A Treatise on the Magnetic Vector Potential [1st ed.] 9783030522216, 9783030522223

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Kristján Óttar Klausen

A Treatise on the Magnetic Vector Potential

A Treatise on the Magnetic Vector Potential

Kristján Óttar Klausen

A Treatise on the Magnetic Vector Potential

123

Kristján Óttar Klausen Reykjavík University Reykjavik, Iceland

ISBN 978-3-030-52221-6 ISBN 978-3-030-52222-3 https://doi.org/10.1007/978-3-030-52222-3

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, 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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the essence of both student and teacher; the spark of curiosity igniting the wonderful guiding force of inspiration.

Foreword by Viðar Guðmundsson

At the University of Iceland, where I have taught various physics courses, the physics students first encounter the electromagnetic vector potential in the second year. This first encounter is rather superficial, but hopefully the best students realize that the potentials at least offer a convenient method to derive the electric and the magnetic fields in many different situations. As Green functions are not introduced at the same time I am not sure if the majority of students really understands this, and a real understanding of the gauge freedom is far away. Luckily, there are sometimes students, who question the role of the potentials and want to learn more about them. Most often this is coupled to their interest in quantum mechanics. Kristján Klausen is the only student I have had in my class who, very soon after the introduction of the vector potential, came to my office with questions if I knew about systems that could lead to a vector potential with certain properties I had not thought of before. I must admit that initially I was not sure why he was asking these questions. I even wondered if some of the questions came from non-scientific literature of any kind, but soon I realized that his questions reflected a sincere interest in phenomena of electromagnetism and his experience with experiments in physics. I am sure that some of my answers at this point in time were very vague or incomplete, so he, as far as I know, started to read some of the original writings about the potentials. Later to keep up with him I also read about some of the founders of electromagnetism and their ideas. These discussions with Kristján and the reading of the background material of electromagnetic theory helped me to get a better understanding of the subject, and at the same time improved the quality of our discussions. After the undergraduate studies at the University of Iceland, Kristján Klausen signed up for an M.Sc. study at the university with an emphasis on pedagogical physics. Soon Kristján came to me and asked if I would be ready to supervise his thesis project, which he wanted to be on the vector potential. As I have been more attracted to application in nanophysical electron systems with a slant to computational methods, I asked him to include one chapter in the thesis to review the computational power inherent in Green functions and linear functional spaces to calculate the vector potential for some simple systems. My intention was to vii

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convince him that calculations based on the fundamental differential equations on a grid are not always the best methods to achieve the required results. The project turned into something much more and I am happy I allowed Kristján to pursue his own interests on the subject of the vector potential. I realized that his interest to treat the differential operators of electromagnetic theory in a more general manner than is usually done in elementary textbooks opened my eyes to certain problems of presentation many authors deal with when, for example, describing induction effects of various kinds in their books. Kristján has now moved on to a Ph.D. project with Andrei Manolescu and Sigurður I. Erlingsson at the Reykjavík University, and I am happy to see that his unrelenting interest in understanding complex phenomena is aiding him in this study. I am happy too that he still comes and discusses computational issues or theoretical with me. I often have to think back and acknowledge with myself how important it is to discuss sincerely with students about their ideas of physical phenomena and methods to describe them. If I had received Kristján initially with a bit more arrogance to hide my ignorance you would probably not be reading this book now, and my knowledge about the foundations of electromagnetic theory would not have improved. Reykjavik, Iceland January 2020

Viðar Guðmundsson

Foreword by Natalia K. Nikolova

The physical significance and the measurability of the electromagnetic potentials, in particular the magnetic vector potential, is a matter of long-standing controversy. Experiments based on the Aharonov-Bohm effect demonstrate the detection of the static magnetic vector potential through an observable shift of the interference pattern of electron waves. Moreover, the effect is predicted by quantum electrodynamics, where the electromagnetic four-vector potential plays a fundamental role in describing the momentum-energy state of an electromagnetic system and in deducing the local interaction with the particle wave. Meanwhile, in classical electrodynamics, the dominant opinion is that the four-vector potential may be a convenient computational tool but it is not measurable and, therefore, carries no physical meaning. This opinion goes back to Heaviside who held Maxwell’s work in high respect, but he felt that the two potentials (scalar and vector) rendered the equations of propagation “unmanageable and also not sufficiently comprehensive.” Heaviside and Hertz (independently) stated the “duplex” field equations (now known as Maxwell’s equations) where the electric and magnetic field vectors are the fundamental quantities, not the electromagnetic potentials. The work of Klausen takes the reader on a short yet thorough journey through both classical and quantum theories of electromagnetism, which highlight the beauty and consistency of the vector-potential model. It also offers a short discussion of experimental work that aims at uncovering the true nature of the magnetic vector potential and its relativistic four-vector counterpart. It encourages critical thinking and re-enforces an opinion shared by many electrical engineers and physicists: our understanding of the electromagnetic phenomenon is incomplete and, indeed, inconsistent in its theoretical models and interpretations. More than 140 years after Maxwell’s Treatise on Electricity and Magnetism was published, we continue to strive to fully understand the electromagnetic phenomenon, as it holds the key to understanding nature. Ancaster, ON, Canada February 2020

Natalia K. Nikolova

ix

Preface

Ever since I played with magnets in my hands as a child I have wanted to understand magnetism, surely many can relate to this. My father often used to say: It feels like something is spinning in them! Intuition is a remarkable thing. My enthusiasm was further increased when I learned about Faraday’s law of electromagnetic induction and understood that it is at the heart of electric power generation and electric motors. I first heard of the magnetic vector potential in a introductory course on electromagnetism during my first year of undergraduate studies in geophysics. I was taking a course of vector calculus at the same time and the similarities of the math caught my interest. That is, if a vector field is rotational, it is said to be non-conservative and cannot be defined as the gradient of a scalar potential, it can however be defined as the circulation of a vector potential. The summer after the course I worked night shifts at Iceland’s Meteorology office where I got a bit of time for independent research to further explore the concept. To my pleasant surprise, the same mathematical relation is used in fluid dynamics. I had many nightlong conversations with meteorologist skilled in vector analysis and learned that the analogy was exact. Then there was no turning back, I had to understand the magnetic vector potential. Researching the concept opened up a world of wonders. The history, conceptual basis, and mathematical formulation of the vector potential were all fascinating. Unfortunately, I found no book focusing on the magnetic vector potential and it dawned on me that I had to write it myself. I got the opportunity to do so as a master’s thesis for my degree in physics education. At the beginning of my Ph.D. program, I refined and expanded the thesis to include superconductivity with the aim to publish it as a book, which you are currently reading. I have found understanding the magnetic vector potential to be a good entry point for many advanced topics in both physics and mathematics.

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Preface

I am aware of the privilege of being able to access centuries of work in physics in a heartbeat through the Internet. However, with every year, the web gains increasing amounts of low-value information or flat-out dis-info, proving the value of classical textbooks with well-known facts. Nonetheless one cannot help feeling that with the bird’s-eye view over science made possible by the World Wide Web, something which was hidden in plain sight will come to light. That is what pushes me forth in continuing the exploration of the age-old concept of the magnetic vector potential. Reykjavík, Iceland January 2020

Kristján Óttar Klausen

Acknowledgements

The writing of this book was made possible by Prof. Viðar Guðmundsson who gladly took on the responsibility of mentoring and supporting me in multiple ways. My deepest gratitude and respect to you Viðar for always being open to discussion of both elementary and controversial aspects of physics, humble and firmly grounded in the laws of nature and mathematics. Thank you Prof. Andrei Manolescu for encouraging me to publish this work and believing in its value, in spite of our different views on the Aharonov-Bohm effect. I have a lot to learn from you. A warm thank you to those who proofread the work: Prof. Ari Ólafsson, Prof. Halldór Pálsson, my father Ingólfur Klausen, Prof. Jón Tómas Guðmundsson, Prof. Ania Sitek, my friend Sigtryggur Hauksson, and my father-in-law Dr. Þorsteinn Kristinn Óskarsson. Special thanks to Dr. Angela Lahee, the Executive Editor of Physics books for Springer, who saw potential in this book and paved the way for the publication. I would like to thank my high school physics teacher and friend Einar Már Júlíusson who encouraged me to pursue physics and set a true example of what good teaching is by never showing the slightest disbelief in students. His view was that every question deserved a thorough and honest response. He would often devote his coffee and lunch brakes to our exploration of fundamental questions in electromagnetism, relativity, and cosmology—never settling for less than wonder. Gratitude and love to my wife for her support and friendship. Thank you mom and dad for disciplining me and bearing with my many flaws and faults. Thank you heavenly father for patience, Iceland’s national university library, and for all the people behind LaTeX typesetting with its wonderful packages and syntax. Finally, I would like to thank every future reader for his time and interest. I hope the reading will prove illuminating and enjoyable.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Mathematical Appearance . . . . . . . . . . . . . . . . . . . 3.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . 3.2 Potential Formulation . . . . . . . . . . . . . . . . . . . 3.3 Spacetime Four-Vector Notation . . . . . . . . . . . 3.4 Variational Approach . . . . . . . . . . . . . . . . . . . 3.5 Gauge Invariance in Quantum Electrodynamics 3.5.1 Local Gauge Covariance . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Three Dimensional Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Line Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Method of Solution for the Current Loop . . . . . . . . . . . . . . . . . .

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2 Conceptual Emergence . . . . . . . . . . . . . . 2.1 Electromagnetic Induction . . . . . . . . 2.1.1 The Electrotonic State . . . . 2.2 Electrokinetic Momentum . . . . . . . . 2.2.1 Vector Potential . . . . . . . . . 2.2.2 Molecular Vortices . . . . . . . 2.3 Elimination for Practical Purposes . . 2.4 Gauge-Invariance . . . . . . . . . . . . . . 2.5 Comeback in Quantized Formulation 2.6 The Aharonov-Bohm Effect . . . . . . . 2.6.1 Aharonov-Casher Effect . . . 2.7 Geometric Phases . . . . . . . . . . . . . . 2.8 Gauge Theories . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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4.2.1 The Dirac Delta Distribution . . . . . . . . . 4.2.2 Green’s Function . . . . . . . . . . . . . . . . . 4.2.3 Eigenfunction Expansion . . . . . . . . . . . 4.2.4 Bilinear Expansion of Green’s Function 4.3 Current Loop . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Theory of Superconductivity . . . . . . . . . . . . . . . . . . . . . . 5.1 Perfect Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 5.2 London Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Penetration Depth and the Meissner Effect . . . . . . . . 5.4 Coherence Length and the Ginzburg-Landau Theory . 5.5 Insulator-Superconductor Junction . . . . . . . . . . . . . . 5.6 Type I Versus Type II Superconductors . . . . . . . . . . 5.7 Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The Josephson Junction . . . . . . . . . . . . . . . . . . . . . 5.9 Microscopic Theory . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Second Quantization Technique . . . . . . . . . 5.9.2 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Bogoliubov-De Gennes Formalism . . . . . . . . . . . . . 5.11 Andreev Reflection and the Proximity Effect . . . . . . 5.11.1 The BTK Model . . . . . . . . . . . . . . . . . . . . 5.11.2 The Superconducting Side . . . . . . . . . . . . . 5.11.3 The Normal Metal Side . . . . . . . . . . . . . . . 5.11.4 The Interface . . . . . . . . . . . . . . . . . . . . . . . 5.11.5 Optical Retro-Reflection . . . . . . . . . . . . . . . 5.12 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Superfluid Vacuum Theory . . . . . . . . . . . . . . . . . . . 5.14 Gauge Invariance in Superconductivity . . . . . . . . . . 5.15 Hole Superconductivity and the H-Index . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Applications and Analogies . . . . . . . . . . . . . . . . . . . . . . . 6.1 Application in Telecommunications . . . . . . . . . . . . . 6.2 Hydrodynamics Analogy . . . . . . . . . . . . . . . . . . . . . 6.2.1 Vorticity, Acceleration and Induction . . . . . 6.2.2 The Navier-Stokes Equation . . . . . . . . . . . . 6.2.3 The Continuity Equation and Incompressible 6.2.4 Irrotational Flow and the Velocity Potential . 6.2.5 Table of Comparison . . . . . . . . . . . . . . . . . 6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Negative Absolute Temperature . . . . . . . . .

