙ is just: 𝑐⊙ = 𝜔𝑟̅ (𝜔 ⃗ ) = 𝜔 𝑐 ⁄𝜔 = 1𝑐. 3.3.5.4. Scaling of the Frequency Energy in The Propagation

The relationship between energy and the angular frequency is ℏ, that often is set to one ℏ=1. ⃗ auto of direction We again look at a single quantum 𝟏 ⃗ ±3 = ±ℏ𝜔 ⃗ auto from the transversal plane. • A quantum of angular momentum 𝐿 ⃗̂ = ±ℏ𝟏 ⃗̂ | with direction 𝜔 ⃗̂ . • A quantum of angular frequency energy ℏ𝜔 = ℏ|𝜔 ⃗ | = ℏ𝜔|𝜔 ⃗̂ ⁄𝜔 ⃗̂ = ℏ⃗𝜔 • A quantum of power as flowing energy into the future ℏ ⃗𝜔 ⃗⃗ ⁄𝜔 ̂ = ℏ𝜔 𝜔 ̂ = ℏ𝜔𝜔 ⃗⃗ , −1 measured per unit [𝜔 ̂ ] of the development parameter 𝑡 = |𝜙|⁄𝜔, which produces an information dimension into the past. (A quantum of power for each creation) ℏ • And we will see, this is also just one quantum of momentum 𝜔 ⃗ throughout space. 𝑐 In all we have the FORWARD momentum with the quality one direction, all parallel, ⃗̂ ∥ 𝜔 𝜔 ⃗ ∥ 𝜔 ⃗̂ = ⃗𝟏 ; but this look different in different unit systems. Later below in chapter II. 4.4 etc. we call such a direction a primary quality of first grade (pqg-1) or a 1-vector direction, what in the meaning of René Descartes would call extension space direction.

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3.4. The Quantum Excited Direction We have here in section 3.3.4 seen that a direction 𝜔 ⃗̂ is linked to the transversal plane whose ⃗⃗⃗̂ is associated with the rotational symmetry of the unitary rotation group 𝑈(1) quality ⊙⊥𝜔 ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈ 𝑈(1) | ∀𝜃 ∈ℝ } (3.137) 3.4.1. The Direction of the First Excitation Described in Cylindrical Coordinates.

In cylindrical coordinates type (𝜌, 𝜑, 𝑧), we can write the angular frequency vector as ⃗̂ ↔ (0,0, 𝜔) = 𝜔(0,0,1), in which the first two coordinates (0,0)→(𝜌, 𝜃)~(𝜌𝜔 , 𝜃) 𝜔 ⃗ = 𝜔𝜔 describe the transversal plane to the vector 𝜔 ⃗ . Looking at the coordinates for the development of the excitation we have, as the new idea, the cylinder axis coordinate: (3.189) 𝑧 = −𝑐𝑡 = −𝑐 |𝜙|⁄|𝜔|, ⃗̂ ↔ (0,0,1), where the opposed development that we have chosen positive out in the direction 𝜔 ⃗̂ . The angular parameter 𝑡 is defined positively as a manufactured past in the direction −𝜔 coordinate is written 𝜑 = 𝜃±𝜔𝑡. In particular, we have the active angle 𝜙=𝜔𝑡 and the start basic angle is 𝜃 for 𝑡 = 0. These two coordinates (𝜑, 𝑧) ↔ (𝜙, 𝑧) follows the information development parameter 𝑡. The radial distribution by 𝜌 is independent of 𝑡. We note the rewriting 𝜌 = 𝜔𝜌𝜔 ⁄𝑐 , therefore, I describe the following options for coordinate arguments for the wavefunction for the excitation of the circle oscillator development 𝜌 𝜌 𝑐 𝜌⁄|𝜔| 𝜌𝜔 𝑐 𝜌⁄|𝜔| 1𝑐 (𝜃) ⊙ ±𝜙 ±𝜙 ) ~ | | 𝜓 ⃗⃗⃗ ( ) ↔ 𝜓±𝜔 ( ). (3.190) 𝜓±𝜔 ⃗⃗⃗ (𝜑 ) = 𝜓±𝜔 ⃗⃗⃗ (𝜃 ± |𝜔|𝑡) = 𝜓±𝜔 ⃗⃗⃗ ( 𝜃 ± 𝜙 ⃗⃗⃗̂ 𝜔 ±ω𝜔̂ −|𝜙| −|𝜙| 𝑧 −𝑐|𝜙|⁄|𝜔| −𝑐𝑡 As an example, I define the entity

AB

𝛹±𝜔 ⃗⃗⃗ by creation in event A, to annihilation at B,

𝜌 (3.191)

AB

𝛹±ω𝜔⃗⃗̂

= { 𝜓±ω (𝜃 ± ωt) ∈ ℂ | ∀𝜌 ∈ [0, ∞[, −|𝜔|𝑡 ⃗⃗𝜔̂

∀𝜃 ∈ [0,2𝜋[, ∀𝑡 ∈ [𝑡A , 𝑡B ] ⊂ ℝ }

I also by intuition consider the entity as a complex scalar field as a function of these cylindrical coordinates and maintains, that the angular coordinate in dualism for intuition; where 𝜙 = 𝜔𝑡 evolves, and 𝜃 is eternal constant distributed on [0,2𝜋[ as a basic background for 𝜙 = 𝜔𝑡, joined in this duality by the resulting angular phase 𝜑 = 𝜃+𝜔𝑡 for a full description. From the formulas (3.148)-(3.149) and (3.163) I write this as the complex scalar field in the excitation transversal plane ⊙⊥𝜔 ⃗ (3.192)

1 2

1

(𝜌,𝜃)

𝜓±𝜔 (𝜙)= 𝑎⊙†±𝜔 |0,0⟩ = 2𝑟̃ (𝜌)⊙𝑒 ±𝑖𝜙 = (2 4 𝜌𝑒 −2𝜌 𝑒 𝑖𝜃 ) 𝑒 ±𝑖𝜙 ∈ℂ, for ∀𝜌 ≥0, ∀𝜃 ∈[0,2𝜋[, ∀𝜙 ∈ℝ, √𝜋

and then expressed by an information development parameter 𝑡 = |𝜙⁄𝜔 |, with the transversal radial rewriting 𝜌𝜔 = 𝑐𝜌⁄𝜔, we now for ∀𝑡 ∈ℝ get (3.193)

𝜔𝜌𝜔

(𝜌

,𝜃) † 𝜓±𝜔𝜔 (𝑡) = 𝑎⊙ ±𝜔 |0,0⟩ = 2𝑟̃ (

𝑐

1 𝜔𝜌𝜔

) ⊙𝑒 ±𝑖𝜔𝑡 = (2 4

√𝜋

𝑐

1 𝜔𝜌𝜔 2 ) 𝑐

𝑒 −2 (

𝑒 𝑖𝜃 ) 𝑒 ±𝑖𝜔𝑡 ∈ ℂ ,

The parameter independent factor for this complex scalar field is symbolised by (3.194)

◎

↔

1

1 2

(2 4 𝜌 𝑒 −2𝜌 𝑒 𝑖𝜃 ) √𝜋

↔

1 𝜔𝜌𝜔

(2 4

√𝜋

𝑐

𝑒

1 𝜔𝜌𝜔 2 ) 2 𝑐

− (

𝑒 𝑖𝜃 )

↔

|𝜓⟩

Thus, hiding the plane coordinates (𝜌, 𝜃) in a classical transcendental substance ◎. What we a priori know about this subject, is that it is transversal ◎⊥𝜔 ⃗̂ . Here it is noted that the idea is that the scale of the isotropic symmetric radial 𝜌 in the transversal ◎ refers to the autonomous norm 1 ≡ |𝜔 ⃗̂ | . ©

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The phase angle parameter dependent version can then be written (compare (2.62)) ⊙ ⊙ (𝜙) ~ 𝜓±𝜔 (𝜙) = 𝜓±⊙ (𝜔𝑡), ̃ 𝜓±𝜔 (3.195) 𝑒 ±𝑖𝜙 ◎ → or traditional written |𝜓(𝑡)⟩= 𝑒 ±𝑖𝜔̂𝑡 |𝜓⟩. ⃗⃗⃗̂ This is involved a 'new' kind of polar cylinder coordinate system (𝜓, 𝑧) ∈ ℂ, ℝ , where 1. 𝜓 ∈ ℂ describe the complex transversal plane and 2. 𝑧 ∈ ℝ describes the linear real axis of the cylinder system This make me rewrite the idea in (3.190) to ⊙ 𝜓 ⊙⃗⃗⃗̂ (𝜙) 2𝑟̃(𝜌) ⊙ 𝑒 ±𝑖𝜙 𝜓 ℂ (𝜔𝑡) ̃ (𝜓±ω ) ↔ ( ±𝜔 ) ~ 𝑒 ±𝑖𝜙 ◎⊥ 𝑐|𝜙|𝜔 (3.196) ( )∈( ) → ) ~( ⃗̂ 𝑧 ℝ −|𝜙| −𝑐𝑡 −|𝜙| ⃗⃗⃗ 𝜔 ̂ ⃗⃗⃗̂ 𝜔 ⃗⃗⃗ 𝜔

⃗ ~ ( 0) ∈ ( ℂ ) Here the autonomous normed angular frequency energy has a unit vector 𝜔 ⃗̂ = 𝟏 ℝ 1 for the direction of development, – a propagation – and the axis { 𝑧 = 𝑐|𝜙|𝜔 ⃗̂ | ∀𝜙 ∈ℝ } of ‘the cylinder'. From this we write (3.191) seen from external in the form AB ±𝑖𝜙 (3.197) 𝛹±𝜔 ⊥𝑐|𝜙|𝜔 ⃗̂ | ◎⊥𝜔 ⃗̂ , ∀𝜙 ∈[𝜙A , 𝜙B ] ⊂ℝ }, existence from A to B. ⃗̂ ⃗⃗ = { ◎𝑒 It is up to the reader to interpret the implications of the formulas (3.190)-(3.197) for physics.

Figure 3.13 The intuition of causality as a primary quality: The in A created entity AB𝛹±𝜔 ⃗⃗⃗̂ , is causally annihilated in B. ̂ ⃗⃗⃗⃗⃗ |𝜔 The development direction AB = 𝑧AB = −|𝜙AB ⃗ is indicated in the reading direction, while the development parameter 𝑡 ≡ |𝜙|⁄𝜔 ̂ has the opposite orientation. The autonomous wavelength is 𝜆̂ = 2𝜋, when |𝜔 ⃗̂ | ≡ 1. (Here is illustrated 2 full periods |𝜙B − 𝜙A | = 2⋅2𝜋).

It is well known from Einstein's theory of relativity that time parameter 𝑡 has a relativistic measure, therefore I prefer besides lights-peed-norming 𝑐=1 and angular frequency to energy standard ratio ℏ=1 to auto-norm the angular frequency |𝜔|=| 𝜔 ⃗̂ |=|𝜔 ̂|=1, so that the development parameter just give the evolution of the angular phase |𝜙|=|𝜔 ̂|𝑡 of the excited state. In short, we set 𝑡 ≡ |𝜙| in the auto-norm picture, hence 𝑥3 = |𝜙| from the extension idea (3.198) 𝑥3 ≡ 𝑧 = −𝑡 = −𝑐𝑡 , when 𝑐=1 , ℏ=1, and |𝜔| = 1. These aspects of the quality a direction from (3.197) is illustrated Figure 3.13, where the intuition is confined to the unitary circle group ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈ 𝑈(1) | ∀𝜃 ∈ℝ }, and we disregard the radial distribution ◎ and concentrate our idea to a rotating unitary helix (like a cylinder). ℂ 𝑒 ±𝑖𝜙 𝑒 𝑖𝜃 𝑒 ±𝑖𝜙 AB )~ ⊙( ) ∈ ( ) | ⊙⊥𝜔 (3.199) {( ⃗̂ , ∀𝜙 ∈[𝜙A , 𝜙B ] ⊂ℝ }. ⊙𝛹±𝜔 ⃗⃗⃗̂ = ℝ −|𝜙| −|𝜙| If we take a classic point of view, we must of causal reasons establish the reference system at the annihilation event B, as, what we call a measurement only can appear here. What can be measured 1 or counted is in principle the number of oscillations 2𝜋 |𝜙B − 𝜙A |. In a traditional interpretation, the real measure has been accepted. Here, the autonomous norm measure is into the past from B, back to the creation A, defined as the extension in a reality from B back to A.100 ©

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The reference of this quantity is the auto-norm direction 𝜔 ⃗̂ as a primary quality. The 'measure mechanics' of this substance is the ‘by us’ contrived helix ±1 of the unitary circle group ⊙= {𝜃→𝑒 𝑖𝜃 |∀𝜃 ∈ℝ}. The quantity of the subject can of course not be measured directly but is an autonomous capacity of the subject as an idea of the existence of the extension.

100

Comment: The idea for us of a subject entity AB𝛹±𝜔 ⃗̂ ⃗⃗ in physics that possess extension, is essential for the idea of natural space in physics. Due to I. Kant: Space itself can never be an object for our intuition. Properly we can promote the idea AB𝛹±𝜔 ⃗⃗⃗̂ to some object for us (das Ding für uns) and make some measurement in space. ⊙

⊙

3.4.1.2. Annihilation of an Excited Circle Oscillator

The excited circle oscillator (3.163) written without the 2 factor,101 by including negative 𝜌, for ∀𝜌 ∈ ℝ\{0} we write (remembering 𝜙 = 𝜔𝑡)

(3.200)

𝜕

𝑖 𝜕

1

1 2

† |0,0⟩⊙ = 𝑒 ±𝑖𝜙 (𝜌 − ∓ 𝜓±𝜔 = |1, ±1⟩𝜔 = 𝑎⊙±𝜔 ) |0,0⟩⊙ = [ 4 𝜌𝑒 −2𝜌 ⊙𝑒 ±𝑖𝜔𝑡 ] 𝜕𝜌 𝜌 𝜕𝜙 ⊙

⊙

√𝜋

∀𝜌∈ℝ

The parity inversion problem we incorporated in the following by considering all proper real radial coordinates as a representation of the odd function dipolar opposition, see (3.120). When the annihilation operator works on the first excitation, we get the ground state back † 𝑎⊙±𝜔 𝑎⊙ ±𝜔 |0,0⟩ = 𝑎⊙±𝜔 |1, ±1⟩𝜔 = ⊙

(3.201)

1 1

= 2 4 (𝜌2 𝜌2 +1+1 )𝑒 √𝜋

1 2

− 𝜌2

𝑒 ∓𝑖𝜙 2

𝜕

𝑖 𝜕

1

1 2

(𝜌 + 𝜕𝜌 ∓ 𝜌 𝜕𝜙) 4 𝜌𝑒 −2𝜌 ⊙𝑒 ±𝑖𝜙

⊙𝑒±𝑖𝜙 𝑒∓𝑖𝜙 =

√𝜋

1

1⋅ 4 𝑒 √𝜋

1 2

− 𝜌2

1

⊙𝑒±𝑖𝜔 𝑡 𝑒∓𝑖𝜔𝑡 =

4

√𝜋

1 2

𝑒 −2𝜌 ⊙.

† |0,0⟩ The creations of excitation A|1, ±1⟩𝜔 = A𝑎±𝜔 B A|1, B A † |0,0⟩ Annihilation 𝑎±𝜔 ±1⟩𝜔 = 𝑎±𝜔 𝑎±𝜔 =

• Having an event A: and |0,0⟩ • subsequent event B: back to the ground state, we have produced a quantity, |𝜙B − 𝜙A | = |𝜔|(𝑡B − 𝑡A ) , (3.202) which gives the difference between the events A and B. In the autonomous image with autonomous norm |𝜔 ⃗̂ AB |=1 we call the phase angle development |𝜙B − 𝜙A | = |𝜔 ⃗̂ AB |(𝑡B − 𝑡A ) a measure of the autonomous time of the entity AB𝛹±𝜔 ⃗̂ ⃗⃗ . Here we recall that the angular frequency energy 𝜔 is given from an external norm 𝜔 ̂ as reference standard. Then, the external extension from A to B is expressed as (3.203) 𝑧AB = −𝑐(𝑡B − 𝑡A ) = (−𝑐 |𝜙B − 𝜙A |⁄|𝜔|| ) [𝑐𝜔 ̂ −1 ]. ⊙

This is measured by our external development parameter 𝑡 = |𝜙|⁄|𝜔| [𝜔 ̂ −1 ], in that the internal phase angle is counted with the external angular frequency 𝜔[𝜔 ̂] of the circle oscillator AB𝛹±𝜔 ⃗̂ ⃗⃗ . 3.4.1.3. Change of Direction of an Excited Circle Oscillator

In another way, we have an excited state AB|1, ±1⟩𝜔 with a direction 𝜔 ⃗̂ AB, and allows a † B ̂±𝜔 = B𝑎±𝜔 combined sequential annihilation- and creation-operation event B𝑁 𝑎±𝜔 AB to act on the entity 𝛹±𝜔 in a penetration through the ground state at B back to a new excited BC|1, state ±1⟩𝜔 with a new direction 𝜔 ⃗̂ BC . Assuming the same angular frequency energy 𝜔 measured by an external reference 𝜔 ̂0 with the norm |𝜔 ̂0 | = 10 as at (3.187) (3.204) 𝜔 ⃗̂ AB ↔ 𝜔 ⃗ AB = 𝜔AB⃗1AB = 𝜔AB 𝜔 ̂0⃗1AB and 𝜔 ⃗̂ BC ↔ 𝜔 ⃗ BC = 𝜔BC 𝜔 ̂0⃗1BC ⃗ XY |= 10 [𝜔 An arbitrary directional basis vectors ⃗1XY has a magnitude |1 ̂0−1 ] of the extension in ⃗̂0 | = 10 [𝜔 relation to the angular frequency reference, 𝜔 ̂0 , that is |𝜔 ̂0 ]. Assuming that there is preservation of frequency energy in the event B, then, that 𝜔BC = 𝜔AB 100 101

A depth is dependent on the transversal (length and breadth) ⊙, as essential for the concept of extension by Descartes. The parity factor 2 occurs when the polar radius coordinate is restricted to positive values 𝜌 >0. (See Section 3.3.1).

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– 3.4.1. The Direction of the First Excitation Described in Cylindrical Coordinates. – 3.4.1.5 The Substance of the Concept of a Photon –

and thus |𝜔 ⃗ BC |=|𝜔 ⃗ AB |. If here 𝜔 ⃗ BC ≠ 𝜔 ⃗ AB , there is a change in B of the direction102 for the entity ABC Ψ±𝜔 . On the other hand, is 𝜔 ⃗ BC = 𝜔 ⃗ AB, the event B become irrelevant for the AC entity 𝛹±𝜔 ⃗⃗⃗ as one and the same direction. As frequency energy 𝜔BC = 𝜔AB can be conserved and the direction is changed in B, we

⃗⃗⃗̂ BC . conclude that the angular momentum change direction ℏ𝐿3AB ≠ℏ𝐿3BC ⇔ ±1𝜔 ⃗̂AB ≠ ±1𝜔 We see that the direction is embedded in the angular momentum as a result of the idea an angular frequency, producing rotation in a plane whereby we get a circle oscillator, which is transversal to the direction with orientation + for forward, and − for the past. This is combined with + for progressive, and − for retrograde rotation. This bipolar orientation of angular momentum defines the transversal plane with direction. We have the concept of a difference between A and B with two orientations AB and BA. My claim is that for the possibility to distinguish two events A and B we shall be able to recognize a transversal plane between them. Yes, it should similarly be possible by intuition to conceive an entity AB⊙𝛹±𝜔 ⃗̂ ⃗⃗ as illustrated in Figure 3.13, which can be used to measure the difference between A and B. – If the subject can be promoted to an object one achieve extension. ±

±

3.4.1.4. The Fundamental Substance for an Entity and the Extensive Difference

By intuition we look at the difference between the two events A and B and write the symbolic expression for the creation of an entity for this AB difference (3.205)

† |0,0⟩. 𝛹 ↔ B𝑎⊙±𝜔 A𝑎⊙±𝜔

AB

The causality for this intuition of the difference presupposes the possibility of excitation of a harmonic circle oscillation transversal to the difference AB with a frequency energy ℏ𝜔 and angular momentum ±ℏ𝜔 ⃗̂ AB . A measurement of such an extensive difference as the number of oscillations will depend on 𝜔[𝜔 ̂] as reference and the measured extensive quantity will then be [𝑐𝜔 expressed by (3.203) 𝑧AB = −𝑐 |𝜙B − 𝜙A |⁄|𝜔| ̂ −1 ]. 3.4.1.5. The Substance of the Concept of a Photon † |0,0⟩ may appear to be associated with a photon which The expression AB𝛹±𝜔 ↔ B𝑎⊙±𝜔 A𝑎⊙±𝜔 object is created and transmitted from the event A with the speed of light and is received and annihilate in B. However, above, we have described a more substantial structure for a difference AB illustrated by Figure 3.13. Therefore, I will introduce the term a subton for this space-time subject. Note that, the conventional space-time 'light-cone’103 is wound up as a spiral helix in a cylinder along |𝜙|= 𝑥3 in Figure 3.13. Thereby it is possible to understand by graphical intuition how simple quantum mechanics performs in space-time. The subton as one quantum excitation count by autonomy its own internal phase angle direction driven external, by its external angular frequency energy 𝜔[𝜔 ̂] as its given quantity. This internal counting measure of the external extension quantity is translated to a space-time measure expressed by the formula (3.203). Annihilation B of a subton under the premise of a prior creation A gives the concept of phase angle plane direction coherent locality transversal propagation extension in space-time.104

† B ̂±𝜔 = B𝑎±𝜔 The change of direction in B requires interaction B𝑁 𝑎±𝜔 with the environment of the entity ABC Ψ±𝜔 . 103 The space-time 'light cone' (Minkowski) is usually defined as { (𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) | 𝑐 2 𝑡 2 = 𝑥12 +𝑥22 +𝑥32 } in Cartesian coordinates, but here in the polar cylinder coordinates is it { (𝑡, 𝜌, 𝜑, 𝑥3 ) | 𝑐 2 𝑡 2 = 𝜌2 +𝑥32 } where 𝜌2 = 𝑥12 +𝑥22 with ⟨𝜌⟩=1. As we view the unit cylinder surface ⃗⃗⃗⃗ ⊙, we get { (𝑡, 1, 𝜙+𝜃, 𝑥3 ) | 𝑐 2 𝑡 2 = 𝑥32 = (𝜙 ⁄𝜔)2 , 𝜑= 𝜙+𝜃 } where we have removed 𝜌=1 on the unit cylinder surface as shown in Figure 3.13. As seen later Figure 5.55 and page 334 that this cylinder it is a null helix curve. 104 This is the idea of something real material: extensa (Descartes), in nature (Spinoza) or in physics for us. This is built on the a priori idea of direction by Immanuel Kant. Now it is an issue for phase information in a received signal from a transmission. 102

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.4. The Quantum Excited Direction –

3.4.2. The Linear Movement 3.4.2.1. The Concept of the Straight Line, through what?105

We now look at the classic Cartesian space with length, breadth, and depth (𝑥1 , 𝑥2 , 𝑥3 ). I look at the meaning of the qualitative concepts: length, breadth, and depth. First, I claim, the a priori synthetic judgment; that the depth belongs to the past and define future as the positive direction toward a receiver. We have the depth (𝑡3 ) as is seen from the receiver, and its measure as from creation FORWARD to annihilation. The transversal is so wide (for us the horizontal) and breadth (for us the vertical height). If we disregard gravity, these two terms are similar (width and breadth is just that which is across), these (𝑥1 , 𝑥2 ) are transversal to a receiver (the viewer) that looks at the transmitter (the object). This means that in the free space (without gravity) the polar coordinates (𝜌, 𝜑) are a better representation of the transversal concept, than the Cartesian (𝑥1 , 𝑥2 ). The coordinates for the view (𝜌, 𝜑, 𝑥3) ↔ (𝑥1, 𝑥2, 𝑥3) .106 Thus, we introduce the concept, that the transversal quality is a rotation represented by the complex number 𝜌𝑒 𝑖𝜑 (instead of linear displacements in Cartesian coordinates)107. Concrete examples of such transversal planes are objects like paper, a screen or a wall tabula for your intuition, or the subject the hemisphere (the stars) as a substance that really rotates relatively in the depth of the deep celestial sky over you. Now we jump right through the transversal defining the primary quality of the substantial concept direction from event A to event B. An entity for such direction subject can be: To define ⃗ 3 ~(0,0, 𝐿3 ) ∈ℝ3 and 𝜔 vectors as objects with the coordinate connections 𝐿 ⃗ ~(0,0, 𝜔3 ) ∈ℝ3 . From the formulation of classical mechanics, we have, 𝐿3 = 𝐼3 𝜔, where 𝐼3 is a factor for moment of inertia. See (3.63), (3.66). Apparently, the angular momentum 𝐿3 is proportional to the angular frequency 𝜔. But quantum mechanical interpreted, a circle oscillation is excited with an angular momentum operator 𝐿̂3 according to (3.63)→(3.66)→(3.88)→(3.104)→(3.110)→(3.167) 𝐿̂3 |1, ±1⟩ ≐ ±1ℏ|1, ±1⟩ has double orientated eigenvalues ±1ℏ that are scalar independent of 𝜔. This fundamental independence of quantities is interpreted as the pure primary quality of one direction, that as in (3.168) and (3.171) is represented of ⃗ +3 = −𝐿 ⃗ −3 = 𝟏 ⃗ ~ 𝐿̂3 where |𝐿 ⃗ ±3 | =1. (3.206) 𝜔 ⃗̂ = +𝐿 By (3.172) we have that this direction is characteristic for every development through a transversal ̂𝜔 |1, ±1⟩ ≐ (1ℏ𝜔+ℏ𝜔)|1, ±1⟩ , (3.166) plane. With Hamiltonian eigenvalue equation 𝐻 𝜔 𝜔 we understand that the excitation of the circle oscillator is given by a quantity of the angular frequency energy, which represents precisely the rotation vector 𝜔 ⃗̂ ↔ 𝜔 ⃗ ~(0,0, ω) . This implies that we measure 𝜔[𝜔 ̂] in terms of our reference angular frequency 𝜔 ̂=1. ⃗ The unit vector 𝜔 ̂ ~(0,0,1[𝜔 ̂]) has the primary quality a direction of the axis of rotation. ⃗̂ ∼ (0,0, 𝜔[𝜔 This allows rotation of the vector 𝜔 ⃗ =𝜔𝜔 ̂]) in coordinates. (𝜔 = ±|𝜔| ∈ℝ)108. The spatial quantity of the extension is measured with the reference [𝑐𝜔 ̂ −1 ]. From this rotation vector direction, a unit vector is written 𝐞3 ~(0,0,1[𝑐𝜔 ̂ −1 ]) . ⃗̂ and 𝐞3 to have the same direction they in our Although we count the two unit vectors 𝜔 ontological intuition belong to two different scaling dimensions: ⃗̂ belong to the rotation axis measured by the angular frequency [𝜔 • 𝜔 ⃗ and 𝜔 ̂]. • 𝑡𝐞3 and 𝐞3 belong to the transmission measured by the development parameter 𝑡[𝜔 ̂ −1 ], 109 −1 which can be transformed to extension 𝑧=𝑐𝑡𝐞3 measured with the unit [𝑐𝜔 ̂ ]. 𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

105

The idea of time as a causal linear direction dated back to Augustine (354-430): Creation - Existence - Doomsday. 𝑇 The coordinate set (𝑥1 , 𝑥2 , 𝑥3 ) represents a column vector coordinates as the transposed of a row set (𝑥1 , 𝑥2 , 𝑥3 ). 107 The factor 𝜌 is not a dilation, but only a stochastic coordinate parameter of the radial distribution in the transversal plane. 108 We let the negative frequencies 𝜔 with a sequential subsequence reception of the individual phase angle signals 𝜃𝑚,𝑘 ∈[0,2𝜋[ from each double±subton 𝑚,𝑘 that can be individually read in B. More about 𝑡𝑚 modulation below. Thus, only single double±subtons which can be modulated with its own quantum mechanical angular phase 𝜃 ′ of (3.238) to be read individually. The idea the linear polarization produces a relative angular information in the creation of a double±subton of event A, which are then transferred to the annihilation of double±subton in event B. I would like to point out, if we can modulate one single double±subton couple with a phase angle difference 𝜃 ′ and read them again without being disturbed by thermal noise, we can phase modulate the transversal plane of a monochromatic plane wave at one single frequency 𝜔𝑘 .125 ⃗⃗⃗ The problem is, that the modulation is quantised to a single entity double±subton ABΨ𝜔𝟐𝑘 . We ask, is this an indivisible elementary-particle or merely a composite particle? For optical photons, room temperature is so low that single double±subtons pair can be detected. The ensemble (3.249) cannot be quantum coded because of simultaneous individual phase detection confusion in the permutation between the identical individualities of respectively + and − rotation orientations. All these individual phases are mutual indecipherable. Superposition of the linearly polarized double±subtons obviously has relevance to the concept of linearly polarized light, as it is an equal number of subtons of positive and negative helicity. 123

The reader, that will investigate this further can study Hilbert couples and transformation. (I don’t have the strength.) This view with transversal plane wave through one direction is also called a transmission channel. 125 One single double±subton is by definition given at one frequency. Opposite this, by ordinary radio technique (which for many years has been my livelihood) we used to say that the phase modulation is synonymous with frequency modulation in contrast to the monochromatic idea. But by radio modulation and communications the temperatures are so high that single photons cannot be registered individually and phase of the double±subton cannot be read. (ℏ𝜔≪𝑘𝑇). So, in the traditional radio wave communication neither this nor the monochromatic idea have relevance. Opposite for light double±photons ℏ𝜔 > 𝑘𝑇. 124

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.4. The Quantum Excited Direction –

Especially we want to write electromagnetic dipole radiation consisting of many 𝑛=1,2 … 𝑁 double±subtons the effect of the phase angles 𝜃𝑛 = 𝜃𝑑 +Δ𝜃𝑛 where any phase noise can be ̅̅̅̅̅ considered negligible mean Δ𝜃 ⃗⃗ 0 ⇒ 𝜃𝑑 ≈ ⃖⃗⃗ ̅̅̅ 𝜃𝑛 . Hence, that 𝜃𝑑 represents the angular 𝑛 ≈ 126 direction in space of the generating dipoles as a macroscopic common. Intuition of (3.249) with polarization direction 𝐞𝜃𝑑 ∥⊙⊥𝐞3 is then (3.250)

⃗

|𝜓𝟐Σ𝑁 ±⊙⊥𝜔𝒆3 (𝑡)⟩

e 𝜃𝑑

= 𝐴𝑁4𝑟̃ (𝜌)⊙ cos(𝜔𝑡+𝜃𝑑 )⊥e3 , for ∀𝜌 ≥ 0 and 𝐴𝑁=½, as ⟨𝜓|𝜓⟩=1.

This is apparently in line with a traditional way of describing linear polarization, but we must note that the paired quantum mechanical angles are the same 𝜃𝑑 as a macroscopicN angular direction in space, resulting from the simultaneous creation of (3.250) as a macroscopic cause. The direction of the causing 𝐞𝜃𝑑 (a oscillating dipole) controls the linear polarization of the macroscopic field with the amplitude 𝐸𝑎 ∈ℝ, the power |𝐸𝑎 |2 (energy flow) is proportional to the number N of subtons, that is, according to (3.236) |𝐸𝑎 |~ √𝑁, and we call the field amplitude of one subton±pair for 𝐸𝑑 , this field is then written127 (3.251) 𝐸⃗ = 𝐸𝐞𝜃𝑑 = 𝐸𝑎 cos(𝜔𝑡) 𝐞𝜃𝑑 = √𝑁𝐸𝑑 2 cos(𝜔𝑡) 𝐞𝜃𝑑 , ⊥e3. The common quantum pairs of the phase angle difference are now omitted and hidden in the dipole direction 𝐞𝜃𝑑 . The physical linear field polarisation lies in that direction 𝐞𝜃𝑑 is multiplied by a real number √𝑁𝐸𝑑 2cos(𝜔 𝑡) ∈ ℝ. The individual quantum phases of the ensemble (3.250) is now stored in the unitary circle group ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈𝑈(1)| ∀𝜃 ∈ℝ}, that here includes the transversal idea in our intuition.(3.250)(3.229)(3.232)(3.222) All subtons in from and is seen created in one direction according to the motto . This requires our intuition for the form of development ⊙⊥e3 → ⊙⊥𝜔e3 → ⊙⊥ 𝜔𝑡e3 ~ ⊙⊥𝜙e3 → ⊙⊥𝑥3 𝐞3 , that the direction 𝜔 ⃗̂ ∥𝐞3 is preserved into the future, given from the past. Looking at a particular ‘place'128 in ' time' 𝑥3 = 𝑐𝑡3 along the line 𝑥3 𝐞3, we get 𝜔𝑥 (3.252) 𝐸⃗ (𝑡, 𝑥3 )= √𝑁𝐸𝑑 2 cos (𝜔𝑡 − 𝑐 3) e𝜃𝑑 = √𝑁𝐸𝑑 2 cos(𝜔𝑡 − 𝑘3 𝑥3 ) e𝜃𝑑 , 𝑘3 =𝜔 𝑐 , e𝜃𝑑 ⊥e3. This is the traditional formula for a transverse field 𝐸⃗ (𝑡, 𝑥3 )∥e𝜃 ⊥e3 propagating along the 𝑑

direction 𝐞3 . For this intuition of the amplitude of the field it is again important to note, that this is essentially different than the transversal extension of the subton beam that is normalized by ⟨𝜓|𝜓⟩=1 to 𝐴𝑁=½ according to (3.250) from (3.188). Therefor intuited as a macroscopic beam with a transversal radius of the order129 ⊙

(3.253)

𝑐 = 𝜆 . 𝑟̅ (𝜔 ⃗)= 𝜔 2𝜋 The 'Beam' created from one point in one direction130 is by definition in this intuition straight as 𝜆 a rectitude line131 with this 'thickness'. (A laser beam, which aperture radius ≧ 2𝜋 of cause). By intuition we see (3.251) a signal field that can be modulated by changing the number 𝑁 of simultaneous created double±subtons. This modulation is called amplitude modulation, in that it changes field amplitude 𝐸𝑎 ∝ √𝑁, but be aware that it does not change the 'thickness' of the beam.

126

A dipole can be considered as a linear harmonic oscillator (in one dimension). It has the polarization direction along this in the transversal plane with the radiation direction perpendicular to this plane. 127 Here is the factor 2 retained even it could be hidden in field the factor 𝐸𝑑 ; to indicate dipole parity opposition + and −. 128 In the substance of this autonomous ontology a place is found by counting radians along the development axis, see Figure 3.13. 129 I have previously (3.194) symbolised the transversal distribution as ◎ 130 This restriction is because we only involve propagation in one direction in all these deliberations. 131 The only way to verify whether a ruler is rectilinear is to aim with light along its egg.

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December 2020

– 3.5.2. The Carrier – 3.4.5.3 Superposition of Linear Polarized double±subtons –

3.5. Modulation of a Quantum Mechanical Field 3.5.1. The Macroscopic Modulation

Here I will draw attention to what we call modulation, as we know it from electromagnetic wave communications by radio, in cables or in optic fibres. In the concept modulation there are two vastly different basic ideas behind the types of macroscopic modulation: amplitude modulation, and phase modulation. In the formula (3.251) the 'radio' modulation is not shown explicitly, this is implicitly included in the number quantity N for the amplitude √𝑁, and the phase angle quantity 𝜙𝑚 = 𝜔𝑡. The phase modulation is as the quantum phase angle divided in the two quantities the angular frequency 𝜔 and development parameter 𝑡. As we have seen through Fourier integral theory in section 1.7.7-1.7.8 these two quantities complement each other. A modulated quantity as a function of the development parameter (3.254) 𝑞𝑚 (𝑡) = ∫ℝ 𝑞̃𝑚 (𝜔)𝑒 −𝑖𝜔𝑡 𝑑𝜔 is complemented by a modulated spectrum as an amplitude function (3.255)

𝑞̃𝑚 (𝜔) =

1 ∫⃗⃗ 𝑞𝑚 (𝑡) 2𝜋 ℝ

𝑒 𝑖𝜔𝑡 𝑑𝑡.

To make the concept of modulation interesting for our conceptual world of information the modulation will have to be dependent on the evolution in the concept of time. This is intuited as a modulation function 𝑚(𝑡𝑚 ), which depends on the modulation development parameter 𝑡𝑚 ; by which the angular frequency spectrum 𝑚 ̃ (𝑡𝑚 , 𝜔) as a function of 𝜔 also will depend of this development. How does this look like in a conceptual world of subtons? We look at a spectrum of angular frequencies 𝜔 and thus at subtons with different eigen frequency energies ℏ𝜔 and as a consequence of this different phase angles 𝜙=𝜔𝑡 ∈ℝ. These quantities, that for us define a spectrum of a full ensemble of subtons, is external to each of the individual subtons. ̂ Here, the eigen frequency for each subton has its own autonomy internal reference ⃗𝜔 ⃗⃗ . ̂ We remember that the subton auto-norm is |𝜔 ⃗ | ≡ 1 [radian], so the quantum phase angle performs the inner development parameter 𝜙 = |𝜔 ⃗̂ |𝑡𝑎 = 𝑡𝑎 = |𝜔 ⃗̂ |𝜙 = 𝜙 . But -. In the modulation of the external spectrum the intuition of the concept permits 𝜔 ∈ℝ as a variable quantity. Relative to this variable input frequency 𝜔 we introduce the idea of an external constant carrier frequency 𝜔𝑐 as a reference for ∀𝜔 ∈ℝ in the spectrum. The external reference for this carrier is the external reference clock [𝜔 ̂], where |𝜔 ̂| ≡ 1. To take the spectrum as a self-consistent ensemble entity Ψ we have to take all the given subton frequencies as internal conserved in their existence ∀ 𝛹𝜔 ∈ Ψ relative to the carrier. A constant relative measure 𝜔⁄𝜔𝑐 in this spectrum Ψ. 3.5.2. The Carrier

Are there physical entities that possesses a quality concerning a constant quantity that could serve as a reference of measuring information in a spectrum? As everlasting constant internal reference of a signal, we introduce the idea of a carrier of angular frequency 𝜔𝑐 ∈ℝ, which real quantity 𝜔𝑐 [𝜔 ̂] is given relative to the external reference |𝜔 ̂| ≡ 1[𝜔 ̂]. For this carrier 𝜔𝑐 we also assign the idea of a development parameter 𝑡𝑐 for the carrier. The idea is, an ideal eternal carrier wave ⃗⃗⃗ 𝑐 ∞ is formed by an external intuition of a transversal plane wave as shown in Figure 3.13, with a constant internal carrier frequency 𝜔𝑐 relative to an ̂ externally referenced |𝜔 ̂| ≡ 1[𝜔 ̂], and preferred in one direction ∀⃗𝜔 ⃗⃗ ∥ 𝐞3 for all carrier subtons with the transversal circle of rotation ⊙⊥𝐞3 of the angular frequency vector ©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 107 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.5. Modulation of a Quantum Mechanical Field –

(3.256)

𝜔 ⃗ 𝑐 =𝜔𝑐 𝐞3 [𝜔 ̂]. ⃗⃗ In addition, we define an eternal evolution parameter 𝑡⃗⃗⃗⃗ 𝑐 ∈ ℝ+ for such noumenon carrier. The first creation, event A0 , (𝑡⃗⃗⃗⃗ 𝑐 = 0), provides carrier subtons0 (all simultaneous). ∞

∞

(3.257)

|𝜓±𝜔 ⃗⃗⃗ 𝑐 ⊥⊙ (𝑡⃗⃗⃗⃗ 𝑐 )⟩0 = ∞

† |0,0⟩ 𝑎⊙±𝜔 ⃗⃗⃗ 𝑐

= ⊙2𝑟̃ (𝜌)𝑒 ±𝑖𝜙𝑐⃗⃗⃗⃗ = ⊙2𝑟̃ (𝜌)𝑒 ±𝑖𝜔𝑐𝑡⃗⃗⃗⃗𝑐 , for ∀𝜌≥0, ∀𝑡𝑐⃗⃗⃗⃗ ≥0. ∞

∞

∞

This eternal (constant) carrier transmits a transversal front into the future with the direction 𝜔 ⃗ 𝑐 = 𝜔𝑐 𝐞3 . By this is produced a parameter for the past 𝑡⃗⃗⃗⃗ 𝑐 = |𝜙⃗⃗⃗⃗ 𝑐 |⁄|𝜔𝑐 | in our thoughts, ⃗⃗ to be back, as a real value 𝑡⃗⃗⃗⃗ 𝑐 ∈ℝ, at the location of the cause of the creation event transmit (0⃗⃗⃗⃗ (𝑡⃗⃗⃗⃗ (3.258) A0 = Atransmit for ∀𝑡⃗⃗⃗⃗ local 𝑐 ) = Alocal 𝑐 ), 𝑐 ≥ 0. ∞

∞

∞

∞

∞

∞

The reason for the creation of the eternal carrier is called a transmitter site location. 132 𝑡⃗⃗⃗⃗ counted in the creative transmitter location. In every𝑐 𝑖𝑠 called the carrier chronometer time day speech, we switch on the radio transmitter A and sets its clock to the start value 𝑡⃗⃗⃗⃗ 𝑐 = 0. front front (0⃗⃗⃗⃗ An imaginary virtual carrier annihilation in event B⃗⃗⃗⃗ = B⃗⃗⃗⃗ 𝑐 ) has always carrier time 𝑐 𝑐 front 133 𝑡⃗⃗⃗⃗ and B⃗⃗⃗⃗ rushes 𝑐 = 0, at the front of first creation. At least one carrier subton0 is eternal 𝑐 with the velocity of light 𝑐𝐞3 in the space of the development parameter 𝑡⃗⃗⃗⃗ through the 𝑐 −1 ]. [𝜔 coordinate 𝑥3,𝑐⃗⃗⃗⃗ = −𝑐𝑡⃗⃗⃗⃗ spanned by (𝑥 )𝐞 ̂ ⃗⃗⃗⃗ 3 𝑐 3,𝑐 front transmitter (0⃗⃗⃗⃗ (𝑡⃗⃗⃗⃗ We remember that the distance to the front B⃗⃗⃗⃗ 𝑐 ) from the first creation Alocal 𝑐 ) 𝑐 just is |𝑥3,𝑐⃗⃗⃗⃗ | = |𝑐𝑡⃗⃗⃗⃗ 𝑐 |. I assume in this ideal naïve thinking that annihilation never occur for at least one first subton0 and that this intuition as in Figure 3.13 can be viewed as always growing and by this defines a carrier transmit ( 𝑡⃗⃗⃗⃗ chronometer time 𝑡𝑐∞ ⃗⃗⃗⃗⃗⃗ in the created location A0 = Alocal 𝑐 ) by 'counting' the local angular phase 𝜙⃗⃗⃗⃗ = 𝜔 𝑡 of for this first subton under its eternal extension and expansion. 0 𝑐 ⃗⃗⃗⃗ 𝑐 𝑐 front (0⃗⃗⃗⃗ The time axis (direction left in Figure 3.13) has a zero point B⃗⃗⃗⃗ 𝑐 ) which is moved into the 𝑐 ̂ future (to the right) with the velocity of light 𝑐𝐞3 ∥ ⃗𝜔 ⃗⃗ . The local time at the front is always 0, 𝑡c,Bfront = 0 and the time axis from this has the direction in to the past.134 The opposite 𝑥3 -axis, 𝑥3 𝐞3 [𝑐𝜔 ̂ −1 ] in the transmission direction 𝐞3 starts with 𝑥3 = 0 at the transmitter A0 = Atransmitter . local Now, the idea that the transmitter A0 is a concrete physical local object with a generator oscillator, 135 the real angular phase 𝜙𝑐 = 𝜙⃗⃗⃗⃗ subton 0 conceivably maintained and counted, 𝑐 from the first hence for this object (3.259) 𝑡𝑐 = 𝑡𝑐,A = |𝜙𝑐 |⁄|𝜔𝑐 | [𝜔 ̂ −1 ] = 𝑡⃗⃗⃗⃗ 𝑐 = |𝜙⃗⃗⃗⃗ 𝑐 |⁄|𝜔𝑐 |. ∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

This real physical transmitter carrier oscillator with the angular frequency 𝜔𝑐 constitutes a chronometer watch called a carrier clock {𝑡𝑐 }, generated from the transmitter location (𝑡𝑐 ), as a propagating development parameter 𝑡𝑐 . A0 = Atransmit local The idea of a constant internal carrier frequency 𝜔𝑐 = 𝜔⃗⃗⃗⃗ is total dependent on the idea of an 𝑐 eternal first carrier subton 0 with the angular frequency 𝜔⃗⃗⃗⃗ 𝑐 , both of course considered relative to our external reference |𝜔 ̂|≡1 [𝜔 ̂]. For an arbitrary point B, with the coordinate 𝑥3,B along the space-axis136 we define the chronometer time as 𝑡B = 𝑡𝑐 − 𝑥3,B ⁄𝑐 , which locally is delayed by the propagation from transmitter A. ∞

∞

⃗. The chronometer time follows the increased order representation over the real numbers, one specific number after another ⃗ℝ Eternity is only a thought, but for it to have meaning at least one subject element is necessary in this substance in thought. 134 Where is the past in a real world? 135 The first subton is eternal in the idealistic thought, therefore 𝑡⃗⃗⃗⃗ 𝑐 ∞ . The term 𝑡𝑐 is the objective chronometer time in A. 136 The space-axis into the future-space from the first creation (3.246) in the direction 𝐞3 (as developed with the past back from the front of expansion. From A created extension.) 132 133

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 108 –

A Research on the a priori of Physics –

December 2020

– 3.5.2. The Carrier – 3.4.5.3 Superposition of Linear Polarized double±subtons –

(𝑡𝑐 ). The idea of the carrier, is continuous to create new carrier subtons 𝑡𝑐 from Atransmit local In order, not to complicate this description unnecessarily a synchronism is assumed in creation with the periodicity interval Δ𝑡𝑐 = 2𝜋⁄|𝜔𝑐 | , and by an imagined numbering of the periods 𝑚 ∈ℕ, we have the creation times 𝑡𝑐,𝑚 = 2𝜋𝑚⁄|𝜔𝑐 | The carrier creation not only thought synchronous but also as phase angle coherent for a macroscopic carrier.137 As subton creation in principle considered countable, the chronological times 𝑡𝑐,𝑚 are countable too. In addition, there can be created several subtons for each period. We say, for each period are created 𝑁± (𝑚) ∈ℕ subtons 𝑡𝑐,𝑚 which contribute to carrier amplitude, at time stamps 𝑡𝑐,𝑚 = 2𝜋𝑚⁄|𝜔𝑐 |. The carrier wave can then be written as a function of the coordinate development parameter 𝑡3 = 𝑡𝑐 − 𝑥3 ⁄𝑐 (3.260)

† macro |0,0⟩𝑡 |𝜓±Σ𝜔 ⃗⃗⃗ 𝑐 ⊥⊙ (𝑡3 )⟩ = 𝐴𝑁(𝑚) 𝑎⊙±𝜔 ⃗⃗⃗ 𝑐 𝑐,𝑚 𝜔 𝑡

𝑐 𝑐 = 𝐴𝑁(𝑚)⊙2𝑟̃ (𝜌)𝑒 ±𝑖(𝜔𝑐𝑡3−2𝜋𝑚) = 𝐴𝑁 ([ 2𝜋 ]) ⊙2𝑟̃ (𝜌)𝑒 ±𝑖𝜔𝑐(𝑡𝑐 −𝑥3⁄𝑐) The normalization of this quantum state requires 𝐴 = 1⁄𝑁(𝑚), as the beam ‘thickness’ holds independent of the number 𝑁(𝑚). From this we write the field of a circularly polarized coherent (monochromatic) carrier wave in the direction 𝐞3

(3.261)

helix (𝑡 , 𝑥3 ) = 𝑨𝑚 𝑨loss (𝑥3 )(𝑒 ±𝑖(𝜔𝑐( 𝑡𝑐−𝑥3⁄𝑐)−2𝜋𝑚) ) = 𝑨(𝑡𝑐 )𝑨loss ((𝑥3 )𝑒 ±𝑖(𝜔𝑐𝑡𝑐−𝑘𝑐,3𝑥3) ) 𝑊±𝜔 𝑐 𝐞3 𝑐 ⊥𝐞

⊥𝐞3

3

where 𝑘𝑐,3 = 𝜔𝑐 ⁄𝑐 is the wave number in the extensive spaces along 𝑥3 𝐞3. The factor 0 ≤ 𝑨loss (𝑥3 ) ≤ 1 indicating the loss of subtons at random annihilation along the development extension 𝑥3 , which can not affect the coherence. To make the transversal plane carrier wave coherent, the creative variations in the transmitter amplitude 𝑨 must be synchronised with the period (3.262)

2𝜋 𝜔𝑐 𝑡𝑐

𝑨(𝑡𝑐 ) = 𝑨 (𝜔 [ 𝑐

2𝜋

]) = 𝑨𝑚 ~√ 𝑁 ([

𝜔𝑐 𝑡𝑐 2𝜋

]),

where [

𝜔 𝑐 𝑡𝑐 2𝜋

∈ℝ] ∈ℤ, and 𝑁(𝑡𝑐,𝑚 ) a function.