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6.3.3 Onsager’s System of Parallel Vortices in a Plane . 6.3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Nomenclature

Vector quantities are symbolized either with arrows over-lined or with boldfaced font depending on context and/or convention therein. Dimensional analysis here assumes all phenomena can be written out in terms of the fundamental quantities of mass (M), length (L), time (T), and charge (Q). Symbol

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A B E q p J

Magnetic vector potential Magnetic field Electric field Charge Momentum Current density Electrostatic (scalar) potential Electromotive force Force Frequency Velocity Vorticity Differential operator Line element Surface element Volume element Electric vacuum permittivity Magnetic vacuum permeability Speed of light Planck constant Gauge function Wave function

N s/C or V s/m T=N s/C m N/C or V/m C Ns A/m2 V=J/C V N Hz m/s 1/s 1/m m m2 m3 C/V m T m/C s m/s kg m2 /s J s/C m 32

ML/TQ M/TQ ML/T2 Q Q ML/T Q/TL2 ML2 /T2 Q ML2 /T2 Q ML/T2 1/T L/T 1/T 1/L L L2 L3 T2 Q2 /ML3 ML/Q2 L/T ML2 /T ML2 /TQ 3 1/L 2

F f v

d dS dV 0 0

c 

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Chapter 1

Introduction

The phenomena of electricity and magnetism must have been know to man from very early on in history since both manifest directly in nature. While electricity is most obvious in the form of lighting, history suggest the first experiments of the Greeks with electricity used amber, fossilized tree resin which can accumulate static charge when rubbed with wool or fur. In fact the Greek word for amber is elektr ¯ on from which the word electricity is derived [1], Fig. 1.1. Magnetism can also be found in the mineral kingdom. Magnetite minerals in basaltic rock can be magnetized when struck by lightning. Fragments from the rock will thus become natural magnets referred to as loadstones. Needles of loadstone were used for navigation as the first compasses. The Greek word for loadstone is magníts líthos meaning stone from the Tekin region in modern Turkey named Magnesia in ancient times, from which the word magnet is derived [2]. Sparse research was done on the phenomena of electricity and magnetism up until the 17th century when William Gilbert argued that the earth itself is a magnet. We will not elaborate on this early research but begin our story at the time when the connection between electricity and magnetism became understood, in the 19th century, thus beginning the science of electromagnetism. As we shall see, at the core of the unification lie subtler fields of energy and momentum giving rise to all electromagnetic interactions. In the following chapter we explore the emergence of this connection in near chronological order with the development of physics. From there we move on to the mathematical descriptions of the story in chapter three. In chapter four we calculate and plot the magnetic vector potential field for two common current distributions. Chapter five explores the connection to superconductivity and chapter six covers patented applications of the field along with exploring the analogy with fluid dynamics. Conclusions are presented in chapter seven. We set out on our journey with the following question in mind: What is the nature of the field uniting electricity and magnetism? I can assure you dear traveler, this path is far from being a dead end. © Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_1

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2 Fig. 1.1 Amber can preserve insects and polished pieces are often used in jewelry. Credit Nammu

References 1. C.J. Brockman, J. Chem. Edu. 6(10), 1726–1752 (1926) 2. P. Wasilewski, G. Kletetschka, Geophys. Res. Lett. 26(15), 2275–2278 (1999)

1 Introduction

Chapter 2

Conceptual Emergence

Abstract The conceptual emergence of the magnetic vector potential is explored historically and mathematically in both classical and quantum physics. It was first defined by Michael Faraday as the electrotonic state and then formulated as a vector potential for the magnetic field by James Clerk Maxwell. We argue that magnetic field lines highlight axes of rotation of electromagnetic momentum and demonstrate that the magnetic vector potential takes the form of a ring vortex around a current element. In light of the Aharanov-Bohm effect, the physical existence of the magnetic vector potential field is defended. Finally we touch on the Aharanov-Casher effect and geometric phases. The first hint of a link between electricity and magnetism may have been the observation that a compass needle is deflected by a lightning strike close by, Fig. 2.1. This effect was experimentally observed by the Danish physicist Oersted during a demonstration in his own lecture April 21st the year 1820. When an electric current runs in a wire, a compass needle in the vicinity will move. This observation is said to have been by accident, like often is the case in scientific endeavor [1]. Oersted’s discovery ignited a bright spark of interest throughout the scientific community at the time, inspiring many to pursue research into the effect. The French mathematician and physicist Ampre found that a force can also exist between two current carrying conductors and presented a mathematical formulation of the law. Ampère remained a key contributor to electromagnetism in its early days, greatly influencing later research. Another Frenchman, the polymath Francois Arago observed that a current carrying wire wound into helical form would function similar to a bar magnet, thus discovering the electromagnet [2], Fig. 2.2. The most extensive research and experimentation on electromagnetism to this day was done by an Englishman named Michael Faraday. Born in London the year 1791 into a relatively poor family, Faraday received only basic education and left school to begin working as an errand boy for a bookshop at the age of 13. Impressing his employer with hard work and focus, Faraday was promoted to an apprentice bookbinder. The job gave him access to many books, including books on science which intrigued him the most. Fascination with the entries on electricity and chemistry in the The Encyclopedia Britannica prompted him to start his own experiments in the shop’s backroom to validate the scientific concepts he read about [4].

© Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_2

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2 Conceptual Emergence

Fig. 2.1 A compass needle will be deflected by a lightning close by. Credit Anastasia Petrova

Fig. 2.2 A current carrying wire wound into a helical form will induce a magnetic field similar to a bar magnet [3]

A successful teacher of music, William Dance, noticed Faraday’s scientific interest and gave him an entry ticket to attend a lecture by the then world-renowned scientist Sir Humphry Davy at The Royal Institution. Faraday took down detailed notes, illustrating and adding explanations. He then bound the 300 pages of notes into a book and sent to Davy, without any expectations. In the same year his seven year apprenticeship ended and Faraday began working with another bookbinder. Around the same time Davy had an accident which damaged his sight and ability to write. Remembering the accurate notes from Faraday, Davy hired him as an assistant and together they toured Europe the next two years. Upon their return in 1814 they carried

2 Conceptual Emergence

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Fig. 2.3 Iron filaments sprinkled on a paper overlying a bar magnet reveal the magnetic field [6]

on Davy’s work at The Royal Institution. Faraday was then 23 years old and now had access to one of the worlds top theoretical and experimental science facilities [5]. His association with The Royal Institution lasted 54 years with Faraday becoming a lecturer and a professor of chemistry. Faraday was a devout Christian and living disproof of the modern dichotomy between religion and science. Faraday’s work on electromagnetism began in 1820 when he heard from colleagues at the Royal Institution about Oersted’s discovery of electric current influencing magnetic needles. His interest in electromagnetism was further ignited with Arago’s discovery of the electromagnet. To visualize the effect a magnet has on its surrounding space, Faraday sprinkled iron filaments on and around it, revealing a distinct structure, Fig. 2.3. From this experiment he coined the term field referring to the invisible web of force surrounding a magnet, the magnetic field. The reader should note the definition of the magnetic field lines in terms of an equilibrium state of iron filaments around a bar magnet for later reference, where we shall see that the magnetic field lines highlight axes of rotation of electromagnetic momentum. The electromagnet, Fig. 2.2, established that magnetism can originate from electricity. From there on the logical next step was to investigate the reverse relation, which was of great value, if electricity could be derived from magnetism. It was Faraday who successfully showed that was indeed possible. Even though Faraday’s work on electromagnetism was put on hold between the years 1825–1830 while working for Davy, Faraday persisted and succeeded in making an electric current flow from magnetic influence in the year 1830 and formulated experimentally what is now known as Faraday’s law of electromagnetic induction [4].

6 Fig. 2.4 Faraday’s law: Moving a magnet perpendicular to the plane of a circular conductor (shaded gray) induces an electric current (dashed lines) within a conducting loop

2 Conceptual Emergence

S

N

2.1 Electromagnetic Induction Figure 2.4 illustrates Faraday’s law of induction. By moving a magnet in the vicinity of a loop of conducting wire, a current will flow within the wire. The bar magnet can be replaced with an electromagnet, making inductance occur between two coils, Fig. 2.5. When asked about the significance of his discovery, Faraday replied along the lines “What good is a new born baby? ”. Faraday’s law of electromagnetic induction is now implemented in the majority of modern day electrical devices. It is through this law that the kinetic energy of wind, water and steam is converted to electrical energy. Vice versa, electrical motors work on this principle to convert electrical energy to motive power. The importance of this discovery can not be overstated. In the modern day technological society, we can not live a single day without encountering electromagnetic induction. Without it, the world would not be the same. Two aspects of the phenomenon stand out: 1. No current will flow unless there is a relative movement of the magnet and the loop, so that the loop experiences a change in the magnetic field. 2. The strongest current is induced with perpendicular movement of the magnet to the plane of the wire loop. Because of the first point, Faraday noted that a wire seems to be in a state of tension when in the vicinity of a magnet. When the magnet moves, the current flows as a result of the changing tension. This observation in conjunction with the second point led Faraday to believe there was another more subtle state or condition of matter underlying the magnetic field which he termed the electrotonic state. He envisioned this field to be parallel to the wires, thus explaining the perpendicular interaction between the movement of the magnet and the current induced in the wire. The mechanism of electromagnetic induction can be likened to that of a humming top, Fig. 2.6. Pressing the handle downwards makes the humming top spin due to screw grooves in the handle. The analogy is that a rotational movement (the current) results from a force applied perpendicular to the plane of rotation. The electrotonic state in this context would correspond to the screw grooves, giving rise to the spin when set in motion.

2.1 Electromagnetic Induction

7

Fig. 2.5 A coil (A) is connected to a battery and has a current flowing. Moving it perpendicular (up/down) relative to the winding of the larger coil (B) will induce a current in the latter which is detected by the galvanometer (G) [7]

Fig. 2.6 A humming top toy [8]. Pressing the handle downwards makes the humming top spin. The mechanism is semi-analogous to that of electromagnetic induction, with the handle referring to the magnet and the rotation of the tin body to the induced rotational current. The electrotonic state corresponds to the grooves and gear mechanism translating the downward movement into spin. Credit for the analogy goes to the author’s father. Image credit: Martin Geupel

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2 Conceptual Emergence

Fig. 2.7 The spread of electrical current through cells is referred to as an electrotonic current. Credit Colin Behrens

2.1.1 The Electrotonic State Through various experimental setups Faraday made many observations about this “peculiar state of matter” [9]. In his own words, the electrotonic state: • • • • • •

Can be taken on by all metals. Shows no known electrical effects whilst it continues. Does not imply force of neither attraction or repulsion. Appears to be instantly assumed. Appears to be a state of tension. May be considered as equivalent to a current of electricity.