As the carrier in its idea is eternal 𝜔 ⃗ 𝑐 = 𝜔𝑐 𝐞3 it demands persistent creation that happens as new events in the transmitter location (continuous eternally). Therefore, regarding (3.258) I ask the reader to consider the inequalities of the events as ontological ideas: (3.263) A0 = Atransmitter (0𝑐 ,03 ) ≠ Atransmitter (𝑡𝑐 , 03 ) ≠ B(𝑡𝑐 , 𝑥3 ) for ∀𝑡𝑐 >0 and 𝑥3 >0 . For a subton entity AB𝛹±𝜔 ⃗⃗⃗ 𝑐 from creation A(𝑡𝑐 , 03 ) to annihilation B(𝑡𝑐 , 𝑥3 ) I point out and claim, that a the valid ontologically for each subtone is A B (3.264) 𝛹±𝜔 𝛹±𝜔 ⃗⃗⃗ 𝑐 (𝑡𝑐 , 03 ) = ⃗⃗⃗ 𝑐 (𝑡𝑐 , 𝑥3 ) . The state of the subton is the same when it is created, as when it annihilates, thus 𝑡𝑐 is the same in both events 𝑡𝑐,A = 𝑡𝑐,B . Therefore, I conclude; the subton can transfer information. – I.e. the subton as feeder brings among other things the clock {𝑡𝑐 }. Remember that the past never reaches the presence nor even the first frontier! 138 The subton 'count' its own internal angular phase, thereby generating the extension |𝑥3 | = 𝑐 | AB𝜙𝑐 |⁄|𝜔𝑐 | = 𝑐|𝑡𝑐,B + 𝑡3 | − 𝑐|𝑡𝑐,A | = 𝑐|𝑡3 | [𝑐𝜔 (3.265) ̂ −1 ]

137

A continuous creation can be explained by the freedom in the circle group ⊙= {𝑈𝜃 : 𝜃→𝑒 𝑖𝜃 ∈𝑈(1)| ∀𝜃 ∈ℝ} , which allows coherency, but as obvious do not require nor cause coherency. 138 The 20th century special theory of relativity has taught us, that the objective time travel to the past is impossible (a taboo). By contrast, the past is disappearing away with the speed of light. Only our memory remains.

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 109 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.5. Modulation of a Quantum Mechanical Field –

If we take the angular frequency of the carrier for the same 𝜔𝑐,A = 𝜔𝑐,B ,139, hence the reference for space and time is the same, therefore, the extension is given by the time it takes for the subton to exist multiplied by the velocity of subtons. The time is to count the angular phase.140 We must ascertain that the information about the clock's value {𝑡𝑐 } is driven FORWARD at the speed of light 𝑐. An intuit existence of a NOW-state-mode of subtons is that they transmit one phase angle value (3.266) 𝜙𝑐,AB = 𝜙𝑐,A = 𝜙𝑐,B = 𝜔𝑐 𝑡𝑐,AB = 𝜔𝑐 𝑡𝑐,A = 𝜔𝑐 𝑡𝑐,B This is called the instantaneous NOW value of the quantum phase angle information. The quantum phase angle as the subton produce is in this way the past through the extension (3.267) 𝜙3 = 𝜔𝑐 𝑡3 . Expressed on an ironic daily manner; the past is light-years away (or just light-seconds away). This is the classical paradox of relativity with light like null line path by (𝑐∆𝑡3 )2 − (∆𝑥3 )2 = 0.141 3.5.2.2. Macroscopic Coherent Amplitude Modulation

We saw at (3.260)-(3.262) that a carrier can be amplitude modulated, if this is done phase angle coherent, without affecting the eternal constant angular frequency 𝜔𝑐 of the carrier. 3.5.2.3. Phase Modulation

Phase modulation is a relative concept. – I.e. the phase angle 𝜙 is varied relative to a reference phase angle 𝜙𝑐 of a phantom carrier with an angular frequency 𝜔𝑐 as a reference. This 𝜔𝑐 is also a reference clock that generates development parameter 𝑡𝑐 for that carrier. The macro-modulating phase difference is called Δ𝜙, hence the modulated variable phase angle gets (3.268) 𝜙 = 𝜙𝑐 +𝛥𝜙 = 𝜔𝑐 𝑡𝑐 +𝛥𝜙 = 𝜔𝑐 𝑡𝑐 +𝛥𝜔∙𝑡𝑐 +𝜔𝑐 𝛥𝑡 Such a phase modulation can then be regarded as frequency modulation FM 𝜔 = 𝜔𝑐 + Δ𝜔, or time modulation TM: 𝑡TM = 𝑡𝑐 + Δ𝑡, where creation of subtons is controlled relative to the carrier clock. Here we shall not elaborate further on a direct time modulation, because with 𝜙 = 𝜔𝑐 𝑡TM = 𝜔𝑐 (𝑡𝑐 + Δ𝑡) it ends up in phase modulation or in special cases in synchronous amplitude modulation. For frequency modulation the phase angle is given by 𝜙FM = (𝜔𝑐 +Δ𝜔)𝑡𝑐 = 𝜔FM 𝑡𝑐 = 𝜔𝑡𝜔 +𝜃. Here it is thus possible to create subtons with different angular frequencies 𝜔FM , but these must all be related to the carrier 𝜔𝑐 as to 𝑡𝑐 to make internal sense for a spectrum. (Those can assumingly be related to one external reference |𝜔 ̂|≡1 [𝜔 ̂]). The signal in our intuition has one direction as a transversal plane wave, it is ∀𝜔 ⃗ FM ∥𝜔 ⃗ 𝑐. If we look at each 𝜔 ⃗ =𝜔 ⃗ FM , each subton |𝜓±𝜔 ⃗⃗⃗ (𝜙𝜔 ⃗⃗⃗ )⟩ generates its own individual development parameter 𝑡𝜔 ⃗⃗⃗ or angular phase angle 𝜙𝜔 ⃗⃗⃗ = 𝜔𝑡𝜔 ⃗⃗⃗ , that will be indefinite in terms of 𝜙𝑐 , in that 𝜙𝜔 =𝜔𝑡𝑐 +𝜃. Since this subton individual concept of time don’t help us in our intuition, we adhere to the carrier clock {𝑡𝑐 } as the reference for the signal giving the development parameter 𝑡𝑐 , where Δ𝑡≡0 in (3.268). This time reference 𝑡𝑐 is linked to the carrier circle oscillator 𝜔 ⃗ 𝑐 . (Ideal From the first subton.) The modulation as a concept can only become meaningful when it varied over time. The time-dependent modulated frequency 𝜔FM (𝑡𝑐 )=|𝜔 ⃗ (𝑡𝑐 )| provides phase modulation 139

𝜔𝑐,A ≠ 𝜔𝑐,B by the Doppler effect when two events A and B are in relativistic different velocity references. (cooling of background radiation by expansion). 140 Here I ask the reader; what is the speed of times? Is it the angular frequency of the carrier 𝜔𝑐 ? Or is it 𝑐 ? 141 Look in chapter II. 5.7 for relativistic issue of different directions in a Minkowski space metric (𝑐∆𝑡3 𝛾0 )2 + (∆𝑥3 𝛾3 )2 = 0

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 110 –

A Research on the a priori of Physics –

December 2020

– 3.5.2. The Carrier – 3.5.2.3 Phase Modulation –

(3.269)

𝜙M (𝑡𝑐 ) = 𝜔𝑐 𝑡𝑐 +Δ𝜙(𝑡𝑐 ) = 𝜔FM (𝑡𝑐 )𝑡𝑐 = (𝜔𝑐 +Δ𝜔FM (𝑡𝑐 ))𝑡𝑐

= 𝜙𝜔 ⃗⃗⃗ (𝑡𝑐 ).

When the signal contain myriads of simultaneous subtons at one specified frequency 𝜔FM (𝑡𝑐 ), the angular phase modulation between them can never be read (demodulated) because they are individual indistinguishable. Therefore simultaneously only one phase angle value 𝜙M (𝑡𝑐 ) for one 𝜔FM (𝑡𝑐 ) at one development stamp 𝑡𝑐 as a possibility in a macroscopic modulated signal. What we really control in the signal, when we modulate the phase angle Δ𝜙(𝑡𝑐 ) after the reference clock {𝑡𝑐 } from the carrier, is the angular frequency energy of the created subtons (3.270) 𝜔FM (𝑡𝑐 ) = 𝜔𝑐 + 𝛥𝜙(𝑡𝑐 )⁄𝑡𝑐 ⇔ 𝛥𝜔FM (𝑡𝑐 ) = 𝛥𝜙(𝑡𝑐 )⁄𝑡𝑐 Here it is very important to remember that individual subtons |𝜓±𝜔 ⃗⃗⃗ (𝑡𝑐 , Δ𝜙𝜔 ⃗⃗⃗ )⟩ only carrier the instantaneous NOW-values of the phase angle difference Δ𝜙𝜔 ⃗⃗⃗ ,A = Δ𝜙𝜔 ⃗⃗⃗ ,B = Δ𝜙M (𝑡𝑐,AB ) or the modulated phase angle 𝜙𝜔 ⃗⃗⃗ ,A = 𝜙𝜔 ⃗⃗⃗ ,B = 𝜙M (𝑡𝑐,AB ), as 𝑡𝑐,A = 𝑡𝑐,B . This is conveyed by all subtons |𝜓±𝜔 ⃗⃗⃗ (𝑡𝑐 , Δ𝜙𝜔 ⃗⃗⃗ )⟩AB just like the reference parameter {𝑡𝑐 } for all carrier subtons |𝜓±𝜔 – That is the signal information from A to B ⃗⃗⃗ 𝑐 ⊥⊙ (𝑡𝑐 )⟩ . AB

The subton not only retains 𝜔 ⃗ FM,A = 𝜔 ⃗ FM,B , but also the phase angle 𝜙M,A = 𝜙M,B relative to the carrier phase angle 𝜙𝑐,A = 𝜙𝑐,B , which too is retained from A to B. All this is here based on the carrier ⃗𝜔 ⃗⃗ c,A = 𝜔 ⃗ c,B is substantial for the described ideology for the carrier subtons. This implies that a relative ratio from A to B is unchanged (preserved).142 We have omitted the effect from that the signal and carrier subtons can interact with the environment.143 We accept loos of part of the subtons in the channel is acceptable. The conclusion is that the development dependent phase modulation PM is pseudonym with a varied frequency FM modulation, that is Δ𝜙(𝑡𝑐 )= Δ𝜙PM (𝑡𝑐 )= Δ𝜔FM (𝑡𝑐 )𝑡𝑐 , when we view by intuition from a chronometer time {𝑡𝑐 } given by a transmitter carrier phase angle 𝜙𝑐 =𝜔𝑐 𝑡𝑐 . For this we Then we consider carrier wave described from the transmitter (3.271)

A

helix

macro ±𝑖(𝜔𝑐 𝑡𝑐 −𝑘𝑐,3 𝑥3 ) ) ̃ 𝑊±⃗𝜔 , where 𝑘𝑐,3 = 𝜔𝑐 ⁄𝑐 . |𝜓Σ±𝜔 ⃗⃗ 𝑐 (𝑡𝑐 , 𝑥3 ) = 𝑨𝐴 𝑨𝑙𝑜𝑠𝑠 (𝑥3 )(𝑒 ⃗⃗⃗ 𝑐 ⊥⊙ (𝑡𝑐 )⟩ → ⃗⃗⃗ ⊥𝜔 𝑐

Here we keep the transmitter amplitude constant 𝑨A = 𝑨(𝑡𝑐 ) = 𝑨(0𝑐 ) for ∀𝑡𝑐 ≥0 When (3.271) is modulated, the transmitter phase angle 𝜙(𝑡𝑐 ) = 𝜔𝑐 𝑡𝑐 +Δ𝜙 = (𝜔𝑐 +Δ𝜔FM (𝑡𝑐 ))𝑡𝑐 . This can also be formulated as FM modulation 𝜔FM (𝑡𝑐 ) = (𝜔𝑐 𝑡𝑐 +Δ𝜙)⁄𝑡𝑐 = 𝜔𝑐 +Δ𝜔FM (𝑡𝑐 ). The wave direction 𝐞3 is conceived to define the direction of all the circle oscillators 𝜔 ⃗ FM = 𝜔FM 𝐞3 . The modulated wave around an imaginary reference carrier is then 𝐴 macro helix ̃ 𝑊±𝜔 (3.272) |𝜓Σ±𝜔 ⃗⃗⃗ FM ⊥⊙ (𝜙PM )⟩ → ⃗⃗⃗ FM (𝜙(𝑡𝑐 ), 𝑥3 ) = 𝑨A 𝑨loss (𝑥3 )(𝑒 ±𝑖(𝜔𝑐𝑡𝑐+𝛥𝜙(𝑡𝑐)−𝑘𝑐,3𝑥3 ) )⊥𝐞 = 𝑨A 𝑨loss (𝑥3 )(𝑒 ±𝑖(𝜔FM (𝑡𝑐)𝑡𝑐−𝑘𝑐,3𝑥3) )⊥𝐞 . 3

This term expresses a circularly polarized modulated wave with helicity +1 or −1 . For a linearly polarized wave from a transmitter dipole we can rewrite the wave as

3

142

Here the Doppler-effect is excluded from consideration, since speeds of events not yet included in this concept. Anyway in the case of Doppler-displacement the mutual multiplicative ratios between the different frequencies is conserved. 143 Whether it is sufficient with the conditions |⃗𝜔 ⃗⃗ FM,A |=|𝜔 ⃗ FM,B | and |𝜔 ⃗ c,A |=|⃗𝜔 ⃗⃗ c,B | under mutual changing in direction in an interaction as an annihilation B with recreation in B towards a new event C for subtons ABC𝛹𝜔 , like multi interaction in a transmission conductor (optical fibres, waveguides and coaxial cables) Subtons AB…C𝛹𝜔 will not be discussed here, but it seems to be the case of the success we already have experienced in 20 th century.

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 111 –

Volume I, – Edition 1, – Revision 3,

December 2020

– I. . The Time in the Natural Space – 3. The Quantum Harmonic Oscillator – 3.5. Modulation of a Quantum Mechanical Field –

(3.273)

helix helix 𝑊+𝜔 ⃗⃗⃗ FM ( 𝜙(𝑡𝑐 ), 𝑥3 , 𝜃𝑑 ) + 𝑊−𝜔 ⃗⃗⃗ FM ( 𝜙(𝑡𝑐 ), 𝑥3 , 𝜃𝑑 )

= 𝑨A 𝑨loss (𝑥3 )(𝑒 +𝑖(𝜔𝑐𝑡𝑐+𝛥𝜙(𝑡𝑐)+𝜃𝑑 −𝑘𝑐,3𝑥3) +𝑒 −𝑖(𝜔 𝑐𝑡𝑐+𝛥𝜙( 𝑡𝑐)+𝜃𝑑 −𝑘𝑐,3𝑥3) ) = 𝑨A 𝑨loss (𝑥3 )2 cos(𝜔𝑐 𝑡𝑐 +𝛥𝜙(𝑡𝑐 )+𝜃𝑑 − 𝑘𝑐,3 𝑥3 ) = 𝑨A 𝑨loss (𝑥3 )2 cos(𝜔FM (𝑡𝑐 )𝑡𝑐 +𝜃𝑑 − 𝑘𝑐,3 𝑥3 ) . Here is 𝜃𝑑 the dipole transversal spatial angle designating a direction 𝐞𝜃𝑑 ∥⊙ ⊥ 𝐞3 ∥𝜔 ⃗. (𝑥 ) Then the macroscopic field with an amplitude 𝐸𝑎 3 as in (3.251) is written (3.274)

𝐸⃗ =

𝐸ω (𝑡𝑐 , 𝛥𝜙) 𝐞𝜃𝑑 = 𝐸𝑎 (𝑥3 )2 cos(𝜔𝑐 𝑡𝑐 +𝛥𝜙(𝑡𝑐 ) − 𝑘𝑐,3 𝑥3 ) 𝐞𝜃𝑑

= 𝐸𝜔FM (𝑡𝑐 , 𝑥3 ) 𝐞𝜃𝑑 = 𝐸𝑎 (𝑥3 )2 cos(𝜔FM 𝑡𝑐 − 𝑘𝑐,3 𝑥3 ) 𝐞𝜃𝑑 The hereby propagated signal has carrier frequency 𝜔𝑐 as the signal's own reference. The common development parameter 𝑡𝑐 obtained by 'counting' the reference phase 𝜙𝑐 for the carrier, that 𝑡𝑐 = |𝜙𝑐 |⁄|𝜔𝑐 |. Both of them 𝜔𝑐 [𝜔 ̂] and 𝑡𝑐 [𝜔 ̂ −1 ] stands preferable in relative reference to an external standard frequency |𝜔 ̂|≡1 [𝜔 ̂]. 3.5.2.4. Amplitude Modulation at Mutual Frequencies

The formulas (3.260) to (3.262) gives the condition for coherent modulation of a macro monochromatic carrier wave |𝜓Σ±𝜔 𝑁(𝑚)⊙2𝑟̃ (𝜌) 𝑒 ±𝑖(𝜔𝑐𝑡3 −2𝜋𝑚) ⃗⃗⃗ 𝑐 ⊥⊙ (𝑡3 )⟩ = However, intuit we have an arbitrary modulation signal as a spectrum after the Fourier method144 (3.275)

𝑚(𝑡) = ∫ℝ 𝑚 ̃ (𝜔)𝑒 −𝑖𝜔𝑡 𝑑𝜔 ,

complemented by

1

𝑖𝜔𝑡 𝑚 ̃ (𝜔) = 2𝜋 ∫ℝ 𝑑𝑡. ⃗⃗ 𝑚(𝑡)𝑒

For the individual modulation frequencies 𝜔m we form a spectrum modulation factor145 (3.276) 𝑚 ̃ 𝑎 (𝑡𝑐 , 𝜔m )𝑒 𝑖𝜔m 𝑡𝑐 , which depends on a carrier chronometer time {𝑡𝑐 }. The modulated wave is written as (3.277)

helix (𝑡 , 𝑥3 ) = 𝑚 𝑊±𝜔 ̃ 𝑎 (𝑡𝑐 , 𝜔𝑚 )𝑒 ±𝑖𝜔𝑚𝑡𝑐 𝑨A 𝑨loss (𝑥3 )(𝑒 ±𝑖(𝜔𝑐𝑡𝑐−𝑘𝑐,3𝑥3) )⊥𝐞 𝑐 ,𝜔𝑚 𝑐

= 𝑚 ̃ 𝑎 (𝑡𝑐 , 𝜔𝑚 )𝑨A 𝑨loss (𝑥3 ) (𝑒 ±𝑖((𝜔𝑐+𝜔𝑚

)𝑡𝑐 −𝑘𝑐,3 𝑥3 )

3

)

⊥𝐞3

For a macroscopic field from a dipole in the angular direction 𝐞𝜃𝑑 , this becomes (3.278)

𝐸⃗ = 𝐸±𝜔𝑐,𝜔𝑚 (𝑡𝑐 , 𝑥3 ) 𝐞𝜃𝑑 =

𝑚 ̃ 𝑎 (𝑡𝑐 , 𝜔𝑚 )𝐸𝑎 (𝑥3 )2 cos ((𝜔𝑐 +𝜔𝑚 )𝑡𝑐 − 𝑘𝑐,3 𝑥3 ) 𝐞𝜃𝑑

Wherein the amplitude factor is given by 𝐸𝑎 (𝑥3 ) = 2𝑨A 𝑨loss (𝑥3 ) and 𝑨− =𝑨+ for the dipole field. We see here that the arbitrary amplitude variation implies an accompanying modulation of frequency. Thus, all randomly modulated waves include a spectrum of subtons with a range of mutual angular frequency energies 𝜔𝑐 +𝜔𝑚 . 3.5.2.5. QAM Modulation

The modern modulation called Quadrature Amplitude Modulation (QAM). Based on the idea of different carrier (OFDM) both phase and amplitude modulated the waves of these subtons. The technical method of controlling the phase modulation is called quadrature. This in an ontological intuition divide it in a Cartesian coordinate system of real and imaginary parts and an amplitude signal modulates these separately, QAM. Modulation done digitally through two DAC (digital analogue converters) and the bandwidth is increased by OFDM FFT−1 method, which is outside of this review.

144 145

See (3.254) and (3.255) and § 1.7.7-0, and later below § 4.1.4.2 The Vector Space of Fourier Integrals. The modulation factor is normally considered arbitrary within the framework 0≪ 𝑚𝑎 (𝑡𝑐 , 𝜔m ) ≤1 and 0 ≤|𝜔𝑚 | 0 ∨ 𝐚 = 𝟎 ⇒ 𝐚⋅𝐚 = 0, Euclidean metric norm. From the difference between two 1-vectors we can deduct the inner product (see Figure 5.7) 𝐝= 𝐚−𝐛 ⇒ 𝐝⋅𝐝= |𝐝|2 = |𝐚–𝐛| ⏟ 2 = (𝐚 𝐛)⋅(𝐚 𝐛)= 𝐚⋅(𝐚 𝐛) 𝐛⋅(𝐚 𝐛)= 𝐚⋅𝐚 𝐚⋅𝐛 𝐛⋅𝐚+𝐛⋅𝐛 = |𝐚| ⏟ 2 +|𝐛|2 − 2𝐚⋅𝐛

(5.52)

⇒

𝐚⋅𝐛 =

1 2

(|𝐚|2 + |𝐛|2 − |𝐚 𝐛|2 )

And from the sum of two 1-vectors 𝐜=𝐚+𝐛 we obtained the inner scalar product 𝐚⋅𝐛 as 𝐜 = 𝐚+𝐛 ⇒ |𝐚+𝐛|2 = (𝐚+𝐛)⋅(𝐚+𝐛)= 𝐚⋅(𝐚+𝐛)+𝐛⋅(𝐚+𝐛)= 𝐚⋅𝐚+𝐚⋅𝐛+𝐛⋅𝐚+𝐛⋅𝐛 = ⏟ |𝐚|2 +|𝐛|2 + 2𝐚⋅𝐛 𝐜⋅𝐜 = |𝐜|2 = ⏟ (5.53) (5.54)

(5.55)

1

⇒ 𝐚⋅𝐛 = 2 (|𝐚+𝐛|2 − |𝐚|2 − |𝐛|2 ) From the quadratic form (5.46) we see that the inner product is 1

𝐚⋅𝐛 ≡ 2 (𝐚𝐛 + 𝐛𝐚) =

1

((𝐚+𝐛)2 𝐚2 𝐛2 ) = 2

1

(𝐚2 +𝐛2 −(𝐚 𝐛)2) = 2

1 4

((𝐚+𝐛)2 −(𝐚 𝐛)2 )

Two geometric 1-vectors together form an inner scalar product. From the ordinary vector geometry, we repeat the simple formula (5.49) 𝐚⋅𝐛 = |𝐚||𝐛| cos 𝜃 ∈ℝ, where 𝜃 = ∢(𝐚, 𝐛) ∈ℝ. The scalar product forms a pure scalar quantity measure ℝpqg-0 for the symmetric colinear internal relations between the two 1-vectors. In general, the scalar is a measure of a colinear internal dependency in a physical entity expressed between two mutual related 1-vectors. In addition to this scalar measure, the anti-symmetry between the two geometric vectors from (5.44) form a plane concept as a primary quality of second grade (pqg-2). This plane substance we intuit as an objective surface that we see from its outside, therefor the anti-symmetry of the product is called an outer quality.

5.2.4. The Geometric Product

We return to the general product of geometric vectors 𝐚 and 𝐛 from (5.44) (5.56)

𝐚𝐛 =

1 1 (𝐚𝐛 + 𝐛𝐚) + (𝐚𝐛 − 𝐛𝐚). 2 2

We have seen (5.46)-(5.54) that the first part is a symmetrical commuting inner product (5.57)

(5.58)

(5.59)

211

1

𝐚⋅𝐛 = 2 (𝐚𝐛 + 𝐛𝐚) (is a real scalar). This symmetrical inner product has also been called the interior product. The last antisymmetric part we will write with wedge angle icon ∧ between the two vectors 𝐚∧𝐛 =

1 (𝐚𝐛 − 𝐛𝐚) 2

=

1

− 2 (𝐛𝐚 − 𝐚𝐛) = −𝐛∧𝐚

(is a bivector).

This part of the product is called the anti-commuting outer product (also called exterior product). In this way, the geometric product of 1-vectors can be written as 𝐚𝐛 = 𝐚⋅𝐛 + 𝐚∧𝐛 This is an example of an of a so-called 2-multivector,211 or just a 2-vector.

2-multivector, 2 stands for the simple product polynomials of two 1-vectors and scalars, e.g. just 𝐚𝐛 or 𝛾𝐚𝐛+𝛽𝐜+𝛼 .

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 165 –

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.2. The Plane Geometric Algebra –

5.2.5. The Outer Product of Geometric Vectors 212 (5.60) (5.61)

(5.62)

(5.63)

We look at the anti-symmetric part 𝐚∧𝐛 by commutating the geometric product (5.60) 𝐛𝐚 = 𝐚⋅𝐛 − 𝐚∧𝐛, as alternative to 𝐛𝐚 = 𝐛⋅𝐚 + 𝐛∧𝐚 We multiply these two equations (5.59) and (5.60) as a product213 of two 2-multivectors 𝐚𝐛𝐛𝐚 = (𝐚⋅𝐛)2 − (𝐚∧𝐛)2 As we defined above (5.42) 𝐛𝐛 = 𝐛2 = |𝐛|2 is a scalar, and214 next 𝐚𝐚 = 𝐚2 = |𝐚|2 is a scalar, then 𝐚𝐛𝐛𝐚 = |𝐚|2 |𝐛|2 , and from (5.49) we have the scalar 𝐚⋅𝐛 = |𝐚||𝐛| cos 𝜃 ∈ℝ, therefor (𝐚∧𝐛)2 = (𝐚⋅𝐛)2 − 𝐚𝐛𝐛𝐚 = (|𝐚||𝐛| cos 𝜃)2 − |𝐚|2 |𝐛|2 = |𝐚|2 |𝐛|2 (cos2 𝜃 − 1) = −|𝐚|2 |𝐛|2 sin2 𝜃 ≤0. This square of the outer product is negative and therefore 𝐚∧𝐛 is not a scalar, but a anticommuting pseudoscalar in the plane, this idea is a subject in the pqg-2 substance. Seen from outside the plane we call this type of object a bivector, as an outer product of two 1-vectors. This is when considered as a subject also called for a 2-blade.215 The letter expression for a bivector A = 𝐚∧𝐛 or B = 𝐛∧𝐚 is here named in large bold Latin letters.216 For the magnitude of the bivector B = −A = 𝐚∧𝐛 we get from (5.62) the area |B| = |−A| = |𝐛∧𝐚| = |𝐚∧𝐛| = |𝐚||𝐛||sin 𝜃| ≥ 0, ∈ℝ+ , where 𝜃 = ∢(𝐚,𝐛) ∈ℝ. This is the formula for the parallelogram area spanned by the two 1-vectors 𝐚, 𝐛 (Figure 5.20 below). The anti-symmetry in (5.58) gives the two orientations of the direction for the rotation idea as an area-segment, that the bivector spans in the plane.

5.2.5.2. The Bivector Orientation as a Sequential Operation

The Bivector B = 𝐛∧𝐚 has a reverse orientated A = −B = 𝐚∧𝐛 in its rotation direction.

Figure 5.10 The quality of the object Bivector 𝐛∧𝐚 spans the area quantity magnitude |𝐛∧𝐚|. The area spanned direction B = 𝐛∧𝐚 is here progressive orientated (contra-clockwise ↺) by the operation order 𝐚,𝐛. It has a reversed ordered 𝐛,𝐚 outer product A = −B = 𝐚∧𝐛 orientated retrograde from 𝐛 to 𝐚 around origo O (clockwise ↻). The bivector concept gives the area quality in a plane with two orientations (−, +) of its quantity ±|𝐛∧𝐚|.

Because of the anti-commutation rule (5.58) 𝐚∧𝐛 = −𝐛∧𝐚 the operation order of the 1-vectors 𝐚 and 𝐛 has an impact. First, we let in our intuition the 1-vector 𝐚 operate on the space substance, then the 1-vector 𝐛 operates through ∧ on the impact of 𝐚 and thus on space, and we have B = 𝐛∧ 𝐚. The vector operator 𝐛∧ pull the 1-vector a through a plane in space, creating a plane-segment B = 𝐛∧𝐚. – We read from left to right, but when we consider 1-vectors as operators the far right operate first then the next left, and so on successively sequential.217 Therefore, orientation of the rotation direction for the plane segment is positive orientated from 𝐚 to 𝐛 around the origo O. This is opposite the reading orientation of 𝐛∧𝐚, the operation sequence orientation of 𝐛∧(𝐚( )) from the inner parenthesis with the successive left operation. 212

These sections is made with grate inspiration from the works of David Hestenes [33] and [10] etc. As full fills the same geometric algebra for products as described by the a priori rules in (5.38)-(5.41). 214 We recall that scalars commute with vectors and vector products (5.41). 215 The name 2-blade for a basis bivector is often used as part of a more general concept of k-blades (𝑎𝑘 ∧…∧𝑎2 ∧𝑎1 ) as subspace of dimension 𝑘 ≤𝑛 basis of 𝑘 1-vectors 𝑎𝑖 ∈𝑉𝑛 for 𝑖=1, … 𝑘 in a geometric algebraic space 𝒢(𝑉𝑛 ) over vector space is 𝑉𝑛 . 216 Here bold capitals in a plane concept of a 𝒢(𝑉3 ) geometric space. For 𝑛 >3 , 𝒢(𝑉𝑛 ) we will use non-bold italic letters 𝐴, 𝐵 ⋯. 217 Refer to written functional principle 𝑓∘𝑔= 𝑓(𝑔)= 𝑓𝑔 ≉ 𝑔𝑓 = 𝑔(𝑓)=𝑔∘𝑓 for the left operation sequential order of functions. 213

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

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A Research on the a priori of Physics –

December 2020

– 5.2.5. The Outer Product of Geometric Vectors – 5.2.5.3 Bivector Quantity and Form Structure Quality Direction –

In Figure 5.10, this is displayed as the progressive orientation ↺+ from 1-vector object 𝐚 to 𝐛 forming a representative subject bivector B = 𝐛∧𝐚 as a plane substance direction quality. The reverse orientated bivector A = −B = 𝐚∧𝐛 is shown as operator 𝐚∧ pulling 1-vector 𝐛 through space and form a plane segment 𝐚∧𝐛 with retrograde orientation from 1-vector object 𝐛 to 𝐚 forming the subject A = −B = 𝐛∧𝐚 with reversed orientation ↻‒ of the area direction of −B. In addition to the two rotation orientations of a plane segment area |B| = |−B| provided by the opposite bivectors 𝐚∧𝐛 and 𝐛∧𝐚, the plane area segment has also a front and a back of a surface (paper\screen seen by an external observer). If the rotation direction as seen from the front is in a clockwise orientation, it is counter-clockwise viewed from the back. Consequently, only the sequential algebraic order of the 1-vector operations has impact to the reversion. This is very important for the intuition of a bivector concept as a pure algebraic object definition B = 𝐛∧𝐚 of a geometric subject. The orientation ↻( ),↺(+) is here irrelevant, since it depends on the outer observer is in front or at back of the paper (f-b-rationale), like seeing the clock face at front or from back. Later below for generalised multivectors we will use † to mark the reversed, e.g. then 𝐁 † = 𝐚∧𝐛. 5.2.5.3. Bivector Quantity and Form Structure Quality Direction

(5.64)

(5.65)

Just as the geometric a pqg-1-vector line-segment object can be written as a scalar quantity 𝛼 multiplied a direction unit 1-vector 𝐚̂ representing the pqg-1 quality 𝐚 𝐚 = 𝛼𝐚̂ = |𝐚|𝐮, where 𝛼 = |𝐚| ∈ℝ+ , and 𝐚̂ = |𝐚| = 𝐮 , that |𝐚̂| = |𝐮| = 1 ∈ℝ+ , ̂ as a plane a bivector B is also written as a real scalar quantity 𝛽 multiplied a unit bivector B direction in space, representing the pure pqg-2 quality ̂ = |B|B ̂ , where 𝛽 = |B| ∈ℝ+ , and B ̂= B ̂ | = 1 ∈ ℝ+ . B = 𝛽B that |B |B| ̂ is the unit for direction of a plane segment as the primary quality of second grade (pqg-2). B ̂ is also a plane pqg-2 quality. The quantity of bivector is its area magnitude Thus B = 𝛽B ̂ | = 1 ∈ℝ+ , where 𝛽 = |B| ∈ℝpqg-2 and the unit bivector has unit area magnitude |B ̂ is the special unit pqg-2 direction quality of a plane in the space concept 𝔊 in physics. B Negative values of the quotient scalar 𝛽− =(−𝛽)2 is an external extension to the plane concept. The magnitude, also called the modulus of multi-vector is generally defined [10]p.46 and [13]p.13 as n n † |𝐴| = √⟨𝐴† 𝐴⟩0, so that |𝐴|2 = ⟨𝐴† 𝐴⟩ = |⟨𝐴⟩𝑟 |2 ≥ 0, written out (5.166) 𝐴 𝐴 = 𝑟 𝑟 0 r=0 r=0



(5.167)



|𝐴|2 = ⟨𝐴† 𝐴⟩ = |〈𝐴〉0 |2 + |〈𝐴〉1 |2 + |〈𝐴〉2 |2 + |〈𝐴〉3 |2 + ⋯ 0

≥ 0.

≥0 is a simplified pure mathematic assumption. – More generally about multi-vectors for physics later below. 253

The direction of a plane object we define as 𝒊=𝛔2 𝛔1 from the two object 1-vectors 𝛔1 , 𝛔2 . For describing the translation through space 𝔊 outside the plane object we need a third 1-vector 𝛔3 ⊥(𝛔2 ∧𝛔1 ) to make an orthonormal basis {𝛔1 , 𝛔2 , 𝛔3 } for the 1-vector 3 combination 𝐭 = 𝑡1 𝛔1 +𝑡2 𝛔2 +𝑡3 𝛔3 ∈(𝑉3 , ℝ)~ℝ1,2,3 representing the translation. 𝐭 ∦(𝛔2 ∧𝛔1 ) ⇔ 𝑡3 ≠0. In the plane 𝑡3 =0. 254 In that plane we have 〈𝐴〉2 ~(𝐫∧𝐮)∥(𝒊=𝛔2 ∧𝛔1 ), then we can express 𝐮 = 𝑢1 𝛔1 +𝑢2 𝛔2 and 𝐫 = 𝑟1 𝛔1 +𝑟2 𝛔2 as 〈𝐴〉1 , choice 𝛔1 = 𝐮.

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

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A Research on the a priori of Physics –

December 2020

– 5.4.2. Reflections – 5.4.2.1 Reflection in a Geometric 1-vector –

5.4. Transformation of Geometric 1-vectors in the Euclidean plan 5.4.1. Parallel Translation of a Vector

As we described earlier in § 4.4.2.13, the points in space 𝔊 are translations invariant. We also decide as a priori that scalar magnitudes associated with the locus situs of geometric points are translations invariant. We have also found that 1-vectors are translation invariant, since the parallel translation concept is based on the substance of the 1-vector concept. We refer to the previous Figure 4.6 (which is shown here). The same will inherently also apply to multi-vectors, as in their idea they are constructed from 1-vectors of the geometric algebra by products and additions. The now well-known simple 2-multi-vector 1-rotor 𝐯𝐮 = 𝐮2 𝐮1 = 𝑈𝜃 ≡ 𝑒 𝒊𝜃 from (5.83) is translation invariant. 5.4.2. Reflections 5.4.2.1. Reflection in a Geometric 1-vector

Given a 1-vector 𝐮 in 𝔊 space. We choose 𝐮 as norm |𝐮|=1. We choose any other object 1-vector 𝐱 in 𝔊 space. These two 1-vector forms a plane spanned by the bivector 𝐮∧𝐱 This plane is the foundation object for Figure 5.41. We make a normalized 1-vector direction 𝐱̂ = 𝐱⁄|𝐱| for 𝐱= |𝐱|𝐱̂. For understanding we have 𝐱̂𝐱̂ = 𝐱̂ 2 =1 and 𝐱𝐱 = 𝐱 2 =|𝐱||𝐱| ∈ℝ+ . The 2-vector 𝐮𝐱̂ forms a rotor 𝑈= 𝐮𝐱̂ = 𝐮𝐱⁄|𝐱| in the plane 𝐮∧𝐱. The 1-rotor was first defined in formula (5.83) and (5.84) above 𝑈𝜃 = 𝐮2 𝐮1 = 𝑒 𝒊𝜃 , and 𝑈𝜃† = 𝐮1 𝐮2 = 𝑒 −𝒊𝜃 , with ∢(𝐮1 , 𝐮2 )=𝜃. Figure 5.41 The plane 𝐮∧𝐱 . When the rotor 𝑈 acts on 𝐱, we get 𝑈𝐱 = 𝐮𝐱̂𝐱 = 𝐮𝐱𝐱⁄|𝐱| = |𝐱|𝐮. Note the unit circle for the We are searching the symmetrically reflected 1-vector 𝐱 ′ to 𝐱 foundation plane idea. around 𝐮, still in the 𝐮∧𝐱 plane. The reverse rotor 𝑈 † = 𝐱̂𝐮 (5.84) acts on 𝐱 ′ and must symmetrically give (5.169) 𝑈 † 𝐱 ′ = |𝐱 ′ |𝐮 ≡ |𝐱|𝐮 = 𝑈𝐱, as |𝐱 ′ | ≡ |𝐱| (dimmed in top middle of Figure 5.41) (5.168)

Let 𝑈 act on this 1-vector 𝑈𝐱 once again 𝑈 2 𝐱 = 𝑈𝑈𝐱 = (5.170) (5.171)

(5.172) (5.173) (5.174) (5.175) (5.176)

©

𝐮𝐱𝐮𝐱𝐱 |𝐱||𝐱|

= 𝐮𝐱𝐮 =

𝐮𝐱𝐱𝐱𝐮 |𝐱||𝐱|

= 𝑈𝐱𝑈 †

As a result, we have the rule of reflection of 1-vectors 𝐱 through (around) a given 1-vector 𝐮 𝐱 ′ = 𝐮𝐱𝐮 = 𝑈𝐱𝑈 †, and reverse ′ † ′ 𝐱 = 𝐮𝐱 𝐮 = 𝑈 𝐱 𝑈, as shown in Figure 5.42. This is the fundamental formulation of reflection in 𝐮 inside this plane of Figure 5.41 and Figure 5.42. (around 𝐮) We can divide any 1-vector into components along 𝐮, like 𝐱 ∥ = (𝐱⋅𝐮)𝐮 , and transverse 𝐱 ⊥ = (𝐱∧𝐮) 𝐮 , as x = x∥ + x⊥ ⇔ x' = uxu = x∥ − x⊥ shown in Figure 5.42 This because we from (5.58)-(5.59) can write a 2-multivector as Figure 5.42 Reflection 𝐮𝐱 = 𝐮⋅𝐱 + 𝐮∧𝐱 ⟺ 𝐱𝐮 = 𝐮⋅𝐱 − 𝐮∧𝐱 = 𝐱⋅𝐮 + 𝐱∧𝐮, around a 1-vektor 𝐮. (through 𝐮 in the plane). and as we have 𝐮⋅𝐱 ⊥ = 𝐱 ⊥ ⋅ 𝐮 = 0, and 𝐮∧𝐱 ∥ = 𝐱 ∥ ∧𝐮 = 0; and 𝐮⋅𝐱 ∥ = 𝐱 ∥ ⋅ 𝐮, and 𝐮∧𝐱 ⊥ = −𝐱 ⊥ ∧𝐮, Hence (5.173) ′ 𝐮𝐱 = 𝐮⋅𝐱 ∥ + 𝐮∧𝐱 ⊥ ⇔ 𝐱𝐮 = 𝐮⋅𝐱 ∥ − 𝐮∧𝐱 ⊥ ⇔ 𝐱 = 𝐮𝐱𝐮 = 𝐮𝐮⋅𝐱 ∥ − 𝐮𝐮∧𝐱 ⊥ = 𝐱 ∥ − 𝐱 ⊥ .

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 193 –

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.4. Transformation of Geometric 1-vectors in the Euclidean plan – 5.4.2.2. Reflection Along a Geometric 1-vector

Multiplying the reflected 1-vector 𝐱 ′ = 𝐮𝐱𝐮 with the factor −1, we get the reflection image 𝐱 ′′ = −𝐮𝐱𝐮 = −𝐱 ′ , called the reflection of a 1-vector 𝐱 along the given 1-vector 𝐮. We see the mirror image 𝐱 ′′ through a line ℓ⊥𝐮 or a plane 𝛾⊥𝐮 perpendicular⊥ (transversal) to the given 1-vector 𝐮. The bivector plane γu∧x spanned by 𝐮∧𝐱 is the paper plane of Figure 5.43, in which all the 1-vectors 𝐮, 𝐱, 𝐱 ′ , 𝐱 ′′ exist as objects. We introduce another plane subject γ⊥u , called a normal plane to the 1-vector 𝐮, a so called reflecting plane. The reflection of 𝐱 around 𝐮 on this reflecting plane γ⊥u is then 𝐱 ′ =𝐮𝐱𝐮, and through the plane γ we see the mirror image (5.177) 𝐱 ′′ = − 𝐮𝐱𝐮 = −𝐱 ′ Figure 5.43 Reflection in a normal plane  to 𝐮 Conversely, given a plane 𝛾 surface object in space 𝔊, this can is equivalent to a reflection along 𝐮. be assigned a normal 1-vector 𝐮 perpendicular 𝐮⊥γ to the plane and normalized |𝐮|=1, thus generating the reflections ±𝐮𝐱𝐮 of any arbitrary 1-vector 𝐱. Note that this simplest algebraic form for a pqg-2 quality reflection has two possible orientations of its outcome ±𝐮𝐱𝐮. 5.4.2.3. Reflection Through a Non-normalized 1-vector

The two opposite orientated reflection formulas ±𝐮𝐱𝐮 both ‘sandwiching’ the arbitrary 1-vector 𝐱 between the given normalized reflection 1-vector 𝐮. If the given 1-vector is not normalized 𝐚 = |𝐚|𝐚̂ = |𝐚|𝐮, the reflection formulas are simply written (5.178) ±𝐚−1 𝐱𝐚 = ±𝐚𝐱𝐚−1 = ±𝐚̂𝐱𝐚̂ ~ ±𝐮𝐱𝐮. (for + see 𝐚 Figure 5.44.) 5.4.3. The Projection Operator From one 1-vector to Another 1-vector

(5.179) (5.180) (5.181) (5.182) (5.183)

(5.184)

Having any given 1-vector 𝐚 with a unit direction as 𝐮 = 𝐚̂ = 𝐚⁄|𝐚|, we will create the projection of each arbitrary 1-vector direction 𝐱 in space on this 1-vector direction 𝐚̂ of 𝐚, just as (6.173). We use the inverse of 𝐚 (5.69) 𝐚−1 = 𝐚⁄𝐚2 ⇒ 𝐚 𝐚−1 =1. The product of these two 1-vectors is 𝐱𝐚 = 𝐱⋅𝐚 + 𝐱∧𝐚 ⟹ 𝐱 = 𝐱 𝐚 𝐚−1 = (𝐱⋅𝐚 + 𝐱∧𝐚)𝐚−1 = (𝐱⋅𝐚)𝐚−1 + (𝐱∧𝐚)𝐚−1 = 𝐱 ∥𝐚 + 𝐱 ⊥𝐚 , where we have divided 𝐚 out again and achieved the parallel component (see §5.2.2.3 scalar product) 𝐱 ∥𝐚 ≡ 𝐚−1 𝐚⋅𝐱 ≡ 𝐚−1 (𝐚⋅𝐱) = (𝐱⋅𝐚)𝐚−1 = 𝐱 − (𝐱∧𝐚)𝐚−1. We note the parallel symmetric, and the orthogonal antisymmetric components | 𝐱 ∥𝐚 𝐚 = 𝐱⋅𝐚 = ½(𝐱 𝐚 + 𝐚 𝐱) = 𝐚⋅𝐱 = ½(𝐚 𝐱 + 𝐱 𝐚) = 𝐚 𝐱 ∥𝐚 , 𝐱 ⊥𝐚 𝐚 = 𝐱∧𝐚 = ½(𝐱 𝐚 − 𝐚 𝐱) = −𝐚∧𝐱 = −𝐚 𝐱 ⊥𝐚. We left multiply with 𝐚−1 in the last form of the symmetry in (5.181) 𝐱 ∥𝐚 = 𝐚−1 𝐚 𝐱 ∥𝐚 = ½ 𝐚−1 (𝐚 𝐱 + 𝐱 𝐚) = ½(𝐱 + 𝐚−1 𝐱 𝐚). This last is illustrated as objects for the intuition in Figure 5.44. Using (5.180) we have now introduced a projection operator 𝑃𝐚 𝐱 ≡ 𝐚−1 𝐚⋅𝐱 = (𝐱⋅𝐚)𝐚−1 = 𝑃𝐚 (𝐱) = ½(𝐱 + 𝐚−1 𝐱 𝐚) ,

that is an linear transformation inside an Euclidean space [10]p.253. Such projection is an idempotent linear transformation operation (5.185) 𝑃𝐚2 = 𝑃𝐚 ⟺ 𝑃𝐚 (𝑃𝐚 (𝐱)) = 𝑃𝐚 (𝐱), Figure 5.44. The projection of 𝐱 along 𝐚 intuited that says, one projection has impact, further as 𝑃𝐚 (𝐱) = 𝐱 ∥𝐚 = ½(𝐱 + 𝐚−1 𝐱 𝐚) = ½(𝐱+𝐱′). projections on 𝐚 has no impact. ©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 194 –

A Research on the a priori of Physics –

December 2020

– 5.4.4. Reflection in a Plane Surface as a Physical Process – 5.4.2.3 Reflection Through a Non-normalized 1-vector –

5.4.4. Reflection in a Plane Surface as a Physical Process

Observing a particle 𝛹𝐩 direction quantity pqg-1-vector 𝐩 (such as light), which we consider as reflected from another physical entity 𝛹γ , that we assign a mirroring plane γ or just a mirroring line ℓ ⊂γ such that we assume that the incident particle is given 𝐩𝑖𝑛 = −𝐮𝐩𝐮, as shown in Figure 5.45, and assuming 𝐮⊥γ ⇒ 𝐮⊥ℓ is the normal 1-vector 𝐮 that perpendicular defines the direction of both ℓ and γ for the surface of the reflecting entity 𝛹γ . We have from the quantum concept that |𝐩𝑖𝑛 | = |𝐩|. We look at the change of the particle 1-vector255 (5.186)

Figure 5.45 Particle reflection inside the plane 𝛾𝐮∧𝐩 = 𝛾𝐮𝐩 by the normal 1-vector 𝐮 to a reflecting line ℓ in the plane 𝛾⊥𝐮 of a reflecting surface 𝛹𝛾 .

Δ𝐩

Δ𝐩 = 𝐩 − 𝐩𝑖𝑛 , which has the direction 𝐮 = |Δ𝐩| .

The simplest linear transformation that can be formed along a finite 1-vector Δ𝐩 = |Δ𝐩|𝐮, (Δ𝐩≠𝟎) or just along its normalized 1-vector 𝐮 is written as (5.187) 𝒰𝐩 = 𝐩in = −𝐮𝐩𝐮 = −Δ𝐩−1 𝐩Δ𝐩 From (5.170) we have 𝐩𝑖𝑛 = −𝐮𝐩𝐮 = −𝑈𝐩𝑈 † = −𝑈𝑈𝐩 = −𝑈 2 𝐩 = 𝒰𝐩, where the 𝐮𝐩 ̂ = |𝐩| = 𝑒 𝒊𝜃 works twice 𝑈𝜃2 in the same plane. 1-rotor 𝑈𝜃 = 𝐮𝐩 The reflection transformation is then expressed as its cause 𝐩in = 𝒰𝐩 = −𝑈𝜃2 𝐩= − 𝑒 𝒊2𝜃 𝐩, which is called an irregular rotation. The incident angle of reflection is equal to the angle of departure (5.189) 𝜃 = 𝜃𝑜𝑢𝑡 = 𝜃𝑖𝑛 , and the total angle is 2𝜃. This entire process in this physics takes place through the plane of one and the same reflection plane 𝛾𝐮𝐩 , spanned by the 2-multi-vector 𝐮𝐩 or just256 the bivector 𝐮∧𝐩. All the 1-vector objects 𝐩, 𝐩𝑖𝑛 , Δ𝐩, and 𝐮⊥ℓ for the intuition257 exist in that foundation plane 𝛾𝐮𝐩 of Figure 5.45. We may say that reflection takes place along the 1-vector 𝐮. For reflection we can introduce a mirror plane γ with 𝐮 as the normal 1-vector direction, as well as the mirroring line ℓ ⊂ γ. This plane must have its cause in the intended physical entity 𝛹γ . That reflecting mirror plane γ for the intuition is perpendicular to the reflection plane γup, γ⊥γup , and therefor outside the paper plane of Figure 5.45. This γ is called the normal plane of the 1-vector 𝐮 direction. We would often prefer to call a plane γ⊥𝐮 for the transversal plane of a 1-vector 𝐮 direction. But by this we have introduced an additional external dimension to the Figure 5.45 plane of the reflection objects 𝐮 etc. This pqg-2 concept of a transversal plane is just dual exterior transversal to a pqg-1-vector, which will be treated later below in section 6.2.4. (5.188)

Throughout this chapter 5.4 we have seen 1-vectors, bivectors and 2-multi-vectors as existing in the same plane, namely the bivector-co-plane γu∧p = γup. Therefor we have intuited the reflection concept purely in this foundation plane of Figure 5.45. Then the reflecting plane is seen purely as a line ℓ ⊂ γ for our intuition. 255

We ignore the recoil changes Δ𝐩γ for the physical entity 𝛹γ , Δ𝐩γ +Δ𝐩 = 0, since the reference frame for the system is 𝛹𝛾 . Since the pqg-0 scalar 𝐮⋅𝐩 don’t affect the pqg-2 direction of the plane. 257 The knowledge 2𝐩⊥ = 𝐩 + 𝐩𝑖𝑛 , 𝐮⋅𝐩⊥ =0 is irrelevant to this interpretation of a reflection by a geometric multiplication algebra. 256

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 195 –

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.4. Transformation of Geometric 1-vectors in the Euclidean plan –

5.4.5. Rotation Inside one and the Same Plane Direction

Two different normalized 1-vectors 𝐮 and 𝐯 where 𝐯≠𝐮, |u|=|v|=1 form a bivector 𝐯∧𝐮, which span a plane 𝜆𝐮𝐯 . The product of these two also form a 2-multi-vector 𝑈= 𝐯𝐮, that works in the rotor plane 𝜆𝐮𝐯 , which is the plane of Figure 5.46 in which, there too are indicated two reflecting planes γ⊥𝐮 ⊥𝐮 and γ⊥𝐯 ⊥𝐯 as smalldotted lines for the intuition. These are transversal normal planes to each of the 1-vectors 𝐮 and 𝐯. These two plane objects γ⊥𝐮 ⊥𝜆𝐮𝐯 and γ⊥𝐯 ⊥𝜆𝐮𝐯 intersect each other in one straight line with the ̂⊥𝜆𝐮𝐯 , normal to the paper pointing towards direction ω the observer, who is seeded in a remote∞ origo and looks at the plane as an intuiting interpreter of the Figure 5.46 plane. We consider any arbitrary 1-vector 𝐱, in space 𝔊 whose projection on the 𝜆𝐮𝐯 plane is showed Figure 5.46. Figure 5.46 ℛ , Rotation in a plane. 𝐱 can be reflected around 𝐮, then we get 𝐱 ′ = 𝐮𝐱𝐮, or reflected along 𝐮, constituted of so that 𝐱"= 𝐮𝐱𝐮 = 𝒰𝐱 (dashed), which is called an irregular rotation. two reflections. Hereafter 𝐱 ′ is reflected around 𝐯, we get 𝐯𝐱 ′ 𝐯 , or 𝐱" reflected along 𝐯, we achieve −𝐯𝐱"v=𝒱 𝐱" , as another irregular rotation. We compose these two reflections to one regular rotation ℛ = 𝒱 𝒰 (by double sandwiching) (5.190)

(5.191)

ℛ𝐱 = 𝑈𝐱𝑈 † =𝐯𝐮𝐱𝐮𝐯 = −𝐯𝐱 ′′ 𝐯 = −𝐯(−𝐮𝐱𝐮)𝐯 alternative = 𝐯𝐱 ′ 𝐯 = 𝐯(𝐮𝐱𝐮)𝐯 . We see that the ambiguity of reflection through a 1-vector is eliminated by this doublet composition, where the multi-vector concept 𝐯𝐮 is the generator of the transformation. We define a 2-multivector as an operator called a rotor 𝑈 = 𝐯𝐮 = 𝐯⋅𝐮 + 𝐯∧𝐮 = 𝑒 +½𝛉 = 𝑒 +𝒊𝜃

(5.192)

𝑈 † = 𝐮𝐯 = 𝐯⋅𝐮 − 𝐯∧𝐮 = 𝑒 −½𝛉 = 𝑒 −𝒊𝜃 Here we introduce the bivector ½𝛉 = 𝒊𝜃 that represents the rotor area with direction, as an argument for this 2-multivector exponential function 𝑒 ±½𝛉 . ̂ , refer to (5.8), (5.90). The unit of plane direction is 𝒊=𝛉 Hence, the regular rotation as a linear transformation is

(5.193)

ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 ½𝛉 𝐱 𝑒 ½𝛉 = 𝑒 𝛉 𝐱 = 𝑒 𝒊𝜃 𝐱𝑒 −𝒊𝜃 = 𝑒 𝒊2𝜃 𝐱 The unitary rotor 2-multivector is rotational invariant in its own plane. This means, its pqg-1 directions by the individual 1-vectors 𝐮 and 𝐯 lose their specific meaning, and instead their relative direction area ½𝛉, which is the pqg-2 direction of that rotation area. The 1-rotor is

(5.194)

Figure 5.47 The rotation is a linear angular pqg-2 transformation 𝑈= 𝑒 +𝒊𝜃 .

𝑈𝜃 = 𝑒 +½𝛉 = 𝑒 +𝒊𝜃 , and the regular rotation is ℛ𝐱 = 𝑈𝐱𝑈 † in Figure 5.47, (by rotor sandwiching) If the start 1-vector 𝐱 is in the rotor plane, we can choose 𝑈= uv = u𝐱⁄|𝐱| the resulting 1-vector 𝑈 2 𝐱 = 𝑒 𝒊2𝜃 𝐱 is in that same plane reduced to a reflection (5.170) 𝐱 ′ =𝐮𝐱𝐮 displayed Figure 5.42 where v ∥ 𝐱 is omitted. – The general formulation (5.193) ℛ𝐱 = 𝑈𝐱𝑈 † apply to all vectors in space 𝔊 outside the rotor 𝑈 plane direction, see more below § 6.3.3, Figure 6.12.

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

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A Research on the a priori of Physics –

December 2020

– 5.4.5. Rotation Inside one and the Same Plane Direction – 5.4.5.5 The Duality of Direction –

This form ℛ𝐱 = 𝑈𝐱𝑈 † is often called the canonical form for any orthogonal transformation ℛ. This is the pqg-2 essence of rotation, the primary quality of the second grade for all space 𝔊 of physics. A rotation plane direction is the same subject all over space 𝔊 (translation invariance). This pqg-2 symmetry is an a priori foundation for all space 𝔊 in physics. 5.4.5.2. The Half Angle Rotor of a Euler Rotation

We see that, when we have a 1-rotor (5.194) 𝑈𝜃 = 𝑒 +𝒊𝜃 the full regular rotation has argument with the double angle as ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑈 2 𝐱 = 𝑒 𝒊2𝜃 𝐱. Therefore, we almost always those to write a 1-rotor with a half angle ½𝜑 argument causing the full regular rotation with the full angle 𝜑 in this (5.195)

𝑈𝜑 = 𝑒 +𝒊½𝜑 ⇒ 𝑈𝜑2 = 𝑒 +𝒊𝜑 ⇒

ℛ𝐱 = 𝑈𝜑 𝐱𝑈𝜑† = 𝑒 +𝒊½𝜑 𝐱 𝑒 −𝒊½𝜑 = 𝑈𝜑2 𝐱 = 𝑒 +𝒊𝜑 𝐱 .

This we will call a Euler rotation, and if the rotor 𝑒 ±𝒊½𝜑 is in one fixed plane 𝒊 we call it a 1-rotor. The purpose of half angles ½𝜑 is clear when it comes to Euler rotations in several planes . 5.4.5.3. The Idea of an Active Rotation

As we know from the treatment in chapter 1, 2 and 3, to experience physics something has to happen. There always is an development, and the measure for this is a development parameter 𝑡 given from a rotating circle oscillator with a frequency energy 𝜔. We now see that this circle oscillation is a rotation along an Euclidean plane. We simply write this unitary as a plane rotation operator (5.196) 𝑒 ±𝒊𝜔𝑡 = 𝑒 ±𝒊𝜑 = 𝑈𝜑2 where 𝒊 is the unit bivector direction of the Euclidean rotation plane, and 𝜔𝑡 = 𝜑 is the active quantum mechanical phase angle of the oscillating entity, that describe the development.258 A 1-rotor for an oscillation is often just written 𝑈𝜑 = 𝑒 ±𝒊½𝜑 = 𝑒 ±𝒊½𝜔𝑡 . 5.4.5.4. The Invariant Direction of a Rotor

The rotational invariance of the rotor object 𝑈= vu= v⋅u+v∧u is illustrated by the difference from Figure 5.46 to Figure 5.47 in accordance with the idea in § 5.2.7.4 and Figure 5.23. By normal duality we can characterise the rotation plane 𝜆𝑈 = 𝜆𝐮𝐯 = 𝜆𝐮∧𝐯 = 𝜆𝛉 = 𝜆𝒊 = 𝜆⊥ω̂ ̂ , |ω ̂ |=1 , which is the normal vector to the rotation plane ω ̂ ⊥𝜆𝑈 . by a new 1-vector ω This pqg-1-vector subject is perpendicular to the 'paper' plane in Figure 5.46 and Figure 5.47 and indicates the direction, that light has facing the observer from the rotation plane. – Imagine that you look perpendicular to the figure plane surface right on.259 5.4.5.5. The Duality of Direction

The light you as reader of this book receive and see has two characteristics: ̂ , or • You receive it as a particle expressed as a momentum pqg-1-vector ℏ 𝑐 𝜔ω • You receive it as a transversal plane wave expressed through a unitary development 1-rotor 𝑈𝜔 (𝑡) = 𝑒 ±𝒊𝜃 = 𝑒 ±𝒊𝜔𝑡 ~ be = b⋅e + b∧e. We presume orthogonality b⋅e=0. Then the field bivector F= b∧e ~∥ 𝒊 gives the plane pqg-2 direction of the wave front. These two is mutual dual in accordance with the complementarity principle between particle pqg-1 direction and the plane wave front pqg-2 direction, where the plane rotor is a combination of a scalar pqg-0 quality without direction and the bivector plane direction pqg-2 quality as an even geometric algebra. The idea of a particle picture of an physical entity will be concerned by the odd part of the geometric algebra. More of this below in the following chapters.

258 259

We will below in section 5.7.5 interpret the development as a direction unit called 𝛾0 as a 1-vector in Space-Time-Algebra. ̂ in perspective, causes that the viewing angle loses its meaning. (An abstraction automatic don in your brain.) You can’t see 𝛚 Later in Section 5.7.4 this will give analytical meaning by the null direction of a Lorentz rotation. The duality concept problem will be further analysed in chapter 6.

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– II. . Geometry of Physics – 5. The Plan Concept – 5.5. Inherit Quantities of the Algebra for the Euclidean Geometric Plane Concept –

All the preceding pages 23-197 has first been written in Danish by the author, and primo 2017 translated to English by the sane author. The following chapters below is written direct in English. First chapter 6 about natural space then afterwards followed by chapter 5.5-5.9 expanding the classical plan geometry, finalised by chapter 7 with Geometric Space-Time-Algebra for relations in natural Space.

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– 5.5.3. The Nilpotent Operation – 5.4.5.5 The Duality of Direction –

5.5. Inherit Quantities of the Algebra for the Euclidean Geometric Plane Concept We define the algebraic foundation of a geometric plane concept from two orthonormal basis vectors {𝛔1 , 𝛔2 }. From these two 1-vectors we define the bivector 𝒊= 𝛔2 ∧𝛔1 = 𝛔2 𝛔1 as the outer product of the two unit 1-vectors. 𝒊 is called the unit pseudoscalar for that plane. We form the 4-dimensional linear algebra for the Euclidean plane 𝒢2,0 (ℝ) = 𝒢2 (ℝ) by the generalised 2-multivector form as (5.161) (5.197) 𝑀 = 𝛼 + 𝑥1 𝛔1 +𝑥2 𝛔2 + 𝛽𝛔2 𝛔1 = 𝛼1 + 𝑥1 𝛔1 +𝑥2 𝛔2 + 𝛽𝒊 2 in the plane space spanned from the mixed 2 -dimensional standard basis for the algebraic plane {1, 𝛔1 , 𝛔2 , 𝒊 ≡ 𝛔2 𝛔1 } (5.198) This orthonormal mixed standard basis possess a multiplication structure expressed in Table 5.1. Table 5.1 Multiplication basis for the Euclidian plane algebra 𝒢2,0 (ℝ) with the pseudoscalar unit 𝒊 ≡ 𝛔2 𝛔1 The orthonormal basis {1, 𝛔1 , σ2 , 𝒊} {1, 𝛔1 , 𝛔2 , 𝛔2 𝛔1 } left

right

1 σ1 σ2

1 1 σ1 σ2

𝒊

𝒊

σ1 σ2 𝒊 σ1 σ1 𝒊 1 −𝒊 −σ2 1 σ1 𝒊 σ2 −σ1 −1

𝛔2 ⋅𝛔1 = 0, σ𝑘2 =

1,

𝛔1 𝛔2 = −𝛔2 𝛔1.

left

right

1 σ1 σ2 𝛔2 𝛔1

1 1 σ1 σ2 𝛔2 𝛔1

σ1 σ2 σ1 σ1 1 −𝛔2 𝛔1 𝛔2 𝛔1 1 σ2 −σ1

𝛔2 𝛔1 𝛔2 𝛔1 −σ2 σ1 −1

The possibility of products of two elements expands to all linear combinations of these in the form (5.197) and these 2-multivectors can be multiplied further inside the algebra 𝒢2,0 (ℝ). 5.5.2. The Auto Product Square in the Euclidean plane

A simple product of 2-multivectors is the square of (5.197) (5.199) 𝑀2 = 𝑀(𝑀) = 𝑀𝑀 = (𝛼+𝑥1 𝛔1 +𝑥2 𝛔2 +𝛽𝛔2 𝛔1 )2 = +(𝛼 2 + 𝑥12 + 𝑥22 + 𝛽 2 )1 + (2𝛼𝑥1 )𝛔1 + (2𝛼𝑥2 )𝛔2 + (2𝛼𝛽)𝛔2 𝛔1 𝑀2 = 𝑀(𝑀), this multiplication is an auto operation of a multivector on itself. 5.5.3. The Nilpotent Operation

In some situation the multiple operations with multivectors is vanishing and nihilates. The simplest case is 𝑀(𝑀) = 𝑀2 = 0, hence we in 𝒢2,0 (ℝ) by (5.199) express it as (5.200) 𝑀2 = (𝛼 2 + 𝑥12 + 𝑥22 − 𝛽 2 ) + (2𝛼𝑥1 )𝛔1 + (2𝛼𝑥2 )𝛔2 + (2𝛼𝛽)𝛔2 𝛔1 = 0. For the nihilation, the scalar part has to vanish, therefore an easy solution is to set (5.201) 𝛼 = 0, and 𝛽 2 = 𝑥12 + 𝑥22 ⇒ 𝛽 = ±√𝑥12 +𝑥22 , that works for ∀𝑥𝑘 ∈ℝ, 𝑘=1,2. A nilpotent multivector in the Euclidean plane 𝒢2,0 (ℝ) is then 𝑋 = 𝑥1 𝛔1 + 𝑥2 𝛔2 ±√𝑥12 +𝑥22 𝛔2 𝛔1. Writing the Cartesian 1-vector x = 𝑥1 𝛔1 + 𝑥2 𝛔2 , we can write the nilpotent multivector as (5.203) 𝑋 = x ± |x|𝒊 = 𝛽(u ± 𝒊), with u = x⁄|x| , and 𝛽 2 = x 2 >0, 𝛽 ∈ℝ. A test of this nihilation is due to the fact 𝒊x + x𝒊 = 0 expressing that x ∥ 𝒊 or x∧𝒊 = 𝒊∧x = 0. (5.202)

(5.204)

2

𝑋(𝑋) = 𝑋 2 = x 2 − |x|2 = 0,

this is equivalent to (𝛽(u ± 𝒊)) = 0.

Using the unit 1-vector direction u = x̂ = x⁄|x|, in the plane direction 𝒊 = 𝛔2 𝛔1 with 𝛽 from (5.201) we write (5.203) for the Euclidean plane the nilpotent multivector as (5.205) 𝛽(u ± 𝒊), where u∧𝒊 = 0. ©

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This fundamental nilpotent element is founded on the two Clifford units u2 = 1 and 𝒊2 = −1 . The special case of a Euclidean plane pqg-2 direction 𝒊 for an acting unit 1-vector direction u in that plane, a nilpotent basis-element is (u ± 𝒊). This is from the units |u|=1, and |𝒊|=1, in all (5.206)

𝒩 = (u ± 𝒊) ⟹ 𝒩 2 = 𝒩𝒩= 𝒩(𝒩) = (u ± 𝒊)(u ± 𝒊) = (u ± 𝒊)2 = 0. Figure 5.48 The nilpotent element in a plane is a 1-vector direction additive combined with its magnitude as the orientated pseudoscalar direction area. This is displayed in the paper plane → where the Euclidian plane pseudoscalar has the signature 𝒊2 = −1 and its 1-vector has u2 = 1.

(5.207)

(5.208) (5.209)

(5.210) (5.211) (5.212) (5.213) (5.214)

(5.215)

The nilpotent element is scalable with all real scalars ∀𝛽 ∈ℝ as (5.205): 𝛽 2 (u ± 𝒊)2 =0. The fundamental nilpotent operator as a multivector has no specific inverse (u ± 𝒊)−1 = (u ± 𝒊)⁄(u ± 𝒊)2 = (u ± 𝒊)⁄0 ~ ∞?, although from (4.76) the inverse of unit 1-vector u is u−1 = u⁄u2 = u , due to u2 =1, and the inverse of the unit pseudoscalar 𝒊 is its reversed 𝒊−1 = 𝒊⁄𝒊2 = −𝒊 , due to 𝒊2 = −1 . We see that the 𝒢2 (ℝ) algebra has zero divisors, e.g. called 𝐷 and there exist a 𝑋 so that 𝐷𝑋 = 0 or 𝑋𝐷 = 0. We also recall that multiplication is not universal commutative in 𝒢2 (ℝ). Regarding the aspects of the plane supported from the bivector idea 𝒊 ≡ 𝛔2 𝛔1 where the 1vector idea of u exist we can resolve it in components u = 𝑢1 𝛔1 + 𝑢2 𝛔2 . We are free to choose a basis {𝛔1 , 𝛔2 } of the plane in concern, therefor we chose 𝛔1 ≡ u, and we define a perpendicular orthonormal 1-vector u⊥ direction to u in the plane as 𝛔2 = u⊥ = 𝒊u = 𝛔2 𝛔1 𝛔1 , achieving {1, 𝛔1 , 𝛔2 , 𝒊 ≡ 𝛔2 𝛔1 } = {1, u, u⊥ , 𝒊 ≡ u⊥ u} From this our nilpotence units is expressed (u ± 𝒊) = (𝛔1 ± 𝛔2 𝛔1 ) = (1 ± 𝛔2 )𝛔1 = 𝛔1 (1 ∓ 𝛔2 ) This is in the chosen 1-vector direction 𝛔1 ≡ u that implicit presume a plane for a perpendicular 𝛔2 = u⊥ direction. An alternative choice is the direction 𝛔2 ≡ u and perpendicular 𝛔1 =u⊥: {1, u⊥ ,u, 𝒊 ≡ uu⊥ } (u ± 𝒊) = (𝛔2 ± 𝛔2 𝛔1 ) = 𝛔2 (1 ± 𝛔1 ) = (1 ∓ 𝛔1 )𝛔2 A new test of these commutating relation (5.209) or (5.210) gives for the nilpotence forms (𝛔1 ± 𝛔2 𝛔1 )2 = (1 ± 𝛔2 )𝛔1 𝛔1 (1 ∓ 𝛔2 ) = (1 ± 𝛔2 )(1 ∓ 𝛔2 ) = 0, (𝛔2 ± 𝛔2 𝛔1 )2 = (1 ∓ 𝛔1 )𝛔2 𝛔2 (1 ± 𝛔1 ) = (1 ∓ 𝛔1 )(1 ± 𝛔1 ) = 0, condensed in the general form of a basic zero division (u ± 𝒊)2 = (1 + u⊥ )(1 − u⊥ ) = 0 From this we can conclude to have two mutually annihilating multivectors of the form (1 + u⊥ ) and (1 − u⊥ ), with the requirement u⊥2 = 1 Often when the mutual annihilating multivectors are needed we choose a direction for our cartesian orthonormal basis {𝛔𝑘 } for a Euclidean space, where 𝛔𝑘2 = 1, of the form (1+𝛔𝑘 ) and its parity inverted (1−𝛔𝑘 ). This mutual annihilation exist in one line direction pointed out by 𝛔𝑘 , with two opposite orientations that simply is each other’s parity inverted. (Think of Newtons Third Law for Nature.)

To make an nilpotent element we have to multiply with an orthogonal basis 1-vector 𝛔𝑗 , 𝑗 ≠ 𝑘, (Euclidian cartesian perpendicular in natural space), then we write (1+𝛔𝑘 )𝛔𝑗 = (𝛔𝑗 +𝛔𝑘 𝛔𝑗 ) ~ (u + 𝒊), (5.216) and its reversed orientation (5.217) 𝛔𝑗 (1+𝛔𝑘 ) = (𝛔𝑗 −𝛔𝑘 𝛔𝑗 ) ©

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~ (u − 𝒊). – 200 –

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– 5.5.3. The Nilpotent Operation – 5.5.3.3 The Spanned Spaces of Mutual Annihilation Zero Divisors –

The nilpotent line concept direction 𝛔𝑗 is complemented by the plane concept direction 𝛔𝑘 𝛔𝑗 . Note the independence between the two line directions of mutual annihilation 𝛔𝑘 ~ u⊥ and the directions of nilpotence 𝛔𝑗 ~ u. 5.5.3.2. The Spanned Spaces of Nilpotence Zero Divisors

First: we look at the basis of nilpotent zero devisor (nzd) from (5.205) {(u + 𝒊)}, wherefrom we span a space of all nzd-multivectors in the line direction u inside a plane direction supported by 𝒊 (5.218) span{ u + 𝒊 } = { 𝛽(u + 𝒊) | ∀𝛽 ∈ℝ, 𝐮2 =1, 𝒊2 = −1 }. When we take two arbitrary elements of this nzd space 𝛽1 (u + 𝒊) and 𝛽2 (u + 𝒊) both are mutual zero divisors because (5.219) 𝛽1 (u + 𝒊)𝛽2 (u + 𝒊) = 𝛽1 𝛽2 (u + 𝒊)(u + 𝒊) = 𝛽1 𝛽2 ⋅0 = 0. The same is the case for the nzd space span{ u − 𝒊 } = { 𝛽(u − 𝒊) | ∀𝛽 ∈ℝ, 𝐮2 = 1, 𝒊2 = −1 }. The combination of a 1-vector with its associated plane bivector gives an auto zero divisor. 5.5.3.3. The Spanned Spaces of Mutual Annihilation Zero Divisors

Second: we look at the basis of mutual annihilating zero divisors (mazd) from (5.215) or (5.214) (5.220) { (1+𝛔𝑘 ), (1−𝛔𝑘 ) } or simply { (1 + u⊥ ), (1 − u⊥ ) }. This 2-dimensional mazd-basis is orthogonal in that (1+𝛔𝑘 )⋅(1−𝛔𝑘 ) = 0, but certainly not an Euclidean plane, and its direction quality is only one and the same line direction 𝛔𝑘 , grade ≤1. All other directions is then transcendental to this issue. From this basis, a spanned space of mazd-multivectors is constructed (5.221) span{ (1+𝛔𝑘 ), (1−𝛔𝑘 ) } = { (𝛼(1+𝛔𝑘 ), 𝛼(1−𝛔𝑘 )) | ∀𝛼, ∀𝛼 ∈ℝ, 𝛔𝑘2 = 1 }. When we take two arbitrary elements of this spanned mazd-space (𝛼𝑎 (1+𝛔𝑘 ), 𝛼𝑎 (1−𝛔𝑘 ))𝑎 and (𝛼𝑏 (1+𝛔𝑘 ), 𝛼𝑏 (1−𝛔𝑘 ))𝑏 Making the mutual zero divisors product of these perform the annihilation (5.223) 𝛼𝑎 (1+𝛔𝑘 ) 𝛼𝑏 (1−𝛔𝑘 ) = 𝛼𝑎 𝛼𝑏 (1+𝛔𝑘 ) (1−𝛔𝑘 ) = 𝛼𝑎 𝛼𝑏 ⋅0 = 0, or (5.223a) 𝛼𝑏 (1−𝛔𝑘 ) 𝛼𝑎 (1+𝛔𝑘 ) = 𝛼𝑏 𝛼𝑎 (1−𝛔𝑘 ) (1+𝛔𝑘 ) = 𝛼𝑏 𝛼𝑎 ⋅0 = 0, etc.. (5.222)

Here we presume the two scalars 𝛼, and 𝛼 to be complete independent real scaling factors. The mutual annihilation absorbs all scaling factors. The conclusion is, that the combination a 1-vector with its own scalar magnitude (|x| + x) is a mutual zero devisor to all scaled versions of its parity inversion 𝛼(|x| − x). The test of this is (|x| + x)𝛼(|x| − x) = 𝛼(|x|2 − x 2 − |x|x + |x|x) = 𝛼⋅0 = 0, under condition |x|2 = x 2 . (5.224) An extra quality of these mutual zero devisors is, that it absorb multiplication with its own direction 1-vector direction unit x̂ = x⁄|x| (|x| + x)x̂ = |x|x̂ + x x̂ = x + x 2⁄|x| = (x + |x|) = (|x| + x). (5.225) Or simply expressed as (1+𝛔𝑘 )𝛔𝑘 = 𝛔𝑘 (1+𝛔𝑘 ) = (1+𝛔𝑘 ), (5.226) and for the its parity inversed we have the absorb (1−𝛔𝑘 )(−𝛔𝑘 ) = (−𝛔𝑘 )(1−𝛔𝑘 ) = (1−𝛔𝑘 ). (5.227) These two (5.226) and (5.227) can be scaled to any size by 𝛼 ∈ℝ. The physical impact of this will be obvious when we multiply with a information development direction dimension basis unit 𝛾0 in chapter and for space-time.

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5.5.4. The Idempotent Operation

(5.228) (5.229) (5.230) (5.231)

(5.232) (5.233)

Operation by a multivector 𝑀 can have some impact on an entity in physics. An extra operation by the same multivector 𝑀 in some situations have no new effect. This opportunity is called idempotence and for multivector operations expressed as 𝑀(𝑀) = 𝑀𝑀 = 𝑀2 = 𝑀. The idempotent operation is often called a projection 𝑃 where 𝑃(𝑃) = 𝑃𝑃 = 𝑃2 = 𝑃. A projection multivector written in components for the plane basis {1, 𝛔1 , 𝛔2 , 𝒊 ≡ 𝛔2 𝛔1 } (5.198) is 𝑃 = 𝛼 + 𝑥1 𝛔1 + 𝑥2 𝛔2 +𝛽𝛔2 𝛔1 , this is for the demand of idempotence 𝑃2 = (𝛼 2 +𝑥12 +𝑥22 −𝛽 2 )+(2𝛼𝑥1 )𝛔1 +(2𝛼𝑥2 )𝛔2 +(2𝛼𝛽)𝛔2 𝛔1 = 𝛼+𝑥1 𝛔1 +𝑥2 𝛔2 +𝛽𝛔2 𝛔1 = 𝑃. The scalar part has to be preserved, as well as each independent direction (𝛼 2 +𝑥12 + 𝑥22 −𝛽 2 ) = 𝛼, (2𝛼𝑥1 ) = 𝑥1 , (2𝛼𝑥1 ) = 𝑥1 , (2𝛼𝛽) = 𝛽. The over simplest solution 𝛼=0 demand 𝑥1 = 𝑥2 = 𝛽 = 0, is nothing. The obvious possibility for idempotence is 𝛼 = ½ and 𝑥12 + 𝑥22 − 𝛽 2 = ¼ ⟹ 𝛽 = ±√𝑥12 +𝑥22 − ¼ . Then we have all the possibilities of projection operators .

𝑃 = ½ + 𝑥1 𝛔1 + 𝑥2 𝛔2 ± √𝑥12 +𝑥22 − ¼ 𝛔2 𝛔1. Again, using the Cartesian 1-vector x = 𝑥1 𝛔1 + 𝑥2 𝛔2 we get for all arbitrary 1-vector x, with |x| ≥ ½ in the plane direction 𝒊 ≡ 𝛔2 𝛔1 the idempotent projection element

𝑃 = (½+x) ± 𝒊√x 2 − ¼ . Taking the arbitrary unit vector direction u, with u2 =1 ⇒ |u|=1 and setting x = ±½u in (5.234) we get writ of the plane part 𝒊 and have the simplest primary idempotent projection operators (5.235) 𝑃u = 𝑃+ = ½(1 + u), and its Clifford conjugated 𝑃u = ̃ 𝑃u = 𝑃− = ½(1 − u), for a unit u2 =1 direction 1-vector u in space independent of any specific plane. – Anyway, defined generally u ≠ ±1, this unit is indeed not a member of any scalar field u ∉ ℝ, ℂ, or 𝕂, but in our primitive definition, u is certainly a member of a 1-vector field with a division algebra. Do we restrict the maximal grade to one for the algebra form ⟨𝐴⟩0 +⟨𝐴⟩1 we can consider u ~⟨𝐴⟩1 as the pseudoscalar u2 =1 of this algebra, with no revered orientation, but the Clifford conjugated260 (5.234)

(5.236)

2

u = −u, with u =1 The primary idempotent member 𝑃u of the multivector algebra of grades ⟨𝐴⟩0 +⟨𝐴⟩1 has no inverse, and any real scaled dilation 𝛼𝑃u with idempotence (𝛼𝑃u )2 = 𝛼 2 𝑃u has no inverse (𝛼𝑃u )−1 , because: 1 1 (1−u) 1−u 1−u 1−u (1 + u)−1 = (5.237) = = = = , that is undefined. 2 2 ( ) ( ) 1+u 1+u 1−u 1− 1 0 1 −u These 𝛼𝑃u is therefore not members in a associative division algebra. Multiplying a dilated primary idempotent projection operator by its own direction unit 1-vector make no change to the projection operator (5.238) 𝛼𝑃u u = u𝛼𝑃u = 𝛼𝑃u = ½𝛼(1 + u)u = ½ 𝛼(u + 1), 𝛼 ∈ℝ. We say 𝛼𝑃u = ½𝛼(1 + u) has the quality of absorbing factors of its own direction unit u. To count u as a pseudoscalar we need to restrict the algebra to one geometric line direction towards infinity. I.e. any proper 1-vector p ~⟨𝐴⟩1 has to fulfil p ∧ u = 0, which result in a line span (5.239) p = 𝜆1 u, for ∀𝜆1 ∈ℝ, from basis {u}. The scalar 1 in (5.235) is defined as the magnitude of u, |u|= u2 =1. See co-linear concept §4.4.4.1. 260

For grade 1-vectors the Clifford conjugated is the same as the parity inversion u ̃ = u = −u, in that u† = u for reversion.