Despite many attempts, Faraday did not manage to measure any direct force reaction (attraction/repulsion) from the electrotonic state alone. He concluded that his observations favored the electrotonic state to be a property of the particles of matter themselves under induction. If so, the state might be present in liquids and even non-conductors [9]. Other scientist as well as Faraday himself were quick to recognize the similarity of the phenomena of induction to the behavior of nerves, which are in a state of tension or readiness similar to a wire hosting the electrotonic state. The word quickly became a standard in the biological vocabulary of the 19th century and has persisted to this day. Electrotonus refers to the spread of charge in a neuron. Therefore, we can say that electrotonic current flows with our every thought, Fig. 2.7. Faraday abandoned the concept of the electrotonic state after a year of experimentation, finding the magnetic field lines sufficient to explain the phenomena of induction. He however felt the need for it again three years later, writing [9]: [...] still there appears to be a link in the chain of effects, a wheel in the physical mechanism of the action, as yet unrecognized. If we endeavor to consider electricity and magnetism as the result of two forces of a physical agent, or a peculiar condition of matter, exerted

2.1 Electromagnetic Induction

9

Fig. 2.8 Michael Faraday in his later years and young James Clerk Maxwell

in determinate directions perpendicular to each other, then, it appears to me, that we must consider these two states or forces as convertible into each other [...] by a process or change of condition at present unknown to us.

Faraday’s struggle with the concept persisted, never being directly able to measure the electrotonic state but nonetheless seeing the need for it—especially with regard to the necessary movement for induction, since “mere motion would not generate a relation, which had not a foundation in the existence of some previous state ” [9]. Toward the end of his investigations, Faraday remarked that he firmly believed in the physical existence of the magnetic field lines and the underlying electrotonic state, which could be seen as a magnetic analog to static electricity. Concluding the review of Faraday’s legacy and foundation for electromagnetism, the story continues with his successor, James Clark Maxwell, who was handed Faraday’s experimental notes in 1855 at the age of 24 from the author himself at the end of his career, then 64 years of age [10], Fig. 2.8. Recently graduated from Cambridge University’s Trinity College, Maxwell was trained in mathematics unlike Faraday. The two developed a friendship and kept in good contact, discussing for example how to express physical and mathematical truths in ordinary language [11].

2.2 Electrokinetic Momentum Starting from the data of Faraday’s observations, Maxwell penetrated the logical structure in the natural laws of electricity, magnetism and the electrotonic state. He equated the electrotonic state to a mathematical quantity which he stated “may

10

2 Conceptual Emergence

even be called the fundamental quantity in the theory of electromagnetism” [12]. Maxwell approached this quantity, A, in multiple ways. It constitutes the momentum in a circuit, which he termed the electrokinetic momentum  A · d, (2.1) p= C

where p is the electrokinetic momentum and d an element of the circuit C. Maxwell found that the inductive- or electromotive force, E, arising from electromagnetic induction was the rate of decrease of the electrokinetic momentum of the circuit dp (2.2) E =− . dt From the above equations it is evident that the electromotive force stems from the change in the quantity A or  d E =− A · d (2.3) dt which is a mathematical formulation of Faraday’s law in terms of the electrotonic state A. The loop integral coincides with the conductor being a closed loop. The electromotive force is the force per charge in the conductor, in modern terms a rotational electric field  E = E · d. (2.4) Together (2.3) and (2.4) give E=−

dA , dt

(2.5)

meaning that an electric field will be induced with a time varying electrotonic state, either in magnitude and/or direction. The relationship to Newtons second law and momentum is as follows, since the electric field is equivalent to force per charge E=

F q

(2.6)

and force is the time derivative of momentum F=−

dp , dt

(2.7)

substituting (2.6) and (2.7) together into (2.5) reveals that A=

p . q

(2.8)

2.2 Electrokinetic Momentum

11

Thus the electrotonic state can be understood as momentum per charge. Furthermore, Faradays’s law of electromagnetic induction is simply Newtons second law, relating the electromotive force to the time derivative of the electrokinetic momentum. By differentiating (2.1) with respect to time and taking into consideration the motion of the circuit itself, Maxwell obtained a general expression for the induction of an electromotive force, with v denoting the particle velocity in the magnetic field B and φ denoting the electrostatic potential, E=v×B−

dA − ∇φ. dt

(2.9)

Maxwell recognized a fundamental difference in the nature of the vectors A and B for the magnetic vector potential and magnetic field respectively [12]. By dimensional analysis with the basic units of length (L), mass (M), time (T) and charge (Q) in SI, starting out with the dimensions of energy U , [U ] =

M L2 , T2

(2.10)

Maxwell found the dimension of the electrokinetic momentum of a circuit element to be   L2 M  = . (2.11) [pek ] = p · q QT The dimensions of A and B can then be written out as    LM pek  [A] = = , qT L    pek  M = . [B] = qT L2

(2.12) (2.13)

The electrotonic state has dimensions of electrokinetic momentum per length but the magnetic field has dimensions of electrokinetic momentum per area, thus belonging to the category of fluxes. In the next section we explore the relationship between the electrotonic state and the magnetic field uncovered by Maxwell. Let us keep in mind that the dimension of the differential operator ∇ is inverse length or 1/L. From now this point onward, the electrotonic state will be referred to as the magnetic vector potential or for short, the vector potential.

2.2.1 Vector Potential Maxwell showed that the magnetic induction in Faraday’s law, Fig. 2.4, depended on the conducting wire loop and not on the surface enclosed by it. Therefore it should

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2 Conceptual Emergence

Fig. 2.9 A magnetic field vector, B, (S-N) arises from a rotational magnetic vector potential, A. Refined original image from Maxwell [13]

be possible to determine the induction by a quantity residing within the wire itself. That could be done by finding a vector, A, related to the magnetic field, B, in such a way that the line integral of A would be equal to the surface integral of B, or in modern notation   A · d = B · dS (2.14) where d and d S denote the line- and surface elements respectively. Invoking Green’s theorem (see the Appendix) one obtains 

 A · d =

∇ × A · d S,

(2.15)

therefore revealing the relation B=∇ ×A

(2.16)

meaning that A is the vector potential for the magnetic field B. This mathematical relationship is often denoted B = rot(A) or B = curl(A) since the geometrical interpretation of the equation is that the vector B is equal to a rotational flow of the field A, Fig. 2.9. Maxwell noted that one can add the gradient of a scalar function χ to the vector potential without changing the value of the magnetic field B [13], since ∇ × (A + ∇χ ) = ∇ × A + ∇ × ∇ χ = ∇ × A

(2.17)

due to the fact that the curl of a gradient of any scalar function f is zero, in general ∇ × ∇ f = 0.

(2.18)

2.2 Electrokinetic Momentum

13

Maxwell furthermore stated [12] that for a given distribution of electric currents there is only one distribution of the values of A where A is everywhere finite and continuous, satisfying the equation ∇ 2 A = −4π μJ,

(2.19)

the operator ∇ 2 being the Laplacian operator which interpreted as concentration by Maxwell, μ being the magnetic permeability and J the total electric current density. One supplementary condition was necessary though, known as the solenoidal condition in vector analysis, ∇ · A = 0, (2.20) stating that the field A is solenoidal and therefore divergence free. The value of the magnetic vector potential is then  A=μ V

J d V. r

(2.21)

This expression was put forth by Maxwell in quaternion form without direct derivation in his treatise [12]. We will analyze it further in Chap. 4, solving for the vector potential of two fundamental current distributions and plotting the results. One of Maxwell’s great insights was equating the magnetic vector potential with Faraday’s electrotonic state [13], describing the result of (2.14): The entire electrotonic intensity round the boundary of an element of surface measures the quantity of magnetic induction which passes through that surface, or, in other words, the number of lines of magnetic force which pass through that surface.

Another key insight was equating and comparing the following ratio to the speed of light in vacuum, 1 , (2.22) c= √ ε0 μ0 where ε0 is the electric permittivity and μ0 the magnetic permeability of vacuum. From there on Maxwell argued that light can be understood as an electro-magnetic wave with “the vacuum and the electromagnetic medium being one the same” [12]. The electromagnetic medium was thus understood to be the substance giving rise to the various phenomena of electricity and magnetism through deformation of it, stress, tension, pressure and density gradients. This viewpoint is further supported by the hydrodynamics analogy analyzed in Chap. 6.2.

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2 Conceptual Emergence

Fig. 2.10 The Faraday effect: A phase shift (β) of the plane of vibration of the electric field component (E) in an electromagnetic wave, passing through a constant magnetic field (B) parallel to the direction of motion

2.2.2 Molecular Vortices In order to explain the Faraday effect, Fig. 2.10 and Fardays’s law, Maxwell turned to a mechanical model comprising of vortices in the medium, inspired by the magnetic vector potential. The main hypothesis of the model was that the rotation consisted of very small unidentified elements of the medium. Both particle spin and virtual particles were of course unknown at the time. Observing the magnetic field lines formed by iron filaments around a magnet, Fig. 2.3, Maxwell noted that the lines “indicate the direction of minimum pressure at every point of the medium” [13], the difference in pressure being caused by molecular vortices with their axis parallel to the magnetic field lines, as depicted in Fig. 2.9. The iron filaments in Fig. 2.3 can thus be seen as being at rest within the vortex center, akin to the stillness in the eye of a storm. To get a feel for this, the reader can refer to the common experience of stirring a cup of clear tea whereby particles in the fluid such as tea leaves will collect in the center of the vortex formed by stirring. Known as the tea leaf paradox, Fig. 2.11, it was explained by Einstein as stemming from velocity differences at the top and bottom of the cup [14]. The analogy is not exact since the cup forms boundary conditions different than in the case of the magnetic field lines. Nonetheless in both instances the particles collect at the center of a vortex. Figure 2.12 shows Maxwell’s original mechanical model to explain mutual induction [13]. Hexagonal cells signify vortices, the hexagonal shape being solely to simplify graphical representation. Between the vortices are hypothesized particles,

2.2 Electrokinetic Momentum

15

Fig. 2.11 Mechanism of the tea leaf paradox [14]

Fig. 2.12 A changing current flowing in the path A–B will induce an opposite current in the path p–q, due to changing angular velocities of the vortices

representing electricity but capable of spin as well. Plus and minus symbols in the vortex centers signify both handedness of rotation and the direction of the magnetic vector normal to the plane of observation, clockwise (−) being into the plane and counter-clockwise (+) pointing outwards, in accordance with the arbitrary directional convention of the curl in (2.16).