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A Research on the a priori of Physics –

December 2020

– 5.5.4. The Idempotent Operation – 5.5.4.3 The Projection of a Paravector on Its Idempotent Basis – 5.5.4.2. The Paravector Concept

The full mixed standard basis for multivectors of grade form ⟨𝐴⟩0 +⟨𝐴⟩1 , then has the form {1, u}. In this basis we span multivectors of grades ≤1 we call paravectors261 expressed in the form (5.240) 𝓅 = 𝜆0 1 + 𝜆u u = 𝜆0 + p, where p = 𝜆1 u = 𝜆1 p̂ , and we use the real component coordinates 𝜆0 , 𝜆1 ∈ℝ. This paravector has a Clifford conjugated (or here just first grade parity inversion) (5.241) 𝓅 = 𝜆0 1 − 𝜆u u = 𝜆0 − p. Such a multivector subject consist in substance of a direction as a primary quality of first grade (pqg-1) together with a scalar as a primary quality of zero grade (pqg-0) with no-direction as a surplus scalar para idea to the Descartes extension pqg-1 idea. As object we have an extended vector p = 𝜆u u that can be drawn on a surface (paper) as an arrow for the intuition, to analogue indicate a natural direction u with extension magnitude |p|=|𝜆1 | ≥0. But we do not have any intuition object of the scalar part 𝜆0 . From the object unit u2 =1 direction u we understand the geometric linear direction (parity inversion) p = −p of orientation for the Clifford conjugation. For intuit interpretation consult drawings above at § 4.4.2.5 at (4.60)-(4.61). Of course the paravector can exist in higher dimensions, e.g. for a plane p = 𝜆1 𝛔1 + 𝜆2 𝛔2 implying the paravector 𝓅 = 𝜆0 + p = 𝜆0 1 + 𝜆1 𝛔1 +𝜆2 𝛔2 from a basis {1, 𝛔1 , 𝛔2 }, but this basis imply further the ⟨𝐴⟩2 pseudoscalar 𝒊 ≡ 𝛔2 𝛔1 as (5.198) which we here try to avoid.262 Therefore, we settle of from the simple basis {1, u} to make the fundamental structure for start off direction as a primary quality of first grade (pqg-1) loud and clear.263 - Anyway, alternative we remember Immanuel Kant 1768 [11] p.361-72, (der Gegenden im Raume). 5.5.4.3. The Projection of a Paravector on Its Idempotent Basis

The paravector structure is also, as by Garret Sobczyk [14] Chap.2, called hyperbolic numbers. The defining foundation of the paravector ide is the basis {1, u} that support each paravector as (5.242) 𝓅 = 𝜆0 1 + 𝜆u u, and its Clifford conjugated 𝓅 = 𝜆0 1 − 𝜆u u For investigating the structure of this paravector idea we will reformulate it in the idempotent orthogonal basis {𝑃u , 𝑃u } defined from (5.235) as the simplest primary idempotent quality basis set (5.243)

𝑃u2 = 𝑃u = ½(1 + u),

and its Clifford conjugated

2

𝑃u = 𝑃u = ½(1 − u),

From this we express its spectral decomposition projection in this basis as 𝓅 = 𝜆+ 𝑃u + 𝜆− 𝑃u , and 𝓅 = 𝜆+ 𝑃u + 𝜆− 𝑃u , with the component coordinates for the paravector projections (5.245) 𝜆+ = (𝜆0 + 𝜆u ) and 𝜆− = (𝜆0 −𝜆u ) ∈ℝ. Conversely from these the coordinates from definition basis {1, u} for (5.242) is (5.246) 𝜆0 = ½(𝜆+ + 𝜆− ) and 𝜆1 = ½(𝜆+ −𝜆− ) ∈ℝ. (5.244)

{𝑃u , 𝑃u } is orthogonal because ½(1+u)½(1‒ u) =0, due to first definition (5.56)-(5.59) we have (5.247)

𝑃u 𝑃u = 𝑃u ⋅ 𝑃u = 𝑃u ∧ 𝑃u = 0. This condition also show that the two basis paravectors are mutual annihilating operations, and we note their sum 𝑃u + 𝑃u = 1 is the unit scalar; and their difference 𝑃u − 𝑃u = u is the direction quality of first grade.

261

The name paravector is taken from William E. Baylis [37], [36], (First named by J. G. Maks, Ph.D. thesis, TU Delft, 1989.) Later below we look into ℨ space with a paravector as 𝓅= 𝜆0 +p = 𝜆0 1+𝜆1 𝛔1 +𝜆2 𝛔2 +𝜆3 𝛔3 , implying higher grades ⟨𝐴⟩2 , ⟨𝐴⟩3 , etc. 263 To set the straight-line direction unit basis u of a paravector basis {1, u} in perspective we take the existence in a possible plane and multiply by a perpendicular unit u⊥ as in (5.208) and get {1, u, u⊥ , 𝒊≡u⊥ u}. Then the paravector 𝓅= 𝜆0 1+𝜆u u ⟶ go to a multivector u⊥ 𝓅 = 𝜆0 u⊥ + 𝜆u u⊥ u, that is nilpotent if 𝜆u = ±𝜆0 . 262

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 203 –

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.5. Inherit Quantities of the Algebra for the Euclidean Geometric Plane Concept –

From all this we achieve the spectral projective properties together with

(5.248)

𝓅𝑃u = 𝑃u 𝓅 = 𝜆+ 𝑃u ,

(5.249)

𝓅𝑃u = 𝑃u 𝓅 = 𝜆− 𝑃u , and 𝓅𝑃u = 𝑃u 𝓅 = 𝜆+ 𝑃u . 2 2 𝜆+ 𝜆− = (𝜆0 +𝜆u )(𝜆0 −𝜆u )= 𝜆0 − 𝜆u ∈ℝ, a scalar given direct from the components (5.245).

(5.250)

and

𝓅𝑃u = 𝑃u 𝓅 = 𝜆− 𝑃u ,

The product of two paravectors 𝓅 = 𝜆+ 𝑃u +𝜆− 𝑃u and 𝓅′ = 𝜆′+ 𝑃u + 𝜆′− 𝑃u is (5.251)

𝓅𝓅′ = (𝜆+ 𝑃u + 𝜆− 𝑃u )(𝜆′+ 𝑃u + 𝜆′− 𝑃u ) = (𝜆+ 𝜆′+ )𝑃u + (𝜆− 𝜆′− )𝑃u, The paravector auto-product square is

(5.252)

𝓅𝓅 = (𝜆+ 𝑃u + 𝜆− 𝑃u )(𝜆+ 𝑃u + 𝜆− 𝑃u ) = 𝜆+2 𝑃u + 𝜆−2 𝑃u. What about the observable magnitude of a paravector? We test the basis auto product square 𝑃u2 = 𝑃u it is certainly not a scalar for the magnitude. Instead we will use the Clifford conjugation quadratic form by using the scalar (5.250)

𝓅𝓅 = (𝜆+ 𝑃u + 𝜆− 𝑃u )(𝜆+ 𝑃u + 𝜆− 𝑃u ) = 𝜆+ 𝜆− (𝑃u + 𝑃u ) = 𝜆+ 𝜆− = 𝜆02 − 𝜆u2 ∈ℝ This is in full agreement with the product of (5.240) and (5.241) in the definition basis {1, u} (5.254) 𝓅𝓅 = 𝓅𝓅 = 𝜆02 − p 2 = (𝜆0 + p)(𝜆0 − p) = (𝜆0 1+𝜆u u)(𝜆0 1−𝜆u u) = 𝜆02 − 𝜆u2 ∈ℝ, that yields the observable pure magnitude for the paravector |𝓅|0 = |𝓅|0 = √|𝜆02 −𝜆u2 | ≥ 0 . (5.255) From this we conclude that the projection basis {𝑃u , 𝑃u } consist of null basis vectors (5.253)

(5.256)

𝑃u 𝑃u = 0 ⟹

|𝑃u |0 = |𝑃u |0 = 0

This (5.254) indicate a Clifford algebra structure with a Minkowski signature metric. – To immediately continue with Minkowski metric jump to section 5.7. 5.5.4.4. Non Measurable Fictive Magnitude of Paravectors

When we follow the definition of multivector magnitude given by Hestenesp.46 as (5.166)(5.167) for the paravector 𝓅 = 𝜆0 1 + 𝜆u u = 〈𝓅〉0 + 〈𝓅〉1 = 〈𝜆0 1〉0 + 〈𝜆u u〉1 we have 2 2 2 2 2 |𝓅| = |〈𝜆0 1〉0 | + |〈𝜆u u〉1 | = 𝜆0 + 𝜆u . (5.257) This is a simple pure mathematic defined magnitude. Hestenes as in [13]p.13(1.51) made the simple assumption that ⟨𝐴† 𝐴⟩0 ≥ 0. (5.257) The question is, will this paravector idea magnitude be measurable in physics? When it comes to Clifford conjugation ⟨𝐴̃ 𝐴⟩0 and special parity inversion ⟨𝐴̅ 𝐴⟩0 we have the ̅p⟩0 = −|p|2 and for the unit ⟨u̅ u⟩0 = u̅ u = −u2 = −1, but simple reality that for 1-vector ⟨p we know that the Euclidian 1-vector has as u2 = 1 the magnitude ⟨p p⟩½ = ⟨p † p⟩½ = |p| = |𝜆u |, that is independent of the scalar quality 𝜆0 magnitude |𝜆0 |. This observable pure magnitude measure of an object 1-vector have extension direction, though its founded in the idea of the formulation in (5.254)-(5.255). But here the outstanding question is, what direction count dos the pure scalar represent in physics? Will the quantity √𝜆02 + 𝜆u2 appear in reality? It is obvious that it is not an intuitive object per se. The simplest way is to use demand the synthetic judgment 𝜆02 ≥ 𝜆u2 ⟹ |𝜆0 | ≥ |p| |𝓅|02 → 〈𝓅𝓅〉0 = 𝜆02 − 𝜆u2 ≥ 0 (5.258) This will treaded further below in section 5.7, and later chapter III. 7. Anyway, when it comes to geometric algebraic bivectors the reversion product is positive ⟨B̃ B⟩0 = ⟨B † B⟩0 = B † B ≥ 0. ©

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– 5.6.1. The Fundamentals of Matrices in a Plane Algebra 𝒢2 – 5.6.1.2 Examples of Matrices of Geometric Multivectors –

5.6. The Real Matrix Representation for the Plane Concept 5.6.1. The Fundamentals of Matrices in a Plane Algebra 𝒢2 5.6.1.1. Matrices for a Cartesian 1-vector Concept for an Euclidean plane ℝ12

(5.259) (5.260) (5.261) (5.262)

(5.263)

(5.264)

(5.265)

(5.266)

(5.267)

The orthonormal Cartesian basis set we name {𝛔1 , 𝛔2 } as the directions in space for our plan. These objects we associate with 2-tuples, first as columns, next as rows 1 0 𝛔1 ↔ ( ), 𝛔2 ↔ ( ) , and 𝛔1 ↔ (1 0), 𝛔2 ↔ (0 1). 0 1 The abstract 2-tuple basis is then with the geometric directions implicit hidden {𝛔1 , 𝛔2 } ↔ { (1) , (0) } {𝛔1 , 𝛔2 } ↔ { (1 0), (0 1) } and 0 1 A general Cartesian 1-vectors in the plane we express as (5.136) x = 𝑥1 𝛔1 + 𝑥2 𝛔2 , with 𝑥𝑘 ∈ℝ, from the perpendicular basis objects 𝛔2 ⊥𝛔1 . 264 As 2-tuples, first as columns, next as rows 1 1 𝑥1 x ⟷ (𝑥 ) = (𝑥 2 ) and x ⟷ (𝑥1 𝑥2 ), where (𝑥1 𝑥2 )T = (𝑥 2 ). 2 𝑥 𝑥 Then the coordinates is performed as an inner product265 of such matrix 2-tuples, row⋅column 1 1 0 0 𝑥1 = x⋅𝛔1 = x⋅ ( ) = (𝑥1 𝑥2 )⋅ ( ) and 𝑥2 = x⋅𝛔2 = x⋅ ( ) = (𝑥1 𝑥2 )⋅ ( ). 0 0 1 1 Orthonormality is expressed as 1 1 0 𝛔2 ⋅𝛔1 = 𝛔1 ⋅𝛔2 = (0 1)⋅ ( ) = (1 0)⋅ ( ) = 0, and e. g. 𝛔12 = (1 0)⋅ ( ) = 1 . 0 0 1 From (5.261)-(5.262) we define two arbitrary 1-vectors in in the {𝛔1 , 𝛔2 } plane 1 𝐚 ↔ (𝑎 2 ) = 𝑎 𝑗 and 𝐛 ↔ (𝑏1 𝑏2 ) = 𝑏𝑘 , for 𝑗, 𝑘 = 1,2. 𝑎 We form the product of the two matrix 2-tuples an see the differences First the inner scalar product is exactly the same as for the 1-vectors 𝐛⋅𝐚 ∈ℝ 1 (𝑏1 𝑏2 )⋅ (𝑎2 ) = 𝑏1 𝑎1 + 𝑏2 𝑎2 = 𝑏𝑘 𝑎 𝑗 ∈ℝ. 𝑎 But we have a totally different tuple epistemology than the geometric products of these 1-vectors 𝐛𝐚 or 𝐚𝐛 as above (5.56), when we form the product of the matrix 2-tuples to a 2×2 matrix as 1 𝑐 11 𝑐 12 𝑎1 𝑏 𝑎1 𝑏2 𝑗 𝑗 (𝑎2 ) (𝑏1 𝑏2 ) = [ 2 1 ] = 𝑎 𝑏 = [ ] = 𝑐 𝑘, for 𝑗, 𝑘 = 1,2. 𝑘 𝑎 𝑏1 𝑎2 𝑏2 𝑎 𝑐 21 𝑐 12

5.6.1.2. Examples of Matrices of Geometric Multivectors

In an Euclidean Cartesian plane, we define a set of 1-vectors (5.268) 𝐚𝑗 = 𝑎1,𝑗 𝛔1 + 𝑎2,𝑗 𝛔2 = 𝑎𝑘𝑗 𝛔𝑘 ∈ 𝒢2 (ℝ), with 𝑎𝑘,𝑗 = 𝑎𝑘𝑗 ∈ℝ, 𝑘=1,2, for Cartesian plane. From this for 𝑚 ∈ℕ we form a m-row matrix (5.269)

[𝐚](𝑚) = (𝐚1 , 𝐚2 , … 𝐚𝑚 )

↔ [

From this we form its transposed m-column matrix a1 2 [𝐚]T(𝑚) = (𝐚1 , 𝐚2 , … 𝐚𝑚 )T = [𝐚](𝑚) = ( a ) (5.270) ⋮ a𝑚

𝑎1 1 𝑎2 1

𝑎11 2 ↔ 𝑎1 ⋮ [𝑎1𝑚

𝑎1 2 𝑎2 2 𝑎21 𝑎22 ⋮ 𝑎2𝑚 ]

⋯ ⋯

𝑎1 𝑚 ]. 𝑎2 𝑚

𝑎1,1 𝑎1,2 ~[ ⋮ 𝑎1,𝑚

𝑎2,1 𝑎2,2 ⋮ ]. 𝑎2,𝑚

264

Generally upper index 𝑥 𝑘 is used for column numbers to indicate contravariance opposite covariance or row numbers 𝑥𝑘 . In the orthogonal case there is no difference 𝑥 𝑘 =𝑥𝑘 (do not mistake 𝑘 in 𝑥 𝑘 for exponents). 265 We presume the reader is familiar with the rules of matrix multiplication. For this, I here have recalled it from [14] Chapter 4.

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December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.6. The Real Matrix Representation for the Plane Concept – 5.6.1.3. The Matrices of the Geometric Algebra 𝒢2(ℝ)

(5.271) (5.272)

(5.273)

(5.274)

How can matrices represent the direction concept just as geometric multivectors do? We recall (5.198) the plane space concept supported by the mixed 22 -dimensional standard basis {1, 𝛔1 , 𝛔2 , 𝛔2 𝛔1 }, ( 𝒊 ≡ 𝛔2 𝛔1 ) The general form of a plane geometric multivector (5.197) is written 𝐺 = 𝛼1 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 ∈ 𝒢2 (ℝ), where 𝛼, 𝜈1 , 𝜈2 , 𝛽 ∈ℝ. The 1-vectors 𝜈𝑘 𝛔𝑘 mutual anticommute, as with the pseudoscalar bivector 𝛽𝛔2 𝛔1 too, but all scalars commute with all elements. We prerequisite 𝛔12 = 𝛔22 = 1 and 𝛔2 ⋅𝛔1 = 0. As a complement to the cartesian form266 x = 𝜈1 𝛔1 + 𝜈2 𝛔2 supported from {𝛔1 , 𝛔2 } we introduce from the paravector idea the mixed 2-tuple matrices as column or row editions 1 1 ( ) or (1 𝛔1 ) , ( ) or (1 𝛔2 ). together with 𝛔1 𝛔2 We take the first two 2-tuple forms as our master and ignore implicit the last two, and instead use the idempotent projection form (5.235) for this projected direction ̃+ = 𝑃+ = 𝑃− = ½(1 − 𝛔2 )| . 𝑃+ = ½(1 + 𝛔2 )| and its Clifford conjugated 𝑃 These two conjugated are mutual annihilating 𝑃+ 𝑃− = 0. (Refer to (5.247) and (5.223)) Inspired from the reflection in a direction 1-vector § 5.4.2.1 we will as in [14] p.79 let 𝑃+ act on the row form (1 𝛔1 ) and then let (1 𝛔1 )T left act on this result in the canonical way

(5.275)

(

𝑃+ 𝑃+ 1 ) 𝑃 (1 𝛔1 ) = ( ) (1 𝛔1 ) = [ 𝛔1 + 𝛔1 𝑃+ 𝛔1 𝑃+

𝑃+ 𝛔1 𝑃+ ]= [ 𝛔1 𝑃+ 𝛔1 𝛔1 𝑃+

𝛔1 𝑃− ], 𝑃−

to get a 2×2 matrix form for a natural plane geometric basis, we as Sobczyk call a spectral basis. This 𝒢2 (ℝ) plane matrix basis represent the support of 4 = 22 -dimensions of real numbers similar to the geometric algebraic form (5.272), but it will have different properties that we certainly not called coordinates but matrix elements, e.g. 𝑔11 , 𝑔12 , 𝑔21 , 𝑔22 ∈ℝ. From this idea we will map the geometric multivector 𝐺 (5.272) to a 2×2 real matrix 𝑔11 𝑔12 (5.276) 𝐺 ⟶ [𝐺] = [𝑔 𝑔22 ] 21 Having the matrix [𝐺] we seeks 𝐺 by the map [𝐺] ⟶ 𝐺. We use the reversed matrix multiplication operation to (5.275) on [𝐺] 𝑔 +𝑔 𝛔 ⏞ 1 (1 𝛔1 )𝑃+ ⏟ (5.277) 𝐺=⏞ [𝐺] ( ) = (𝑃+ 𝛔1 𝑃+ ) (𝑔11 +𝑔12 𝛔1 ) = 𝑔11 𝑃+ + 𝑔12 𝛔1 𝑃− + 𝑔21 𝛔1 𝑃+ + 𝑔22 𝑃− , 𝛔1 ⏟ 21 22 1 where we use the fact: 𝑃− = 𝛔1 𝑃+ 𝛔1 , and 𝑃+ 𝛔1 = 𝛔1 𝑃− , or 𝛔1 𝑃+ = 𝑃− 𝛔1 . By 𝛔12 =1 we have the unitarity 𝑃 1 (1 𝛔1 )𝑃+ ( ) = (1 𝛔1 ) ( + ) = 𝑃+ + 𝛔1 𝑃+ 𝛔1 = 𝑃+ + 𝑃− = 1. (5.278) 𝛔1 𝑃+ 𝛔1 To transform the geometric multivector (5.272) 𝐺 = 𝛼1 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 we will as Garret Sobczyk [14] p.80, [15] multiply on both sides with this unit (5.278) to get the matrix form .

.

(5.279)

⏞1 1 𝐺 = (1 σ1 )𝑃+ ( ) ⏟ 𝐺 (1 σ1 ) 𝑃+ ( ) = ⏟ σ1 σ1 By (5.281) resulting in:

266

= 𝐺=

⏞ 𝐺 (1 σ1 ) 𝑃+ [ ⏟ σ1 𝐺

𝐺σ1 1 ]𝑃 ( ) σ1 𝐺σ1 + σ1

⏞ 𝛼+𝜈2 (1 σ1 ) 𝑃+ [ ⏟𝜈1 −𝛽

𝜈1 +𝛽 1 ]( ) . 𝛼−𝜈2 σ1

Here we use the scale coefficients 𝜈𝑘 for the line extension variation 𝜈𝑘 𝛔𝑘 instead of 𝑥𝑘 𝛔𝑘 for the position coordinate 𝑥𝑘 .

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– 5.6.1. The Fundamentals of Matrices in a Plane Algebra 𝒢2 – 5.6.1.3 The Matrices of the Geometric Algebra 𝒢2(ℝ) –

This result we get by, for each place in the matrix⏞ by use of the form⏟ and the fact that (5.280)

𝑃+ σ1 𝑃+ = 0, 𝑃+ σ2 σ1 𝑃+ = 0, and 𝑃+ σ2 = 𝑃+ 1. Then | ⏞ 𝑃+ 𝑃+ 𝐺 = 𝑃+ (𝛼 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 )𝑃+ = 𝑃+ (𝛼+𝜈2 ) ⏞1 𝑃+ = 𝑃+ (𝛼 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 )𝛔1 𝑃+ = 𝑃+ (𝜈1 +𝛽) 𝑃+ 𝐺𝛔 (5.281) ⏞ 𝑃+ 𝛔 = 𝑃+ 𝛔1 (𝛼 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 )𝑃+ = 𝑃+ (𝜈1 −𝛽 ) 1 𝐺 𝑃+ {

𝑃+ ⏞ 𝛔1 𝐺𝛔1 𝑃+ = 𝑃+ 𝛔1 (𝛼 + 𝜈1 𝛔1 +𝜈2 𝛔2 + 𝛽𝛔2 𝛔1 )𝛔1 𝑃+ = 𝑃+ (𝛼−𝜈2 )

}

A plane geometric algebraic element 𝐺 ∈ 𝒢2 (ℝ) (5.272) is by the projection spectral basis (5.275) mapped to a real matrix as (5.276) 𝛼+𝜈2 𝜈1 +𝛽 (5.282) 𝐺 ⟶ [𝐺] = [ ] 𝜈1 −𝛽 𝛼−𝜈2 How will this look when we do not specify the coefficients in (5.272) and just want to find an expression from an abstract multivector 𝐺 ∈ 𝒢2 (ℝ) for a matrix [𝐺] representing the same plane quality of our physical entity? First, we do not know the expansion (5.272), then: For the projector 𝑃+ for the form expressed in (5.277) will stuck in first line of (5.279) at ⏞ 𝐺 𝐺σ1 1 1 (5.283) 𝐺 = (1 σ1 )𝑃+ [𝐺] ( ) = (1 𝛔1 ) 𝑃+ [ ] 𝑃+ ( ). σ1 σ1 ⏟ σ1 𝐺 σ1 𝐺σ1 From this we extract that 𝐺 𝐺σ1 (5.284) 𝑃+ [𝐺] = 𝑃+ [ ]𝑃 σ1 𝐺 σ1 𝐺σ1 + But we can as well use the projection 𝑃− in our deduction. The change is that we go over in a conjugated picture by an inner automorphism. We as Sobczyk [14], [15] invent a 𝛔1 -conjugation 𝛔 𝛔 (5.285) 𝐺0 1 = σ1 𝐺σ1 ⇒ 𝑃+ 1 = σ1 𝑃+ σ1 = 𝑃− By this we transform (5.284) to σ 𝐺σ σ1 𝐺 (5.286) 𝑃− [𝐺 𝛔1 ] = 𝑃− [ 1 1 ]𝑃 𝐺σ1 𝐺 − 𝑔11 𝑔12 We seek the same real matrix of the form (5.276), therefore we presume [𝐺 𝛔1 ] = [𝐺] = [𝑔 𝑔 ], 21 22 with 𝑔𝑘𝑗 ∈ℝ for the real matrix [𝐺]. By the unitarity (5.278) 𝑃+ + 𝑃− = 1 we form the sum 𝑃+ [𝐺] + 𝑃− [𝐺 𝛔1 ] = (𝑃+ + 𝑃− )[𝐺] = [𝐺]. This sum of (5.284) and (5.286) then give us the desired real matrix 𝐺 𝐺σ1 σ 𝐺σ σ1 𝐺 (5.288) [𝐺] = 𝑃+ [ ] 𝑃+ + 𝑃− [ 1 1 ]𝑃 σ1 𝐺 σ1 𝐺σ1 𝐺σ1 𝐺 − This is the map of the unspecified abstract geometric algebraic multivector 𝐺 ∈ 𝒢2 (ℝ) supported from the spectral basis (5.275), given from our intuitive object standard basis (5.271), using the mutual annihilating projection operators (5.274) 1 𝑃+ σ1 𝑃− 1 {1, σ1 , σ2 , σ2 σ1 }, (5.289) [ ], (σ ), 𝑃± = ½(1 ± σ2 ). σ1 𝑃+ 𝑃− 1 𝑔11 𝑔12 The real matrix form [𝐺] = [𝑔 𝑔22 ] we cannot intuit as an geometric object 21 on an natural surface in space, but we writ it out as (5.287)

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– II. . Geometry of Physics – 5. The Plan Concept – 5.6. The Real Matrix Representation for the Plane Concept –

= 𝑔11 𝑃+ + 𝑔12 σ1 𝑃− + 𝑔21 σ1 𝑃+ + 𝑔22 𝑃− = ½(𝑔11 +𝑔22 ) + ½(𝑔12 +𝑔21 )σ1 + ½(𝑔11 −𝑔22 )σ2 + ½(𝑔12 −𝑔21 )σ2 σ1 , and its σ1 -conjugation σ 𝐺0 1 = 𝑔11 𝑃− + 𝑔12 σ1 𝑃+ + 𝑔21 σ1 𝑃− + 𝑔22 𝑃+ (5.291) = ½(𝑔11 +𝑔22 ) + ½(𝑔12 +𝑔21 )σ1 − ½(𝑔11 −𝑔22 )σ2 − ½(𝑔12 −𝑔21 )σ2 σ1 . Then we have the form 𝛔 (5.292) 𝐺 = 𝛼1 + 𝜈1 σ1 + 𝜈2 σ2 + 𝛽σ2 σ1 and 𝐺0 1 = 𝛼1 + 𝜈1 σ1 − 𝜈2 σ2 − 𝛽σ2 σ1. (5.290)

𝐺

5.6.1.4. An Example of a Matrix in 𝒢2(ℝ)

A simple example is the special orthogonal rotation group 𝑆𝑂(2) of real 2×2 matrices of the type 𝑔11 𝑔12 cos 𝜙 − sin 𝜙 (5.293) [ ] = [𝑔 ~[𝑒 𝑖𝜙 ], 𝑔22 ], sin 𝜙 cos 𝜙 21 We see that anti-symmetry cancel when (𝑔12 +𝑔21 ) = 0 and (𝑔11 −𝑔22 ) = 0, further (5.294) ½(𝑔11 +𝑔22 ) = 𝛼 = cos 𝜙 and ½(𝑔12 −𝑔21 ) = 𝛽 = sin 𝜙 . In this way we get the 1-rotor form as (5.83) for a rotation (5.295) 𝐺rotor = 𝑈𝜙 = cos 𝜙 + σ2 σ1 sin 𝜙 = 𝑒 σ2σ1𝜙 , and (5.296)

𝛔

1 𝐺rotor = 𝑈𝜙† = cos 𝜙 − σ2 σ1 sin 𝜙

= 𝑒 −σ2σ1𝜙 .

This of course can be dilated by a factor 𝜌 to a 1-spinor in the plane. When we have 𝜈1 = ½(𝑔12 +𝑔21 ) ≠ 0 and\or 𝜈2 = ½(𝑔11 −𝑔22 ) ≠ 0, there is also involved some extension translation variation 𝐭 = 𝜈1 σ1 + 𝜈2 σ2 ∈𝒢2 (ℝ) along the plane supported by σ2 σ1 ∈𝒢2 (ℝ). remember the unitarity (5.85) 𝑈𝜃† 𝑈𝜃 = 𝑈𝜃 𝑈𝜃† = 1 for the 1-rotor in the geometric algebraic plane. For the real 𝑆𝑂(2) matrix we have the transposed 𝑔11 𝑔12 T 𝑔11 𝑔21 cos 𝜙 sin 𝜙 (5.297) [𝑔 ] = [ 𝑔22 𝑔12 𝑔22 ] = [− sin 𝜙 cos 𝜙] 21

= [𝑈𝜃† ]

The product of these two (5.297) and (5.293) 𝑔11 𝑔12 T 𝑔11 𝑔21 cos 𝜙 sin 𝜙 cos 𝜙 − sin 𝜙 (5.298) [𝑔 ] [ ] = [ ][ ] = cos2 𝜙 + sin2 𝜙 = 1 𝑔22 𝑔12 𝑔22 − sin 𝜙 cos 𝜙 sin 𝜙 cos 𝜙 21 Something similar for the determinant of (5.293) due to the anti-symmetry cos 𝜙 − sin 𝜙 (5.299) | | = cos2 𝜙 + sin2 𝜙 = 1 sin 𝜙 cos 𝜙 We say that the rotation matrix (5.293) is unitary. Here we will not go further with the matrix formalism for the plane idea.

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5.7. Plane Concept Idea of an Non-Euclidean Clifford Algebra First a natural 1-vector object u or p = 𝜆1 u can be drawn on a surface (paper) as an arrow for the intuition to indicate a natural direction u with extension magnitude u2 =1 or |p| = |𝜆1 | ≥0. The fundamental idempotent multivector ½(1 ± u) or the paravector 𝓅 = 𝜆0 + p of the grade form ⟨𝐴⟩0 + ⟨𝐴⟩1 cannot be drawn direct for the intuition because the scalar part has no extension. To remedy this we will make use of the Minkowski space concept, inspired from [16], [17], [6]. 5.7.1. Plane Geometric Clifford Algebra with Minkowski Signature for Measure Information

Besides the Euclidean plane concept 𝒢2,0 expressed in (5.198) we make an non-Euclidean plane 𝒢1,1 . For this we invent an external unit 1-vector 𝛾0 for the information development direction, as a first grade quality (pqg-1) with a positive causal orientation towards the future.267 We demand a positive signature Clifford metric (+) for this causal direction (5.300) 𝛾02 = +1, and ̃ 𝛾0 = 𝛾0 = 𝛾0 = 𝛾0. Conversely to this creative information direction 𝛾0 we invent unit 1-vectors 𝛾𝑘 for each one Descartes extension pqg-1 directions. For these directions we demand a negative signature (−) (5.301) 𝛾𝑘2 = −1, for 𝑘 =1,2,3, and ̃ 𝛾𝑘 = 𝛾𝑘 , but 𝛾𝑘 = −𝛾𝑘 . The purpose for this is, that quadratic norm for the information development is in balance with the quadratic norm for the extension (5.302) 𝛾02 + 𝛾𝑘2 = 0 0 |. For each 𝑘 the basis set {𝛾0 , 𝛾𝑘 } is a orthonormal basis (5.303) 𝛾0 ⋅ 𝛾𝑘 = 0 and |𝛾0 | = |𝛾𝑘 | = 1, Figure 5.49 The -bivector  ≡ 𝛾1 𝛾0 , (𝑘=1) forming for an abstract plane concept, we call it a -plane,268 a -plane from the 1-vectors 𝛾0 for development that has Clifford algebra 𝒢1,1 (ℝ), signatures (+, −). with positive signature γ02 =1, and for extension 𝛾1 with equiangular signature 𝛾12 =−1. Forming any unit

From this abstraction of this 1-vector basis {𝛾0 , 𝛾1 } -bivector amoeba  =, in -plane, with signature we form a mix of two new units 2 =1. This intuit display object is an abstraction of measure substance of information about extension. (5.304) 1 ≡ 𝛾0𝛾0 = 𝛾02 = +1, the real scalar unit. (5.305)  ≡ 𝛾1 𝛾0 = 𝛾1 ∧ 𝛾0 , the -plane unit pseudoscalar -bivector, with the reversion ̃ = − = 𝛾0 𝛾1 = 𝛾0 ∧𝛾1 , (5.306) = in that 𝛾1 ∧ 𝛾0 = −𝛾0 ∧𝛾1 For the signature square of this -plane pseudoscalar -bivector unit (5.305) we have (5.307)

(5.308)

2 = 1 = 𝛾1 𝛾0 𝛾1 𝛾0 = −𝛾1 𝛾1 𝛾0 𝛾0 = 1. The geometric substance structure of the -bivector direction plane is displayed in Figure 5.49. From the defining basis {𝛾0 , 𝛾1 } we form the mixed basis {1,  }, a scalar and a pseudoscalar unit. This we combine to a full mixed basis for the Minkowski -plane algebra 𝒢1,1 (ℝ) {1,| 𝛾0 , 𝛾1 ,  ≡ 𝛾1 𝛾0 }. The action of the  multiplication operations gives the exchange properties

(5.309)

 𝛾0 = 𝛾1 ,

 𝛾1 = 𝛾0 ,

𝛾0  = −𝛾1 ,

𝛾1  = −𝛾0 .

2 = 1, 1 ∈ℝ, is the neutral multiplication identity.|

267

This primary quality of first grade (pqg-1) as direction towards the future has no Descartes extension. It is a quality of counting times of occurrence in a process of development; one count is the unit 1-vector 𝛾0 , with 𝛾02 =1 for FORWARD. We use 𝜏𝛾0 , 𝜏∈ℝ. – In a tradition of classical mechanics this count is often interpreted as a continues floating river of time. (as a mysterious concept.) 268 The name -plane is used instead of the obvious name Minkowski-plane to prevent confusion to other conceptual interpretations.

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The closed algebra for this basis fulfils the multiplication scheme 5.2a. below.

shown in Table

Besides these basis we further form the non-zero null basis { 𝓃, 𝓃 } for the -plane in 𝒢1,1 (ℝ) defined by: (5.310)

𝓃 ≡ √½(𝛾0 + 𝛾1 ), and 𝓃 ≡ √½(𝛾0 − 𝛾1 ), with the nilpotent spectral basis properties 2

𝓃2 = 𝓃 = 0, and 𝓃⋅𝓃 = 𝓃⋅𝓃 = 1. This basis idea do not possess orthogonality, though it display perpendicular directions in Figure 5.50. The pseudoscalar for { 𝓃, 𝓃 } is the same as (5.305)269 (5.312) 𝓃 ∧ 𝓃 = , (5.311)

still with 2 =1, and reversed like (5.306) as bivector ̃ = − = 𝓃 ∧ 𝓃, (5.313) = We have the product of the two null-basis-1-vectors (5.314) 𝓃𝓃 = 𝓃⋅𝓃 + 𝓃∧𝓃 = ⏟ 1 +  , and 𝓃𝓃 = ⏟ 1 − , The full mixed null-basis for the -plane is (5.315) {1,| 𝓃, 𝓃,  = 𝓃∧𝓃 }

Figure 5.50 The null basis { 𝓃 , 𝓃 } with the pseudoscalar -bivector unit  = 𝓃 ∧ 𝓃 = 𝛾1 𝛾0 . Multivectors expressed as 𝑀= 𝜏𝛾0 ±𝜏𝛾1 , ∀𝜏∈ℝ fulfils the balance (5.302) (𝜏𝛾0 )2 +(𝜏𝛾1 )2 = 0 and falls on the null lines 𝑀=𝜏√2𝓃 or 𝑀=𝜏√2𝓃.

The operation of  gives the direction absorptions  𝓃 = 𝓃,  𝓃 = −𝓃, (5.316) 𝓃 = −𝓃, 𝓃 = 𝓃 . The closed algebra for this basis fulfils the multiplication scheme shown in Table 5.2b.

Table 5.2 Multiplication basis for the -plane algebra 𝒢1,1 (ℝ) with the pseudoscalar unit 2 = 1,  ≡ 𝛾1 𝛾0 .

a: {1, 𝛾0 , 𝛾1, } orthonormal left

right

1 𝛾0 𝛾1 

1 𝛾0 𝛾1  1 𝛾0 𝛾1  𝛾0 +1 − −𝛾1 𝛾1  −1 −𝛾0  𝛾1 𝛾0 1

Two cases of mixed basis 𝛾1 𝛾0 =  =  = 𝓃∧𝓃, 2 = +1,

b: {1, 𝓃, 𝓃,  } the null-basis left

2

𝓃2 = 𝓃 =0,

𝛾0 𝛾1 =  = − = 𝓃∧𝓃

right

1 𝓃 𝓃 

1 𝓃 𝓃 1 𝓃 𝓃 𝓃 0 1+ 𝓃 1− 0  𝓃 −𝓃

  −𝓃 𝓃 1

Conversely to (5.310) we get the {𝛾0 , 𝛾1 } basis from the { 𝓃, 𝓃 } null basis (5.317)

𝛾0 = √½(𝓃 + 𝓃), and 𝛾1 = √½(𝓃 − 𝓃). The nilpotence (5.311) of the null basis { 𝓃, 𝓃 } make it possible to scale the two basis as 1 {𝜆𝓃, 𝓃} in the way that we preserve the unit pseudoscalar -bivector and the inner product 𝜆 1 1 1 (5.318)  = 𝜆𝓃 ∧ 𝜆 𝓃, and 𝓃 ⋅ 𝜆𝓃 = 𝜆𝓃 ⋅ 𝜆 𝓃 = 1, for ∀λ ∈ℝ, λ ≠ 0. 𝜆 We see that the null basis 1-vectors have no specific magnitudes, it is the direction -bivector pseudoscalar and the inner product scalar that is preserved in the multivector (5.314) 𝓃𝓃 = ⏟ 1+. From this we form the hyperbola transformed of (5.317) that is displayed in Figure 5.51 1 1 (5.319) 𝛾0 (𝜆) = √½ (λ𝓃 + 𝜆 𝓃) , and 𝛾1 (𝜆) = √½ (λ𝓃 − 𝜆 𝓃) , for ∀λ ∈ℝ, λ ≠0. Further from this we also have the preserved unit pseudoscalar area as (5.305) 269

The reason is: 𝓃 ∧ 𝓃 = ½(𝓃𝓃 − 𝓃𝓃) = ½(½(𝛾0 +𝛾1 )(𝛾0 −𝛾1 ) − ½(𝛾0 −𝛾1 )(𝛾0 +𝛾1 )) = ¼(4𝛾1 𝛾0 ) = 𝛾1 𝛾0 = .

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(5.320) (5.321)

 = 𝛾1 (𝜆)𝛾0 (𝜆), and still for the basis {1, 𝛾0 (𝜆), 𝛾1 (𝜆), } algebraically orthonormality is preserved as (5.303)270 𝛾0 (𝜆)⋅ 𝛾1 (𝜆) = 0,

and

2

|𝛾0 (𝜆)| = |𝛾1 (𝜆)| = 1

2

⟸ |(𝛾0 (𝜆)) | = |(𝛾1 (𝜆)) | = 1.

The basis {𝛾0 (𝜆), 𝛾1 (𝜆)} is the hyperbola transformed from the null basis { 𝓃, 𝓃 }. The former is orthonormal but remark that its display is certainly not perpendicular nor normal in Figure 5.51. We try to display this transformation invariant structure from the a priory direction  = 𝓃∧𝓃 basis { 𝓃, 𝓃 }, displayed as perpendicular 1-vectors 𝓃⊥𝓃, but as null basis not orthogonal 𝓃⋅𝓃=1. { 𝓃, 𝓃 } in Figure 5.51 is turned by ⎼45° from defined directions displayed in Figure 5.50 above: Figure 5.52 Projection of the -bivector in to a 1-vector σ1 as object on a surface

for the intuition. Figure 5.51 The invariant hyperbola transformed 𝛾0 (𝜆) and 𝛾1 (𝜆) is displayed in the null basis { 𝓃 , 𝓃 } diagram. We have chosen perpendicular display of the null basis 1-vectors { 𝓃 , 𝓃 } representing the founding -bivector =𝓃∧𝓃 defining the direction of the -plane of this figure. The null line 𝓃2 =0 representing a possible light signal of information 2 is displayed from left to right in the reading direction. The Clifford conjugated null line 𝓃 =0 is chosen displayed vertical perpendicular 𝓃⊥𝓃. Due to the quality of the full mixed null basis {1,| 𝓃, 𝓃, =𝓃∧𝓃} of the -plane (5.315) these are not orthogonal 𝓃⋅𝓃=1, (5.311), but multiplied invariant constant 𝓃𝓃 = 𝓃⋅𝓃+𝓃∧𝓃 = 1+, (5.314). The word ‘auto’ refer to an autonomous entity that has a locality-now center, performed by the intersection of the locality development line direction 𝛾0 and the now extension line direction 𝛾1 , with the autonomous full mixed basis {1, 𝛾0 , 𝛾1 , }, (5.308). The invariant hyperbola transformed (5.319) 𝛾0 (𝜆) and 𝛾1 (𝜆) is displayed for the argument, e.g. 𝜆~2.5. The product  = 𝛾1 (𝜆)𝛾0 (𝜆) (5.320) and the orthonormality (5.321) is invariant preserved for all values ∀λ ∈ℝ, λ ≠0. λ 1 the tangent speed of the oscillator rotating null cone in the display will grow and exceed speed of information 𝑐1. It is obvious that the extended circumference arcus speed of the circle oscillator do not exceed 𝑐1, (𝑐=1), therefor an unit circle oscillator |r̂𝜃 |=|𝛾𝜃 | = |𝛾0 |=1 is the binding. The plane rotation 𝑒 𝒊𝜃 = 𝑒 σ2σ1𝜃 initiate a third normal extension direction σ3 = ⃖⃗⃗ 𝛾3 𝛾0 274 as an axis of rotation in ℨ-space. The a priori idea is, that information development isometry is isotropic distributed over all extended directions in ℨ-space, whence we judge |𝛾𝜃 | = |𝛾1 | = |𝛾2 | = |𝛾3 | = |𝛾0 | = 1. (5.337) From this isotropic isometry we have ambiguity in what space extension direction to intuit for one quantum of information as a unit direction -bivector. The information isometry can only follow one nilpotent pair null lines direction  = 𝓃∧𝓃 inside its Minkowski -plane direction. One intuit picture is oscillation rotation of the information 𝜃 -plane has one persistent oscillating 1-vector extension direction 𝛾𝜃 always orthogonal to the future oriented development direction 𝛾0 principal isotropic in all spatial directions a priori orthogonal to all these extensions ∀𝛾𝜃 ⋅ 𝛾0 = 0, it can be perpendicular to some extensions 𝛾0 ⊥𝛾𝜃 and parallel to one other direction, see , e.g. 𝛾0 ∥𝛾3, but still always orthonormal 𝛾0 ⋅ 𝛾𝑘 = 0 and |𝛾𝜇 | = 1, 𝜇 = 0,1,2,3, as . Actively 𝛾0 can be parallel with one straight line extension direction e.g. σ3 = ⃖⃗⃗ 𝛾3 𝛾0 , 𝛾3 ⋅𝛾0 = 0 and ambiguously simultaneous bee curved275 along the circumference arcus of the oscillating unit circle radius |r̂𝜃 | = |𝛾𝜃 |=1 so that 𝛾0 ⊥𝛾𝜃 and substantial 𝛾0 ⋅ 𝛾𝜃 = 0, (see below ). 274 275

In chapter 6 we will look further in the structure of the three dimensional natural ℨ-space The ideology of a tangent vector is non necessitated due to no demand of straitens of non-extensive development 𝛾0 direction.

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– 5.7.2. The Traditional Display of the Minkowski -plane – 5.7.2.2 The Traditional Display of the Minkowski space –

Only one direction in the form of a 𝜃 -plane can participate in a propagation of one quantum active information at the time. This quantum direction is the nilpotent unit (5.312) 𝜃 = 𝓃∧𝓃. To possess the quality of autonomous extension for an entity it has to give its own measure in form of an oscillator giving the a priori count of development. The frequency energy of this oscillator we use as norm i.e. |𝜔|=1. The direction of oscillation we describe by the Euclidean plane bivector 𝛾1 𝛾2 = ⃗⃗ σ2 σ1 = 𝒊. The direction of development 𝛾0 is orthonormal to both 𝛾1 and 𝛾2 expressed by (5.303). But the bivector idea (𝛾1 𝛾2 ) is not orthogonal to 𝛾0 in that inner product (5.338) 𝛾0 ⋅(𝛾1 𝛾2 ) = ½(𝛾0 𝛾1 𝛾2 + 𝛾1 𝛾2 𝛾0 ) = 𝛾1 𝛾2 𝛾0 ≠ 0 .276 Hereby we decide a shortcut and promote the subject idea (𝛾1 𝛾2 ) to the object picture σ2 σ1 = 𝒊 (5.339) 𝛾1 𝛾2 = σ2 σ1 = 𝒊 = 𝒊3 and name 𝒊3 for the plane bivector seen from a third extension direction (see below , ). The plane circle oscillator for the autonomous isometric measure is short (5.340)  = 𝑒𝛾1𝛾2 𝜃 = 𝑒𝒊𝜃 = 𝑒𝒊3𝜃 . This circular oscillating 1-rotor of the form 𝑒 𝒊𝜃 for the Euclidean plane leave all the multivectors we have treated above invariant, but off course change their directions. Special the 𝜃 -bivector direction is rotated in this oscillation. Inside this 𝜃 -plane the nilpotent null directions 𝓃 and 𝓃 is invariant preserved also by the hyperbola transformation (5.322).

276

When it comes to the bivector idea (𝛾1 𝛾2 ) multiplied by an independent 1-vector e.g. 𝛾0 we get a 3-vector as later below III. (7.15) 𝒾𝛾3 = 𝛾1 𝛾2 𝛾0 , where this three-vector concept is dual to the 1-vector idea 𝛾3 by the grade-4 pseudoscalar 𝒾 of STA.

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– II. . Geometry of Physics – 5. The Plan Concept – 5.7. Plane Concept Idea of an Non-Euclidean Clifford Algebra – 5.7.2.3. Minkowski Space with Display of Three Extension Dimensions

In the tradition of Feynman diagrams nilpotent null development of subtons are displayed as wave lines for photons and helixes for gluons.

We follow the display with the development direction 𝛾0 vertical and the null directions turned ±45° from this in the 1 -bivector-plane. In Figure 5.54 we display the projection of the extension directions 𝛾1 and 𝛾2 with the hyperbola transformation parameter λ close to one in the defining formula (5.319) as 𝛾1 (𝜆≅1) and 𝛾2 (𝜆≅1). The same we do in Figure 5.55 where we further introduce the third extension direction 𝛾3, with λ→0; projection of 𝛾3 (λ=0̃ ) is displayed antiparallel ⇃↾ to 𝛾0 , 𝛾3 ∥𝛾0. Chosen with opposite orientation of the development direction. But attention; 𝛾0 ⋅ 𝛾3 = 0; there are all algebraically orthogonal 𝛾0 ⋅ 𝛾𝑘 = 0, for 𝑘 =1,2,3, (5.303), and we too presume 𝛾1 ⋅ 𝛾2 = 𝛾2 ⋅ 𝛾3 = 𝛾3 ⋅ 𝛾1 = 0. The projection mapping of the Minkowski basis {𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 } with signatures (+, −, −, −) into the Euclidean basis {σ1 ,σ2 ,σ3 } with signatures (+, +, +) is performed by the multiplication operations (5.328) 𝛾1 ⟶ σ1 = ⃖⃗⃗ 𝛾1 𝛾0 , σ1 ⟶ 𝛾1 = ⃖⃗⃗ σ1 𝛾0 (5.341) 𝛾2 ⟶ σ2 = ⃖⃗⃗ 𝛾2 𝛾0 , convers σ2 ⟶ 𝛾2 = ⃖⃗⃗ σ2 𝛾0 𝛾3 ⟶ σ3 = ⃖⃗⃗ 𝛾3 𝛾0 , σ3 ⟶ 𝛾3 = ⃖⃗⃗ σ3 𝛾0 In Figure 5.55 we will imagine that the third Minkowski 3 -bivector-plane 32 =1, 3 ≡ 𝛾3 𝛾0 has the same start direction as 1 ≡ γ1 𝛾0 → σ1 , (3 as 2 un-displayed). The invariant nilpotent null directions 𝓃 and 𝓃 in the 1 -bivector-plane is translation invariant moved one unit along the ±𝛾2 direction of the Figure 5.55 display. The idea is that the entity represented by this frame is rotating with an oscillation in the extension plane 𝛾1 𝛾2 = 𝑒𝛾1𝛾2 𝜃 = 𝑒𝒊𝜃 , relative to an external laboratory. By this oscillating rotation of both 3 and 1 -bivector direction the nilpotent isotropic directions is twisted up in two opposite equal orientated null helixes stretched out in the past as the oscillator is propagating into the future. It is the nilpotence of the null basis { 𝓃 , 𝓃 } 2 (5.310) 𝓃2 = 𝓃 = 0 (5.311) that makes this invariant stretching possible carrying an information signal through the development from the passed we call the subton memory space of extension. This signal of information through space is what we in chapter I. 3.4 called a subton, that was displayed in Figure 3.13. where we did not have the idea a nilpotent concept, but implicit had, that carrier time always is 𝑡⃗⃗⃗⃗ 𝑐 = 0 throughout the propagation as described in section I. 3.5.2, and which transfer of information in principle is displayed in Figure 3.14. The subton idea is the a priori foundation concept of a 𝑈(1) spin-1-rotor oscillator propagating with the speed of information, which we set isometric equal to one, where the autonomous measure is one count of quantum in Figure 5.55 The four dimensional Minkowski space the oscillation of the subton entity development. projected into the figure plane. The extension space is The primitive subton idea is the foundation for generated by a plane cyclic oscillating rotation producing the concept of a photon (and properly also gluons). a helix formed nilpotent null curve of development. ∞

The frame {σ𝑘 } preform a rotating oscillation around σ3 .

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– 5.7.3. The Paravector Space and the Minkowski 1-vector in STA – 5.7.2.3 Minkowski Space with Display of Three Extension Dimensions –

5.7.3. The Paravector Space and the Minkowski 1-vector in STA

In § 5.5.4.2 (5.240) we defined the foundation of a paravector as a scalar plus a 1-vector (5.342) 𝓅 = 𝜆0 + p = 𝜆0 1 + 𝜆1 u. The 1-vector can have different directions p = 𝜆1 𝛔1 + 𝜆2 𝛔2 in a Euclidean plane {𝛔1 , 𝛔2 } or in space {𝛔𝑘 }, p = 𝜆𝑘 𝛔𝑘 , for 𝑘=1,2,3.277 The para-multi-vectors of grade form ⟨𝐴⟩0 +⟨𝐴⟩1 is then (5.343) 𝓅 = 𝜆0 + p = 𝜆0 1 + 𝜆𝑘 σ k . The pure magnitude of the paravector is given from (5.253)-(5.255) as |𝓅| = |𝓅| = √|𝓅𝓅| = √|𝜆02 −𝜆12 −𝜆22 −𝜆32 | ≥ 0. The paravector space is supported from the mixed basis {1, σ1 ,σ2 ,σ3 } of the form ⟨𝐴⟩0 +⟨𝐴⟩1. To see how the paravector concept look likes in Minkowski space we map by multiplication operation from right with the quantum count of development expressed as the 1-vector unit 𝛾0 (5.345) 𝑝 = 𝓅𝛾0 = 𝜆0 𝛾0 + p𝛾0 = 𝜆0 𝛾0 + 𝜆𝑘 σk 𝛾0 = 𝜆0 𝛾0 + 𝜆𝑘 𝛾𝑘 = 𝜆𝜇 𝛾𝜇 . Use (5.328) σ𝑘 𝛾0 = 𝛾𝑘 . (5.344)

This is a 1-vector in a four dimensional Minkowski space supported as a span from the orthonormal 1-vector basis {𝛾𝜇 }, for 𝜇 = 0,1,2,3.278 Here the multivector form is a pure ⟨𝐴⟩1 , i.e. a primary quality of first grade (pqg-1) with the development count as the quantity unit. The geometric algebra founded on this basis {𝛾𝜇 } we as Hestenes [6] 1966 call Space-Time-Algebra (STA). The 1-vector basis {𝛾𝜇 } is sometimes called a Dirac basis of STA -or just- a Minkowski basis. The magnitude of such a 1-vector 𝑝 = 𝜆𝜇 𝛾𝜇 in the Minkowski metric is given by the quadratic form (5.346)

2

𝑝2 = (𝜆𝜇 𝛾𝜇 ) = 𝜆02 −𝜆12 −𝜆22 −𝜆32 ⟹ |𝑝| = √|𝑝2 | = √|𝜆02 −𝜆12 −𝜆22 −𝜆32 | ≥ 0.

The two pictures describe the same structure of a physical entity therefore the magnitude |𝑝|=|𝓅|. Expressed from the supporting 1-vector basis {𝛾𝜇 } the Cartesian basis {σ𝑘 } = {σ1 ,σ2 ,σ3 } look like a Minkowski -bivector basis {𝑘 } by the map (5.326) 𝑘 ≡ 𝛾𝑘 𝛾0 ⟷ σ𝑘 The Cartesian basis {σ𝑘 } supporting the Euclidean space by 1-vectors describing the Cartesian extension of locality for an entity in physics. Adding a scalar dimension to the Cartesian frame gives us a paravector (5.343) with an mixed basis {1, u}, {1, σ1 ,σ2 } or {1, σ1 ,σ2 ,σ3 }. This paravector picture split the picture of STA developing space with four direction dimensions into a scalar part without direction and one, two or three Cartesian extension directions σ𝑘 . This agrees with the classical way to interpret space extension with a distance measure separated from the development of time as a pure scalar measure from a clock without extensive direction. Contrary if the local cyclic oscillating rotation clock produce the direction of quantum counts 𝛾0 and we can right multiply it to the paravector basis using (5.328) and get the STA basis { 1𝛾0 , σ1 𝛾0 , σ2 𝛾0 , σ3 𝛾0 } = { 𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 }, (5.347) then 𝛾0 by inheritance to the common measure of extensions gives the performed development. Having a single 1-vector direction u in a Euclidean space u2 =1, we can choose it as a frame basis 1-vector e.g. σ1 = u. We multiply this with 𝛾0 𝛾0 = 𝛾02 = 1 and get the mapping operation (5.348) u = u𝛾0 𝛾0 = σ1 𝛾0 𝛾0 = 𝛾1 𝛾0 = 1 =  We hereby see that the Euclidean 1-vector direction u is synonymous with a Minkowski -bivector including an autonomous development quantum measure count of its own unit extension. We note that these two concepts have the same Clifford signature (+): u2 = 1 and 2 =1. Physical entities that we represent by Euclidean 1-vectors is in Space Time Algebra (STA) represented by -bivectors with positive signature (+). 3 Here we use Einstein sum convention 𝜆𝑘 𝛔𝑘 = ∑2𝑘 or =1 𝜆𝑘 𝝈𝑘 over double indices, and low indices for the covariant coordinates in the orthonormal frame of the Euclidian space, to keep intuition simple. More about 𝑘=3, {𝛔𝑘 } in chapter 6. 278 Note we as common use Greek letters indices for four dimensional Minkowski space and Latin indices for Euclidian 1,2,3 support. 277

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5.7.4. Lorentz Rotation in the Minkowski -plane

In Figure 5.53 we display the hyperbola curves for invariant transformations by the scaling parameter 𝜆 of the nilpotent null directions (5.318) in a geometric Clifford algebra 𝒢1,1 (ℝ). Using the idea from Kepler’s second law, that angular area to the curve is essential as a parameter for a transformation. The area unit element in Minkowski -plane is just the -bivector  ≡ 𝛾1 𝛾0, where 2 ≡ 1. From this we choose a real parameter 𝜁 ∈ℝ often called rapidity for the area argument ½𝜁 in a hyperbolic multivector exponential function we call a Lorentz 1-rotor279 (5.349) 𝑈𝜁 = 𝑒 ½𝜁 = exp(½𝜁) = cosh(½𝜁) + sinh(½𝜁). The essential argument in this 1-rotor has positive Clifford signature (+) ∶ 2 = +| | ≡ 1 (½𝜁)2 = ¼𝜁 2 ≥0 (5.350) The multivector value of the exponential function (5.349) exist in the mixed algebra 𝒢1,1 (ℝ) of the Minkowski -plane. The algebra for these functions is analysed in next section 5.8, but here we will use that cosh is even in its power series of 2 =1, causing that cosh it is a even scalar, and that sinh is odd, so that a factor  can be separated from the scalar function, then (5.349) is (5.351) 𝑈𝜁 = 𝑒 ½𝜁 = cosh(½𝜁) + sinh(½𝜁) = cosh(½𝜁) +  sinh(½𝜁) , 𝜁 ∈ℝ. To investigate the possible impact of such a Lorentz rotor we will intuit a Euclidean space 𝒢2 (ℝ) or 𝒢3 (ℝ) 1-vector unit direction object u = 𝑢𝑘 σ𝑘 that has a measure 𝛾02 =1 (5.348) mapping it to a subject 1-vector in a full Minkowski space 𝒢1,2 (ℝ) or 𝒢1,3 (ℝ) substance. The mapping is performed by right multiplying by the development measure direction unit quantum 𝛾0: 𝛾u ≡ u𝛾0, producing a unit Minkowski -plane subject u ≡ 𝛾u 𝛾0 . From the Euclidean direction object u we can dilate to an arbitrary 1-vector pu = 𝜆u u, 𝜆u ∈ℝ with magnitude |pu | = |𝜆u |, that in the basis {σ𝑘 } is pu = ∑2𝑘 or=13 𝜆𝑘 σ𝑘 , where 𝜆𝑘 = 𝜆u u⋅σ𝑘 . From this we construct a paravector like (5.342) with an arbitrary development parameter 𝜆0 (5.352) 𝓅u = 𝜆0 + pu = 𝜆0 1 + 𝜆u u. Letting this paravector left operate on the development direction unit quantum 𝛾0 we achieve by invention a generalised Minkowski 1-vector in a pure ⟨𝐴⟩1 grade form in STA as (5.345) ( 𝜇 = 0, 𝑘). (5.353) 𝑝u = 𝓅𝛾0 = 𝜆0 𝛾0 + pu 𝛾0 = 𝜆0 𝛾0 + 𝜆u 𝛾u = 𝜆0 𝛾0 + 𝜆𝑘 𝛾𝑘 = 𝜆𝜇 𝛾𝜇 , 𝜆𝜇 ∈ℝ, The extension Clifford conjugated of this is (5.354) 𝑝u = 𝓅𝛾0 = 𝜆0 𝛾0 − pu 𝛾0 = 𝜆0 𝛾0 − 𝜆u 𝛾u = 𝜆0 𝛾0 − 𝜆𝑘 𝛾𝑘 = 𝜆𝜇 𝑔𝜇𝜇 𝛾𝜇 ,

1

𝑔𝜇𝜇 = ( −1−1 ).

Now we have a subject 𝑝 in the Minkowski space to act on by the u -plane Lorentz rotor

−1

𝑈𝜁,u = 𝑒 +½𝜁u = cosh(½𝜁) + u sinh(½𝜁). ̃ = † = − The conjugated of this is achieved by the reversed -bivector (5.306)  =  † ̃𝜁,u = 𝑒 −½𝜁u = cosh(½𝜁) − u sinh(½𝜁) = 𝑈𝜁,u (5.356) 𝑈 . (5.355)

The u -plane Lorentz rotor is unitary ̃𝜁,u = 𝑈 ̃𝜁,u 𝑈𝜁,u = 𝑒 0 = cosh2 (½𝜁) − sinh2 (½𝜁) = 1. (5.357) 𝑈𝜁,u 𝑈 We form the Lorentz rotation by the canonical form (5.358)

2 ̃𝜁,u = 𝑒 ½𝜁u 𝑝u 𝑒 −½𝜁u = 𝑒 𝜁u 𝑝u = 𝑈𝜁,u 𝑝u′ = 𝒰𝜁 𝑝u = 𝑈𝜁,u 𝑝u 𝑈 𝑝u = (cosh 𝜁 + u sinh 𝜁)𝑝u .

We split this into the two parts, first the development like hyperbolic rotation 2 (5.359) 𝜆0 𝛾0′ = 𝑈𝜁,u 𝜆0 𝛾0 = 𝑒 ½𝜁u 𝜆0 𝛾0 = 𝜆0 (𝛾0 cosh 𝜁 + u 𝛾0 sinh 𝜁) = 𝜆0 (𝛾0 cosh 𝜁 + 𝛾u sinh 𝜁), 279

This area argument differ from the cyclic angular form of (5.192) 𝑒 𝒊𝜃 where the argument has negative signature (−), 𝒊2 = −1 .

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In Figure 5.53 green for 𝜆0 ≥1. Then second the extension like hyperbolic rotation 2 (5.360) 𝜆u 𝛾0 = 𝑈𝜁,u 𝜆u 𝛾u = 𝑒 ½𝜁u 𝜆u 𝛾u = 𝜆u (𝛾u cosh 𝜁 + u 𝛾u sinh 𝜁) = 𝜆u (𝛾u cosh 𝜁 + 𝛾0 sinh 𝜁), yellow for 𝜆u ≥1 in Figure 5.53. (The extended area where the autonomous information is void) The white area 𝜆0 ≤1 and 𝜆u ≤1 in Figure 5.53 is inside the limit of one quantum count in the cyclic oscillation that gives the development unit 𝛾0 of the entity that possesses the isometry |𝛾0 | = |𝛾u | = 1 = |𝛾1 | = |𝛾2 | = |𝛾3 | = 1 ⇔ |𝛾𝜇 | = 1. (5.361) The information of any Minkowski STA 1-vector 𝑝u to and from the entity follow the Lorentz rotation invariant null line directions { 𝓃, 𝓃 } in the u -plane. The directions of 𝛾0 and 𝛾u are altered by the Lorentz rotation, but there orthonormality (5.303) is invariant preserved just as (5.321) (5.362) 𝛾0′ ⋅𝛾u′ = 0 and |𝛾0′ | = |𝛾u′ | = 1. As orthonormal subjects, these directions are never perpendicular objects for us! (Kein Ding für uns) The drawn 1-vector direction of the development 𝛾0 in Figure 5.50, Figure 5.51 and Figure 5.53 as an intuition perpendicular object to the extension direction 𝛾1 is an illusion! They just display in a diagram an instrumentalization of a development measure substance as an subject of one quantum count 𝛾0. The magnitude as (5.346) of any STA 1-vector is invariant preserved by the unitary Lorentz rotation given by the canonical form (5.358), in the u -plane simply 𝑝u′ = 𝑒 𝜁u 𝑝u 2 |𝑝u′ | = |𝑒 𝜁u 𝑝u | = |𝑒 𝜁u ||𝑝u | = |𝑝u |, (5.363) because |𝑒 𝜁u | = 𝑒 𝜁u 𝑒 −𝜁u = 𝑒 0 = 1. This is in agreement with the component coordinates (𝜆0 , 𝜆u ) from (5.359) and (5.360) that is preserved. It is the orthonormal basis 1-vectors that change directions {𝛾0 , 𝛾u } ⟷ {𝛾0′ , 𝛾u′ } inside the u -plane ′u = u = 𝛾0 𝛾u = 𝛾0′ 𝛾u′ . What implication dos this have on the physical world? We view intuit two identical entities that are distinguishable Ψ and Ψ′, that each has their individual frames Ψ~{𝛾0 , 𝛾u } and Ψ′~{𝛾0′ , 𝛾u′ }. These identical entities have the same primary qualities possessing equal quantities: 𝜆0 and 𝜆u . It is the orthonormal frame directions inside the Minkowski u -plane that the Lorentz rotation change in the information transmission boost. An example is two Hydrogen spectrums 11H and 11H' that for each local autonomy is the same but look different due to the boost of frame reference 11H'. 5.7.4.2. The Lorentz Transformation of a paravector

The Lorentz rotor (5.355) is written with an implicit knowledge of the development unit quantum 𝛾0, when multiplying with 𝛾0 𝛾0 = 𝛾02 = 1 we can by (5.348) write (for details see below section 5.8) (5.364) 𝑈𝜁,u = cosh(½𝜁) + u sinh(½𝜁) = cosh(½𝜁u) + sinh(½𝜁u) = 𝑒 ½𝜁u = exp(½𝜁u) Going back to the paravector form (5.352) 𝓅u = 𝜆0 + pu = 𝜆0 1 + 𝜆u u = 𝑝u 𝛾0. We find that the Lorentz rotation by the canonical form (5.358) can be written (5.365) 𝓅u′ = 𝑒 ½𝜁u 𝓅u 𝑒 −½𝜁u = 𝑒 𝜁u 𝓅u = (cosh 𝜁 + u sinh 𝜁)𝓅u = (cosh 𝜁 + u sinh 𝜁)(𝜆0 +𝜆u u). Here we lose knowledge, the rotation plane is undefined, we only have one 1-vector direction u. We need a -bivector to define a rotation plane direction. This problem is obvious when we split into the two parts, first the hyperbolic rotation of the development scalar is not a scalar 𝑒 ½𝜁u 𝜆0 = 𝜆0 (cosh 𝜁 + u sinh 𝜁), but we get a new Lorentz transformed scalar (5.366) 𝜆′0 = 𝜆0 cosh 𝜁 + 𝜆u sinh 𝜁, and a invariant Euclidean 1-vector direction u with a Lorentz transformed magnitude (5.367) 𝜆′u u = (𝜆u cosh 𝜁 + 𝜆0 sinh 𝜁)u. The magnitude of the paravector 𝓅u′ = 𝜆′0 1 + 𝜆′u u is as (5.363) preserved |𝓅u′ | = |𝓅u |. We conclude that a unit Euclidean 1-vector u2 ≡ 1 defining the extension direction u of a local entity is invariant preserved by the Lorentz rotation transformation for the autonomous entity. In traditional language, when you see a galaxy, your looking direction towards that galaxy is ©

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invariant by change of radial speed of that galaxy. This point of view is primitive and may seems naïve, but it is in fact a priory fundamental to a model of the extension of our spatial universe. Remember that the invariant extension direction u is an Cartesian and Euclidean pqg-1-vector object that has only to do with the orthonormal subject 1-vectors directions units 𝛾0 and 𝛾u through their outer product bivector u ≡ 𝛾u 𝛾0 = 𝛾u ∧𝛾0, where the isomorphic map u ↔ u represent the one and the same direction in our spatial universe. This invariant direction is equivalent to the invariant null line direction { 𝓃, 𝓃 } by the outer product 𝓃 ∧ 𝓃 = u . When you try to compare the magnitudes of two colinear Euclidean pqg-1-vectors x and u you are forced to construct a measure by a development unit 𝛾0 and then you inherit a primary quality of second grade (pqg-2) for the isometry u -bivector for the measurement process. 5.7.4.3. The Lorentz boost

(5.368)

(5.369) (5.370) (5.371)

In the tradition of Lorentz transformations, we are looking at two equal but distinguishable entities ΨR and ΨS with relative velocities to each other. We presume the velocity of interest is in the direction from R to S. We name the speed 𝛽 ∈ℝ (± magnitude) for the velocity, where 𝛽 >0 for orientation away from each other. We presume further that defining oscillators of each entity is equal and that therefore their local development units 𝛾0 is comparable. We take the autonomous viewpoint of entity ΨR . The information received to ΨR about the source ΨS stays in the null line directions. Orthonormal to the development unit 𝛾0 we have the extension unit 𝛾u . (not perpendicular) We interpret their direction 𝛾u 𝛾0 = u ↔ u as isomorph to direction between R and S. This is the same as the direction of the null lines 𝓃 ∧ 𝓃 = u in the direction of the STA Minkowski u -bivector-plane.280 The traditional classical interpretation the direction is a Cartesian Euclidean 1-vector u starting in point R of entity ΨR pointing towards ΨS . The idea is, that u represents the autonomous object direction of entity ΨR receiving a signal from source ΨS . For ΨS to send a signal towards ΨR it has to use the direction of u with negative orientation. We presume we can choose some perpendicular transverse 1-vector u⊥ ⊥u in ΨRS , u⊥2 ≡ 1 . Then we can choose to imagine the display of the null-basis 1-vectors directions as 𝓃∥u and 𝓃∥u⊥ . The information of the signal has direction represented by the null lines of 𝓃∧𝓃 = u , i.e. the u -bivector. We conclude the demand u ∥u , u⊥ ∥u , u∧u⊥ = u , u⋅u⊥ = 0, and u2 = u⊥2 = 1.281 When it comes to transmission of information (classically called ‘forces’282) we gain knowledge from using STA 1-vectors with Minkowski metric that’s generated by the -plane supported by the orthometric -bivector unit, e.g. u ≡ 𝛾u 𝛾0. We presume that the simplest information can be represented by the STA 1-vector like (5.353) 𝑝u = 𝜆0 𝛾0 + 𝜆u 𝛾u Now we are ready to look at the communication that due to the relativistic speed 𝛽 has to include the Lorentz boost transformation as the rotation in the u -plane (5.358) 𝑝u′ = 𝑒 𝜁u 𝑝u = (cosh 𝜁 + u sinh 𝜁)𝑝u. We presume the speed of information set as (5.330) 𝑐= |𝛾u |⁄|𝛾0 | = 1 and the relative speed is |𝑣| 𝛽 = tanh 𝜁 = sinh 𝜁 ⁄cosh 𝜁 ∈ℝ. 𝛽 = ± 𝑐 , the traditional relative speed. From the literature we have for cosh 𝜁 the Lorentz factor 1 𝛾 = √1−𝛽 2 , the Lorentz factor. ∈ℝ. 𝛾 = cosh 𝜁 = (1 − 𝛽 2 )−½ ∈ℝ. Then we write the Lorentz rotation (5.358), (5.369) as

280

The two null line directions { 𝓃 , 𝓃 } is invariant and parallel to the u -plane in the u -bivector direction. When they display for us, they appear advantageously perpendicular 𝓃⊥𝓃 , but they are neither orthogonal nor normal units (5.311), note . 281 The object idea u⊥ from an origo R, perpendicular to RS direction u, is in principal superfluous here, but it help us to intuit that there is something perpendicular transverse to the transmission direction of information along the null direction 𝓃, i.e. 𝓃 . Later we will consider that the extension unit u⊥ represent a oscillating rotation in a transversal plane to u. (background in I. 3.4). 282 E.g., gravitation, strong, weak, and electromagnetic forces, and with of course memory knowledge as modulation (e.g. OFDM).

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𝑝u′ = 𝛾(1 + 𝛽u )𝑝u. The relativistic radial speed factor 𝛽 can take the quantitative real values −1 ≤ 𝛽 ≤ 1. The radial velocity of ΨS is then vS = 𝛽u seen from ΨR , where the information from ΨS is Lorentz transformed by (5.369), (5.372). Consulting (5.359)-(5.363) we have for the separated directions: First for the development like Lorentz rotation |𝛾0′ | = |𝛾0 | = 1 (5.373) 𝛾0′ = 𝑒 ½𝜁u 𝛾0 = 𝛾0 cosh 𝜁 + 𝛾u sinh 𝜁 = 𝛾(1 + 𝛽u )𝛾0 , Secondly for the extension like Lorentz rotation |𝛾u′ | = |𝛾u | = 1 (5.374) 𝛾u′ = 𝑒 ½𝜁u 𝛾u = 𝛾u cosh 𝜁 + 𝛾0 sinh 𝜁 = 𝛾(1 + 𝛽u )𝛾u , The magnitudes of the rotated direction units is invariant preserved just as the dilation coordinates (𝜆0 , 𝜆u ) for the information content for both the transmitter ΨS and the receiver ΨR represented by the STA 1-vector 𝑝u = 𝜆0 𝛾0 + 𝜆u 𝛾u autonomous for both the entities ΨS and ΨR . The transformation seen from R is (5.375) 𝑝u′ = 𝛾(1 + 𝛽u )𝑝u = 𝜆0 𝛾0′ + 𝜆u 𝛾u′ The magnitude of the STA 1-vector as well as its development and extension coordinates is preserved |𝑝u′ | = |𝑝u | = 𝜆02 −𝜆u2. – The quantities of the information is invariant preserved! It is the STA frame orthonormal basis {𝛾0 , 𝛾u } → {𝛾0′ , 𝛾u′ } that is distorted in the mixed basis {1, 𝛾0 , 𝛾u , u ≡ 𝛾u 𝛾0 }, where the part {1, u }, u2 = (𝛾u 𝛾0 )2 = 1 is invariant. We remember that the subjects 𝛾0′ and 𝛾u′ indeed not fulfil the idea of perpendicular squareness but is anyway orthogonal. What we find is the frame distortion of the STA frame {𝛾0′ , 𝛾u′ } in the u -plane by the Lorentz rotation boost (5.368)-(5.375) is displayed in Figure 5.51.283 Dialectic opposite, the null basis { 𝓃, 𝓃 } has the quality in its perceivable objective display the opportunity to represent the physical perpendicular directions of information, even though 2 not orthonormal (5.310) 𝓃⋅𝓃 = 𝓃⋅𝓃 = 1, and 𝓃2 = 𝓃 = 0.284 All four components of the full mixed null-basis {1, 𝓃, 𝓃, u = 𝓃∧𝓃} is invariant in the u -plane. (5.372)

The traditional apparently contraction is an illusion (𝛾𝜆0 )|𝛾0 | = 𝜆0 (𝛾|𝛾0′ |) and (𝛾𝜆u )|𝛾u | = 𝜆u (𝛾|𝛾u′ |). All this invariance in the structure of the transformed entities and in all information that are exchanged, lead to the question, what is it that really is changed in a Lorentz boost rotation? The answer is the foundation of the measurement reference is inevitably justified locally. (everywhere) 5.7.4.4. The Doppler Effect of the Lorentz Boost

We have through this book tried to justify that an a priory measure is founded on counting quanta of radian development in an circular oscillating entity. The task is to define a reference entity quality that oscillate as reference chronometer clock, that defines frequency energies 𝜔 of other identical entities in consideration. A local R known entity ΨR has a known quality oscillating frequency energy 𝜔0 = 𝜔0,R quantity measured stationary in the local observing laboratory. A far away boosted entity ΨS identical with ΨR has in its local autonomy the same oscillating frequency energy 𝜔0,S = 𝜔0 as ΨR . What measured difference in received frequency will we perceive at R from the distant boosted entity ΨS ? To answer this, we use the relativistic Doppler formula for the longitudinal speed 𝛽∥ ∈ℝ in the direction u of the boost in the u -bivector-plane 𝛽∥ =𝛽 (1−𝛽∥ )𝜔0 (1−𝛽)𝜔0 1−𝛽 (5.376) 𝜔received from S = 𝛾(1 − 𝛽∥ )𝜔0 = → = 𝜔0 √ . 2 √(1−𝛽)(1+𝛽)

√ 1−𝛽

1+𝛽

For 0 < 𝛽 ≤ 1 we have the redshift, that we interpret as galaxies boosting further away. – 283

I am sorry to tell that I have tried to find an analytic expression between the arguments 𝜆 and 𝜁 in my formula books and Wikipedia etc. without success, some reader may find this. – My decision is that I do not have the lifetime capacity for this. 284 We see the double null helixes structure direction in Figure 5.55 perform physical perpendicular in its opposition in that the unit circular speed measure propagation speed from the objective extension direction u = σ3 .

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The decrease in the received oscillating frequency energy dilate the chronometric count along the locally defined development parameter relative to the local equivalent identical entity ΨR , [𝜔0−1 ]. 5.7.4.5. The Space-Time Algebra STA from a 4-dimensional 1-vector basis

Often the Minkowski -plane is called the Lorentz plane because it is the hyperbolic plane for the Lorentz rotation. What happens external outside this plane. The tradition is to make a time dimension and three space dimensions with opposite signatures, we use in this book (+, −, −, −) for a Clifford algebra 𝒢1,3 (ℝ) the Minkowski metric space foundation basis {𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 }. The direction 𝛾0 is the unit cyclic measure of development, that is an isometric measures for the three dimensional directions units 𝛾1 , 𝛾2 , 𝛾3 of Descartes extension: length, breadth, depth in the so called natural space. The STA basis {𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 } we define as orthonormal, from (5.300)-(5.304) we have (5.377) 𝛾0 ⋅ 𝛾1 = 𝛾0 ⋅ 𝛾2 = 𝛾0 ⋅ 𝛾3 = 0, |𝛾0 | = |𝛾1 | = |𝛾2 | = |𝛾3 | = 1, 𝛾1 ⋅ 𝛾2 = 𝛾2 ⋅ 𝛾3 = 𝛾3 ⋅ 𝛾1 = 0. ← Orthogonal extensions, therefor ⊥. As we have seen above 𝛾0 ⋅ 𝛾𝑘 =0 are in substance not perpendicular, in that their directions can be changed by a Lorentz rotation. Anyway, we display the 𝛾0 direction as perpendicular to all NOW lines in Figure 5.56the NOW plane 𝛾1 𝛾2 of , where 𝛾0 ∥(𝛾1 𝛾2 ). The three extension direction unit 1-vectors 𝛾𝑘 display a three dimenFigure 5.56 The STA basis {𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 } sional Cartesian basis, that local are both perpendicular and orthonormal. displayed as perpenThe development dimension is not spatial, an therefor we display the dicular directions in direction unit 𝛾0 as cyclic oscillating around in an unit circle following all four dimensions. the arc of its circumference repeating itself periodically for every 2𝜋𝛾0 We chose the 𝛾3 The founding unit 𝛾0 reference is then the radian measure of the unit direction for the relativistic velocity circle performing an oscillation with an autonomous frequency energy 𝜔 that causes the Lorentz 𝛾2 𝛾1 𝜔𝑡 of the local entity. The oscillating rotation is then performed by 𝑒 , rotation 𝑒 𝛾3 𝛾0 𝜁 in a traditional prescribed 𝑒 −𝑖𝜔𝑡 ∈ℂ, where 𝑡 is the development chronometer. Minkowski 3 -plane The reader is encouraged to compare this chronometric timing concept direction 3 ≡ 𝛾3 𝛾0 with Figure 1.2 in chapter I. 1.6.2.3. When it comes to the impact on that’s not displayed. The Euler oscillating rotation extension space compare with the idea of Figure 3.13 in chapter I. 3.4.1. 𝑒 𝛾2 𝛾1 𝜔𝑡 by the arc Interpreted as a tangent 1-vector, 𝛾0 support a classical (Augustinian) direction 𝛾0 in the timeline going from the first beginning to the final end (Α→Ω). Here in extension plane supthis book we (similar to the ancient Greeks) like to wrap this concept into ported by 𝛾1 𝛾2 = 𝒊3 . a oscillating circular rotation for each local autonomous defining entity. The null direction is shown as null helixes When it comes to two entities ΨA and ΨB separated in the extension collapsed around the 𝛾3 direction 𝛾3 where ΨB gets information about ΨA , we need a third signal extension direction. entity Ψ bringing the information about ΨA to ΨB . We set the speed of Undecided orientation this subton Ψ to one unit 𝛽=1 (e.g. speed of light 𝑐=1) observed from both −𝑘 =𝛾0 𝛾𝑘 or 𝑘 =𝛾𝑘 𝛾0 ΨA and ΨB . Seen from ΨB the transmission direction of the signal Ψ will . fall in the null direction of the receiving entity ΨB that is the interpreting observer. In principle Figure 5.55 show the same situation as Figure 5.56 for a subton Ψ intuit from external for the receiving situation at B, the differences is in the interpretations of the displays.

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5.7.5. The planes of Space-Time Algebra and the Euclidean Cartesian plane 5.7.5.1. Founding Summary of Minkowski Space with Euler and Lorentz Rotations

We have in this section 5.7 introduced the Minkowski metric space as a Clifford algebra with mixed signatures, (+) for development and (−) for extensions, by defining a Minkowski -plane bivector in § 5.7.1 with signatures (+, −) for the algebra 𝒢1,1 (ℝ) and introducing null directions for information, and in § 5.7.1.3 the development count 𝛾0 as an isometric measure. The traditional Minkowski display is interpreted in section 5.7.2, first for the plane display (+, −) in Figure 5.53, then in § 5.7.2.2 this plane display is Euler rotated around in a two dimensional extension plane 𝛾1 𝛾2 direction resulting in a Minkowski space algebra 𝒢1,2 (ℝ) with signature (+, −, −) displayed in Figure 5.54, with a null cone. In §5.7.2.3 this is supplemented with a third dimension of extension 𝛾3 , forming the full STA Minkowski signature (+, −, −, −). From the paravector concept 𝓅 in section 5.7.3 we get from the development count 𝛾0 the STA Clifford algebra 𝒢1,3 (ℝ) with its 1-vector concept 𝑝=𝓅𝛾0 (5.345). The Lorentz rotation is discussed in section 5.7.4 as a hyperbolic rotation in a -bivector-plane, then as a Lorentz transformation § 5.7.4.2 of the paravector, and the STA Lorentz boost in § 5.7.4.3 with the relativistic speed 𝛽, and invariant quantities of information with a hyperbolic same folding of development to a helix extension directions approaching the null direction for 𝛽→1. 5.7.5.2. Mapping Operation Between STA planes and the Euclidean Cartesian plane

As the ideological foundation for the plane concept we choose to use the Euclidean approach by the Cartesian orthonormal basis {σ1 , σ2 } that we immediately expands to the standard basis (5.198) {1, σ1 , σ2 , 𝒊 ≡ σ2 σ1 } for the Euclidean plane algebra. Our experience has taught us, that is an over simplified static view, where nothing happens, and no information is transmitted. Anny knowledge of the physical situation in this is a priori transcendental to our intuition, as Immanuel Kant told us. We have to endow the situation with a measure concept, that make something happens. We need a reference for this, and this is a cyclic oscillator with a stable reliable angular frequency energy 𝜔, that make the one quantum count for us. This oscillator possesses an information development direction unit 𝛾0. We expand the sequential order of counts to a real number continuous monotone growing ⃗⃗ , 𝛾02 ≡1 ∈ℝ ⃗⃗ } as a linear Euclidean algebra parameter span of development direction {𝜆0 𝛾0|∀𝜆0 ∈ℝ 𝒢1 (ℝ) of one dimension for a continuous 1-vector concept 𝜆0 𝛾0 of a development direction, a primary quality of first grade (pqg-1), that indeed not possess any Descartes extension in space. We choose this continuous count to be the measure for the Euclidean plane object defined by the Cartesian orthonormal basis {σ12 = σ22 = 1, σ1 , σ2 }. The measure 𝛾02 =1 can be the measure for these two σ12 = σ22 = 1. The information signal generated from the oscillating generator for 𝛾0 has to be isometric for both σ1 and σ2 transmitted from their common bottom to their tips, vice versa. The measurement mapping is established by the multiplication operation (5.328): {1, σ1 , σ2 } 𝛾0 {1𝛾0 , σ1 𝛾0 , σ2 𝛾0 } = {𝛾0 , 𝛾1 , 𝛾2 } . (5.378) Then we have an autonomous isometric orthonormal basis for a plane in STA. We remember the negative signature for 𝛾12 = 𝛾22 = −1, confirmed by σ𝑘 𝛾0 σk 𝛾0 = −σ𝑘 σ𝑘 = −σ𝑘2. The Euclidean plane unit bivector 𝒊≡ 𝛔2 𝛔1 is mapped by 𝛾02 =1 (5.379) 𝒊 = 𝒊𝛾02 = −𝛔2 𝛾0 𝛔1 𝛾0 = −𝛾2 𝛾1 = 𝛾1 𝛾2 . Now we have done the map for the Euclidean plane concept represented by their standard bases {1, 𝛔1 , 𝛔2 , 𝒊 ≡ 𝛔2 𝛔1 } ⟷ {𝛾0 , 𝛾1 , 𝛾2 , 𝛾1 𝛾2 } (5.380) and connected the extension in the Cartesian plane with a oscillating reference measure 𝛾0. Then we have the isometric measure in what David Hestenes call Space-Time-Algebra (STA) [6]

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We now have two different epistemological pictures of the physical space 𝔊 where plane surfaces 𝔓 by analogy can be drawn as two object 1-vectors 𝛔1 and 𝛔2 directions, from which we form the standard basis {1, 𝛔1 , 𝛔2 , 𝒊 ≡ 𝛔2 𝛔1 } for the Euclidean plane, that gives opportunity for an Euler rotation as (5.195) or oscillation as (5.196). The 1-rotor of this Euler rotation is (5.381) 𝑈𝜙 = 𝑒 𝒊½𝜙 = 𝑒 𝛔2𝛔1 ½𝜙 . This is equivalent to the same rotation in the extension plane of the Space-Time-Algebra (STA) (5.382) 𝑈𝜙 = 𝑒 𝛾1 𝛾2 ½𝜙 . We see that the Euler rotation is the founding essence for the plane concept 𝔓, that we know for a local classic two dimensional Euclidean picture surface internal in the physical space 𝔊. Both models of the STA basis and the traditional Euclid Cartesian basis is compared in Table 5.3 below. When we include the measure of information development to our picture of the physical space 𝔊, where we are familiar with the Euclidean plane concept 𝔓 containing the Euler rotation idea, we endow our model with an enrichment of the concept of the Minkowski 𝑘 -bivectors supporting an information plane like concept of balance (5.302): (5.383) 𝛾02 + 𝛾𝑘2 = 0. When the measure 𝛾0 is the action, the isometric extension 𝛾𝑘 is the reaction! This plane like isometry in balance is constructed orthogonal 𝛾0 ⋅ 𝛾𝑘 = 0 as an Minkowski 𝑘 -plane supported by the -bivector 𝑘 ≡ 𝛾𝑘 𝛾0 . The choice 𝛾02 ≡ 1 demand 𝛾𝑘2 = −1 for the balance. This balance also gives the null directions by their sum (𝛾0 + 𝛾1 ), or difference (𝛾0 − 𝛾1 ), (5.310). This construction make the Lorentz rotation from section 5.7.4 possible inside such a 𝑘 -plane by the Lorentz rotor (5.349), together with its full regular rotation by the canonical form (5.358), (5.384)

𝑈𝜁,𝑘 = 𝑒 ½𝜁𝑘 = exp(½𝜁𝑘 ), where 𝑘2 =1

𝛾𝑘′ = 𝑒 ½𝜁𝑘 𝛾𝑘 𝑒 −½𝜁𝑘 = 𝑒 𝜁𝑘 𝛾𝑘 .

⇒

This rotation is parameter augmented by the rapidity 𝜁 ∈ℝ, that is approaching infinity 𝜁→∞ when the relativistic speed 𝛽= tanh 𝜁 → 1 is approaching the speed of information 𝑐=1, which is the case when we look at things by light, that it transmited towards us. The foundation of the two structures is shown in Table 5.3 by the mapping of these two (5.380).

Table 5.3 Comparison of the two standard basis pictures of the plane concept: {𝛾0 , 𝛾1 , 𝛾2 , 𝛾1 𝛾2 } ⟷ {1, 𝛔1 , 𝛔2 , 𝒊≡ 𝛔2

Lorentz - Minkowski - Hestenes, STA [6]

Euclid - Descartes - Euler → Pauli

STA planes

Euclidean Signature

𝒢1,2 (ℝ)

Development 1-vector Extension 1-vectors Minkowski -bivector Extension bivector Oscillation rotor (, −, −, ) Lorentz rotor (+, −)

Signature

STA, 𝑘=1,2

𝛾02 = +1

𝛾0 = ⃖⃗⃗ σ𝑘 𝛾𝑘

𝛾𝑘2 = −1 𝑘2 = +1

𝛾𝑘 = ⃖⃗⃗ σ𝑘 𝛾0 𝑘 ≡ 𝛾𝑘 𝛾0

(𝛾1 𝛾2 )2 = −1 𝛾1 𝛾2 = 𝒊 𝑈𝜙 𝑈𝜙† =1 ̃𝜁,𝑘 =1 𝑈𝜁,𝑘 𝑈

Jens Erfurt Andresen, M.Sc. Physics,

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unknown reason

idea as cause

static eternally existence immutable

σ𝑘 = ⃖⃗⃗ 𝛾𝑘 𝛾0

σ𝑘2 = +1

1-vectors extension

𝒊 ≡ σ2 σ1

𝒊2 = −1

bivector extension

𝑈𝜙 =𝑒 𝛾1 𝛾2 ½𝜙 𝑈𝜙 = 𝑒 𝒊½𝜙 𝑈𝜙 𝑈𝜙† =1

(+, +)

rotor oscillator

𝑈𝜁,𝑘 =𝑒 ½𝜁𝑘 Anny chance is transcendental to Cartesian space.