16 Fig. 2.13 Plane view of the magnetic vector potential for a current element

2 Conceptual Emergence

 ∇×A

J

Fig. 2.14 Magnetic field around a current carrying conductor

 B J

When current flows from A to B, angular momentum will be formed on both sides of the current with opposite orientation of spin, Fig. 2.13. With the row of vortices g–h being set in motion, the row k–l row above will start rotating, inducing a current in the opposite direction to the original one in the path q–p. By applying the model to static electricity, Maxwell was led to his famous correction to Ampère’s law by adding the displacement current to it. The mechanical model also seems to have been a crucial step for Maxwell towards the realization that light is a transverse wave of magnetic and electric oscillations [15]. The model highlights the fact that Maxwell’s line of thought in approaching electromagnetism was greatly influenced by the concept of the magnetic vector potential, Faraday’s electrotonic state, understood as vacuum vortices with angular momentum, having the magnetic field lines as their axes of rotation. The mechanical model gives an explanation of the rotational magnetic field formed around a current carrying conductor, Fig. 2.14, which Faraday had observed with iron filings and was a well established phenomena at the time. The flow of the current J sets in motion vortices with opposite spin direction, Fig. 2.13, which give rise to the magnetic field. Like the magnetic field, the geometrical shape of the magnetic vector potential has rotational symmetry around the conductor. This implies a spinning torus shape called a ring vortex, Fig. 2.15, known from flow in liquids [16], the smoke ring being the most familiar example. A natural question to ask in this context is what substance is set in motion by the flow of the current? To the majority of scientist in the 19th century the answer was the aether, an all pervading fluid like substance, the deformations of which were the cause of electromagnetic vibrations.

2.2 Electrokinetic Momentum

17

Fig. 2.15 Sketch of the magnetic vector potential due to a linear current [17]

The concept was dismissed in the favour of a static, empty, and incompressible vacuum in the beginning of the 20th century. Nonetheless, the vacuum has properties of magnetic permeability and electric permittivity since both a magnet and a charge keep their qualities in empty space. In the final words of his Treatise on Electricity and Magnetism Vol. 2, Maxwell suggests future research should be aimed at the properties of the electromagnetic medium itself [12]. In concluding the review of Maxwell’s work on the magnetic vector potential, one must bear in mind that at the time of his writings only primitive atomic models had been put forth, the electron was yet to be discovered and the spin property of particles was to be considered more than half a century later.

2.3 Elimination for Practical Purposes Even though Maxwell had successfully unified the magnetic and electric field, B and E, via the magnetic vector potential A, no practical benefit was evident from the unification. In fact, Maxwell himself later put forth the electromagnetic theory of light without the potentials. The physicists Heaviside and Hertz, went a step further and felt the potentials did not even have to be mentioned due to their elusive nature and lack of direct measurability and eliminated them completely from electromagnetic

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2 Conceptual Emergence

theory in the beginning of the 20th century [18]. This downplay of the importance of potentials can still be seen in some modern textbooks on electromagnetism. It’s worth noting that the reformulation of the equations of electrodynamics goes hand in hand with a crisis in mathematics at the same time. Mathematical description of dynamics in three dimensions was in development with two promising systems, Hamilton’s quaternion analysis and Grassman’s system of multivectors. Modern vector analysis can be seen as a simplified and reduced mixture of the two systems. For example, the convention of writing i, j, k as the vector basis in cartesian coordinates comes from the quaternionic notation, where i, j and k are all complex numbers. This has to be kept in mind when reading Maxwell’s papers. Also, the divergence and curl operators with their differential notation of ∇· and ∇× have their roots in quaternion analysis [19]. William Clifford united the two systems at the end of the 19th century and put forth a complete vector algebra for all dimensions known as Clifford algebra. Due to his early death the system was quickly forgotten but has resurfaced in later years and proven to be a most excellent mathematical language for electromagnetism along with other fields of physics. The reader can refer to the Appendix for a short introduction to Clifford’s geometric algebra.

2.4 Gauge-Invariance One of the main reasons for the viewpoint that the potentials have no physical significance is the freedom in their definition. An addition of a scalar or gradient field to the electrostatic or magnetic vector potential respectively, will result in the same electromagnetic field. Changing the potentials in this way is referred to as a gauge transformation and implies a certain symmetry of the theory called gauge symmetry or gauge-invariance. The magnetic field will be unaltered by the following transformation, as before in (2.17), (2.23) A → A = A + ∇χ . In order that the electric field will be the same, the electrostatic potential φ, will have to be transformed as well with φ → φ = φ −

1 ∂χ c ∂t

(2.24)

where χ is a nonsingular smooth differentiable scalar function. This hints at an intrinsic spacial/temporal relationship between the potentials A and φ, which will be clarified in Chap. 3.3.

2.5 Comeback in Quantized Formulation

19

2.5 Comeback in Quantized Formulation In quantum mechanics, probability amplitudes replace particle trajectories in the equations of motion. Instead of working with forces, interactions are described in terms of momentum and energy. The force fields of the electric and magnetic fields are therefore not readily worked with nor quantized. However, the electromagnetic potentials, A and φ, can be directly incorporated into the framework of quantum mechanics in order to implement electromagnetic interactions, with A as the electromagnetic momentum and φ as the energy. For this reason the importance of the potentials was revived with the formulation of quantum mechanics [20]. The Hamiltonian for purely electromagnetic interactions takes the following form [21] 2 1  pˆ − qA + qφ. Hˆ = 2m

(2.25)

Schrödinger’s equation, which describes the wave function ψ of a particle or system, i

∂ ψ = Hˆ ψ, ∂t

(2.26)

then becomes   2 ∂ 1  i ψ = pˆ − qA + qφ ψ. ∂t 2m

(2.27)

The soviet physicist Vladimir Fock found that in order for the quantum dynamics of electromagnetic interactions of charged particles to be invariant under gauge transformations, the wave function ψ has to be transformed as well with the following rotation or phase transformation, implying a relationship between the phase of the wave function and the electromagnetic potentials, ψ → ψ  = ψ · exp



 iq χ , 

(2.28)

where i is the unit complex number, q is the charge,  the reduced Planck constant and χ is the gauge function [22]. The wavefunction phase θ and the gauge function have the relation q (2.29) θ = χ. 

2.6 The Aharonov-Bohm Effect The relationship between the electromagnetic potentials and the phase of the wave function was further hypothesized by Yakir Aharonov and David Bohm [23] where they proposed an experimental setup for interference, Fig. 2.16.

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2 Conceptual Emergence

Interference A Δθ

Electron stream B

Fig. 2.16 Experimental setup showing a phase shift of the interference pattern in a double slit experiment. A magnetic coil/solenoid is centrally located behind the slits such that the magnetic field B is perpendicular to the path of the incoming electrons, pointing out of the page. The interference pattern is shifted by a phase θ in the presence of the magnetic field, even if it is completely shielded, which implies that the magnetic vector potential field A is the cause

A stream of electrons is imposed on a metal plate with two slits, in front af a magnetic solenoid. The magnetic field is mainly concentrated in the center of the solenoid and can be made effectively zero outside of it. The interference pattern is shifted when a current flows through the solenoid, even though the magnetic field strength along the possible paths of the electrons is effectively zero. The magnetic vector potential around the solenoid is however non-zero, Fig. 2.17, and lies in the same plane as the electron stream. Referring to Fig. 2.16, for each electron wavefunction the phase (2.29) will be influenced by the vector potential, θ=

q 

 A · d,

(2.30)

since by the gradient theorem (see the Appendix) 

 ∇χ · d =

A · d.

(2.31)

Integrating along a closed path around the magnetic solenoid, one can write the relative phase difference as a closed path integral q θ = 

 A · d.

(2.32)

2.6 The Aharonov-Bohm Effect

21

 B

Fig. 2.17 Rotational magnetic vector potential field A around a coil with a magnetic field B within it

 A

Even though the magnetic field is zero along both paths there is nonetheless an electromagnetic influence since the Hamiltonian includes the magnetic vector potential (2.25). The closed path integral of the magnetic vector potential is equal to the magnetic flux of the solenoid, from (2.16) and Green’s theorem, 

 A · d =

 ∇ × A · dS =

B · d S = B ,

(2.33)

with B symbolizing the magnetic flux through the area enclosed by the line integral. Changing the current in the solenoid, and therefore the magnetic flux, should then shift the interference pattern. This is termed the magnetic1 Aharonov-Bohm (AB) effect. It was first experimentally observed in 1960 by Chambers [24] but the result received criticism since the magnetic field was not completely zero outside the region. In 1986, Tonomura et al. shielded the electron beams completely from the magnetic field using toroidal ferromagnets, shielded with both a superconducting layer and a copper layer but still observed the effect, thus basing it on a solid experimental foundation [25]. The shift of the interference pattern could be due to non-local interaction of the magnetic field, however, that conflicts with the relativity principle. In order that the interaction be a local one, the physicality of the vector potential becomes a requirement [26]. Despite this argument and validation from experiment, interpretation of the effect remains controversial to this day with many claiming that it is a purely classical phenomenon or that the potentials can be eliminated all together. Interest1 Aharonov and Bohm [23] also proposed an similar experimental setup showing the interference effect of the electric scalar potential in a region with an electric field free region. The relative phase change is then given by a time integral of the potential differences.

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2 Conceptual Emergence

Fig. 2.18 Phase difference of wavefronts (white) in water (black) moving left to right passing a clockwise rotating vortex. Image from [27] with permission from IOP Publishing and M. Berry

ingly, Berry along with Chambers et al. [27] proposed an analogous system with wave fronts passing a vortex in water, reminiscent of Maxwell’s model, thus demonstrating the phase change in a classical setting, Fig. 2.18. In this fluid analogy, the phase shift of the wavefront is simply due to the fact that the wave is accelerated on one side of the vortex and slowed down on the other side. The electromagnetic-fluid analogy is further explored in Chap. 6.2.

2.6.1 Aharonov-Casher Effect In the year 1984, Aharonov and Casher [28] proposed a similar effect, which can be seen as dual (not in the strict mathematical sense) to the AB-effect, Fig. 2.19. A neutral particle with a magnetic moment μ diffracting around a line of charge will obtain a phase shift  1 (2.34) ϕ = 2 (E × μ) · d. c In their paper, Aharonov and Casher draw attention to the viewpoint that the phase shift is in essence a topological effect, stemming from movement through a multiply connected region, due to the hole made by the line charge [28]. In the same year, Sir Michael V. Berry published his seminal paper in which he showed that the AharonovBohm effect can be interpreted as a geometrical phase factor [29].

2.7 Geometric Phases

-

23

 B

q

μ -

 E

Fig. 2.19 The duality between the AB-effect, upper diagram, in which a phase shift results from the movement of charge in a magnetic vector potential field (around a magnetic field source) and the AC-effect, lower diagram, where a phase shift results from the movement of a magnetic moment through a scalar potential gradient (around an electric field source)

2.7 Geometric Phases Consider a quantum mechanical system with a Hamiltonian Hˆ (r) that is changed by varying the spatial parameters r(t) = rt in the time interval from t = 0 to T = 0 such that r(0) = r(T ). This means that the system has been transported in in a loop/circuit, C, in its parameter space [29]. At any instant the system has the eigenstates |n(rt ), where (2.35) Hˆ (r)|n(r) = E n (r)|n(r). Schrödinger’s equation describes the time evolution of the state vector, |ψ(t) of the system ∂ (2.36) Hˆ (rt )|ψ(t) = i |ψ(t) . ∂t In spite of the system having the same initial and end point in the parameter space, the state picks up a phase factor |ψ(t) = eiα |n(rt ).

(2.37)

The phase factor consist of two distinct phases [29], α = β + γ , the temporal dynamical phase  −1 T E n (rt ) dt, (2.38) βn =  0 and the spatial geometric phase, also known as the Berry phase,

24

2 Conceptual Emergence

 γn (C) = i

n(rt )|∇r n(rt ) · dr.

(2.39)

C

By defining the Berry connection as An (r) = in(rt )|∇r n(rt ),

(2.40)

the geometric phase can be expressed as  γn (C) =

An · dr,

(2.41)

C

which has the same form as the Aharonov-Bohm phase (2.32). In the same way that the the magnetic field is the curl of the magnetic vector potential, we can define the Berry curvature as n (r) = ∇r × An (r)

(2.42)

and write the geometric phase in terms of it, in the same manner as in (2.33). The correlation to electromagnetism holds even further as the time derivative of the Berry connection or curvature in lattice models can give rise to a sort of electromotive force, in analogy with Faraday’s law [30]. Geometric phases have been heavily studied in multiple branches of physics in the last 40 years in relation to concepts such as the quantum Hall effect, electric polarization, the Foucault pendulum, exchange statistics and more [31]. Barry Simon showed that the geometric phase is in fact the holonomy, Fig. 2.20, of the vector (fiber) bundle over the parameter space of the Hamiltonian [32]. By using the mathematics of connections on fiber bundles, the relationships between curvature, geometric phases, topology and gauge transformations can be expressed and generalized [33]. Defining all the necessary mathematical preliminaries in order to elaborate on fiber bundles is too big of a task for our current endeavor. The interested reader can for example refer to Ref. [34] for a rigorous mathematical approach or Ref. [35] for a more conceptual approach.

2.8 Gauge Theories Invariance under a gauge transformation (2.23) or a phase transformation (2.28) sparked a new approach to theories in physics. By insisting on invariance under a specific transformation with a certain symmetry, physical interactions could be uncovered. These transformations became known as gauge transformations and the corresponding theories gauge theories. Thus began a new paradigm in theoretical physics of symmetry dictating interactions [15]. The seminal paper of Emmy Noether

2.8 Gauge Theories Fig. 2.20 As a tangent vector (blue) is parallel transported along the closed path shown, it obtains a phase θ due to the curvature of the sphere which is an example of holonomy

25

θ

(1918) on the connection between symmetries and conserved quantities had set the stage for this line of thought [36]. For example the conservation of charge stems from global phase symmetry [37]. From being understood as the electrotonic state, the comprehension of the magnetic vector potential further evolved to the concept of a gauge field. Gauge theories incorporate the mathematics of group theory, already fully developed before its application. The gauge theory of electromagnetic interactions rests on the symmetry of the phase transformation in (2.28) or U(1) symmetry groups. By exploring more complex symmetries, Yang & Mills in 1954 formulated a gauge theory of the strong nuclear force interaction with the SU(2) symmetry group and later others extended gauge symmetry to SU(3) symmetry, resulting in quantum chromo-dynamics. In 1998 a team from Cambridge university presented a gauge theory of gravity using Clifford’s geometric algebra [38]. Furthermore the Standard Model, describing all known elementary particles, can be formulated by the internal symmetries of the group product U(1) × SU(2) × SU(3). The Higgs mechanism, based on spontaneous symmetry-breaking of the gauge field, is needed in order to incorporate the masses of particles [39]. In this way, the magnetic vector potential can be said to continue to play a key role in the development of physics to this day as the generalized concept of a gauge field and is a good entry point for physics students into modern physics.

26

2 Conceptual Emergence

References 1. American Physical Society, APS News 17(7) (2008). https://www.aps.org/publications/ apsnews/200807/physicshistory.cfm 2. A.K.T. Assis, J.P.M.C. Chaib, Ampère’s Electrodynamics (Montreal, Apeiron, 2015) 3. C.S. Potts, H.C.P. Richards, H. Khunrath, Electricity: Its Medical and Surgical Applications, Including Radiotherapy and Phototherapy (Lea & Febiger, Philadelphia, New York, 1911) 4. J.M. Thomas, Michael faraday and The Royal Institution: The Genius of Man and Place (Taylor & Francis, Oxford, 1991) 5. M. Faraday, The Correspondence of Michael Faraday, vol. 1 (London, The Institution of Engineering and Technology, 1991), pp. 1811–1831 6. W.H. Oskay, Creative Commons: Attribution 2.0 Generic—creativecommons.org/licenses/ by/2.0/. www.evilmadscientist.com (2010) 7. A.W. Poyser, Magnetism and Electricity: A Manual for Students in Advanced Classes (Longmans, Green, & Co., New York, 1892) 8. M. Geupel, Deviant Art (2011). https://racoonart.deviantart.com/art/humming-top-198925449 9. M. Faraday, Experimental Researches in Electricity, vols. 1 and 2 (Taylor and Francis, London, 1839) 10. S.P. Israelsen, The Purdue Historian 7(1), 1–16 (2014) 11. J. Bence, The Life and Letters of Faraday (Longmans, Green and Co, London, 1870) 12. J.C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1873) 13. J.C. Maxwell, Philos. Mag. Ser. 23(151), 12–24 (1861) 14. A. Einstein, Die Naturwissenschaften 14(11), 223–224 (1926) 15. C.N. Yang, Phys. Today 67(11), 45–51 (2014) 16. H. Lamb, Hydrodynamics (C.J. Clay and Sons, London, 1895) 17. P. Tait, Lectures on Some Recent Advances in Physical Science (MacMillan & Co., London, 1876) 18. A.C.T. Wu, C.N. Yang, Int. J. Mod. Phys. A 21(16), 3235–3277 (2006) 19. M.J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (University of Notre Dame Press, London, 1967) 20. R.P. Feynman, R.B. Leighton, M.L. Sands, The Feynman Lectures on Physics, Vol. I and II (Addison-Wesley, 1963) 21. D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, New Jersey, 1995) 22. J.D. Jackson, L. Okun, Rev. Mod. Phys. 73(3), 663–680 (2001) 23. Y. Aharonov, D. Bohm, Phys. Rev. 115(3), 485–491 (1959) 24. R.G. Chambers, Phys. Rev. Lett. 5(1), 3–5 (1960) 25. Tonomura, Phys. Rev. Lett. 56(8), 792–795 (1986) 26. Y. Aharonov, D. Bohm, Phys. Rev. 123(4), 1511–1524 (1961) 27. M.V. Berry, R.G. Chambers, M.D. Large, C. Upstill, J.C. Walmsley, Eur. J. Phys. 1(3), 154–162 (1980) 28. Y. Aharonov, A. Casher, Phys. Rev. Lett. 53, 319–321 (1984). https://doi.org/10.1103/ PhysRevLett.53.319 29. M.V. Berry, Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 392(1802), 45–57 (1984). https:// doi.org/10.1098/rspa.1984.0023 30. S. Chaudhary, M. Endres, G. Refael, Phys. Rev. B 98, 064310 (2018). https://doi.org/10.1103/ PhysRevB.98.064310 31. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, E. Karimi, Nat. Rev. Phys. 1(7), 437–449 (2019). https://doi.org/10.1038/s42254-019-0071-1 32. B. Simon, Phys. Rev. Lett. 51, 2167–2170 (1983). https://doi.org/10.1103/PhysRevLett.51. 2167 33. C.N. Yang, Tsinghua Sci. Technol. 3(1), 861–970 (1998) 34. M. Nakahara, Geometry, Topology and Physics (Taylor & Francis Group, 2003) 35. R. Healey, Gauging What’s Real: The Conceptual Foundations of Gauge Theories (Oxford University Press, 2007)

References 36. 37. 38. 39.

E. Noether, Transp. Theory Stat. Phys. 1(3), 186–207 (1971) K.A. Brading, Stud. History Philos. Mod. Phys. 33, 3–22 (2002) A. Lasenby, C. Doran, S. Gull, Philos. Trans. R. Soc. Lond. 356(1737), 487–582 (2004) D.J. Griffiths, Introduction to Elementary Particles (John Wiley & Sons, New York, 1987)

27

Chapter 3

Mathematical Appearance

Abstract We venture on to explore the various roles taken on by the magnetic vector potential in the mathematical description of light and electromagnetic interactions in general. Starting from the relations of the electric and magnetic field to the corresponding vector and scalar potentials, we derive Maxwell’s equations. In the process we uncover a dependence of the displacement current on the divergence of the magnetic vector potential, also known as the gauge condition. We argue that the gauge can not be chosen arbitrarily and that gauge conditions have physical implications for momentum conservation, which becomes more evident in a four-vector spacetime formulation. We briefly show how the magnetic vector potential enters the Hamiltonian for electromagnetic interactions using a variational approach and how it relates to gauge invariance in quantum electrodynamics.

3.1 Electromagnetism As the name suggests, the theory of electromagnetism covers the sources, interactions and applications of electric and magnetic fields as well as electromagnetic waves. Visible light makes up only a little part of the frequency spectrum, the band most abundant in solar radiation, which our eyes are well equipped to receive and interpret, Fig. 3.1. The classical mathematical description of electromagnetic phenomena consist of four well known equations termed Maxwell’s equations, which arguably are more directly from Heaviside and Hertz as noted in Sect. 2.3. However, as is the case with most human advances, they are the result of a combined effort of many. The differential formulation of the equations in vacuum in SI-units is as follows, ∇ ×E=−

∂B , ∂t

∇ × B = μ0 J + μ0 ε0 ρ , ε0 ∇ ·B=0.

(3.1) ∂E , ∂t

∇ ·E=

© Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_3

(3.2) (3.3) (3.4) 29

30

3 Mathematical Appearance

Fig. 3.1 Interference of electromagnetic waves bringing out colors on the surface of a soap bubble. Credit Lanju Fotografie

In plain words the equations state the following relations: I. A rotational electric field will be induced by a time varying magnetic field. II. A rotational magnetic field will be induced by a current density J and/or a time varying electric field. III. A source for the electric field is a charge density, ρ. IV. The magnetic field is without sources or in other words divergence-free. Without sources, ρ = 0 and J = 0, the equations become symmetric with respect to interchangeable roles of the magnetic and electric field. Thus it becomes evident that one field will induce the other and wave propagation is implied. The integral formulation holds more information about the relative dimensions of its components, with N signifying an arbitrary three dimensional volume with the boundary ∂ N and M an arbitrary two dimensional surface with the boundary ∂ M, ˛ ∂M

E · d = −

˛ ∂M

¨ B · d = μ0

¨ B · dS ,

E · dS =

1 ε0

(3.5)

M

d J · dS + μ0 ε0 dt M

" ∂N

d dt

¨ E · dS ,

(3.6)

M

˚ ρ dV ,

(3.7)

N

" ∂N

B · dS = 0.