The geometric Clifford algebra 𝒢1,2 (ℝ) with a mixed signature basis (+, −) or (+, −, −, … ) make it possible to establish the fundamental idea of a isometric measurement concept for physics. We use the extended circular Euler rotating oscillator ( , −, −, ) to produce the development direction 𝛾0 as an extra dimension of quantum timing counts. A Lorentz rotation (+, −) express relative external speed.

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We use the local Cartesian orthonormal basis, that by analogy objects can be drawn on a surface and by that illustrate the plane concept for the intuition. From this we construct the geometric Clifford algebra 𝒢2 (ℝ) with the simple quadratic form as signature (+, +), and the bivector 𝒊2 = −1 from which we form the Euler rotor circle oscillator plane.

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The null direction in STA space can be interpreted as the direction where we see the Euclidean object of the Cartesian basis {σ1 , σ2 } drawn on our practical analogy surface 𝔓 in our physical space 𝔊. Compare this object with Figure 5.14 as intuition of the standard basis {1, σ1 , σ2 , 𝒊 ≡ σ2 σ1 } for the plane concept, where we have the defining unit scalar measure |σ1 | = |σ2 | = |𝒊| = 1 = 𝛾02. The founding chronometric unit 𝛾0 is given as one quantum radian circumference phase angular measure 1 = 𝛾02 of the reference oscillating Euler circular rotation that perform a Descartes extended plane for an entity in physical space of one universal Nature. 5.7.5.3. Exponential Function in the Plane Concepts

We have now introduced the two forms of exponential functions with bivectors argument (5.385) exp(𝒊𝜑) = 𝑒 𝒊𝜑 ∈ 𝒢2 (ℝ), where 𝒊2 = −1, for the Euler form for the rotation in a Euclidean plane supported from 𝒊 ≡ σ2 σ1 ∈ 𝒢2 (ℝ). And (5.386)

exp(𝜁) = 𝑒 𝜁 ∈ 𝒢1,1 (ℝ), where 2 = 1 for the Lorentz rotation in a Minkowski -plane supported from 𝑘 ≡ 𝛾𝑘 𝛾0 ∈ 𝒢1,1 (ℝ). These functions is both multivector valued and given by multivector augments. Before we proceed with geometric algebra for higher grades of dimensions, e.g. for the natural ℨ-space, we will introduce a generalised concept of the exponential function in geometric algebra. We benefit from a general knowledge about the exponential power expansions for multivectors. --------------------

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December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.8. The Exponential Function of Arbitrary Multivectors –

5.8. The Exponential Function of Arbitrary Multivectors The generalised multivector 𝐴 can be dissolved in its finite grade components 〈𝐴〉𝑟 (5.387) 𝐴 = ∑𝑛𝑟=0〈𝐴〉𝑟 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 + ⋯ + 〈𝐴〉𝑛 ∈ 𝒢𝑛 = 𝒢𝑛 (ℝ). Note that each 〈𝐴〉𝑟 of grade-𝑟 for 2 < 𝑟 ≤ 𝑛 is an external extension to the plane concept. We now define a generalisation of the exponential function for multivectors by a power series (5.388)

exp(𝐴) = 𝑒 𝐴

=



A  k! k

𝐴2

𝐴

k

= 1 + 1! +

=0

2!

+

𝐴3 3!

+ ⋯+

𝐴𝑘

+⋯

𝑘!

This is an algebraic definition of the exponential function and if the multivector 𝐴 ∈ 𝒢𝑛 stays in the algebra 𝒢𝑛 the result 𝑍 = 𝑒 𝐴 ∈ 𝒢𝑛 stays inside the same algebra. The reason is that for all grades higher than 𝑛 disappears 〈𝐴〉>𝑛 = 0, and the full geometric algebra 𝒢𝑛 is closed. The same closed consistency for the even algebras 𝒢𝑛+ , with 𝐴+ = 〈𝐴〉0 + 〈𝐴〉2 + ⋯. For two different multivector 𝐴, 𝐵 ∈ 𝒢𝑛 or 𝒢𝑛+ we formulate the restrictive additive rule for the exponentials (5.389)

(5.390)

𝑒 𝐴 𝑒 𝐵 = 𝑒 (𝐴+𝐵) = 𝑒 (𝐵+𝐴) = 𝑒 𝐵 𝑒 𝐴 The restriction is that products commutes 𝐴𝐵 = 𝐵𝐴, just as addition always do 𝐴+𝐵 = 𝐵+𝐴. This is seen by the binomial series for the added multivectors with the integer exponents n

 (n−n!k )!k!A

(𝐵 + 𝐴)𝑛 =

n−k

B k , with the demand 𝐴𝐵 = 𝐵𝐴, as [10]p.73 resulting in

k =0

(5.391)

𝑒 𝐴𝑒𝐵 =





m=0

Am m!





  Bn! =

n



n

n =0

n =0 k =0

An−k Bk ( n−k )! k!



 ( A+nB! )

=

n

= 𝑒 (𝐴+𝐵) = 𝑒 (𝐵+𝐴) .

n =0

For the non-commutating multivectors 𝐴𝐵≠𝐵𝐴 we will always be able to write (5.392) 𝑒 𝐴𝑒𝐵 = 𝑒𝐶 , but then we have 𝐶 ≠ 𝐴+𝐵, and we do not have a general solution. Later in this book we will study the case with anticommuting 𝐴𝐵 = −𝐵𝐴 bivector exponentials for rotations in an even algebra 𝒢3+ ~𝒢0,2 containing several independent plane directions. 5.8.1.2. The Hyperbolic Functions of Multivectors

We separate the exponent power series (5.388) in even and odd powers by the definitions  A −A 2k 𝐴2 𝐴4 𝐴6 (5.393) cosh 𝐴 ≡ e + e =  (A2k )! = 1 + 2! + 4! + 6! + ⋯ 2

(5.394)

A

sinh 𝐴 ≡ e − e

−A

2

=

k =0 

 (A2k +1)! 2 k +1

= 𝐴+

k =0

𝐴3 3!

+

𝐴5 5!

+

𝐴7 7!

+⋯

The sum of the even and the odd hyperbolic part is the exponential series (5.388) (5.395) 𝑒 𝐴 = cosh 𝐴 + sinh 𝐴. The input multivector 𝐴 is the argument in the functions 𝑒 𝐴 = exp(𝐴), cosh(𝐴) and sinh(𝐴) 5.8.1.3. The cosine and sine Functions of Arbitrary 𝒢𝑛 Multivectors

Introducing the generalised multivector unit pseudoscalar 𝐼 with the founding property 𝐼 2 = −1. Take it and (left) operate by multiplying it to a multivector 𝐴 we get a new multivector 𝐵 = 𝐼𝐴. We will use this to transform (5.393), (5.394) to the general cosine and sine multivector functions (5.396) (5.397)

cos 𝐴 = cosh 𝐼𝐴 = 1 + 1

sin 𝐴 = 𝐼 sinh 𝐼𝐴 = 𝐴 +

(𝐼𝐴)2 2! (𝐼𝐴)3 3! 𝐼

+ +

(𝐼𝐴)4 4! (𝐼𝐴)5 5! 𝐼

+ +

(𝐼𝐴)6 6! (𝐼𝐴)7 7! 𝐼



+⋯=

 k =0 

+⋯=

( −1)k A2 k

( 2k )!

 (−1(2) kA+1)! k =0

k

2 k +1

= 1− = 1−

𝐴2 2! 𝐴3 3!

+ +

𝐴4 4! 𝐴5 5!

− −

𝐴6 6! 𝐴7 7!

+ ⋯. + ⋯.

We compare with the traditional definition formulas, that also is valid for multivectors ©

Jens Erfurt Andresen, M.Sc. Physics,

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– 5.8.2. Exponential and Hyperbolic Functions in the Plane Concept – 5.8.2.3 The Lorentz 1-Spinor in the Line Direction Paravector Space –

𝑒𝐼𝐴 + 𝑒−𝐼𝐴 𝑒𝐼𝐴 + 𝑒−𝐼𝐴 and sin 𝐴 = , 2 2𝐼 and back to the exponential function with the multivector argument 𝐵 = 𝐼𝐴 (5.399) 𝑒 𝐼𝐴 = cos 𝐴 + 𝐼 sin 𝐴 . Here it is essential to distinguish the different pseudoscalar units 𝐼=〈𝐼〉𝑛 all with the same quality 𝐼 2 = − 1 of the different direction primary quality for each maximal grade 𝑛 for the geometric algebras 𝒢𝑛 of physics. For the Euclidian plane direction unit 𝒊 we have 𝒊2 = − 1. More about these pseudoscalars for higher maximal grades later below chapter 6, 7, etc… (5.398)

cos 𝐴 =

5.8.2. Exponential and Hyperbolic Functions in the Plane Concept

In the plane concept the multivector 𝐴 can be dissolved in its grade components 〈𝐴〉𝑟 (5.400) 𝐴 = ∑2𝑟=0〈𝐴〉𝑟 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 ∈ 𝒢2 = 𝒢2 (ℝ). (5.161)-(5.162). 5.8.2.2. The 1-Spinor in the Euclidean Cartesian plane

First, we look at the even part of the plane algebra, as 𝐸 = 〈𝐸〉0 + 〈𝐸〉2 ∈𝒢2+ (5.401) 𝐸 = 𝛿 + 𝛽𝒊 = 𝛿 + B, where B 2 = −𝛽 2 , and 𝒊2 = −1, where the Euclidean bivector B is in the plane direction of 𝒊 ≡σ2 σ1 , as the bivector span, B = 𝛽𝒊. The scalar 𝛿 ∈ℝ, multiplicative commute with B, B𝛿 = 𝛿B, therefor by (5.389) exp(𝐸) (5.402) (5.403)

𝒵 = 𝑒 𝐸 = 𝑒 𝛿+𝛽𝒊 = 𝑒 𝛿 𝑒 𝛽𝒊 = 𝜌𝑒 𝛽𝒊 ∈ 𝒢2+ , with 𝜌 = 𝑒 𝛿 ∈ℝ , 𝛽 ∈ℝ, and 𝛿 = ln(𝜌) ∈ℝ. Here in the plane the power expansion in the even multivector 𝐸 of max grade 𝑛=2 is 𝒵= 𝑒 𝐸 = 1 +

𝐸 1!

+

𝐸2 2!

+ ⋯+

𝐸𝑘 𝑘!

+⋯

And the 1-spinor form as in section 5.2.9 (5.106) we have the 𝜌 dilated Euler 1-rotor 𝑈𝛽 = 𝜌𝑒 𝒊𝛽 (5.404)

ru1 = r⋅u1 + r∧u1 = 𝒵 = 𝜌𝑒 𝒊𝛽 = 𝜌 cos 𝛽 + 𝒊𝜌 sin 𝛽, where the last expression is the cartesian component coordinates to the multivector (5.138) using the polar coordinates (𝜌, 𝛽) as input arguments, again the direction of this plane is 𝒊 ≡ σ2 σ1.

5.8.2.3. The Lorentz 1-Spinor in the Line Direction Paravector Space

Second, we look at the simple mixed paravector algebra, as 𝓅 = 〈𝓅〉0 + 〈𝓅〉1 written as (5.240) (5.405) 𝓅 = 𝛼1 + 𝜁u = 𝛼 + p, where p2 = 𝜁 2 , and u2 ≡ 1. In that 𝛼 ∈ℝ, commute with u as 𝛼u = u𝛼, we have the commuting product factors of exp(𝓅) (5.406) 𝓅Λ = 𝑒 𝓅 = 𝑒 𝛼+𝜁u = 𝑒 𝛼 𝑒 𝜁u = 𝜌Λ 𝑒 𝜁u . The first factor in the last term is a dilation by a real scalar without direction (5.407) 𝜌Λ = 𝑒 𝛼 = cosh 𝛼 + sinh 𝛼 ∈ℝ. The second factor is a Lorentz rotor paravector of the grade form 〈𝐴〉0 + 〈𝐴〉1 (as (5.358)) (5.408)

2 𝑈𝜁,u = 𝑒 𝜁u = cosh(𝜁u) + sinh(𝜁u) =

cosh 𝜁 + u sinh 𝜁.

The reason for this is that u2 =1, first the even expansion cancel the 1-vector direction (5.409)

cosh(𝜁u) = 1 +

(𝜁u)2 2!

+

(𝜁u)4 4!

+⋯=1+

𝜁2 2!

+

𝜁4 4!

+ ⋯ = cosh 𝜁

∈ℝ.

Then the odd expansion separate the 1-vector direction (5.410)

sinh(𝜁u) = 𝜁u +

(𝜁u)3 3!

+

(𝜁u)5 5!

+ ⋯ = u (𝜁 +

𝜁 3 u2 3!

+

𝜁 5 u4 5!

+ ⋯ ) = u sinh 𝜁.

We find that (5.406) is a paravector as (5.242) containing a scalar part plus a 1-vector part (5.411) 𝓅Λ = 𝑒 𝛼+𝜁u = 𝜌Λ 𝑈𝜁,u = 𝜌Λ cosh 𝜁 + u(𝜌Λ sinh 𝜁). The exponential function exp(𝓅) of a paravector is a paravector 𝓅Λ with preserved line direction. ©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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December 2020

– II. . Geometry of Physics – 5. The Plan Concept – 5.8. The Exponential Function of Arbitrary Multivectors – 5.8.2.4. The Lorentz 1-Spinor in the Minkowski -plane

To find a correspondence to the Space-Time-Algebra (STA) we have to introduce an information development unit 1-vector direction 𝛾0, and to map the paravector (5.405) to the a Minkowski -plane we simply operator right multiply with 𝛾0 , 𝛾02 ≡ 1 (we use (5.328) with u = σ𝑘 ) (5.412) 𝓅 = 𝓅𝛾0 𝛾0 = ( 𝛼1 + 𝜁u)𝛾0 𝛾0 = 𝛼 + 𝜁u𝛾0 𝛾0 = 𝛼 + 𝜁𝛾u 𝛾0 = 𝛼 + 𝜁u. To achieve the STA 1-vector we map this by right multiply operation once again with 𝛾0 (5.413) 𝑝u = 𝓅𝛾0 = 𝛼𝛾0 + 𝜁u 𝛾0 = 𝛼𝛾0 + 𝜁𝛾u . (using (5.309) or) Using this STA 1-vector 𝑝u = 𝛼𝛾0 +𝜁𝛾u as an argument in an exponential function we write (5.414) exp(𝑝u ) = 𝑒 𝛼𝛾0 +𝜁𝛾u . The result of this will of course stay in the STA algebra due to the exponential power series (5.388). The two different directions 𝛾0 and 𝛾u anticommute 𝛾0 𝛾u = −𝛾u 𝛾0 and are orthogonal 𝛾0 ⋅𝛾u =0 and therefor independent. General for 𝐴𝐵 ≠ 𝐵𝐴 we will be able to write 𝑒 𝐴 𝑒 𝐵 = 𝑒 𝐶 ⇏ 𝐶=𝐴+𝐵, but for 𝐶 ≠ 𝐴+𝐵 we do not have a general solution for 𝐶 from 𝐴 and 𝐵 [10]p.74. The two exp forms 𝑒 𝛼𝛾0 and 𝑒 ζ𝛾u have independent 1-vector direction arguments and their product factors are mostly not commuting ?

𝑒 𝛼𝛾0 𝑒 ζ𝛾u ≠ 𝑒 ζ𝛾u 𝑒 𝛼𝛾0 . Anyway, by using the paravector concept 𝓅 = 𝛼 + 𝜁u where the development measure parameter is a pure real scalar 𝛼 ∈ℝ. (5.416) 𝓅Λ = 𝑒 𝛼+𝜁u = 𝑒 𝛼 𝑒 𝜁u = 𝑒 𝜁u 𝑒 𝛼 = 𝑒 𝛼 (cosh 𝜁 + u sinh 𝜁). (5.415)

Mapped to a STA 1-vector space {𝛾0 , 𝛾u } ~ {𝛾0 , 𝛾1 , 𝛾2 , 𝛾3 }285, (5.417) 𝑝u,Λ = 𝓅Λ 𝛾0 = (𝑒 𝛼 cosh 𝜁 + u 𝑒 𝛼 sinh 𝜁)𝛾0 = (𝑒 𝛼 cosh 𝜁)𝛾0 + (𝑒 𝛼 sinh 𝜁)𝛾u In (5.416) we associate the Cartesian length direction with the measure bivector u ≡ 𝛾u 𝛾0 , u2 ≡1 that in the tradition is pseudonym with the Euclidean 1-vector u ≡ 𝛾u 𝛾0 , u2 ≡ 1, in (5.411). When we think of a physical length direction quantity unit u, we cannot deny a measure unit 𝛾0. 5.8.3. A Philosophical Conclusion on All This Exercise

When you look on the mathematical constructions you may wonder, what will work when it is used on physics, and give consistence with lab measurements? One a priori rule, that had been mandatory for science since Baruch Spinoza is that Nature is the master in the name of Physic. Any Low of mathematic has to obey the a priori structures of Physics in a universal context. When you (we) as humans construct a mathematic system, the Laws of this are dependent on Physics, in the last instance our thoughts are physical processes. An simple example: The capability to count numbers as one and one more… of equal identical but distinguishable entities is possible, not only by humans but all entities that possess information. When you let the human science of mathematics be the master, you have lost to religion, and any knowledge about Nature is segregated beyond the transcendental barrier. The human mathematical idea of continuity has to be a map of moving two entities relative to each other in an extensive way without abruption by annihilation and creation. To accept the existence of scalars without directional extension to be a quantity of an entity, do not need a religious super-instance to be the background for the whole universe. Anyway, this philosophy is performed by humans and not by animals or any other physical entities neither super-computers with artificial intelligence. Consciousness is not an art, but an a priori fundamental concept idea emerged from Nature. 285

Where the communication extension direction is 𝛾u = 𝜐1 𝛾1 + 𝜐1 𝛾2 + 𝜐1 𝛾3 for 𝜐𝑘 ∈ℝ, and 𝜐12 + 𝜐22 + 𝜐32 = 1 = |𝛾u | .

©

Jens Erfurt Andresen, M.Sc. Physics,

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December 2020

– 5.9.3. General Exponential Series – 5.8.2.4 The Lorentz 1-Spinor in the Minkowski -plane –

5.9. Concluding Summary on the Algebra for the Geometric Plane Concept 5.9.1. The Euclidean Plane Concept

From the first start for a plane concept we have Euclid’s Elements chapter 5.1 where we emphasised the angular concept feature between 1-vectors. In chapter 5.2 from the linear additive algebra of geometric vector space we form the bilinear Clifford algebraic quadratic form from which we develop a vector product space called geometric algebra, where the geometric product is classified in an inner 5.2.3 and an outer product 5.2.5, that in the plane concept is an inner scalar and an outer bivector. The geometric product of two 1-vectors result in a 1-rotor concept in the plane 5.2.7, with a bivector argument exponential function 5.2.8, followed by the plane 1-spinor concept. In section 5.3 we synthesises some qualities of the plane concept in the view of geometric algebra with multivectors of grades 0,1, and 2, i.e. scalars, 1-vectors, and bivectors, together with 1-rotors as exponential functions. In section 5.4 we describe the transformations that geometric algebraic elements can do to the plane concept for physical entities as reflections 5.4.2 and rotations 5.4.5 with introduction of the canonical form for sandwich operations. In section 5.5 inherit qualities of nilpotence 5.5.3 and idempotence 5.5.4, paravectors and mutual annihilating projection spectral basis. Followed 5.6 by the 2×2 real matrix concept representation of the Euclidean plane related to the geometric algebra 𝒢2 (ℝ). 5.9.2. The Non Euclidean Plane Concept

In section 5.7 definition of the Minkowski isometric measure balance in a -plane algebra 𝒢1,1 (ℝ) with mixed signature (+, −) resulting in null basis directions for the information propagation. A possible length quantity for an extension direction shall inherit its measure from a isometric balance with a development measure. The Lorentz invariant rotations secure the conservations and the traditional relativistic Lorentz transformation make an apparently deformation from boosts. 5.9.3. General Exponential Series

In section 5.8 we generalise the exponential power series to concern all type of geometric multivectors, where the product of two multivector valued exponential functions only commute if their two arguments commute. Simply if there is independent basic directions of the generating multivectors (1-vectors, bivectors, etc.) for the exponential arguments that support the acting rotations, the sequential order of the multiplication operations has causal impact.

©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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– 5.9.3. General Exponential Series – 5.8.2.4 The Lorentz 1-Spinor in the Minkowski -plane –

6. The Natural Space of Physics 6.1. The Classic Geometric Extension Space ℨ

(6.1) (6.1a) (6.1b) (6.1c)

We recall from section 4.2 the name 𝔊 for the idea of a total natural space with a geometry. First for the plane concept 𝔓 we have three points A, B, C ∈𝔊, where the third C ∉ℓAB is not on the line A, B ∈ℓAB . We form in our minds a triangle △ABC with the circumscribed circle ⊙ABC that define the plane γABC . We will from this plane define the natural space ℨABCD: We presume four points A, B, C, D ∈𝔊, where we can pass judgments A, B ∈ℓAB and C ∉ℓAB making a plane A, B, C ∈γABC and D ∉γABC . Point D external to this plane defining an extension space ℨABCD, as a subject to the concept of the extension space substance ℨ. ℨABCD ⊂ 𝔊, ℨABCD ∈ℨ , where we apply the a priori synthetic judgments: A, B, C, D ∈𝔊 ⇒ ℓAB , ℓBC , ℓCA , ℓAD , ℓBD , ℓCD ⊂ ℨABCD ; AB, BC, CA, AD, BD, CD ⊂ ℨABCD ; γABC ,γABD, γBCD ,γCAD ⊂ ℨABCD ; △ABC,△ABD,△BCD,△CAD ⊂ ℨABCD Tetrahedron(ABCD) ⊂ ℨABCD ; Circumscribe Sphere(ABCD) ⊂ ℨABCD The geometric extension of space per se is a platonic idea and therefore transcendental for the recognition, but for the intuition, it is possible to recognize or construct extended structures in natural space 𝔊 of physics. The simplest symmetry of four different points ABCD is represented by the Platonic solid regular tetrahedron from which the circumscribed sphere is inherit.

3 dimensions and the Concept of Geometric Extension (pqg-3) Quote [12] : “Euclid’s Elements: E I.De.5. A surface is that which has length and breadth only. E XI.De.1. A solid is that which has length, breadth, and depth. E XI.De.2. A face of a solid is a surface. E XI.De.3. A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane. E XI.De.4. A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. E XI.De.5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. E XI.De.6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes. E XI.De.7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another. E XI.De.8. Parallel planes are those which do not meet. E XI.De.9. Similar solid figures are those contained by similar planes equal in multitude. E XI.De.10. Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude. E XI.De.11. A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines, that is, a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point. E XI.De.12. A pyramid is a solid figure contained by planes which is constructed from one plane to one point. ©

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– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.1. The Classic Geometric Extension Space ℨ –

E XI.De.13. A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms. E XI.De.14. When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere. E XI.De.15. The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned. E XI.De.16. The center of the sphere is the same as that of the semicircle. E XI.De.17. A diameter of the sphere is any straight line drawn through the center and terminated in both directions by the surface of the sphere. E XI.Pr.1. A part of a straight line cannot be in the plane of reference and a part in plane more elevated. E XI.Pr.3. If two planes cut one another, then their intersection is a straight line. E XI.Pr.4. If a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them. E XI.Pr.5. If a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane. E XI.Pr.6. If two straight lines are at right angles to the same plane, then the straight lines are parallel. E XI.Pr.8. If two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane. E XI.Pr.9. Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. E XI.Pr.10. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles. E XI.Pr.11. To draw a straight line perpendicular to a given plane from a given elevated point. E XI.Pr.12. To set up a straight line at right angles to a give plane from a given point in it. E XI.Pr.13. From the same point two straight lines cannot be set up at right angles to the same plane on the same side. E XI.Pr.14. Planes to which the same straight line is at right angles are parallel. E XI.Pr.15. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel. E XI.Pr.16. If two parallel planes are cut by any plane, then their intersections are parallel. E XI.Pr.17. If two straight lines are cut by parallel planes, then they are cut in the same ratios. E XI.Pr.18. If a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane. E XI.Pr.19. If two planes which cut one another are at right angles to any plane, then their intersection is also at right angles to the same plane. “

©

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– 6.1.2. Additional A Priori Judgments to the Euclidean Stereo Space Geometry – 5.8.2.4 The Lorentz 1-Spinor in the Minkowski -plane –

6.1.2. Additional A Priori Judgments to the Euclidean Stereo Space Geometry

r. Postulate: A space object (solid) is uniquely defined by four points A,B,C,D , which does not lie in the one and the same plane. The plane γABC through A,B,C, as in 5.1.1.2,k., is seen as a foundation for a fourth point D ∉ 𝛾ABC on a line ℓAD through A outside and perpendicular to the plane γABC (E XI.De.3, E XI.Pr.12). Such lines from all points X∈γABC generate the entire stereo space outside the plane. s. A straight line perpendicular to the plane is called a normal to the plane. From any point outside the plane can be raised just one normal to the plane. Contradiction, when a line through D and A not normal the line ℓAD is inclined to the plane γABC , (E XI.De.5.). t. Two planes have a mutual inclination (E XI.De.6.) defined as the angle between their normal lines or between their two normal 1-vectors |n2 | = |n1 | = 1. For the object planes γABC and γACD we have the angular measure arc1 =∡BAC and arc2 =∡CAD for the two angles internal in the two planes with radius reference |AB| =1. We have the two normal 1-vector objects n1 and n2 support spanning the two normal lines through A to the planes γABC and γACD . The mutual angle arcn2∧n1 =∡(n2 , n1 ) define the angle between the planes. That angle exists in the plane of n2 ∧n1 perpendicular to the other planes. These unit normal 1-vectors n1 and n2 represents the two axes of arcus circular rotations in their respective planes arc1 ⊂γABC and arc2 ⊂γACD. u. Two plane that intersects in a line has both their normal 1-vectors perpendicular to that line. This line is spanned from a third unit 1-vector n3 that is perpendicular n3 ⊥n1 and n3 ⊥n2. There is now a symmetry to choose n2 ⊥n1 . In geometric algebra we rename such an orthonormal basis σ1 = n1 ,σ2 = n2 and σ3 = n3, when σj ⋅σk = 𝛿𝑗𝑘 , 𝛿𝑗𝑘 =1 when 𝑗=𝑘 else 𝛿𝑗𝑘 =0 for 𝑗, 𝑘=1,2,3. Their three supported lines ℓ𝛔𝑗 form 12=3⋅4 mutual equal right angles in a symmetry. Their three transversal normal planes also have mutual perpendicular angles. v. Four points forming a tetrahedron fix a solid in space. The circumscribed sphere to the four points A,B,C,D form a center O from which four v. 1-vectors called a tetraon point out the four points on the sphere that define the stereo ℨ-space symmetry 𝑆 2 → 𝑆 3 outwards. The regular tetrahedron is a Platonic ideal. ©

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Figure 6.1 Concept of a Space: r-w.

r. Four points→ one space object. s. One external point→ one normal.

t. The inclination of a plane object γACD to a plane object γABC , with definition of a mutual angle.

u. The orthonormal basis set and their transversal planes.

Volume I, – Edition 1, – Revision 3,

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– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.1. The Classic Geometric Extension Space ℨ –

w. When the fourth defining point D in a solid in space is moved to the plane γABC , it collapses to that plane (see 5.1.1.2,o) and the circumscribed sphere collapse to a circle, (compare with chapter 5.3 Figure 5.31 and Figure 5.34 ). 6.1.2.1. The concept of Spatial Angular Structure

The idea of an angular circle rotation subject is representing a plane and gives raise to rotational symmetry in this plane, which we represent with a normal 1-vector for the rotation axis perpendicular to this circle plane. This symmetry plane direction primary quality of grade 2 (pqg-2) we call the transversal plane to the axial normal direction pqg-1. In Figure 6.1,t we see two objects of these planes with an inclination. The angle between these inclining planes is defined as the angle between their normal 1-vectors ∡(n2 , n1 ). The intersection between these two inclining transversal planes form a line ℓAC ⊥(n2 ∧n1 ) perpendicular to their normal 1-vector axes that form a third transversal plane γn2∧n1 , that a priori is perpendicular to two original planes γ⊥n2 and γ⊥n1 . The causal third normal 1-vector σ3 to γ⊥σ3 = γn2 ∧n1 is parallel to the intersection line σ3 ∥ℓAC . In this way the idea of an axial structure in space along σ3 as the intersection line between the two inclining planes is the angular generator for a cylindrical rotation around this intersection axis. This axial rotation symmetry is essential for the transversal plane waves of light. 6.1.3. The Euclidean 1-vector Space for Natural Space.

(6.2)

In the tradition since Descartes the space extensions of Figure 6.2 Axial cylinder symmetric solid is that which has length, breadth, and depth. (E XI.De.1.), rotation of a transversal plane field 𝛾n2∧n1 which had led to the Cartesian 1-vector space (𝑉3 , ℝ) with a Euclidean metric (5.51) defined by an orthonormal basis set, e.g. {σ1 , σ2 , σ3 }. An arbitrary 1-vector in this space can then be formed by 3 x = 𝑥1 σ1 +𝑥2 σ2 +𝑥3 σ3 ∈(𝑉3 , ℝ) ~ ℝ11 ⊕ℝ12 ⊕ℝ13 = ℝ1,2,3 (or = ℝ3xyz ). A local point X position in space pointed out from an origo O by given coordinates (𝑥1 , 𝑥2 , 𝑥3 ) in the three directions of the 1-vectors in a basis set {σ1 , σ2 , σ3 }

6.1.3.2. Covariant Cartesian Coordinates

(6.3)

Given the orthonormal basis set of 1-vectors {σ1 , σ2 , σ3 }. Each covariant coordinate is defined as normal distance to the transversal planes through O, from X in Figure 6.3, defined by σ3 σ2 = σ3 ∧σ2 , σ1 σ3 = σ1 ∧σ3 , and σ2 σ1 = σ2 ∧σ1.

6.1.3.3. Contravariant Coordinates

(6.4) (6.5)

When the basis is not Cartesian there are oblique angles between the normal transversal planes to the 1-vector axis 𝑥1 n1 , 𝑥 2 n2 , 𝑥 3 n3. Then the contravariant coordinates 𝑥 𝑘 are defined as the coordinate-axis intersected by the parallel planes (hyper surfaces) formed by the two other axes respectively (n3 ∧n2 ) ∠n1 , (n1 ∧n3 ) ∠n2 and (n2 ∧n1 ) ∠n3 . Then the addition form works, but not the Pythagorean. Figure 6.3 2 2 2 𝐱 = ⃗⃗⃗⃗⃗ OX = 𝑥1 n1 + 𝑥 2 n2 + 𝑥 3 n3 , 𝐱 2 ≠ 𝑥1 + 𝑥 3 + 𝑥 3 . The Cartesian space coordinate system. It is left to the reader to figurate this by an oblique prism.

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

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A Research on the a priori of Physics –

December 2020

– 6.1.4. A Curiosum, the Concept of a Tetraon in a Tetrahedron – 6.1.3.4 The Classical Cartesian Coordinate System for Position Points in ℨ-Space – 6.1.3.4. The Classical Cartesian Coordinate System for Position Points in ℨ-Space

(6.6)

(6.7)

The tradition says that we are pointing out three directions in natural physic ℨ-space by giving a orthonormal basis set of 1-vectors {σ1 , σ2 , σ3 }. The main problem here is that the coordinate-axis for these directions not necessary intersects due to their translation invariance of the 1-vector idea. This demand us to choose an arbitrary origo O for the position coordinate system,286 we name this cartesian system for {O, σ1 , σ2 , σ3 }. When we have a central origo O in our world of locality of what we call ℨ-space and we point out three perpendicular object directions σ1 , σ2 , σ3 , then we can point out a position ⃗⃗⃗⃗⃗ OX = x = 𝑥1 σ1 + 𝑥2 σ2 + 𝑥3 σ3. A point X in ℨ-space relative to origo O, demanded by the orthogonal 1-vector directions σ1 , σ2 , σ3 and given by the real scalar coordinates that meets 𝑥1 = σ1 ⋅ x, 𝑥2 = σ2 ⋅ x, 𝑥3 = σ3 ⋅ x . We see that the three orthogonal pqg-1-vector directions for any physical ℨ-space entity not automatic point out an origo, we must do it our self in this simple Cartesian view. Instead when we consider three non-parallel planes, we automatic get an intersecting origo. First two inclining planes (E XI.De.6.), 6.1.2,t and 6.1.2.1 will intersect in a straight line. This line will intersect the third plane in just one point, that automatic will be an origo for these three inclining planes. Do we have a physical entity, that can be characterised by three independent plane pqg-2 direction qualities there will always be one intersection point, that will form a locality center as an origo point for that entity. For the idea of three perpendicular planes we can let their Figure 6.4 Three perpendicular planes normal 1-vectors be the orthonormal basis {σ1 , σ2 , σ3 }. intersects in just one point O. These planes The intersections of these three planes will then implicit be are represented by their three orthogonal normal 1-vectors as basis {σ1 , σ2 , σ3 } for the origo as a center of locality for these planes as shown 287 their directions. The axis for the drawn Figure 6.4. The translation of each plane will result in objects 1-vectors do not intersect. translation of the locality center origo for the belonging This figure has reference to Figure 6.3. entity through ℨ-space in that same direction.

6.1.4. A Curiosum, the Concept of a Tetraon in a Tetrahedron

(6.8)

Four points A, B, C, D are defining a solid span in space § 6.1.2, Figure 6.1,v. Classically the Platonic tetrahedron is the ideal subject for the simplest solid object. We will concentrate on the center of the locus situs and choose the circumscribed center O. We form the 1-vectors from center to the four points ua = ⃗⃗⃗⃗⃗⃗ OA, ub = ⃗⃗⃗⃗⃗⃗ OB, uc = ⃗⃗⃗⃗⃗⃗ OC, and ud = ⃗⃗⃗⃗⃗⃗ OD, |u | |u | |u | |u |. where due to the circumscribed circle a = b = c = d In stereo ℨ-space the fourth 1-vector is a linear combination of the three others, e.g. ud = 𝜆a ua + 𝜆b ub + 𝜆c uc, where 𝜆a , 𝜆b , 𝜆c ∈ℝ, As a performance in The 1-vector space (𝑉3 , ℝ) for ℨ is 3-dimensional dim(𝑉3 ) = 3.

286

It is mandatory to say: There exist no ‘GOD’ for the 1-vector idea that make these three intersect in one and the same point that we call an origo for any locality in ℨ space. It is us that by an ingenious work construct an idea of an origo center for location, from which we can span coordinate-axis from what we call it an object 1-vector basis {O, σ1 , σ2 , σ3 }. 287 We do not have to play ‘GOD’ for the 1-vector basis and make an external choice of origo. The idea of their transversal planes form an automatic intersection that perform a center of locality for our idea of a physical entity that possess plane qualities. An everyday example is: Two walls in a room meeting the floor making a corner vertexes, that’s an origo in practice.

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Volume I, – Edition 1, – Revision 3,

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– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.1. The Classic Geometric Extension Space ℨ –

(6.9)

(6.10)

These four outwards 1-vector directions from an arbitrary locus situs point O, we call a tetraon. The tetraon points out the four vertexes of a tetrahedron and by that the circumscribed sphere. Is the tetrahedron regular symmetric the 1-vectors in the tetraon comply to ua + ub + uc + ud = 0, ua = (ub + uc + ud ), ub = (uc + ud + ua ), uc = (ud + ua + ub ), ud = (ua + ub + uc ). This symmetry is well-known for the four valent carbon atom in a methane molecule. In general, given a locus situs center O and three arbitrary linear independent pqg-1 directions given by three unit 1-vectors ua , ub , uc, then a fourth pqg-1 direction in ℨ-space can be spanned from these r = 𝛼 a ua + 𝛼 b ub + 𝛼 c uc for by contravariant coordinates ∀𝛼 a , 𝛼 b , 𝛼 c ∈ℝ. 288 𝛼a 𝛼b 𝛼c 2 For non-orthogonality we note 𝑟=|r| ≠ √𝛼 a 2 +𝛼 b +𝛼 c 2. Anyway for 𝜆a = , 𝜆b = , and 𝜆c = , 𝑟

𝑟

𝑟

from these we form unit radius-1-vector u = r⁄𝑟 = 𝜆a ua + 𝜆b ub + 𝜆c uc, that from all possibilities span a unit sphere, so that the fourth direction from an center point out by these r = 𝑟u in ℨ-space. In this 𝑆 2 spherical symmetric289 in space the fourth 1-vector is linearly dependent on the other three 1-vectors as a basis {ua , ub , uc }. To imagining this ℨ symmetry, the reader can refer to the plane object in Figure 5.32 and Figure 5.33 and extrapolate the fourth direction out of the figure plane. 6.1.4.2. The Six Bivector Angular Planes of the Regular Tetraon

(6.11)

Like in (5.115) we will look to the 1-rotor planes made by the mutual pair products of the four 1-vector directions ua , ub , uc , ud, each consisting of two 1-vectors representing the 1-rotors, that we can split into scalars and bivector uc ub = uc ⋅ub + uc ∧ub , ua ub = ua ⋅ub + ua ∧ub , ud uc = ud ⋅uc + ud ∧uc , ua uc = ua ⋅uc + ua ∧uc , ub ud = ub ⋅ud + ub ∧ud , ua ud = ua ⋅ud + ua ∧ud . For the regular central symmetric tetraon Figure 6.5 where we demand 𝐮a + 𝐮b + 𝐮c + 𝐮d = 0 (6.9), and achieve equal mutual angles 𝛽 (~109.5°) with the 1-rotor split uc ub = − 13 + uc ∧ub , ⧫, ua ub = − 13 + ua ∧ub , ,

(6.12)

ud uc = − 13 + ud ∧uc , ⧫,

ua uc = − 13 + ua ∧uc , ,

ub ud = − 13 + ub ∧ud , ⧫,

ua ud = − 13 + ua ∧ud , .

Figure 6.5 Regular tetraon demand by The scalar number − 13 = cos 𝛽 given from the four mutual angles between the 1-vectors are mutual covariant coordinates 1-rotor split in scalar and bivector (6.12). for the basis set {ua , ub , uc , ud } itself. These are the normal distances from O to the faces of the endowed regular tetrahedron. Faces, that are transversal planes290 to this 1-vector basis. 1 −1 We use the projection operator (5.184) 𝑃a x = (x⋅a)a−1 → e.g. 𝑃ua ub = (ub ⋅ua )u−1 a = − 3 ua . Multiplying this by ua just give the mutual covariant coordinate 𝐮a 𝑃𝐮a 𝐮b = − 13 for this basis. By normalising this regular tetraon basis ua2 = ub2 = uc2 = ud2 = 1, we get that e.g. ua = u−1 a . 1 1 The covariant sum in direction ua simply is 𝑃𝐮a 𝐮a +𝑃𝐮a 𝐮b +𝑃𝐮a 𝐮c +𝑃𝐮a 𝐮d = +1 − 3 − 3 − 13 = 0 The contravariant sum in direction ua is 1ua +0ub +0uc +0ud = ua , and the full contravariant sum for the regular tetraon basis is 1ua + 1ub + 1uc + 1ud = 0 just as the demand (6.9).

Only three of the six bivector planes defined by the 1-rotors (6.11) are necessary to give an unique intersection definition of a locus center origo O. – Below we set 𝐮x ⋅𝐮y = 0, this gives a Cartesian basis. 288

Upper indices are used to emphasising contravariant coordinates, where lower indices indicate covariant coordinates. The name 𝑆 2 for the spherical symmetric has its origin in two angular spherical coordinates (1, 𝜃, 𝜙) for a unit sphere. 290 We will gradually realise that such idea of transversal plane directions is a very fundamental concept for physics! 289

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Denmark

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– 6.2.2. The Trivector concept – 6.1.4.2 The Six Bivector Angular Planes of the Regular Tetraon –

6.2. The Geometric Algebra of Natural Space In the tradition natural space has been represented by the 1-vector space (𝑉3 , ℝ), dim(𝑉3 )=3 of 3-dimensions for any extension length, breadth, and depth.157 We demand the natural 3-dimensional space 𝑉3 of physics as Euclidean ℰ3 , where the auto product v 2 = v⋅v ≥0 for all 1-vectors ∀v ∈ 𝑉3 are positive definite, setting the metric signature ϵ𝐴 = +1, referring to § 5.2.1.5. For ℨ-space we will expand this 3-dimensional view of 1-vectors with a linear geometric algebra of higher dimensions as we did for the pure plane concept with the scalar concept and the bivector concept that extensively in the plane idea was imagined as an anticommuting pseudoscalar concept. 6.2.1. Addition of Bivectors

(6.13)

(6.14)

In ℨ-space we accept bivectors from several independent planes. We take start in three linear independent 1-vectors a, b, c and make two linear independent bivectors out of these b∧a and c∧a. The addition of these bivectors is defined by the distributive rule b∧a + c∧a = (c + b)∧a This is in Figure 6.6 shown as the object (c+b)∧𝐚, Figure 6.6 Bivector addition from where the sum of two bivectors is again a bivector. the foundation on the external product definition of 1-vectors, In this 3-dimentional 1-vector structure the interpretation of the that comply the distributive law. bivector B=𝛽𝒊 as a plane pseudoscalar loos its specific meaning. The two independent plane directions given by the two unit bivectors 𝒊ba = b∧a and 𝒊𝐜𝐚 = c∧a can by linear combination give any plane direction concerning the common 1-vector direction a, X 𝐚 = 𝛽𝒊ba + 𝛼𝒊ca Anny two plane pqg-2 directions form together an intersection pqg-1-vector direction. That can be intuited by their 1-vector objects as in Figure 6.6, compare § 6.1.2,t. and Figure 6.2, and is given from E XI.De.6. and especially E XI.Pr.3. We see the translation of the objects a represent the subject pqg-1-vector direction a. The same for b and c, and further for the bivector subjects b∧a, c∧a, and (c+b)∧a, we will intuit as translation invariant objects concerned in their respective supported subject planes. Anyway, as a foundation we shall take start in 3-dimensional set of three linear independent geometric 1-vector directions as basis set for a Euclid vector space (𝑉3 , ℝ) ↔ ℰ3 , representing a classical local extension (e.g. as a Descartes system).

6.2.2. The Trivector concept

(6.15)

(6.16)

©

From three linear independent geometric 1-vector objects a, b, c we form a solid prism as shown in Figure 6.7. First, we let 1-vector b operate on 1-vector a to form the bivector b∧a . Then we let the 1-vector c operate on this bivector and get a trivector T= c∧(b∧a) representing a oriented volume spanned by these three 1-vectors. In Figure 6.7 this solid volume object is marked c∧b∧a. This outer multiplication shall obey the associative rule c∧(b∧a) = (c∧b)∧a = T . Therefor we often just use the nomenclature c∧b∧a for a trivector made from the three 1-vectors left sequence. By commutation of the 1-vectors we from (5.58) have c∧a∧b = c∧(a∧b) = c∧(−b∧a) = −T .

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Figure 6.7 A trivector object c∧𝐛∧𝐚 is formed and spanned by the three 1-vector objects a, b, c. This prism is an example on a more general formless but directional trivector volume subject in the substance idea of the ℨ space concept. – OBS: This displayed geometric object is sinistral, so that T = c∧𝐛∧𝐚 has a negative chiral orientation.