(3.8)

3.1 Electromagnetism

31

The equivalence of the differential and integral formulations becomes clear by using the boundary theorem in the appropriate dimension, see the Appendix. Only when the integral manifolds N and M are fixed can the time derivative be brought under the integral. In case of a time varying manifold one has to use the Leibniz Integral theorem, also known as Reynold’s transport theorem in fluid dynamics. For a time varying vector field F(r, t), d dt



¨

¨ F · dS = M(t)

M(t)

 ˛ ∂ F + [∇ · F] v · dS − [v × F] · d ∂t ∂ M(t)

(3.9)

where v is the velocity of the contour ∂ M. Applying this to (3.5) along with ∇ · B = 0 results in ˛ ¨ ˛ ∂B  E · d = − (3.10) · dS + [v × B] · d , ∂ M(t) M(t) ∂t ∂ M(t) denoting the induced electric field with E . From (3.1) a time varying magnetic field by itself will induce a rotational electric field so ¨ − M(t)

˛

¨

∂B · dS = ∂t

∇ × E · dS = M(t)

∂ M(t)

E · d.

(3.11)

Inserting (3.11) into (3.10) and writing the closed loop integral of the electric field on the left hand side of (3.10) in terms of force per charge we obtain ˛

˛ ∂ M(t)

F/q · d =

∂ M(t)

˛ E · d +

∂ M(t)

[v × B] · d.

(3.12)

Rearranging the charge q, we uncover the Lorentz force law F = q (E + v × B) ,

(3.13)

which is often said to be a necessary supplement in order to make the equations complete [1]. The above analysis however shows that the force law is embedded in the time varying integral and corresponding surface.

3.2 Potential Formulation Starting with Maxwell’s two original equations relating the magnetic and electric field to the potentials, ∂A , (3.14) E = −∇φ − ∂t

32

3 Mathematical Appearance

B=∇ ×A,

(3.15)

let us see if we can recover (3.1)–(3.4) above. By taking the curl of (3.14) we recover (3.1), ∂B ∂ , (3.16) ∇ ×E=− ∇ ×A=− ∂t ∂t since ∇ × (∇ f ) = 0 for any scalar field f and the space and time derivatives commute. Taking the curl of (3.15) gives ∇ × B = ∇ × ∇ × A.

(3.17)

For an arbitrary vector field f, we have the vector identity ∇ × (∇ × f) = ∇(∇ · f) − ∇ 2 f ,

(3.18)

∇ × B = −∇ 2 A + ∇(∇ · A).

(3.19)

therefore

Comparing with (3.2), restated here for convenience ∇ × B = μ0 J + μ0 ε0

∂E , ∂t

we find that ∇ 2 A = −μ0 J

(3.20)

which is the same as (2.19) that Maxwell had uncovered [2]. The second term, known as the displacement current μ0 ε0

∂E = ∇(∇ · A) , ∂t

(3.21)

depends on the value of ∇ · A, called the gauge. The Coulomb gauge, ∇ · A = 0 is thus not in accordance with (3.2). However with the Lorenz gauge ∇ ·A=−

1 ∂φ , c2 ∂t

(3.22)

we find that (3.21) adds up;   1 ∂φ 1 ∂ ∂E ∂E μ0 ε0 =∇ − 2 = 2 (−∇φ) = μ0 ε0 , ∂t c ∂t c ∂t ∂t

(3.23)

3.2 Potential Formulation

33

since c−2 = μ0 ε0 and E = −∇φ (in case of a time-invariant vector potential). In order to have (3.2) consistent with the Coulomb gauge, we have to add to the current density (3.20) the term 1 ∂ (∇φ) = JL , (3.24) c2 ∂t which, as Jackson notes, is the irrotational or longitudinal part of the current with ∇ × JL = 0 [3, 4]. The value of the divergence of A is therefore not arbitrary, having implications for the displacement current term ∂E/∂t, which was famously added by Maxwell to Ampère’s law [5], allowing him to derive the electromagnetic wave equations [6]. Continuing with our analysis, we have yet to recover (3.3) and (3.4) which describe the divergence of the electric and magnetic field. The result that the magnetic field is divergence free is directly implied in (3.15) since the divergence of a purely rotational field is zero, ∇ · B = ∇ · (∇ × A) = 0. (3.25) For the electric field, taking the divergence of (3.14) gives  ∇ · E = −∇ 2 φ − ∇ ·

∂A ∂t

 ,

(3.26)

since the divergence of a gradient is the same as the Laplacian operator, ∇ · ∇ = ∇2 . Assuming a static vector potential, we then recover (3.3) if ∇2φ = −

ρ . ε0

(3.27)

If we were not to assume a static vector potential and use the Lorenz gauge as above, by rearranging derivatives (3.26) becomes ∂ ∇ · E = ∇ · (−∇φ) − ∂t



1 ∂φ c2 ∂t

 .

(3.28)

In the light of (3.27) we then have a wave equation for the electric scalar potential ∇2φ −

1 ∂ 2φ ρ =− . c2 ∂t 2 ε0

(3.29)

We can also obtain a wave equation for the magnetic vector potential in a similar manner, putting together (3.19) and (3.2) gives

34

3 Mathematical Appearance

− ∇ 2 A + ∇(∇ · A) = μ0 J + μ0 ε0

∂E . ∂t

(3.30)

Replacing E with the potentials (3.14), rearranging and again using that c−2 = μ0 ε0 for a more classical form we get   1 ∂ 2A 1 ∂φ ∇ A− 2 2 −∇ ∇ ·A+ 2 = −μ0 J. c ∂t c ∂t 2

(3.31)

The Lorenz gauge cancels the third term on the left hand side leaving 1 ∂ 2A = −μ0 J , c2 ∂t 2

∇2A −

(3.32)

which has the general solution [4] μ0 A(r, t) = 4π

˚

3 

dr V

ˆ

   J(r , t  ) 1    δ t + r−r −t . dt |r − r | c 

(3.33)

We have thus seen how the Lorenz gauge seems to be a natural expression of the divergence of the magnetic vector potential, as in (3.22) which, along with the wave equations for the potentials (3.29) and (3.31), forms a complete description of the classical electromagnetic field [7].

3.3 Spacetime Four-Vector Notation By working within in four dimensional spacetime, (3.29) and (3.32) combine into a single equation describing the behaviour of the electromagnetic field, Aα = μ0 J α ,

(3.34)

where the electric scalar potential φ and the magnetic vector potential A are joined together in the electromagnetic four-potential, α ∈ 0, 1, 2, 3 with A0 = φ/c, Aα =



 φ ,A . c

(3.35)

Similarly the charge and current density form the four-current J α = (cρ, J).

(3.36)

The superscript refers to the fact the the four-vectors can be seen as rank one contravariant tensors, obtained by a projection onto the manifold with the metric, as

3.3 Spacetime Four-Vector Notation

35

opposed to the dual covariant vectors. The time and space derivatives come together in the operator  known as the d’Alembertian , ≡

1 ∂2 − ∇2. c2 ∂t 2

(3.37)

The four-potential, four-current and the d’Almbertian all turn out to be Lorentz invariant, meaning that they stay the same in reference frames moving relative to each other. Underlying is the postulate of relativity that the speed of light is a finite constant with the same value for all reference frames [8]. One interesting aspect of the four-vector formulation is that the electric scalar potential becomes the time component of the four-potential whereas the magnetic vector potential is the space component. Recalling that the magnetic vector potential signifies momentum per charge in space, the electric scalar potential can then be seen as momentum in time. In fact, this might be generalized to say that energy is momentum in time, or vice versa, that momentum is energy in space. The potential energy of a static charge can then be seen to stem from its movement at the speed of light through the dimension of time. In the same way a stationary charge density can be viewed as a current through time. Another interesting feature of the spacetime formulation is that both the Lorenz gauge condition (3.22) and the continuity equation become simple statements of both quantities being divergence free in spacetime [8]. Defining the covariant derivative operator   1∂ ,∇ , (3.38) ∂α = c ∂t the divergence of the four-potential becomes the invariant ∂α Aα =

1 ∂φ + ∇ · A = 0. c2 ∂t

(3.39)

In the same way the continuity equation or the law of charge conservation

can be written as

∂ρ +∇ ·J=0, ∂t

(3.40)

∂α J α = 0.

(3.41)

To see more clearly how (3.40) describes the conservation of charge, let’s rewrite (3.40) with the volume integrals ˚

˚ ∇ · J dV = −

Invoking the divergence theorem gives

∂ρ d V. ∂t

(3.42)

36

3 Mathematical Appearance

"

˚ J · dS = −

∂ρ dV , ∂t

(3.43)

stating that any change of charge density in time within a volume will result in a equal and opposite current density flux over the boundary of the volume. Since (3.40) and (3.39) are identical in structure, the same applies to the magnetic vector potential, meaning that any change in time of an electric potential within a volume will result in a flux of magnetic vector potential (electrokinetic momentum) over the boundary, " ˚ 1 ∂φ A · dS = − 2 d V. (3.44) c ∂t Equation (3.44) holds only in the Lorenz gauge. Given that the scalar potential represents energy per charge and the magnetic vector potential the momentum per charge, as supported by dimensional analysis, (3.44) states that a change of energy within a volume will result in a flux of momentum through the boundary of the volume. An arbitrary choice of the divergence of the magnetic vector potential or gauge, would likewise mean an arbitrary choice of the source of charge momentum. Therefore we stress the argument that the gauge conditions have physical interpretations and that the Lorenz gauge is a natural one.

3.4 Variational Approach Calculus of variations allows for finding equations of motion by determining the minimal extrema of the action integral ˆ

t2

L dt = 0 ,

(3.45)

L (qi , q˙i , t) = T − U

(3.46)

S=δ t1

where

is the Lagrangian energy density equal to the difference between kinetic and potential energies T and U respectively, qi are generalized coordinates with the dotted q symbolizing the time derivative. The equations of motion come from solving the Euler-Lagrange equation   ∂L d ∂L − = 0. (3.47) dt ∂ q˙i ∂qi Electromagnetic interactions are dictated by the potentials with the Lagrangian LEM = qv · A − qφ .

(3.48)

3.4 Variational Approach

37

The total Lagrangian for a charged particle in an electromagnetic field has an added term from the free particle [9] LTot =

1 2 mv + qv · A − qφ. 2

(3.49)

By formulating the Lorentz force law in terms of potentials, the form of the Lagrangian can be found and vice versa, as solving the Euler-Lagrange equation results in the Lorentz force law. The canonical or general momentum, p is given by p=

∂L ∂L = mv + qA , = ∂ q˙i ∂v

(3.50)

in agreement with (2.8). The Hamiltonian can be found with a Legendre transformation H =p·v−L. (3.51) Using the Lagrangian from (3.49) results in 1 2 mv + qφ. 2

(3.52)

1 (p − qA) . m

(3.53)

H= Rewriting (3.50) shows that v=

Plugging back into (3.52) gives the Hamiltonian H=

1 (p − qA)2 + qφ , 2m

(3.54)

same as (2.25) discussed in Sect. 2.5.

3.5 Gauge Invariance in Quantum Electrodynamics The following gauge transformations leave the electric and magnetic fields invariant, ∂χ , ∂t A → A = A + ∇χ . φ → φ = φ −

(3.55) (3.56)

Inserting them into the Lagrangian (3.49), one obtains an extra term of the gauge function χ (r, t) [10],

38

3 Mathematical Appearance

L=

1 2 d mv + qv · A − qφ + qχ , 2 dt

(3.57)

with the added total time derivative dχ ∂χ = + v · ∇χ . dt ∂t

(3.58)

The transformation makes no change to the equations of motion and is essentially a translation that amounts to the unitary transformation 

iq χ T = exp 

 ,

(3.59)

which is the same phase factor mentioned in (2.28). The observable’s O, before the gauge transformation symbolized with O1 and after it with O2 , have the relation ˆ 2T † ˆ1 = TO O

(3.60)

where † denotes the conjugate transpose, whereas the state vectors relate directly with the same transformation |ψ1 = T |ψ2 . (3.61) The transformation (3.59) is then the link between different formulations of quantum electrodynamics with its roots in the gauge invariance of the potentials. Aitchinson and Hey put forth a trivial yet interesting observation [11], writing the wave function in real and complex parts ψ = ψR + i ψI ,

(3.62)

one can take a closer look at the effect of the global phase transformation. By Euler’s formula we have ψ  = ei α ψ = (cos α + i sin α) (ψR + i ψI ) ,

(3.63)

with α being the argument of (3.59) as above. Therefore, ψR + i ψI = ψR cos α − ψI sin α + i (ψR sin α + ψI cos α)

(3.64)

and so the transformed parts of the wave function are given by ψR = ψR cos α − ψI sin α , ψI = ψR sin α + ψI cos α .

(3.65) (3.66)

3.5 Gauge Invariance in Quantum Electrodynamics

39

This can be seen as a rotation in the internal space of the wave function, in the same way that a rotation in the complex plane by an angle θ is described by the transformation z  → eiθ z. In the phase transformation, the angle corresponds to q χ which has not been given a physical meaning, but clues can be obtained from the hydrodynamics analogy in Sect. 6.2.4. Observing that

T T † = eiα · e−iα = cos2 (α) + sin2 (α) = 1 ,

(3.67)

the phase transformation is said to be unitary, with the set of all such transformations forming a group of U(1) symmetry, corresponding to the rotational symmetry of a circle. Furthermore the group is abelian meaning that applying two consecutive phase transformations will give the same result, independent of the order, since eiα eiβ ψ = ei(α+β) ψ = eiβ eiα ψ.

(3.68)

The transformations eiα and eiβ are then said to commute, with the commutator relation (3.69) [eiα , eiβ ] = eiα eiβ − eiβ eiα = 0. This symmetry extends to both the Pauli and Dirac equations, the former being the non-relativistic limit of the latter [12]. The Pauli equation adds to the Schrödinger equation a description of electromagnetic interactions for spin 21 particles, fermions in general and the photon in particular, i

2 1  ∂ψ = σ · (p − qA) ψ + qφψ , ∂t 2m

(3.70)

where σ = (σ1 , σ2 , σ3 ) are the Pauli matrices describing the spin. As a side-note, the algebra of the Pauli matrices is isomorphic to the Clifford algebra C(3) [13] which is yet another case of mathematics being reinvented for explanation of physical behaviour. Jackson notes in Ref. [4] that for a quantum-mechanical description of the photon, only the magnetic vector potential has to be quantized, further supporting the viewpoint that the vector potential is the root cause of electromagnetic interactions. In order that (3.70) be invariant for the phase transformation ψ → ψ  = exp



 iq χ ψ = eiα ψ , 

(3.71)

we need to make the following changes to the electromagnetic potentials, same as before

40

3 Mathematical Appearance

A → A = A + ∇χ , ∂χ , φ → φ = φ − ∂t

(3.72) (3.73)

since i

  ∂ψ  ∂α iα ∂χ iα ∂ψ ∂ψ = i eiα +i e ψ = eiα i − iq e ψ ∂t ∂t ∂t ∂t ∂t

(3.74)

and pˆ eiα ψ = −i∇(eiα ψ) = iq∇χ · eiα ψ − eiα i∇ψ = (pˆ + iq∇χ ) eiα ψ. (3.75) Feynman in Ref. [12] pointed out that the magnetic vector potential enters into the Hamiltonian of the Pauli equation above as a perturbation potential for transitions between states. This could be a clue to the ill-known mechanism for photon emission and absorption, with momentum exchange being governed by the vector potential in a process similar to electromagnetic induction.

3.5.1 Local Gauge Covariance To approach gauge invariance from the opposite point of view is to start with a description of a free particle, impose gauge invariance and make the necessary corrections. For simplicity we use Schrödinger’s equation throughout this section, a comparable analysis can be applied on both the Pauli and Dirac equation. Schrödinger’s equation for a free particle reads  2 ∂ψ =− ∇ ψ. (3.76) i ∂t 2m By a global phase transformation ψ → ψ  = eiα ψ , with α being a constant, it is trivial to show that the equation is globally phase invariant. If however α is a function of local coordinates, the equation is not invariant for the transformation since one obtains extra terms of derivatives of the phase function α as shown before in (3.74) and (3.75). We can however make the equation locally covariant by introducing the gauge covariant derivative Dμ = ∂μ + with the defined four-gradient

iq Aμ , 

(3.77)

3.5 Gauge Invariance in Quantum Electrodynamics

41

∂μ = (∂t , −∇).

(3.78)

Keeping in mind that ∂0 = ∂t and A0 = φ, with the substitution ∂μ → Dμ , Eq. (3.76) becomes    2 −2 iq ∂ iqφ ψ= ∇ − A ψ. i (3.79) + ∂t  2m  Here we have Schrödinger’s equation for a particle in an electromagnetic field, with the same Hamiltonian as in (3.54). Rewriting with the momentum operator p = −i∇ , we have i where

∂ψ = Hˆ ψ , ∂t

1 Hˆ = (p − qA)2 + qφ . 2m

(3.80)

(3.81)

(3.82)

Imposing the local phase transformation  iq ψ → ψ = exp χ (x, t) ψ ,  



(3.83)

the symmetry holds only if the fields A and φ transform in the familiar manner φ → φ = φ −

∂χ , ∂t

A → A = A + ∇χ . By starting out with an equation for a free particle and imposing U(1) phase/gauge symmetry we have uncovered the fundamental interaction of electromagnetism. The boson following the symmetry is the photon described by the magnetic vector potential A, with charge being the single corresponding quantum number. The covariance is manifested in the observationally equal effect of changing the phase locally and the influence of the field A in which the particle moves [11]. The particle is no longer free however, since we have introduced the interaction with the potentials. This relation between the phase and matter fields is known in the literature as minimal coupling. From this line of thought one could try imposing more complex symmetries and see if that leads to other fundamental interactions, which indeed they to. This can be said to be the paradigm of 20th century physics; that symmetries of a theory dictate the interactions. In the same way that U(1) symmetry can be understood as the rotational symmetry of a circle, the SU(2) symmetry can be understood as the rotational symmetry on a sphere. More complex symmetries such as SU(2) turn out

42

3 Mathematical Appearance

to be non-commutative and are said to be non-abelian. Non-commutative operations can be visualized by rotating a book on a table. Rotation of a quarter of a circle on to the top edge of the book and again sideways will result in a different final state then if the rotations were done in the reverse order. Diving into the realm of nonabelian gauge symmetries would be sidetracking from our story about the magnetic vector potential. Nonetheless it is worth while to stop and ponder the richness of the concept, first intuitively sensed by Faraday when experimenting with magnets.

References 1. D.K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Massach., 1983) 2. J.C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1873) 3. J. Schwinger, L.L. Deraad, K. Milton, W. yang Tsai, J. Norton, Classical Electrodynamics (Massach.: Perseus Books, 1998) 4. J.D. Jackson, Classical Electrodynamics, 3rd edn. (John Wiley & Sons, New York, 1999) 5. J.C. Maxwell, Philos. Mag. Ser. 4 23(151), 12–24 (1861) 6. C.N. Yang, Phys. Today 67(11), 45–51 (2014) 7. D.J. Griffiths, Introduction to Electrodynamics (Prentice Hall, New Jersey, 1999) 8. B. Thidé, Electromagnetic Field Theory (Upsilon Books, Uppsala, 2004) 9. L. Landau, E. Liftshitz, The Classical Theory of Fields (Pergamon Press, 1971) 10. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley-Interscience, New York, 1989) 11. I. Aitchison, A.J.G. Hey, Gauge Theories in Particle Physics Volume 1: From Relativistic Quantum Mechanics to QED (Oxford: Taylor & Francis Group, 2003) 12. R.P. Feynman, Quantum Electrodynamics (W. A. Benjamin, New York, 1961) 13. D. Hestenes, Space-Time Algebra (Springer International Publishing, Birkhäuser Basel, Switzerland, 2015)

Chapter 4

Three Dimensional Solutions

Abstract In this chapter we solve the Laplace equation for the magnetic vector potential of two fundamental current distributions and plot the result. The solution for the line current is relatively straight forward. For the current loop we use the method of eigenfunction expansion. The mathematical preliminaries such as the Dirac Delta distribution and Green’s function are succinctly defined. We uncover how the geometry of the magnetic vector potential follows the geometry of the current distribution and how the magnetic field lines are equipotential lines of the vector potential field.

4.1 Line Current To obtain the value of the magnetic vector potential for a given current distribution we solve (3.20) which has the integral expression (2.21), A(r) =

μ0 4π

 V

J(r ) dr . |r − r |

(4.1)

Here we let the current flow in the z-direction of a Cartesian coordinate system (x, y, z), Fig. 4.1, with unit vectors (ˆex , eˆ y , eˆ z ) respectively. By aligning the coordinate system such that the current lies along the z-axis, the current density can be described with J = I eˆ z ,

(4.2)

ignoring the thickness of the wire. Only the z-component of the integral in (4.1) contributes and since the integrand is even and both the unit vector and current are constant, we get

© Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_4

43

44

4 Three Dimensional Solutions

Fig. 4.1 Line current in the z-direction defined in a Cartesian coordinate system

z I · eˆz |r − r | 0 y

 μ0 ∞ I eˆ z dz √ 4π −∞ r 2 + z2  ∞ dz I μ0 eˆ z = √ 2π r 2 + z2 0  L I μ0 dz eˆ z = lim √ 2 L→∞ 2π r + z2 0 √  I μ0 r2 + L2 + L eˆ z ln = lim . L→∞ 2π r

A(r) =

x

r

(4.3) (4.4) (4.5) (4.6)

We see that as L → ∞, the value of the magnetic vector potential becomes infinite as well. However, at a distance r orthogonal to a wire of finite length L, we have the expression √  r2 + L2 + L I μ0 eˆ z ln A(r) = , (4.7) 2π r and in the vicinity of the wire where r TC

Surface currents Be

Bi = Be

Be

Bi = 0 λL

(a)

(b)

Fig. 5.1 (a) An external magnetic field Be applied to a sample at temperature T above the critical temperature TC permeates the sample. (b) Meissner effect: An external magnetic field applied to a superconductor is expelled by surface currents flowing in the outer layer of thickness λ L

Experiments have shown that the superconducting penetration depth of various samples is always larger than λ F (0) which leads to the definition of another key concept in the theory of superconductors, the coherence length [3].