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.2. The Geometric Algebra of Natural Space –

(6.17)

For a commutation of the three 1-vector outer product sequence in a trivector we have c∧b∧a = a∧c∧b = b∧a∧c = −c∧a∧b = −b∧c∧a = −a∧b∧c = T. The two opposite sequential cyclical permutation orientations correspond to the two opposite orientations ± of the spatial volume direction ±T.Figure 6.7

6.2.2.2. The Magnitude of a Trivector

(6.18)

The trivector T = c∧ (b∧a) = c∧B is the outer product of a 1-vector c to a bivector B = (b∧a). The square auto product of this bivector is as (5.66) B 2 = − |B|2= − |b|2 |a|2 sin2 𝜃 ≤0, where the arc in the intuited bivector object is 𝜃=∢(b∧a), which parallelogram area |B| = |b∧a| = |b||a||sin 𝜃| ≥ 0 ∈ℝ, then represent the bivector magnitude. We intuit the third 1-vector c in the angle arc 𝜑=∢(c∧B) in the sense of E XI.De.5. and get the height of the prism |c⊥(b∧a) | = |c||sin 𝜑| ≥ 0 ∈ℝ , with the volume |T| = |c∧b∧a| = |c||b||a||sin 𝜃||sin 𝜑| ≥ 0 ∈ℝ This volume object for our intuition represents the magnitude for a trivector idea.

6.2.3. The Trivector and the ℨ-space Chiral Pseudoscalar (6.19)

From this we define the trivector volume square 2 T 2 = T⋅T = |c⊥(b∧a) | B2 = −|B|2 |𝐜|2 |sin 𝜑|2 = −|c|2 |b|2 |a|2 sin2 𝜃 sin2 𝜑 ≤0. Conclusion is T 2 = −|T|2 ≤0, and we call the trivector a chiral pseudoscalar for the ℨ-space. This simple trivector T = c∧b∧a is also often called a 3-blade. The external product represents a primary quality of third grade (pqg-3) that gives the ℨ-space chiral direction of a volume. The orientation of this pqg-3 direction volume can be reversed as expressed in (6.15)-(6.17) by the two states T = ±|T|. This is the fundamental principle of the permutation orientation of three 1-vector directions.

6.2.3.2. The Cartesian Orthonormal Basis 1-vector Set of Primary Quality of Third Grade (pqg-3)

A three dimensional Euclidean geometric 1-vector space ℰ3 →(𝑉3 , ℝ) can be generated from an dextral orthonormal set of geometric basis {σ1 , σ2 , σ3 } for a Cartesian system where σ2 ⊥σ1 , σ3 ⊥σ2, σ1 ⊥σ3 are perpendicular and |σ1 |=|σ2 |=|σ3 |=1. By the geometric vector product algebra this is expressed as (6.20)

(6.21)

1 for 𝑗=𝑘

1

σ𝑗 ⋅σ𝑘 = 2 (σ𝑗 σ𝑘 + σ𝑗 σ𝑘 ) = 𝛿𝑗𝑘 where 𝛿𝑗𝑘 = {0 for 𝑗≠𝑘 From this Euclidean ℰ3 orthonormal basis of three 1-vectors we form a unit dextral chiral pseudoscalar trivector for the ℨ-space

𝓲 = σ3 ∧σ2 ∧σ1 = 𝓲 ≡ σ3 σ2 σ1 This chirality is the outer product σ3 ∧σ2 ∧σ1 shown in Figure 6.8 as an external view of a dextral object basis for us. Further the orthonormality (6.20) for this dextral object basis gives the scalar products in this to zero 0, hence we archive this simple operator product for the chiral volume unit pseudoscalar

(6.22)

𝓲 ≡ σ3 σ2 σ1 . This is a simple left order sequential operator product: First we operate with σ1, then to the left multiplication with operator σ2 and further left multiplying with the operator σ3. This is equal to the outer dextral chiral product σ3 ∧σ2 ∧σ1 . In all we have the unit chiral pseudoscalar volume operator 𝓲 ≡ σ3 σ2 σ1 = σ3 σ2 σ1 in Figure 6.9, an unit amoeba.

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 238 –

Figure 6.8 The unit trivector direction formed on the dextral orthonormal basis {σ1 , σ2 , σ3 } as an object in ℨ space.

𝓲=

Figure 6.9 The unit chiral pseudoscalar subject as the pqg-3 direction quality of the ℨ space substance: 𝓲 ≡ σ3 σ2 σ1 . Any amoeba |𝓲| ≡ 1 is a representative.

A Research on the a priori of Physics –

December 2020

– 6.2.3. The Trivector and the ℨ-space Chiral Pseudoscalar – 6.2.3.2 The Cartesian Orthonormal Basis 1-vector Set of Primary Quality of Third Grade (pqg-3) –

This unit pseudoscalar represent a chirality direction as the primary quality of third grade (pqg-3) of the ℨ-space with a dextral (righthanded) orientation, that is expressed through the left operated sequential order 1,2,3 of the algebraic multiplication operations of the 1-vectors σ1 , σ2 and then σ3. The reversed sinistral (lefthanded) orientation of the volume pqg-3 direction we express by the reversed commutated order (6.23)

= σ ̃ 3 σ2 σ1 .

−𝓲 ≡ σ1 σ2 σ3

where the left operated sequential order is 3,2,1.291 The idea of this trivector chiral pseudoscalar concept for our intuition always a priori depends on the orthonormal basis {σ1 , σ2 , σ3 } object for the ℨ-space substance direction considered in our minds. See this dextral orientated sequential object Figure 6.9, (right hand rule: 1-thumb, 2-index, 3-long) Although this chiral directed unit volume by definition is linked to the unit cube as an object, orientated by (6.22) or (6.23), the unit chiral pseudoscalar subject 𝓲 can take any amoeba 𝓲 form in the ℨ-space substance, as long as its volume is one unit |𝓲| ≡ 1, and orientation is preserved. To simplify intuition, we must demand the dextral object basis orthogonal: σ1 ⋅σ2 = σ2 ⋅σ3 = σ3 ⋅σ1 = 0, and normalisation: σ12 = σ22 = σ32 = 1 ⇒ |σ1 | = |σ2 | = |σ2 | = 1. (6.24) (6.25)

(6.26)

The chiral unit pseudoscalar characteristic is given by the fact 𝓲2 = −1 because 𝓲𝓲 = 𝓲2 = (σ3 σ2 σ1 )2 = σ3 σ2 σ1 σ3 σ2 σ1 = σ3 σ3 σ2 σ1 σ2 σ1 = σ2 σ1 σ2 σ1 = − σ2 σ2 σ1 σ1 = −1 , or alternative expressed 𝓲2 = (σ3 σ2 σ1 )2 = − (σ1 σ2 σ3 )(σ3 σ2 σ1 ) = −σ1 σ2 σ3 σ3 σ2 σ1 = −σ1 σ2 σ2 σ1 = −σ1 σ1 = −1 . This leads to the normalized pqg-3 volume magnitude |𝓲| = |−𝓲| = 1. We remember from (5.72) the plane unit anticommuting pseudoscalar

𝒊 = σ2 ∧σ1 =

1 2

(σ2 σ1 − σ2 σ1 ) = σ2 σ1

̂ = 𝒊 = σ2 σ1. The unit trivector chiral this is the unit bivector for the plane pqg-2 direction B pseudoscalar can then be written 𝓲 = σ3 𝒊. Geometric interpreted σ3 ⊥𝒊, where σ3 is normal to 𝒊, that is the transversal unit bivector to the normal 1-vector σ3. We remember from § 5.2.6.4 that the form of the plane subject 𝒊 has any arbitrary shaped plane amoeba as object 𝒊 as long as this has unit magnitude |𝒊| = 1, illustrated Figure 5.14. The same will be the case for any shaped solid volume amoeba subject 𝓲 with magnitude |𝓲| = 1 that will represent the idea of the pqg-3 direction subject 𝓲 of the ℨ-space substance. We have both chiral orientations 𝓲 and −𝓲 that are the two stats of the unitary trivector idea. ̂ has two eigenstates The chiral direction of a unit space solid volume T (6.27)

291

̂ 2 = 𝓲2 = −|𝓲|2 = −|T ̂|2 = −1 T ̂ has two eigenvalues 1 and −1. We say that the unit-space-solid-volume direction T Comparing with quantum mechanics we intuit 𝓲 as a direction operator for a unit-space-solid with two chiral orientation eigenstates. ̂ for a space-solid Any arbitrary space volume 𝜐 = |T| ≥0 provided by a trivector T = 𝜐T + − pqg-3 direction, thus has two eigenstates T = + 𝜐𝓲 or T = − 𝜐𝓲 and the quantitative eigenvalues +𝜐 and −𝜐 for the volume. When you have a solid volume, you should seriously consider its orientation and which of the two trivectors T or −T you use in the chiral intuition. The operator 𝓲 acts on the space concept 𝔊 and create one ℨ-space direction. ̂ = ±𝓲 = ±1𝓲, T

in that

In this book we use left multiplication of the sequential operational order. This is the reversed order of that first defined by David Hestenes [6](6.3)p.16, [10](3.2), etc. i= σ1 ∧σ2 ∧σ3 . The standard of this book is therefore 𝓲= −i, opposite the Hestenes tradition.

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Implicitly 𝓲 = σ3 σ2 σ1 is given by the three orthonormal geometric 1-vector-operators. First the object σ1 operates on space and sets a linear direction, second object σ2 operates perpendicular to 𝛔1 through space spanning a plane pqg-2 direction by the unit segment 𝒊 = σ2 σ1 , then the third object σ3 operates on that plane substance through 𝔊 space spanning a ℨ-space pqg-3 direction by the chiral unit segment 𝓲 = σ3 σ2 σ1 forming a subject given by the three object 1-vectors from the dextral orthonormal basis set {σ1 , σ2 , σ3 }. 6.2.3.3. The Hodge Coordinate for the Pseudoscalar Span in ℨ Space

(6.28)

All chiral trivector pseudoscalars in ℨ-space is proportional to the basic unit trivector volume T = 𝜐𝓲 For all ∀𝜐 ∈ℝ we have the Hodge map: 𝜐 → (*𝜐) = T = 𝜐𝓲 . This map is a linear one-to-one map from the real numbers to the chiral pseudoscalar of the directional primary quality of third grade (pqg-3) for the ℨ-space concept. These pseudoscalars represent the directional volume quantity of space, where the negative parameter coordinates 𝜐 0. (Compare with § 5.2.6.2 for plane concept)223. 𝜐 =0 represent every pqg-0 point in ℨ space without any direction.292

6.2.4. The Geometric Algebraic Basis of a ℨ-space 6.2.4.1. The 1-vector Basis

(6.29)

(6.30)

We form the dextral orthonormal basis set {σ1 , σ2 , σ3 } of 1-vectors. From this we can span a classical 1-vector space as (6.2) by 3 x = 𝑥1 σ1 +𝑥2 σ2 +𝑥3 σ3 ∈(𝑉3 , ℝ) ~ ℝ11 ⊕ℝ12 ⊕ℝ13 = ℝ1,2,3 Where 𝑥1 , 𝑥2 , 𝑥3 ∈ℝ are the scalar field pqg-1 coordinates. This traditional Cartesian span is shown in Figure 6.10, where the covariant coordinates are 𝑥1 = x⋅σ1 , 𝑥2 = x⋅σ2 , 𝑥3 = x⋅σ3 External to this we can form the outer orthogonal product σ𝑘 σ𝑗 = σ𝑘 ∧σ𝑗 =

1 2

(σ𝑘 σ𝑗 − σ𝑘 σ𝑗 ), for 𝑗, 𝑘 = 1,2,3.

Figure 6.10 The linear pqg-1 span 𝐱 ∈(𝑉3 , ℝ) with Cartesian coordinate from a pqg-1 basis {σ1 , σ2 , σ3 } in ℨ Space.

6.2.4.2. The Transversal Bivector Basis as a Dual Basis of a ℨ Space

(6.31)

The dextral orthonormal 1-vectors basis set {σ1 , σ2 , σ3 } gives direct three orthogonal plane directions represented by the three linear independent orthogonal unit bivectors shown in Figure 6.11: 𝒊3 = 𝒊21 = 𝒊𝛔2 𝛔1 = 𝛔2 𝛔1 = 𝓲σ3 = σ3 𝓲, 𝒊2 = 𝒊13 = 𝒊𝛔1 𝛔3 = 𝛔1 𝛔3 = 𝓲σ2 = 𝛔2 𝓲, } where 𝓲 = σ3 σ2 σ1 . 𝒊1 = 𝒊32 = 𝒊𝛔3 𝛔2 = 𝛔3 𝛔2 = 𝓲σ1 = σ1 𝓲, These are three orthogonal 1-rotor planes, that is transversal dual to the 1-vector directions. From these {𝒊1 , 𝒊2 , 𝒊3 } as basis we can span every arbitrary bivector plane after the concept (6.14) in § 6.2.1

(6.32)

(6.33)

292

Figure 6.11 The dual basis in ℨ Space: From the pqg-1 basis set we form the pqg-2 bivector basis {𝒊1 , 𝒊3 , 𝒊1 } for three orthogonal plane directions.

X = X123 = 𝛽1 𝒊1 , +𝛽2 𝒊2 , +𝛽3 𝒊3, in ℨ-space. This creates a new rotor plane direction expressed by the unit bivector 𝒊 = X⁄|X| ⇒ X = |X|𝒊. Substituting(6.31) into (6.32) we get the bivector as product of a 1-vector and the pqg-3 chiral pseudoscalar volume trivector 𝓲 = σ3 σ2 σ1 in ℨ-space, X = 𝛽1 σ1 𝓲 + 𝛽2 σ2 𝓲 + 𝛽3 σ3 𝓲 = (𝛽1 σ1 + 𝛽2 σ2+ 𝛽3 σ3 )𝓲 = 𝐱𝓲 = 𝓲𝐱.

This agrees with Descartes idea of the concept no extension, contra the pqg-3 pseudoscalar extension of solid subjects. We remember that a Descartes extension has length, breadth, and depth as a pqg-3 structure. (depths is length into the fare dep).

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(6.34)

Do we have the same bivector coordinates 𝛽𝑗 = 𝑥𝑗 , for 𝑗=1,2,3 in (6.32) as in (6.29) we have the same 𝐱. From the principle in(6.31) we conclude that the bivector X = 𝓲x always stands transversal perpendicular to x. We have in this way constructed a dual linear bivector space to the 1-vector space inside the ℨ-space. We can also call this the transversal bivector space. From the standard orthonormal dextral 1-vectors basis set {σ1 , σ2 , σ3 } we construct the orthonormal dual bivector basis {𝒊1 , 𝒊2 , 𝒊3 } = {σ3 σ2 , σ1 σ3 , σ2 σ1 } ⊆ {σ𝑘 σ𝑗 } for 𝑗≠𝑘 where 𝑗, 𝑘 = 1,2,3.

6.2.5. The Geometric Algebra for Euclidean ℨ-space

(6.35)

(6.36)

For the ℨ-space we a priori judge: the primary quality of fourth grade (pqg-4) has no existence. Therefor an outer product with a fourth 1-vector vanish. Special we demand x∧𝓲 = x∧σ3 ∧σ2 ∧σ1 = 0 in a pure ℨ-space.293 From the general separation in symmetric and antisymmetric products (5.33) and (5.44) we then have the symmetric product between a 1-vector and the pqg-3 unit chiral pseudoscalar 𝓲x = x𝓲 = x⋅𝓲 + x∧𝓲 = x⋅𝓲. This inner product with a 1-vector x contract the pqg-3 trivector quality 𝓲 to a pqg-2 bivector quality X = x⋅𝓲 = x𝓲 = 𝓲x , which has the transversal plane quality to the dual pqg-1 direction.

6.2.5.2. The Commuting Pseudoscalar for ℨ-Space

(6.37)

We know by definition (5.19) that scalars commute with all multivector components. We note that all chiral pseudoscalars T = 𝜐𝓲 = 𝓲𝜐 commute with all 1-multivector and bivectors 𝓲𝐱 = 𝐱𝓲 = X and 𝓲X = 𝓲𝓲𝐱 = −𝐱 = 𝐱𝓲𝓲 = X𝓲. A simple test is: σ1 𝓲 = σ1 σ3 σ2 σ1 = −σ3 σ1 σ2 σ1 = σ3 σ2 σ1 σ1 and σ2 σ3 σ2 σ1 = σ3 σ2 σ1 σ2, etc..

6.2.5.3. The Mixed Product Between a 1-vector and a Bivector

(6.41)

For the outer product between a 1-vector a and a bivector B = b∧c we expand the pqg-2 bivector to a pqg-3 trivector quality and (6.17) gives that it is symmetric a∧B = a∧b∧c = b∧c∧a = B∧a Because of this symmetry we express this extension grade lift (area to a volume) by the form a∧B = 12(aB + Ba). We now look the mixed product aB and split it in an inner and an outer part aB = a⋅B + a∧B and an antisymmetric plus a symmetric part aB = 12(aB − Ba) + 12(aB + Ba)

(6.42)

We now see that the new constructed inner mixed product is anti-symmetric a⋅B = 12(aB − Ba) = −B⋅a.

(6.38) (6.39) (6.40)

(6.43)

(6.44)

293 294

This is a 1-vector as shown by Hestenes294 expressed by a linear 1-vector expansion a⋅B = 𝐚⋅(b∧c) = (a⋅b)c − (a⋅c)b Here the scalar products a⋅b and −𝐚⋅c are the coordinates for a along the 1-vectors c and 𝐛. The dot product with a 1-vector a⋅ is an extension grade contraction, here from pqg-2 to pqg-1 In the opposite way of (6.40) the reversed mixed product is Ba = B⋅a + B∧a = −a⋅B + a∧B From the a priori definition of a bivector B = b∧c we write the mixed product as

See more about extension to ℨ-space later below, where we define four-vectors ≠ 0. Indication above § 5.7.1 (5.300)-(5.301). David Hestenes has a algebraic prove of this [10]p.33,(6.15), and in general I have taken the principal of this section from his book [10] and his other articles.

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(6.45)

aB = a(b∧c) or Ba = (b∧c)a These by (6.40), (6.43) and (6.38) consist of a 1-vector part 〈aB〉1 plus a trivector part 〈aB〉3.

6.2.5.4. The Simple Product of Three 1-vectors

(6.46) (6.47) (6.48)

Further by the associative law (5.38) we define a 3-multivector product of three 1-vectors abc = a(bc) = (ab)c From the origin 2-vector product (5.59) of two 1-vectors ab = a⋅b + a∧b, we get 𝐚(b⋅c + b∧c) = (a⋅b + a∧b)𝐜 = abc and by applying (6.40) with the distributive rules (5.39) and (5.40) we have (a⋅b)c + (a∧b)⋅c + (a∧b)∧𝐜 a(b⋅c) + a⋅(b∧c) + a∧(b∧c) = ⏟ = abc ∈ 𝒢3− (ℝ) . ⏟ ⏟ ⏟ ⏟ pqg-1 pqg-3 qualities of directions. 〈abc〉1 〈abc〉3 For this simple product abc the scalar 〈abc〉0 = 0 and the bivector 〈abc〉2 = 0 part vanish. At orthogonality 〈abc〉1 = 0, then we just have abc = 〈abc〉3 = ± |abc|𝓲 with chiral direction.

6.2.5.5. Even and Odd Multivector in general

(6.49) (6.50) (6.51)

(6.52)

For the 𝔓 plane concept we from (5.162) have resolve a general 2-multivector as 𝐴 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 ∈ 𝒢2 (ℝ). For the ℨ-space concept we resolve a general 3-multivector as 𝐴 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 + 〈𝐴〉3 ∈ 𝒢3 (ℝ). A generalised a 𝑛-multivector in 𝒢𝑛 (ℝ) we resolves in grades 𝑟 and write it as 𝐴 = ∑𝑛𝑟〈𝐴〉𝑟 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 + 〈𝐴〉3 + ⋯ + 〈𝐴〉𝑛 ∈ 𝒢𝑛 (ℝ). A multivector that is simple graded as 𝐴 = 〈𝐴〉𝑟 = 𝐴𝑟̅ is called homogeneous of grade 𝑟, and often named a 𝑟-blade or just a simple 𝑟-vector with a primary quality of 𝑟’th grade. A 𝑟-blade or a simple 𝑟-vector representing a pqg-𝑟 direction. E.g. simply: trivector as 3-blade, bivector as a 2-blade, 1-vector as a 1-blade, and scalar as 0-blade Later we will introduce a 4-vector as a 4-blade, etc. The multivector 𝐴 is called • odd when 〈𝐴〉𝑟 = 0 for all even 𝑟, 〈𝐴〉− = 〈𝐴〉1 + 〈𝐴〉3 + ⋯ or • even when 〈𝐴〉𝑟 = 0 for all odd 𝑟, 〈𝐴〉+ = 〈𝐴〉0 + 〈𝐴〉2 + ⋯ In general, all multivectors can be separated 𝐴 = 〈𝐴〉+ + 〈𝐴〉− Now we see that (5.59) ab = a⋅b + a∧b is even, ab = 〈ab〉+ = 〈ab〉0 + 〈ab〉2, and that (6.46)-(6.48) is odd, abc = 〈abc〉−, as well as the 1-vector a = 〈a〉− is odd.

6.2.5.6. Product of Two Bivectors

We look at two bivectors B and A = a 2 ∧a1 = a2 a1 (where a 2 ⋅a1 =0). We form the product (6.53)

(6.54)

AB = a 2 a1 B = a 2 (a1 ⋅B + a 1 ∧B) = a2 ⋅(a1 ⋅B) + a 2 ∧(a1 ⋅B) + a 2 ⋅(a1 ∧B) + a 2 ∧a1 ∧B = A⋅B + 12(a2 a1 B − Ba2 a1 ) + A∧B = A⋅B + 12(AB − BA) + A∧B = A⋅B + A×B + A∧B. Where we have used (6.42) and introduced the commutator product A×B = 12(AB − BA). This deduction works for any simple grade 𝑟-blade 𝐵= 〈𝐵〉𝑟 = 𝐵𝑟̅ , special B= 〈B〉2, [13]p.10,(1.37). The inner product of bivectors is a scalar A⋅B =〈AB〉0 ∈ℝ, just as two equal grade-𝑟-blades.295 In a pure ℨ-space due to (6.35) the outer product of two bivectors vanish A∧B = 0.

295

This is taken from [13]p.6,(1.21) for the general definition of the inner product of blades 𝐴𝑟̅ ⋅𝐵𝑠̅

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(6.55)

(6.56) (6.57)

(6.58)

The commutator product of two multivectors 𝐴 and 𝐵 we in geometric algebra define as 𝐴 × 𝐵 = 12(𝐴𝐵 − 𝐵𝐴) For further details for higher grades please consult the literature, e.g. [10], [13], [18]. Here we mention the Jacobi identity 𝐴 × (𝐵 × 𝐶) + 𝐵 × (𝐶 × 𝐴) + 𝐶 × (𝐴 × 𝐵) = 0 Special for a bivector 𝐵 and a general 1-vector 𝑎 we as (6.42) have the commutation 𝐵 × 𝑎 = 12(𝐵𝑎 − 𝑎𝐵) = 𝐵⋅𝑎 = −𝑎⋅𝐵 = −12(𝑎𝐵 − 𝐵𝑎) = −𝑎 × 𝐵 When we later use this (6.55) commutating product in a quantum mechanical context we use the nomenclature [𝐴×𝐵] = 12(𝐴𝐵 − 𝐵𝐴) or 2[𝐴×𝐵] = 𝐴𝐵 − 𝐵𝐴 for multivector commutator relations to keep the correspondence with the quantum mechanical commutator relation we first defined in chapter I. (2.50) [ b,d ] = b d − d b. It is worth noting the difference by the factor one-half and think, what impact this have to physics?

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6.3. The ℨ-space Structure Quality Described by Multivectors

(6.59)

A general arbitrary 3-multivector is constructed from four primary qualities of grades pqg-0 + pqg-1 + pqg-2 + pqg-3. We will really try to add these different qualities together. A classical space solid object we have the tradition to define in a Euclidean vector space (𝑉3 , ℝ)~ℝ13 fixed to a standard orthonormal dextral basis {𝛔1 , 𝛔2 , 𝛔3 } of three 1-vector objects. All 1-vectors in this Euclidean vector space possess the primary qualities of first grade (pqg-1). This basis implies the transversal bivector basis {𝒊1 , 𝒊2 , 𝒊3 } = {σ3 σ2 , 𝛔1 𝛔3 , 𝛔2 𝛔1 }, (6.31), (6.34) as a generator of all planes possessing pqg-2 quality. On top of that we have trivector chiral pseudoscalar unit 𝓲 ≡ σ3 σ2 σ1 generating the chiral volume possessing pqg-3 quality. In all we expand a multivector form from the geometric algebra 𝒢3 = 𝒢3 (ℝ)=𝒢(𝑉3 , ℝ) as 𝐴 = 𝛼 +⏟ 𝑥1 σ1 + 𝑥2 σ2 + 𝑥3 σ3 + ⏟ 𝛽1 𝒊1 + 𝛽2 𝒊2 + 𝛽3 𝒊3 + 𝜐𝓲 𝐴 = 〈𝐴〉0 + Refer to:

(6.60) (6.61) (6.62) (6.63)

(6.64) (6.65)

〈𝐴〉1 (5.59) |

〈𝐴〉2

+ (6.29)

|

+ 〈A〉3

(6.32), (6.33), (6.36)

| (6.15) in 6.2.2

We have separated the multivector concept in four different primary qualities grades 〈𝐴〉0 = 𝛼, pqg-0, scalar, dim(ℝ) = 1, 〈𝐴〉1 = a = 𝑥1 σ1 + 𝑥2 σ2 + 𝑥3 σ3, pqg-1, 1-vector, dim(𝑉3 ) = 3, 〈𝐴〉2 = b𝓲 = (𝛽1 σ1 + 𝛽2 σ2 + 𝛽3 σ3 )𝓲 = 𝛽1 𝒊1 + 𝛽2 𝒊2 + 𝛽3 𝒊3 , pqg-2, bivector, dim(𝑉3 ) = 3, 〈𝐴〉3 = 𝜐𝓲, chiral volume pseudoscalar, pqg-3, trivector, dim(ℝ) = 1. The multivector idea over a geometric product algebra has a linear addition structure of multiple primary qualities of grades we call it 𝒢3 (ℝ) = 𝒢(𝑉3 , ℝ) over a Euclidean vector space 𝑉3, with which we try to describe the local topological form structure of the physical ℨ-space. This linear addition structure for multivectors over a real field ℝ we express 𝐴 = 𝛼 + a + b𝓲 + 𝜐𝓲. The linear algebra 𝒢3 = 𝒢3 (ℝ) has more dimension from the different grade qualities, in all 𝑛 dim(𝒢3 )= 1 + 3 + 3 + 1 = 8. In generell, dim(𝒢𝑛 ) = ∑𝑛𝑟=0(𝑛 𝑟) = 2 The mixed basis for the hole linear algebra 𝒢3 is {1, σ1 , σ2 , σ3 , σ3 σ2 , σ1 σ3 , σ2 σ1 , σ3 σ2 σ1 }.

6.3.2. The Even and the Odd Geometric Algebra (6.66)

When we split this algebra in even 𝒢3+ and odd 𝒢3− , so 𝒢3 = 𝒢3− + 𝒢3+ , we write the multivector − 𝐴 = 〈𝐴〉+ + 〈𝐴〉− = 〈𝐴〉+ ⏟+ b𝓲 + a⏟+ 𝜐𝓲 , 0,2 + 〈𝐴〉1,3 = 𝛼 +

−

where the even algebra 𝒢3+ is a closed multiplication spinor algebra, while 𝒢3− is open, not closed. The spinor multivector subject can be expressed as a 1-rotor multiplied with a real dilation factor (6.67)

𝒊𝜃 〈𝐴〉+ = 〈𝐴〉+ 0,2 = 〈𝐴〉0 + 〈𝐴〉2 = 𝛼 + b𝓲 = 𝐫⋅𝐮 + 𝐫∧𝐮 = 𝐫𝐮 = 𝜌𝑈 = 𝜌𝑒

∈ 𝒢3+ (ℝ),

as in (5.163) origin in (5.97), where the rotation is in the plane of the bivector b𝓲 = 𝐫∧𝐮. You may define two object 1-vectors whose product spinor 𝐫𝐮 determines the rotation, and dilation 𝜌=|𝐫𝐮|. (We prefer to choose|𝐮|=1 so that 𝜌=|𝐫| if possible.)296 If you determine the 1-vector b = −𝓲(𝐫∧𝐮) as an object it will represent the rotation-axis for 〈𝐴〉+ . This rotation-axis has a transversal plane direction in which 𝐫 and 𝐮 exist. − The odd algebra 𝒢3− (ℝ) part substance of ℨ-space gives us subjects 〈𝐴〉1,3 = a + 𝜐𝓲 . The geometric 1-vector can give us a straight rectilinear translation as in §4.4.2.13 and § 5.3.7.2 by the subject 𝐭 = a, or by the 1-vector object a representing a pqg-1 direction and a straight line-segment magnitude |a| for a physical entity. The new pqg-3 subject 𝜐𝓲 gives us a chiral directed volume of a solid spatial entity object, as a chiral quality pqg-3. 296

I think that there could be a little problem here; if we find 𝐫 and 𝐮 colinear 𝐫∧𝐮 = 0 ⇔ 𝐛 = 0 then the spinor is a pure scalar 𝐫⋅𝐮, and by that, does a pure scalar spinor gives any sense, even if it has a final magnitude |𝐫𝐮| >0? Oscillations will resolve this.

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Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.3. The ℨ-space Structure Quality Described by Multivectors – − The odd 3-multivector 〈𝐴〉1,3 represents a physical extended157, 292 entity with a pqg-1 length quantity a with magnitude |a| and a direction â=a⁄|a|, that is joined with a chiral pseudoscalar pqg-3 volume quantity 𝜐𝓲 with a scalar volume 𝜐= ±|𝜐| factor (coordinate) as magnitude |𝜐𝓲| and direction 𝓲. A negative 𝜐0. We now have a chiral pseudoscalar unit 𝓲 representing the primary quality of third grade, a chiral pqg-3direction that gives the necessary extended structure of ℨ-space of physics.

6.3.3. Operational Structure of the Trivector Chiral Volume Pseudoscalar 𝓲 of ℨ Space

In the text above we have invented a unit subject 𝓲, that when it with a multiplication operation on a 1-vector object b create a transversal plane subject bivector 𝓲b. To instrumentalise this we choose locally an orthonormal standard frame basis set {σ1 , σ2 , σ3 }, so that b= 𝛽σ3. Then we can write 𝓲b = 𝛽σ2 σ1 = 𝛽𝒊, where 𝒊 = 𝒊3 = 𝓲σ3 is the transversal unit. When we further choose the object direction 𝛔1 = 𝐮 ⊥b and consider another 1-vector 𝐫 ⊥b, 𝐫∦𝐮, 𝒊𝜃 we can construct a 1-spinor multivector 〈𝐴〉+ 0,2 = 𝐫⋅𝐮+𝐫∧𝐮 = 𝐫𝐮 = 𝜌𝐯𝐮 = 𝜌𝑈 = 𝜌𝑒 , with 𝐫=𝜌𝐯. Here we repeat that the scalar quantity 𝐫⋅𝐮= 𝜌cos 𝜃, specifies the inner covariant scalar part from the 1-rotor 𝐯𝐮 arc angle 𝜃 between the two chosen object 1-vectors 𝐮 and 𝐯. External to this scalar the bivector 𝐫∧𝐮 = 𝒊𝜌 sin 𝜃 = σ2 σ1 𝜌 sin 𝜃 exist extended in the plane 𝒊 = σ2 σ1 . The 1-spinor 〈𝐴〉+ 0,2 = 𝐫𝐮 has an amplitude, magnitude 𝜌 = |𝐫𝐮| as a transversal plane quantity. We now look at the normalized 1-spinor as an exponential expanded (5.90),(5.191) 1-rotor (6.68)

〈𝐴〉+ 0,2 |〈𝐴〉+ 0,2 |

= 𝐯𝐮 = 𝑈 = 𝑒 𝒊𝜃 = exp(𝒊𝜃) = 1 +

(𝒊𝜃) 1!

+

(𝒊𝜃)2 2!

+

(𝒊𝜃)3 3!

+

(𝒊𝜃)4 4!

+

(𝒊𝜃)5 5!

+⋯

The magnitude of this 1-rotor (a unitary spinor) 𝐯𝐮 = 𝑈 is always one 𝑈𝑈 † = |𝐯𝐮|2 = 1. The even exponents (5.92) of the bivector (𝒊𝜃)𝑛 contribute to the pure scalar 𝐯⋅𝐮 = cos 𝜃, and ̂ = σ3 and forget the odd exponents (5.93) contribute to the bivector 𝐯∧𝐮 = 𝒊 sin 𝜃. Here take ω ̂⊥𝒊 as frame for directions. From this we define implicit the basis {σ1 , σ2 , σ3 } and instead use ω ̂ , from this we scale the transversal bivector 𝛉 = 𝓲b = 𝒊𝛽 for a rotation the 1-vector b = 𝛽ω angular area around b. For the 1-rotor we have 𝜃= ½𝛽 and ½𝛉= 𝒊𝜃, with a magnitude 𝜃= |𝒊𝜃| as the radian arc measure 𝜃 of the 1-rotor angle, see Figure 5.47 where we have that angular sector areal is the half |½𝛉| of this with the pqg-2 direction bivector 𝛉 = 2𝒊𝜃, with magnitude 2𝜃= 𝛽. Hereby we express the regular rotation with this angular bivector as (5.193) from a 1-vector object 𝐱 ∈ 𝒢3 (ℝ) (6.69)

(6.70)

ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 ½𝛉 𝐱 𝑒 ½𝛉 = 𝑒 𝛉 𝐱 = 𝑒 𝒊𝜃 𝐱𝑒 −𝒊𝜃 = 𝑒 𝒊2𝜃 𝐱 In comparing this with the magnitude |𝓲b|=|b|=|𝛽| the reader is encouraged to consider the idea: at the first choose 𝛽 = 2𝜃, so that the rotation axis direction 1-vector b has magnitude |b|= |2𝜃| so that the 1-rotor is 𝑈= 𝑒 ½𝓲b and the regular rotation is ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 ½𝓲b 𝐱 𝑒 ½𝓲b = 𝑒 𝓲b 𝐱 ∈ 𝒢3 (ℝ) Now we have defined an operator ℛ= 𝑒 𝓲b that rotate a 1-vector 𝐱 direction in ℨ-space around another 1-vector b direction, where the magnitude |b| =|2𝜃|, equals the rotation angle. Then the transversal bivector 𝛉 = 𝓲b expose the rotation ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 𝓲b 𝐱 as shown in Figure 6.12, which is a perspective that has its projection on plane 𝒊 in Figure 5.47.

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

– 246 –

=

𝓲𝐛

Figure 6.12 Regular rotation around 𝐛 by an angle 𝛽=2𝜃=|𝐛|, driven by the unitary spinor rotor 𝑈= 𝑒 ½𝓲𝐛 , through the operator ℛ= 𝑒 𝓲𝐛 . 1-vector rotation: 𝐱 → 𝐲 = ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 𝓲b 𝐱 . The rotation 1-vector 𝐛 and its bivector 𝛉 = 𝓲b ̂ . Then 𝑈= 𝑒 ½𝓲𝝎̂𝛽 = 𝑒 ½𝓲b. direction 𝒊 = 𝓲ω

A Research on the a priori of Physics –

December 2020

– 6.3.4. Rotation in ℨ-space – 6.2.5.7 Commutator Product of Multivectors in Geometric Algebra –

̂ as our angular ̂ = σ3 for the unit pqg-1 direction. When we use ω We have chosen the name 𝝎 ̂ | ≡ 1 and choose the phase development parameter 𝛽=(𝑡 −𝑥3 ⁄𝑐 ) reference frequency count |ω we have the bivector exponential oscillation operator 𝑒 𝓲ω̂(𝑡−𝑥3⁄𝑐) = 𝑒 𝓲𝛔3𝛽 in the transversal ̂ . This express the classical plane wave form, that is transversal to the propagation. plane to ω ̂ the normal to the rotation oscillation plane. We therefor led the unit Our desire is to make ω ̂ ∈ 𝒢3+ (ℝ), and by that bivector 𝒊 represent this transversal plane, then we have 𝒊 = 𝓲ω −1 ̂ = −𝓲𝒊, where we simply have chosen [|ω ̂| =1] as unit for the phase angle parameter 𝛽. ω For ∀𝛽 ∈ℝ we have a bivector argument in the rotation exponential 𝑒 𝒊𝛽 = 𝑒 𝓲ω̂𝛽 ∈ 𝒢3+ (ℝ). ̂ , as the 1-vector The tradition uses the designation n for a plane normal therefor b̂ ≡ n = ω − (ℝ), ̂ direction for a rotation axis object b = 𝛽b ∈ 𝒢3 where the magnitude 𝛽 = |b| represent an angle in the transversal plane pqg-2direction for the bivector 𝓲b = 𝛽𝓲b̂ = 𝒊𝛽 subject in 𝒢3+ (ℝ). This is in everyday world illustrated by pointing you index finger or arm towards the paper, a screen, a wall or even the celestial sky and you will realise that there is something transversal and by that, a space around your pointing, and we describe this connecting structure by the pqg-3 directional chiral volume pseudoscalar unit 𝓲 as a primary quality of third grade (pqg-3). The idea 𝓲 has obtained the name pseudoscalar because 1-dimensionality as (6.28), and in 𝒢3 (ℝ) commutes with both 1-vectors and bivectors: 𝓲𝐱 = X = 𝐱𝓲 and 𝓲B = −𝐛 = B𝓲, due to (6.33), (6.36) and the definition (6.21) just like a scalar multiplication. The reverse order orientation definition (6.23) 𝓲̃ = −𝓲 gives the pseudo quality, together with the fact that 𝓲2 = −1. In this context we remember that ordinary scalars commute with every kind of multivector. 6.3.4. Rotation in ℨ-space

We now got the insight that essential concept of an ℨ-space entity is rotation oscillation in a plane given by at least two unit 1-vectors directions u and v forming a rotor concept by a product vu. Here we normalize the even spinor in 𝒢3+ by a rotor 〈𝐴〉+ 0,2 = vu. (with spinor radius |𝐫u|=|vu|=1.) It is obvious, that a permutation reverses the orientation of the rotor plane pqg-2 direction. This we from (5.191) and (5.192) express as unitary operators (6.71) (6.72)

(6.73) (6.74)

(6.75)

̂

𝑈 = vu = v⋅u + v∧u = 𝑒 +½𝛉 = 𝑒 +½𝓲b = 𝑒 +½𝓲b𝛽 = 𝑒 +½𝒊𝛽

= 𝑒 + 𝒊𝜃

̂

𝑈 † = uv = v⋅u − v∧u = 𝑒 −½𝛉 = 𝑒 −½𝓲b = 𝑒 −½𝓲b𝛽 = 𝑒 −½𝒊𝛽 = 𝑒 − 𝒊𝜃 Here we introduce the bivector ½𝓲b = ½𝛉 that represents the 1-rotor area with direction, as an ̂. argument for the 2-multi-vector exponential function 𝑒 ±½𝓲𝐛 , around the axis of rotation b = 𝛽ω ̂ is Hence, the regular rotation is a linear transformation along the transversal plane 𝓲ω ℛ𝐱 = 𝑈𝐱𝑈 † = 𝑒 ½𝓲b 𝐱 𝑒 ½𝓲b = 𝑒 𝓲b 𝐱 = 𝑒 𝓲ω̂𝛽 𝐱 = 𝑈𝑈𝐱 = 𝑈𝛽2 𝐱 ∈ 𝒢3 (ℝ). Refer to section 5.4.5. The unitary 𝑈 demand that |𝑈|2 = 1, by (6.71), (6.72) we have |𝑈|2 = 𝑈𝑈 † = vuuv = 𝑒 +½𝓲b 𝑒 −½𝓲b = 1. One essential thing here is that rotors 𝑈 does not commutate with 1-vectors 𝐱. In (6.73) we have the two-sided multiplication operation (sandwiching) ℛ𝐱 = 𝑈𝐱𝑈 † ∈ 𝒢3 (ℝ) We can also have situations where we use Left or Right multiplication operations: ℛ𝐿 𝐱 = 𝑈𝐱 or ℛ𝑅 𝐱 = 𝑈 † 𝐱 = 𝐱𝑈 ∈ 𝒢3 (ℝ). Anyway 𝑈, 𝑈 † , 𝑈 2 , exist in the same plane direction, a transversal plane to one and the same ̂ for the 1-vector direction, therefor we may always implicit presume a unit 1-vector direction ω † 2 ̂ . To be explicit we can write 𝑈𝜃ω̂ , 𝑈𝜃ω̂ , 𝑈𝛽ω̂ , and therefor transversal plane 𝓲ω † 2 ℛ𝛽ω̂ 𝐱 = 𝑈𝜃ω̂ 𝐱𝑈𝜃ω ̂ 𝐱, ̂ = 𝑈𝛽ω

(6.76)

©

with 𝛽= 2𝜃. As operator we have explicit example

2 ℛ𝛽ω̂ = 𝑈𝛽ω ̂ = ℛ 𝐿,𝜃ω ̂ from left, and ̂ , where we have 𝑈𝜃ω

Jens Erfurt Andresen, M.Sc. NBI-UCPH,

– 247 –

† 𝑈𝜃ω ̂ from right. ̂ = ℛ 𝑅,𝜃ω

Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.3. The ℨ-space Structure Quality Described by Multivectors –

6.3.5. Multiplication Combination of Rotors 6.3.5.1. The Unitary group 𝑈(1) for Plane Combination of Rotors

(6.77)

The unitary plane 1-spinor rotor 𝑈 belong to the simple unitary group 𝑈(1). We have its elements as complex functions 𝑢: 𝜙 → 𝑒 𝑖𝜙 ∈ℂ of one parameter 𝜙 ∈ℝ, that is commutative in multiplication inherit from the additive commutative of the parameter, due to the rule from (1.55): 𝑢(𝜙1 )⋅𝑢(𝜙2 ) = 𝑢(𝜙1 +𝜙2 )~𝑒 𝑖𝜙1 𝑒 𝑖𝜙2 = 𝑒 𝑖(𝜙1+𝜙2) , thus 𝑈(1) is an Abelian, unitary since 𝑢∗ 𝑢=1, as well as a group, since 𝑢 and 𝑢∗ are each other's multiplicative inverse, by the neutral element. This one parameter group is cyclic identical as (1.56): 𝑢(2𝜋n) = 𝑒 2𝜋n = 𝑒 0 = 1 , for ∀𝑛 ∈ℤ. We will call these Abelian elements isomorph with 1-rotors of the geometric algebra. The circular rotation substance has a primary quality of even grades 0 and 2, given by 𝑈(1). The elements in the group 𝑈(1), 𝑢: 𝜙 → 𝑈𝜙 , gives subjects in the geometric algebra 𝒢3+ (ℝ) by a rotor 〈𝐴〉+ 0,2 = 𝑈𝜙 = vu, with the one parameter 𝜙=2∢(v,u), for ℨ-space of physics. The mandatory issue here is that the 𝑈(1) multiplicative Abelian group is substantial connected to ̂ for the transversal the plane idea. In ℨ-space idea we are forced to look at an idea of a bivector 𝓲ω ̂ with the plane operator plane direction to a (implicit) 1-vector direction ω ̂ ½𝜙 𝓲ω 𝑈𝜙ω̂ =𝑒 , with the transversal plane Abelian rule 𝑈(𝜙1+𝜙2)ω̂ = 𝑈𝜙1 ω̂ 𝑈𝜙2ω̂ = 𝑈𝜙2ω̂ 𝑈𝜙1ω̂ .

6.3.5.2. Multiplication Combination of Direction Different 1-rotors in ℨ-space

(6.78)

(6.79)

̂, Different rotation directions can be combined. First, we look at different 1-vector directions ω ̂ 1 and ω ̂ 2, thus ω ̂ 1≠ ω ̂ 2, geometrically ω ̂ 1 ∦ω ̂ 2 . From these we have plane directions 𝓲ω ̂1 ω ̂ 2, in which the angular 1-rotors have different pqg-2 directions and 𝓲ω 𝑈1 = 𝑈𝜃1 ω̂1 = 𝑒 𝓲ω̂1½𝜃1 , and 𝑈2 = 𝑈𝜃2 ω̂2 = 𝑒 𝓲ω̂2½𝜃2 ∈ 𝒢3+ (ℝ) ⊂ 𝒢3 (ℝ). These two subject 1-rotors we intuit as products of three unit 1-vector objects u, v and w and write 𝑈1 = vu →⊙ and 𝑈2 = wv →⊙ displayed in Figure 6.13. From these we make the rotor product 𝑈3 = 𝑈2 𝑈1 = 𝑈2 𝑈1 = (wv)(vu) = wu → ⊙ We see that these two circular rotors 𝑈1 and 𝑈2 intersects along the 1-vector object v, representing the a priori intuition for the interaction of the product of the two rotors.

Figure 6.13 Rotor multiplication in ℨ space gives 𝑈3 = 𝑈2 𝑈1 = (wv)(vu) = wu, as an arc object on the unit sphere that result in a new circular plane direction ⊙. v represents the intersecting 1-vector interaction.

©

Jens Erfurt Andresen, M.Sc. Physics,

Denmark

Figure 6.14 Non-commuting rotors 𝑈2 𝑈1 ≉ 𝑈1 𝑈2 𝑈4 = 𝑈1 𝑈2 = (u'v)(vw')= u'w'= (vuv v)(v vwv)=v uw v. Different directions for these two resulting planes. Four regular circular rotations ℛ𝑘 = 𝑈𝑘2 = 𝑒 𝓲𝛚̂𝑘𝜃𝑘 .

– 248 –

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December 2020

– 6.3.5. Multiplication Combination of Rotors – 6.3.5.4 The Abstract Generalised Rotor Form –

(6.80) (6.81)

The rotors 𝑈𝑘 are rotation-invariant in each their own circular plane (around in a hole circle ⊙). Although circular rotors commute with others in that same plane, external between different planes they do not commute: 𝑈2 𝑈1 ≉ 𝑈1 𝑈2 as illustrated in Figure 6.14. To understand this, we auto rotate the invariant rotors by using the unit vv = v 2 = 1 𝑈1 = vu = vuvv = u'v and 𝑈2 = wv = vvwv = vw', where we have reflected u and w in v and achieved the new objects u'= vuv and w'= vwv. 297 𝑈4 = 𝑈1 𝑈2 = 𝑈1 𝑈2 = (u'v)(vw') = u'w' = (vuv v)(v vwv) = v uw v. Each regular circular rotation we represent as a squared rotor ℛ𝑘 = 𝑈𝑘2 = 𝑒 𝓲𝛚̂𝑘𝜃𝑘 . (no 𝑘 sum). In Figure 6.14 we see in all four circular plane directions of these (Two for commutation of two). When acting on a physical entity as object represented by a 1-vector direction we use the fundamental canonical form (sandwiching) for the operation ℛ𝑘 𝐱 = 𝑈𝑘2 𝐱 = 𝑈𝑘 𝐱𝑈𝑘† . For the combined rotation by multiplication of rotor operators we write

(6.82) (6.83)

† 2 x1,2 = ℛ1,2 𝐱 = 𝑈1,2 𝐱 = 𝑈1,2 𝐱𝑈1,2 = (𝑈2 𝑈1 )𝐱(𝑈2 𝑈1 )† = 𝑈3 𝐱𝑈3†.

For the permutated rotor operator product 𝑈2 𝑈1 we write † 2 x2,1 = ℛ2,1 𝐱 = 𝑈2,1 𝐱 = 𝑈2,1 𝐱𝑈2,1 = (𝑈1 𝑈2 )𝐱(𝑈1 𝑈2 )† = 𝑈4 𝐱𝑈4† ≉ x1,2 ! The 2-rotors do no longer belong to the Abelian group 𝑈(1), but is rather isomorph to 𝑆𝑈(2).298

6.3.5.3. Comment on the Ontology of Directions and Possibility of Location

Are we caught in a trap? We know, we have the idea of a 1-vector subject v as a translation invariant direction over all ℨ-space. For the intuition we define v ≡ ⃗⃗⃗⃗⃗ AB as an object of points we mark on a surface. The rotor objects we for intuition define as 𝑈1 ≡ vu and 𝑈2 ≡ wv have translation invariant subjects 𝑈1 and 𝑈2 with two independent arc-circular plane directions. These two pqg-2 directions 𝑈1 and 𝑈2 intersects in a pqg-1-vector subject v direction in ℨ-space sphere volume. When we imagine the two arc-circular plane rotor subjects 𝑈1 and 𝑈2 qualities, we auto inherit through the intersection the 1-vector v direction quality. (as caught in a trap). Dependent on the two arc quantities of 𝑈1 and 𝑈2 we inherit two extra independent pqg-2 direction subjects defined by 𝑈3 ≡ 𝑈2 𝑈1 and 𝑈4 ≡ 𝑈1 𝑈2. Each of these two new arc-circular planes intersects the pqg-1-vector v direction in just one point as a center of locality for this situation. This subject center of locality is translations invariant just as all the subjects 𝑈1 , 𝑈2 , 𝐯 and 𝑈3 or 𝑈4 . The situated center of locality is caught in this trap we call a 2-rotor. (isomorph with 𝑆𝑈(2)) • We as thinking observers are excluded to the external and can only point out some object mark point as center on a chosen surface to symbolise the locus situs for this intuition. – 6.3.5.4. The Abstract Generalised Rotor Form

We have here above described the directional circular rotors (6.78) etc. as 𝑈𝑗 = 𝑒 𝓲ω̂𝑗½𝜃𝑗 . Each of these for 𝑗 ∈ℕ describe its own independent plane pqg-2 direction by the unit bivectors ̂𝑗 , endowed with an angular parameter 𝜃𝑗 . Then each rotor is just written299 𝒊𝑗 = 𝓲ω (6.84)

(6.85)

𝑈𝑗 = 𝑒 𝒊𝑗 ½𝜃𝑗

∈ 𝒢3+ (ℝ) ⊂ 𝒢𝑛 (ℝ) .

These rotors operators have directions that acts on what stands to the right in the writing and change the direction of these operands. Be careful, the multivectors of the simple form 𝑈𝜃 = 𝑒 𝒊½𝜃 ∉ ℂ, but ∈ 𝒢2 (ℝ) ⊂ 𝒢3+ (ℝ) ⊂ 𝒢3 (ℝ) ⊂ ⋯ ⊂ 𝒢𝑛 (ℝ), are indeed not a complex scalars but represent geometrical direction in physical ℨ-space ⊂ 𝔊.

§ 5.4.2.1 II. 5.4.2.1 Reflection in a Geometric 1-vector . A practical physical example of this problem has been constructed in Rubrik’s Cube, when we rotate in two or three planes (perpendicular) things get that complicated. The reader may look in [19] Figure.4.9-10p.164, or [10]Fig.3.5p.285. 299 with the pure unitary complex number analogy 𝑈 = 𝑒−𝑖½𝜃 ∈ℂ as a scalar without any physical direction! 297 298

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Jens Erfurt Andresen, M.Sc. NBI-UCPH,

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Volume I, – Edition 1, – Revision 3,

December 2020

– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.3. The ℨ-space Structure Quality Described by Multivectors –

6.3.6. Rotation of Multivectors

After this, we rotate a bivector (5.58) B = 𝐜∧𝐚 = 12(𝐜𝐚 − 𝐚𝐜) by rotating each 1-vector (6.86)

(6.87) (6.88) (6.89) (6.90) (6.91)

1

ℛB = 𝑈B𝑈 † = 𝑈(𝐜∧𝐚)𝑈 † = 2 𝑈(𝐜𝐚 − 𝐚𝐜)𝑈 † = 12(𝑈𝐜𝑈 † 𝑈𝐚𝑈 † − 𝑈𝐚𝑈 † 𝑈𝐜𝑈 † ) = 𝐜 ′ ∧ 𝐚′ = B ′ . here we used that 𝑈 † 𝑈 =1.300 That is simple, ℛ rotates a bivector to a bivector just as a 1-vector rotates to a 1-vector. The unit pqg-3 trivector 𝓲 as chiral volume pseudoscalar 〈𝐴〉3 commute with all terms in the 𝒢3 (ℝ) algebra therefor 𝑈𝓲𝑈 † = 𝑈𝑈 † 𝓲 = 𝓲 𝓲 is rotation invariant. The scalar 〈𝐴〉0 is of cause too rotation invariant ℛ〈𝐴〉0 = 𝑈〈𝐴〉0 𝑈 † = 𝑈𝑈 † 〈𝐴〉0 = 〈𝐴〉0 We here see that rotation ℛ preserve grades in the 𝒢3 (ℝ) geometric algebra ℛ〈𝐴〉𝑟 = 𝑈〈𝐴〉𝑟 𝑈 † = 〈𝐴〉′𝑟 All these grades are connected in rotation by e.g. ( 𝓲ℛB = 𝓲𝑈B𝑈 † = 𝑈𝓲B𝑈 † = 𝓲B′ and 𝓲B = −b ) ⇔ ℛb = 𝑈b𝑈 † = b′. We then conclude that all multivectors constructed of a polynomial of all grades 𝐴 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 + 〈𝐴〉3 + ⋯ rotates in the same manner without mixing the grades. In the traditional matrix representation in frame coordinates this is called an orthogonal rotation.

6.3.7. Framing a Field for a Geometric Algebra in ℨ-space

(6.95)

Giving a dextral (righthanded) orthonormal basis {e𝑗 , 𝑗=1,2,3}= {e1 , e2, e3 } as a founding object for a Cartesian coordinate system301 for a straight-line field structure in ℨ-space. We can obtain any local orthonormal (standard) frame by an orthogonal rotation in the canonical form σ𝑗 = 𝑈e𝑗 𝑈 † This operation302 is a mapping of the frame {e1 , e2 , e3 } ⟶ {σ1 , σ2 , σ3 }. The inverse mapping {σ1 , σ2 , σ3 } ⟶ {e1 , e2 , e3 }, with the inverse operation e𝑘 = 𝑈 † σ𝑘 𝑈,

(6.96)

due to 𝑈𝑈 † = 𝑈 † 𝑈 = 1. Alternatively, the local frame can be expressed by a rotation matrix e𝑘 = 𝛼𝑗,k σ𝑗 = ∑𝑗 𝛼𝑗,k σ𝑗

(6.92) (6.93) (6.94)

The matrix elements can be solved as a scalar function of 𝑈 𝛼𝑘,𝑗 = σ𝑘 ⋅e𝑗 = (𝑈e𝑘 𝑈 † )⋅e𝑗 = 〈𝑈e𝑘 𝑈 † e𝑗 〉0.

(6.97)

or (6.98)

𝛼𝑗,𝑘 = σ𝑗 ⋅e𝑘 = σ𝑗 ⋅(𝑈 † σ𝑘 𝑈) = 〈σ𝑗 𝑈 † σ𝑘 𝑈〉0. To do this the reader can study this further in the literature, e.g. [10]p.286ff.

300

This deduction is inspired from [18] p48. Where e2 ⊥e1 , e3 ⊥e2 , e1 ⊥e3 , and |e1 | = |e2 | = |e3 | = 1, as orthonormal e𝑗 ⋅e𝑘 = 12(e𝑗 e𝑘 + e𝑗 e𝑘 ) = 𝛿𝑗𝑘 , 𝑗, 𝑘 = 1,2,3. And where translation invariance is presumed obvious as well as we have Galileo translation invariance over time. Different local points P in ℨ space relative to an origo O for the basis {O, e1 , e2 , e3 }, is a problem we already know. 302 This is written conversely to [5]p.23-24 and [10]3.31p.286. 301

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– 6.3.7. Framing a Field for a Geometric Algebra in ℨ-space – 6.4.1.3 The simple Euclidean Plane Geometric Clifford Algebra 𝒢2,0 –

6.4. The Geometric Clifford Algebra Due to Hestenes, Clifford himself called his algebra for a geometric algebra, so here we call the real Clifford algebra for geometric algebra with the term 𝒢𝑛 = 𝒢𝑛 (ℝ)= 𝒢(𝑉𝑛 , ℝ) ~ 𝐶ℓ𝑛 (𝑉, ℝ). This type of linear algebra can be equipped with different type of basis vectors. E.g.: 𝒢3 (ℝ) has the intuit object standard basis {σ1 , σ2 , σ3 } of 1-vectors, where σ12 = σ22 = σ32 = 1. Further the dual basis {𝒊1 , 𝒊2 , 𝒊3 } = {σ3 σ2 , σ1 σ3 , σ2 σ1 } of bivectors, where 𝒊12 = 𝒊22 = 𝒊32 = −1, is intuited as a subject orthonormal basis for the substance idea of planes in ℨ-space of physics. 6.4.1.1. The Quadratic Form in general

It is now time to expand the metric quadratic form from (5.37) and (5.42) v 2 = 𝑸(v) ϵ𝐴 , Where we have the possible signatures ϵ𝐴 = 1,0, −1. We can construct a linear space of dimension 𝑛 = dim(𝑉𝑛 ) where the generating 1-vector spaces 𝑉𝑝 has positive signed quadrats and the rest dimensions 𝑉𝑞=𝑛−𝑝 has negative signed quadrats. We combined both by addition to a linear space 𝑉𝑛 = 𝑉𝑝 ⨁𝑉𝑞 . Multiplication of these 1-vectors 𝑣𝑘 ∈𝑉𝑛 , 𝑘=1,2, … ,𝑛, forming polynomial multivectors generating the linear spaces of the geometric algebra 𝒢𝑝,𝑞 ← 𝒢𝑛 where 𝑛=𝑝+𝑞, we define the quadratic form (6.99)

2 2 𝑄(𝑣) = 𝑣12 + ⋯ + 𝑣𝑝2 − 𝑣𝑝+1 − ⋯ − 𝑣𝑝+𝑞 ∈ℝ

This geometric algebra 𝒢𝑝,𝑞 (ℝ) = 𝒢(𝑉𝑛 , ℝ) is equal to a Clifford algebra 𝐶ℓ𝑝,𝑞 (∀𝑣∈𝑉𝑛 ,ℝ, 𝑄(𝑣)). 6.4.1.2. The Clifford Algebra for Complex Numbers

Short, the quadratic form also works for ℂ: 𝑄(𝑧) = 𝑧12 + 𝑧22 + ⋯ + 𝑧𝑛2 , then we e.g. write 𝐶ℓ0 (ℂ) ~ ℂ, 𝐶ℓ1 (ℂ) ~ ([ℂ], ℂ), 𝐶ℓ2 (ℂ) ~([ℂℂ ℂℂ],ℂ), … We will not go further into this right here, but history is rich in this. Anyway: The complex number ℂ is god for complex plane idea, as the transversal plane concept. – But: We here stick to the real field ℝ for a general geometric algebra 𝒢𝑝,𝑞 = 𝒢𝑝,𝑞 (ℝ) = 𝒢(𝑉𝑛 , ℝ).303 6.4.1.3. The simple Euclidean Plane Geometric Clifford Algebra 𝒢2,0

A plane concept 𝔓 we traditional span by Cartesian coordinate system from the orthonormal basis set {σ1 , σ2 } as a 2-dimensional 1-vector space (𝑉2 , ℝ) the geometric algebra for this is 𝒢2,0 and for this we have the 22 = 4-dimensional linear mixed grades. The multivector for this has the grade structure 𝐴 = 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 in 𝒢2,0 (ℝ). We name a orthonormal basis {1, σ1 , σ2 , σ21 ≡ σ2 σ1 } for this, that have Table 6.1 Multiplication basis for 𝒢2,0 . the group multiplication structure Table 6.1: \right 1 σ1 σ2 𝛔21 Multiplication of all elements with −1 close the left\* 1 1 σ1 σ1 𝛔21 multiplication group for this plane 𝒢2,0 = 𝒢2 (ℝ) algebra. σ σ 1 −𝛔 −σ 1 1 21 2 The 〈𝐴〉2 (pqg-2) unit bivector σ21 ≡ σ2 σ1 squares to σ2 σ2 𝛔21 1 σ1 2 σ21 = −1, reverses σ12 = −σ21 and anticommute with 𝛔21 𝛔21 σ2 −σ1 −1 all 1-vectors in its own plane σ21 x = −xσ21. From this we span the full multivector algebra 𝐴= 〈𝐴〉0 + 〈𝐴〉1 + 〈𝐴〉2 for 𝒢2,0 . First the scalar 〈𝐴〉0 = 𝛼1, where 𝛼 ∈ℝ and the general bivector 〈𝐴〉2 = 𝛽3 σ21, where 𝛽3 ∈ℝ. − Then we have that any 1-vector 〈𝐴〉1 is expressed in the odd algebra 𝒢2,0 as σ1 〈𝐴〉1 = x = 𝑥1 σ1 + 𝑥2 σ2 (𝑥1 , 𝑥2 ) (σ ) in a matrix formulation. (6.100) ↔ 2 For this Cartesian plane we have the quadratic metric xx = x 2 = 𝑥12 +𝑥22 and general for an Euclidean space x 2 = 𝑥𝑘 𝑥𝑘 = ∑𝑘 𝑥𝑘2 with the orthonormal basis σ𝑘 ⋅σ𝑗 =𝛿𝑘𝑗 . To enrich the plane concept with the complex numbers ℂ, will be awkward when it comes to + ℨ-space with 𝒢3,0 , instead we will stick to the form 〈𝐴〉0 +〈𝐴〉2 for plane spinors of 𝒢0,2 (ℝ)~ 𝒢3,0 . 303

Just as David Hestenes [6], [10], [5], [33], etc. uses the real field in his new foundation of geometric algebra for physics.

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– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.4. The Geometric Clifford Algebra – 6.4.1.4. Plane Subjects in Euclidean ℨ Space Clifford Algebra 𝒢3,0

For any plane substance defined by two orthonormal (σ𝑘 ⋅σ𝑗 =𝛿𝑘𝑗 ) basis 1-vector objects {σ𝑘 , σ𝑗 } in an Euclidean ℨ-space we write the full product of these basis 1-vectors (6.101) σ𝑘𝑗 = σ𝑘 σ𝑗 = 〈σ𝑘 σ𝑗 〉0 + 〈σ𝑘 σ𝑗 〉2 = σ𝑘 ⋅σ𝑗 + σ𝑘 ∧σ𝑗 . Where special for (6.102)

𝑘≠𝑗 ⇒ σ𝑘𝑗 = σ𝑘 ∧σ𝑗 = 𝓲σ𝑙 = 𝒊𝑙 , where 𝑘≠ 𝑙 ≠𝑗, and alternative 𝑘=𝑗 ⇒ σ𝑘𝑘 = σ𝑘 σ𝑘 =1. From this we first write the matrix form where the elements are the separated products of σ𝑘 σ𝑗 , σ1 σ

σσ

σ1 ⋅σ

σ ⋅σ

σ𝑜 ∧σ

σ ∧σ

σ𝑘𝑗 ↔ [ 𝑘 𝑘 𝑗 𝑘 ] = [ 𝑘 𝑘 𝑗 𝑘 ] + [ σ𝑘 ∧σ𝑘 σ𝑗 ∧σ𝑘 ] = 1 (1 0) + σ𝑘 ∧σ𝑗 (0 −1). σ𝑘 σ𝑗 σ𝑗 σ𝑗 σ𝑘 ⋅σ𝑗 σ𝑗 ⋅σ𝑗 0 0 1 1 0 𝑘 𝑗 𝑗 𝑗 2 1 −𝒊𝑙 ( ) = 𝟏 (1 0) + 𝒊𝑙 (0 −1) (6.104) σ𝑘𝑗 ↔ 𝒊𝑙 1 0 1 1 0 For the three perpendicular planes in ℨ-space we choose matrix indices 𝑗 = 1,3,2 {𝑘 = 2,1,3 } as column sequential cyclic order by colour in following framework ⟶ ⟶ 𝑙 = 3,2,1 This matrix structure gives the foundation for the Pauli matrices (see below) + . We see the equivales to a basis of an even multivector algebra 𝒢3,0 for one plane, e.g. for indices 𝑙 (6.103)

(6.105)

𝐴 → 〈𝐴〉+ 0,2 = 〈𝐴〉0 + 〈𝐴〉2 = 𝛼𝑘𝑗 1 + (𝛽𝑘𝑗 )σ𝑘𝑗 = 𝛼𝑙 + (𝛽𝑙 )𝒊𝑙 ,

where 𝛼𝑙 , 𝛽𝑙 ∈ℝ .304

To full understand this rotor as (6.101) for the plane concept we recall from the original definition (5.56)-(5.60), that a reversion change the orientation of the rotor direction. In general, the reverse order in geometric algebra is expressed as (𝐴𝐵)† = 𝐵 † 𝐴† , from this it is obvious that for 1-vectors a† = a or 〈𝐴〉1† = 〈𝐴〉1 because there is essential only one 1-vector in 〈𝐴〉1 . Scalars has no direction (pqg-0), therefor 〈𝐴〉0† = 〈𝐴〉0 too. (6.106) (6.107) (6.108) (6.109)

For the simplest product we have (ac)† = c † a† = ca. For the reverse of (6.105) we have 𝐴† → 〈𝐴〉+† 0,2 = 〈𝐴〉0 − 〈𝐴〉2 ~ 𝛼𝑘𝑗 1 − (𝛽𝑘𝑗 )σ𝑘𝑗 = 𝛼𝑙 − (𝛽𝑙 )𝒊𝑙 . From this complicated matrix concept idea , we get +† 2 2 2 2 〈𝐴〉+ and σ𝑘𝑗 2 = − 1 for 𝑘≠𝑗, =1,2,3. 0,2 〈𝐴〉0,2 = 〈𝐴〉0 + 〈𝐴〉2 = 𝛼 + 𝛽 , We repeat the 1-rotor concept from (6.77) combined with (6.71), (6.72) ← (5.191), (5.192) ̂ sin 𝜙 ̃ = vu = v⋅u + v∧u = 𝑒 +½𝛉 = 𝑈𝜙𝛚̂ = 𝑒 + 𝓲𝛚̂𝜙 = 1 cos 𝜙 + 𝓲ω − 𝑈 = uv ̃ = uv ̃𝜙𝛚̂ = 𝑒 −𝓲ω̂𝜙 = 1 cos 𝜙 − 𝓲ω ̂ sin 𝜙 ̃ = vu = v⋅u − v∧u = 𝑒 −½𝛉 = 𝑈 𝑈†= 𝑈 where we have the rotor angle between two 1-vectors 𝜙=∢(𝐮,𝐯), and the bivector ̂𝜙 as argument in the multivector exponential function 𝑒 ½𝛉 . We have unitarity ½θ = 𝓲ω

̃ = 𝑈𝑈 † = cos 2 𝜙 + sin2 𝜙 = 1. 𝑈𝑈 ̂ =ω ̂ 𝑙 ≡ σ𝑙 = − 𝓲σ𝑘𝑗 From the orthogonal basis matrix structure (6.103) where we choose ω 1𝑘 0𝑙 0𝑘 −1𝑙 ̂𝑙 [1 (6.111) σ𝑘𝑗 ↔ 1 [ 0 1 ] + 𝓲ω where 𝑙≠𝑗≠𝑘≠𝑙, and 𝑗, 𝑘, 𝑙=1,2,3 as (6.101). 0𝑗 ], 𝑙 𝑗 𝑙 For the trigonometric functions from the rotation angle for the real 2×2 rotation matrix we have cos 𝜙 − sin 𝜙 (6.112) 𝜙 ⟶ cos 𝜙 [1 0] + sin 𝜙 [0 −1] = [ ], (without specified direction). 0 1 1 0 sin 𝜙 cos 𝜙 The set of these functions is called the special orthogonal group 𝑆𝑂(2) mentioned (1.54), with cos 𝜙 − sin 𝜙 the determinant | | = 1, equivalent to (6.110) sin 𝜙 cos 𝜙 When we instead use the complex number plane picture for the circle group 𝕋⊂ ℂ, from I. (1.51) (6.113) ℝ → 𝕋 ∶ 𝜙 → 𝑒 𝑖𝜙 = cos 𝜙 + 𝑖 sin 𝜙 this group is isomorph equivalent to the unitary 𝑈(1) where 𝑒 𝑖𝜙1 𝑒 𝑖𝜙2 = 𝑒 𝑖(𝜙1+𝜙2) in one plane. (6.110)

304

Here we don’t use Einstein sum over double indexes, ∑𝑗𝑘 (𝛽𝑘𝑗 )σ𝑘𝑗 . The sum is valid, but not the essence for one plane.

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– 6.4.2. The Pauli Basis for combined direction structures of ℨ-space – 6.4.2.3 The Pauli Basis Generated from 1-vectors –

In this the complex numbers pure imaginary unit 𝑖 ∈ℂ, with 𝑖 2 = ‒1, is indeed not a bivector narrative for the intuition, but intentionally abstract as in an a priory transcendental tradition. Anyway, the group 𝑈(1) structure is isomorph equivalent to the multivector concept of a 1-rotor. ̂ sin 𝜙, (In the transversal plane direction of ω ̂ ), (6.114) 𝑢: 𝜙 → 𝑈𝜙𝛚̂ = 𝑒 𝓲ω̂𝜙 = 1 cos 𝜙 + 𝓲ω ̂. as one real parameter multivector function for the angular rotation in a directional plane 𝓲ω + These elements as in the group 𝑈(1), gives subjects in the geometric algebra that is even 𝒢3,0 in + 305 〈𝐴〉 ̂ ̂ the total 𝒢3,0 as rotor ̂ = 𝑈𝜙 = 𝐯𝐮⊥ω, around an object ω in ℨ-space of physics. 0,2 = 𝑈𝜙ω 6.4.2. The Pauli Basis for combined direction structures of ℨ-space

We again choose the standard 1-vector basis {σ1 , σ2 , σ3 } for the geometric algebra 𝒢3,0 (ℝ). ̂ = σ3 we can rewrite the 〈𝐴〉2 unit part of (6.103) and (6.111) Do we choose the direction so ω −𝓲σ3 = σ1 σ2 = −𝒊3 , 〈𝐴〉2 ↔ σ2 ∧σ1 (0 −1) = σ21 (0 −1) = 𝓲ω ̂ (0 −1) = 𝓲σ3 (0 −1) ↔ { (6.115) 1 0 1 0 1 0 1 0 +𝓲σ3 = σ2 σ1 = +𝒊3 . We see (as seen so often before) that the transversal pqg-2 rotation orientation around a pqg-1 ̂ has two states. As we may know306 there are just three categorical independent direction ω 1-vector directions in ℨ-space of physics, locally represented by different objects σ1 , σ2 and σ3 . 6.4.2.2. The Pauli Matrices as generator operators

Wolfgang Pauli expressed this differences and dependency by three fundamental matrices: −1), 𝜎̂ = (1 0), and the neutral Identity 𝜎̂ = (1 0). (6.116) 𝜎̂1 = (0 1), 𝜎̂2 =𝑖 (0 3 0 1 0 1 0 0 −1 0 1 2 2 2 2 The identity matrix expresses that unitary structure by 𝜎̂1 = 𝜎̂2 = 𝜎̂3 = 𝜎̂0 = 𝜎̂0 . 2 And 𝑖 =√−1 , i.e. 𝑖 = −1 is a quality multiplication operation, that when used two times reverses the orientation of a direction with two stats of orientation (±) From this we construct the closed Pauli group that consist of 24 =16 elements . 𝜎̂2 𝜎̂3 𝜎̂3 𝜎̂1 𝜎̂1 𝜎̂2 (6.117) {±𝜎̂0 , ±𝜎̂1 , ±𝜎̂2 , ±𝜎̂3 , ±𝑖𝜎̂1 = { , ±𝑖𝜎̂2 = { , ±𝑖𝜎̂3 = { , ±𝑖𝜎̂0 }. 𝜎̂3 𝜎̂2 𝜎̂1 𝜎̂3 𝜎̂2 𝜎̂1 The three generating multiplication matrix operators 𝜎̂1 , 𝜎̂2 , 𝜎̂3 of the Pauli group produce 23 = 8 different direction qualities with double ± orientation in physics. A closer look on this closed group show a closed subgroup we call the Lifted Pauli Group | { ±𝜎̂0 , ±𝑖𝜎̂1 , ±𝑖𝜎̂2 , ±𝑖𝜎̂3 } (6.118) = { ± (1 0) , ± (0 𝑖 ) , ± ( 0 1) , ± ( 𝑖 0) }. 0 1 −1 0 𝑖 0 0−𝑖 3 2 With 2 = 8 elements representing 2 = 4 direction qualities of double ± orientation. We return to the more intuitive picture of multivectors in a geometric algebra. 6.4.2.3. The Pauli Basis Generated from 1-vectors

From the dextral orthonormal basis {σ1 , σ2 , σ3 } of 1-vectors for the geometric algebra 𝒢3,0 (ℝ) for ℨ-space we have the mixed basis of 23 = 8 direction qualities that we call a Pauli basis307 {1, σ1 , σ2 , σ3 , 𝓲σ1 = σ3 σ2 , 𝓲σ2 = σ1 σ3 , 𝓲σ3 = σ2 σ1 , 𝓲 ≡ σ3 σ2 σ1 } (6.119) Of these 8 direction qualities there are one pqg-0, three pqg-1, three pqg-2 planes, and one pqg-3 quality for volume. The first is a scalar and has as such no direction in ℨ-space of physics. All of these 8 qualities can quantitively be scaled with both positive and negative reals ℝ, so, we do not ned the ± for orientations as in (6.117) for this generating mixed basis elements to be complete in the geometric algebra 𝒢3,0 (ℝ). (In all with 24 =16 generators as (6.117).) For every multivector in 𝒢3,0 (ℝ) from (6.59) we have a linear combination 305

̂ , this is the transversal rotation plane 〈𝐴〉2 ~ 𝓲ω ̂ , as an impact of 〈𝐴〉3 ~ 𝓲 . In physics, there is something outside 〈𝐴〉1 ~ ω Immanuel Kant was the first to reason this177 by the Kepler-Newton square law from Descartes original extension idea. 307 David Hestenes call the algebra for this group the Pauli algebra [6]p.16, §’6 The Algebra of Space’ 1966\2015. 306

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December 2020

– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.4. The Geometric Clifford Algebra –

(6.120)

𝐴= ⏟ 𝛼 +⏟ 𝑥1 σ1 +𝑥2 σ2 +𝑥3 σ3 + ⏟ 𝛽1 𝓲σ1 +𝛽2 𝓲σ2 +𝛽3 𝓲σ3 + ⏟ 𝜐𝓲 pqg-0

3D pqg-1

3D pqg-2

pqg-3,

= 𝛼 + 𝐱 + 𝐛𝓲 + 𝜐𝓲 . ⏟ 𝐴 = 〈𝐴〉0 +〈𝐴〉1 +〈𝐴〉2 + 〈𝐴〉3

where 𝛼, 𝑥𝑘 , 𝛽𝑘 , 𝜐 ∈ℝ. We see by comparing (6.119) with (6.117) why we call the local orthonormal basis for the geometric algebra 𝒢3,0 with sigma names σ1 ↔ 𝜎̂1 , σ2 ↔ 𝜎̂2 , σ3 ↔ 𝜎̂3 , just as the Pauli matrices as generators for the algebra of the Pauli group. There is a formal difference in the orientation of these two algebraic structures 𝓲 ≡ σ3 σ2 σ1 ↔ −𝑖 , due to the left sequential multiplication operational definition of 𝓲 in the context of this book. We can find the local Pauli frame from a dextral reference basis {e𝑗 } by rotation as (6.92), (6.96) (6.121) σ𝑗 = 𝑈e𝑗 𝑈 † . The general pqg-1-vectors is like (6.29) given by local coordinates 𝐱 = 𝑥1 σ1 +𝑥2 σ2 +𝑥3 σ3 In 𝒢3,0 we have the positive norm with σ12 = σ22 = σ32 = 1 where the quadratic form (6.99) (6.122) x 2 = (𝑥1 σ1 )2 +(𝑥2 σ2 )2 +(𝑥3 σ3 )2 = 𝑥12 + 𝑥22 + 𝑥32 , gives a quadratic measure for the distance, length, or magnitude 𝑑= |x| = √𝑥12 + 𝑥22 + 𝑥32 . 6.4.3. The Quaternion Picture 6.4.3.1. An Anti-Euclidean Geometric Algebra 𝒢0,2

We now look at the dual basis {𝒊1 , 𝒊2 , 𝒊3 } = {σ3 σ2 , σ1 σ3 , σ2 σ1 } defined from (6.31) in 𝒢3 (ℝ). From these we have further the fundamental multivector product interconnectivity 𝒊1 ≡ σ3 σ2 = −𝒊3 𝒊2 = 𝒊2 𝒊3 𝒊2 ≡ σ1 σ3 = −𝒊1 𝒊3 = 𝒊3 𝒊1 (6.123) for the unit basis elements in 𝒢0,2 (ℝ) ⊂ 𝒢3 (ℝ). 𝒊3 ≡ σ2 σ1 = −𝒊2 𝒊1 = 𝒊1 𝒊2 These orthonormal bivector basis elements have the quality, that an auto multiplication operation as a quadratic form of the orthogonal plane directions with negative signature (−) in 𝒢0,2 (ℝ) 𝒊12 = 𝒊22 = 𝒊32 = 𝒊1 𝒊2 𝒊3 = −1 A question, what inversible direction represent the triple product 𝒊1 𝒊2 𝒊3 = −1 ? Written out in 1-vector basis 𝒊1 𝒊2 𝒊3 ≡ σ3 σ2 σ1 σ3 σ2 σ1 = 𝓲𝓲 = 𝓲2 = −1, as the square of the unit chiral volume 𝓲 having two orientations of its direction for the ℨ-space of physics. For intuition see Figure 6.11&8. Every bivector plane subject in ℨ-space can be spanned from a basis {𝒊1 , 𝒊2 , 𝒊3 } as (6.62) (6.125) B = 𝛽1 𝒊1 + 𝛽2 𝒊2 + 𝛽3 𝒊3 = 𝛽1 𝓲σ1 + 𝛽2 𝓲σ2 + 𝛽3 𝓲σ3 = b𝓲 = 〈B〉2 ∈ 𝒢0,2 (ℝ) ⊂ 𝒢3 (ℝ). Such even pqg-2 elements 〈𝐴〉2 can represent a rotation direction by the unit 𝒊B = B̂ = B⁄|B|. The strong interconnectivity in (6.123) makes us rationalise and choose one direction of ̂ , where we see the transversal plane 𝛔1 , σ2 or σ3 as a view for the intuition. E.g. σ3 = ω ̂ of this 1-vector direction. For the orthogonal space to this transversal plane we 𝒊3 = σ2 σ1 = 𝓲ω have two independent orthogonal plane basis bivectors 𝒊1 and 𝒊2 , that support this space through ℨ-space of physics. We look through the picture of this transversal plane 𝒊3 = 𝒊2 𝒊1 of ℨ-space and have defined a mixed basis (6.124)

(6.126)

|

{1, 𝒊1 , 𝒊2 , 𝒊3 ≡ 𝒊1 𝒊2 }, that due to (6.124) form an anti-Euclidean geometric algebra 𝒢0,2 (ℝ) in ℨ-space of physics.308 This is generated from the bivector basis {𝒊1 , 𝒊2 }, where the 𝒊3 direction is implicit given as orthogonal transversal to the pqg-1 intersection direction between the two plane directions 𝒊1 and 𝒊2 . The idea of this concept is to manage rotation around σ3 , in the 𝒊3 plane.309

308 309

There are two independent plane directions 𝒊1 and 𝒊2 in (6.126), therefor 0,2 in the designation 𝒢0,2 (ℝ) for this algebra. To get an intuition of this pqg-2 dual space to the pqg-1 direction the reader can look at Figure 6.11, Figure 6.3 and Figure 6.1,u (−𝓲𝒊1 =σ1 =σ1 =n1 , − 𝓲𝒊2 =σ2 =σ2 =n2 , − 𝓲𝒊3 =σ3 =σ3 =n3 ). To intuit this rotation the reader can take a view at Figure 6.12 and compare to the inclination of the two planes (𝓲n1 , 𝓲n2 ) in Figure 6.1,t.

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Jens Erfurt Andresen, M.Sc. Physics,

Denmark

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December 2020

– 6.4.3. The Quaternion Picture – 6.4.3.2 Quaternions ℍ –

In duality to the Euclidean space spanned from a standard 1-vector basis {σ1 , σ2 , σ3 } for 𝒢3,0 (ℝ) this generalised anti-Euclidean even geometric algebra 𝒢0,2 (ℝ) for space spanned by the planes of the supporting bivector basis {𝒊1 , 𝒊2 , 𝒊3 }. For this we have the two opposite chiral orientations (6.127) 𝒊1 𝒊2 𝒊3 = 𝒊3 𝒊1 𝒊2 = 𝒊2 𝒊3 𝒊1 = −1 for the sinistral, inverse to the reversed sequence 𝒊3 𝒊2 𝒊1 = 𝒊1 𝒊3 𝒊2 = 𝒊2 𝒊1 𝒊3 = +1 for the dextral orientation. 𝒊3 𝒊2 𝒊1 = ̃ 𝒊1 𝒊2 𝒊3 . The foundation of the duality is in the defined pseudoscalar (6.22) 𝓲 ≡ σ3 σ2 σ1 of the Pauli basis (6.119). Here the reader should note that the quality of the unit chiral volume pseudoscalar possess a commutative quantity in the idea of the formulation (6.129) 𝓲 = √𝒊1 𝒊2 𝒊3 = √−1, in that 𝒊1 𝒊2 𝒊3 = 𝓲𝓲 = −1, The anti-commuting three plane bivector basis elements {𝒊1 , 𝒊2 , 𝒊3 } of the anti-Euclidean even real algebra 𝒢0,2 (ℝ) form a multiplicative group with, in all 23 = 8 group elements | {+1= 𝒊3 𝒊2 𝒊1 , 𝒊1 =𝒊2 𝒊3 , 𝒊2 =𝒊3 𝒊1 , 𝒊3 =𝒊1 𝒊2 , −𝒊1 =𝒊3 𝒊2 , −𝒊2 =𝒊1 𝒊3 , −𝒊3 =𝒊2 𝒊1 , −1=𝒊1 𝒊2 𝒊3 }. (6.130) (6.128)

This group is often called the Quaternion Group, that is isomorph with the Lifted Pauli Group. We can write a table of multiplication structure for the possible products inside this closed group: |

Table 6.2 Multiplication table for the basis elements for the Quaternion Group 1 𝒊1 𝒊2 𝒊3 −1 −𝒊1 −𝒊2 −𝒊3 1 1 𝒊1 𝒊2 𝒊3 −1 −𝒊1 −𝒊2 −𝒊3 𝒊1 𝒊1 −1 𝒊3 −𝒊2 −𝒊1 1 −𝒊3 𝒊2 𝒊2 𝒊2 −𝒊3 −1 𝒊1 −𝒊2 𝒊3 1 −𝒊1 𝒊3 𝒊3 𝒊2 −𝒊3 −1 −𝒊3 −𝒊2 𝒊1 1 −1 −1 −𝒊1 −𝒊2 −𝒊3 1 𝒊3 𝒊2 𝒊3 −𝒊1 −𝒊1 1 −𝒊3 𝒊2 𝒊1 −1 𝒊3 −𝒊3 −𝒊2 −𝒊2 𝒊3 1 −𝒊1 𝒊2 −𝒊3 −1 𝒊1 −𝒊3 −𝒊3 −𝒊2 𝒊1 1 𝒊3 𝒊2 −𝒊1 −1 left\*

\right

The multiplicative neutral is the scalar 1 and it has the additive inverse scalar factor −1. Linearity with four real mixed dimensions, as the other four is the additive inverse of the these. In all; two independent generalised plane directions as specified (6.126), see Figure 6.1,t and (E XI.De.6.). These planes imply a mutual perpendicular third plane. In the tradition these planes have two Cartesian ℝ dimensions each. These are equivalent to one complex number ℂ dimension for each complex plane. Two planes ℝ4 ~ℂ3 imply three planes in strange mixed 3-dimensional way. Instead we here try quaternions ℍ. 6.4.3.2. Quaternions ℍ

From this basis group (6.130) we form a linear space of multivectors over the real field ℝ. This we as Hamilton call quaternions310 (6.131) 𝑄 = 𝑞0 + 𝑞𝑘 𝒊𝑘 = 𝑞0 1 + 𝑞1 𝒊1 + 𝑞2 𝒊2 + 𝑞3 𝒊3 ∈ ℍ, where ∀𝑞0 , 𝑞𝑘 ∈ℝ and 𝑘 = 1,2,3. The reversed orientated (or Clifford conjugated) of a quaternion direction we define as (6.132) 𝑄̃ = 𝑞0 − 𝑞𝑘 𝒊𝑘 ∈ ℍ. Here in ℨ-space with the even algebra ℍ~ 𝒢0,2 (ℝ) we use 𝑄 † = 𝑄̃ . The auto product square is 𝑄 2 = 𝑄𝑄 = 𝑞02 − 𝑞12 − 𝑞22 − 𝑞32 ∈ℝ, and quaternion norm is |𝑄|2 = 𝑄𝑄̃ = 𝑄𝑄 † = 𝑞02 + 𝑞12 + 𝑞22 + 𝑞32 > 0. (6.133) 310

Pseudonymous Hamilton named the quaternion basis 𝒊 ≡ 𝒊1 , 𝒋 ≡ 𝒊2 , 𝒌 ≡ 𝒊3 , where 𝒊2 = 𝒋2 = 𝒌2 = 𝒊𝒋𝒌 = −1 , as (6.124). Throughout history the Hamilton names 𝒊, 𝒋, 𝒌 for his quaternions basis ‘vectors’. Hamilton was the first to use the term ‘vector’ as an mathematical-geometrical concept. The object names 𝒊, 𝒋, 𝒌 have also been used for 1-vector basis in the Gips ‘vector’ tradition where 𝒌 = 𝒊×𝒋. E.g. for waves, 𝒌 had been used for the autonomous wavevector as a 1-vector. Maxwell used the quaternion idea to develop his electromagnetic equations, but Lord Kelvin then Gips reformulated it, and the chiral information was lost in the transcendental, and the unproductive idea of axial vectors was born. The importance of chiral direction was first problematised by I. Kant 1768 [11]p.361-372, see note90. – Therefor we will use geometric algebra.

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– II. . Geometry of Physics – 6. The Natural Space of Physics – 6.4. The Geometric Clifford Algebra –

Therefore, the magnitude of the quaternion is 𝛼 = |𝑄| = √𝑞02 + 𝑞12 + 𝑞22 + 𝑞32 ∈ℝ . The symmetric bilinear form for the two generating bivectors (6.126) gives (𝒊1 )2 = −1 and (𝒊2 )2 = −1, therefor a geometric algebra named 𝒢0,2 just as a Clifford algebra 𝐶ℓ0,2.311 The third bivector direction is then implicit given 𝒊3 ≡ 𝒊1 𝒊2 , where 𝒊32 = −1, and the scalar quality is introduced by the product 1 = 𝒊3 𝒊2 𝒊1 , hence a full closed even algebra 𝒢0,2 . In our practise the negative part87 of the real field of quaternions coordinates 𝑞𝑘