5.4 Coherence Length and the Ginzburg-Landau Theory In the literature on superconductors there are two concepts termed the coherence length [6], they differ in their temperature dependence. We begin by introducing the temperature independent coherence length due to Pippard [7]. Temperature independent Pippard coherence length/range Equation (5.9) is valid in the case of uniform flow or slowly varying drift velocity of the electrons, when electron velocities have negligible variance over a specific distance, ξ0 [4]. If that is not the case, a generalized model in needed. There is an accurate analogy with Ohm’s law (5.10) which brakes down when the classical penetration depth becomes similar in magnitude to the mean free path, , of electrons in the metal. Then, one has to take into account the mean value of the electric field around each point in a volume with radius . The generalized equation according to Ref. [7] is J(r) =

3σ 4π 

 V

R[R · E(r )] −R/  e dr R4

where R = r − r and the integral is over the total volume of the metal.

(5.18)

5.4 Coherence Length and the Ginzburg-Landau Theory

61

In the same manner the London equation (5.9) can be generalized with 3 J(r) = 4π ξ0 

 V

R[R · A(r )] −R/ξ  e dr R4

(5.19)

where ξ is the general coherence length for impure metals or alloys, given in terms of the coherence length of a pure metal ξ0 and the mean free path  [4], 1 1 1 = + . ξ ξ0 

(5.20)

If the magnetic vector potential A(r) is constant in a volume of radius ξ0 we obtain the London equation (5.9) from (5.19). There is an argument for the coherence length based on the uncertainty principle: The superconducting phenomena comes into play at the critical temperature TC and the electrons have momentum of the order p ≈ k BVTFC where k B is the Boltzmann constant and v F is the Fermi velocity. Then according to the uncertainty principle

x ≥

v F  =a =: ξ0 2 p k B TC

(5.21)

with a being a material constant [3]. As a final note on the conceptual meaning of the coherence length, it can be shown that the wavefunction of a single Cooper pair, which will be discussed in Sect. 5.9, has a spatial extent of the same order of magnitude as ξ0 [6]. The theories of London and Pippard contain neither wavefunctions nor Cooper pairs and are both based purely on electromagnetism and phenomenology. Temperature dependent Ginzburg-Landau coherence length Before a microscopic theory of superconductivity was formulated, Ginzburg and Landau put forth a theory based mostly on intuition [6]. The need for a more general model than that of the London equations was evident due to their prediction of a negative surface energy at the interface between the superconducting and normal state, in contradiction to the observed positive surface energy [8]. They introduced an effective wavefunction describing superconducting electrons locally with |ψ(r)|2 = n s (r)

(5.22)

where n s is the superconducting electron number density. The semi-classical Ginzburg-Landau (GL) theory was a precursor of the quantum mechanical BCS theory. For future reference we note as in Ref. [4] that there is a formal analogy with the BSC pairing potential to this effective wavefunction (5.22) and that they are in fact directly proportional. The equations governing the behavior of the wavefunction are obtained by defining the free energy of the superconductor in a magnetic field and minimizing it with respect to the magnetic vector potential [3]. For a superconductor with constant

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5 Theory of Superconductivity

electron density and no external field, the free energy can be expanded as E = α |ψ|2 +

β |ψ|4 2

(5.23)

where α(T ) ∝ (T − TC ) and β(TC ) are coefficients of the expansion and functions of the temperature and critical temperature respectively [9]. If ψ is assumed to be slowly varying near the critical temperature TC another term has to be added to the free energy [8] β 2 |∇ψ|2 , E = α |ψ|2 + |ψ|4 − (5.24) 2 2m ∗ where m ∗ refers to an unknown phenomenological mass. In order to have the last term invariant with respect to the gauge of the vector potential, the gradient is made covariant with E = α |ψ|2 +

 2 β 1  ∗ |ψ|4 − −i∇ − e A ψ 2 2m ∗

(5.25)

where e∗ is the phenomenological charge, which later turned out to be the charge of an electron pair e∗ = 2e. Finally, an external magnetic field B = ∇ × A is added giving [9] E = α |ψ|2 +

 2 β 1  1 ∗ |ψ|4 − |∇ × A|2 . −i∇ − e A ψ + 2 2m ∗ 2μ0

(5.26)

Minimizing this free energy with respect to the magnetic vector potential A and assuming the Coulomb gauge, one obtains the following set of equations,  2  1  i∇ + e∗ A ψ = α + β |ψ|2 ψ, ∗ 2m Js =

 e∗2 −ie∗  † † ψ − ∗ |ψ|2 A. ∇ψ − ψ∇ψ 2m ∗ m

(5.27)

(5.28)

These are known as the Ginzburg-Landau (GL) equations [4], first formulated in 1950 [10]. By considering a trivial case of the first equation in one dimension in terms of reduced variables one gets the temperature dependent GL coherence length [4]  − 1 ξG L (T ) =  2m ∗ α(T ) 2 , (5.29) describing the space in which ψ varies slowly with negligible energy increase. In the case of a pure superconductor well below the critical temperature, the GL coherence length and the temperature independent Pippard coherence length are equal [3]. By considering the second GL equation (5.28) with the approximation that the value of ψ in weak fields is the same as the constant equilibrium value in the absence

5.4 Coherence Length and the Ginzburg-Landau Theory

63

of an external field ψ0 , one obtains a London type equation Js = −

e∗2 |ψ0 |2 A. m∗

(5.30)

We should then be able to derive a penetration depth as in Sect. 5.3. For the sake of diversity, let us derive it in terms of the magnetic vector potential with the classical correspondence to the current, (5.31) ∇ 2 A = −μ0 J. Then ∇2A =

μ0 e∗2 |ψ0 |2 1 A= 2 A m∗ λG L 

and so λG L =

m∗ μ0 e∗2 |ψ|2

(5.32)

(5.33)

which has the same form as (5.15) given the condition of (5.22). The magnetic vector potential therefore penetrates the superconductor up to the penetration depth λ L like the magnetic field. The induced surface currents follow the magnetic vector potential field lines, Fig. 5.1. Once again we see the primary effect in the magnetic vector potential, rather then the magnetic field. It turns out that the temperature dependence of the GL penetration depth λG L near TC is the same as that of the GL coherence length ξG L and thus a dimensionless temperature independent parameter can be defined as [4] κ=

λG L . ξG L

(5.34)

This parameter is known simply as the GL parameter and its value for a pure standard superconductor is much less than unity which implies λG L ξG L . Solving the GL equations with the proper boundary conditions, a positive surface energy for the junction of a normal and superconducting material is obtained, in alignment with experiments [3]. Shortly after the formulation of GL theory, when discussing surprising experimental results of Zavaritskii, Abrikosov considered the case when λG L > ξG L [11]. He found that at the exact value 1 κ=√ , 2 the surface energy between a superconducting and a normal layer becomes zero and above that value it becomes negative. Superconductors with

64

5 Theory of Superconductivity

1 κ>√ 2 were first called superconductors of the second group but are today known as type-II superconductors [11]. Conversely, for type-I superconductors 1 κ ,

1  2 1 + E − 2 > 2 2E

1  2 1 = − E − 2 < 2 2E

uk +2 = vk + 2 whilst

1  2 1 − E − 2 < 2 2E

1 1  2 = + E − 2 > 2 2E

uk −2 = vk − 2

1 2 → electron-dominant, 1 2 1 2 → hole-dominant. 1 2

(5.78)

(5.79)

So in light of the quasiparticle character of the excitations according to (5.68), we can deduce that the positive solution in (5.76) gives an electron dominant excitation

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5 Theory of Superconductivity

Ek

Fig. 5.11 Degeneracy and type of the wave vector k for each energy value between

and μ in a homogeneous superconductor

Δ ke

kh

0

kh

ke

k

whereas the negative one is hole dominant [30]. For clarity we will write ke for k+ and kh for k− . Since ±k is a solution for both ke and kh , the energy-momentum relation yields a fourfold degeneracy of states for each energy above as depicted in Fig. 5.11. In the presence of an applied external magnetic field, an additional twofold spin degeneracy has to be considered.

5.11 Andreev Reflection and the Proximity Effect When a superconductor is brought in contact with a metal or semiconductor, superconductivity is induced in the latter. This is the superconducting proximity effect, also known as the Holm-Meissner effect [31], discovered in 1932. The effect has emerged as a key element in modern nanoscale technologies, notably in the fabrication of topological quantum bit devices [32, 33]. The mechanism of the effect is understood in terms of Andreev-reflection [34], where Cooper pairs form in a superconductor when electrons incident from the normal conductor meet the boundary. Conversely, Cooper pairs are said to leak into the metal from the superconductor [3]. The latter process has been criticized as being only vaguely clarified [35] and despite having been known for close to a century now, there is no generalized model of the effect but multiple theoretical models exists for various special cases of the proximity effect [36]. In this section we will take a look at the popular BTK-model of Andreev reflection [30] and its optical analogy.

5.11.1 The BTK Model In the year 1982, Blonder-Tinkham & Klapwijk (BTK) [30] solved the BogoliubovDeGennes equations, (5.71), for a normal metal-superconductor (N-S) junction, Fig. 5.12. Assuming constant values of the chemical potential μ and the superconducting

5.11 Andreev Reflection and the Proximity Effect

77

E Δ

Superconductor

Normal metal

1 e

N (E) N (0)

h

Fig. 5.12 A normal metal-superconductor junction, showing the density of states in the superconductor. An electron incident at the boundary, with energy lower than , will be reflected as a hole whilst forming a Cooper pair within the superconductor

gap parameter for an incident electron from the metal side, the solutions of the incoming and reflected waves are ψinc = and ψr e f l = a

  1 ike x e 0

    0 ikh x 1 −ike x +b , e e 1 0

(5.80)

(5.81)

for the composite wavefunctions ψ±ke/ h

  u = 0 e±ike/ h x , v0

(5.82)

where u 0 and v0 are the probabilities for an electron and hole, respectively. The coefficients of reflection, a and b, for the hole and electron are given by a=

(u 20 − v02 )(Z 2 + i Z ) u 0 v0 and b = , u 20 + (u 20 − v02 )Z 2 (u 20 − v02 )Z 2

(5.83)

where Z = h/v F is a dimensionless parameter describing the magnitude of a delta potential barrier of height h at the interface of the normal metal and superconductor. For the simplest case, let Z = 0. Then a = v0 /u 0 and b = 0, meaning that an incident electron (5.80) is purely reflected as a hole (5.81), in a process known as Andreev reflection [37]. An additional electron is absorbed by the superconductor, which

78

5 Theory of Superconductivity

causes the reflected hole. If an electron is incident on the boundary at an angle, the reflected hole path is parallel to the incident path, which is termed retro-reflection. Only in the case that the electron is incident orthogonal to the boundary, Fig. 5.12, do reflection and retro-reflection look the same. Let us now explore in more detail how (5.80) and (5.81) are obtained.

5.11.2 The Superconducting Side The starting point of the BTK model is to solve two time dependent Schrödinger equations coupled by the superconducting energy gap for a normal metalsuperconductor junction or interface, Fig. 5.12,  2 2   ∇ ∂f = − − μ(x) + V (x) f (x, t) + (x) g(x, t), i ∂t 2m  2 2   ∇ ∂g =− − − μ(x) + V (x) g(x, t) + (x) f (x, t). i ∂t 2m

(5.84)

Plane wave solutions of the following form, where u 0 and v0 are taken to be real and positive, f (x, t) = u 0 eikx−i Et/ , (5.85) g(x, t) = v0 eikx−i Et/ , allow for the time dependence to be factored out, resulting in the Bogoliubov-De Gennes equations, (5.71). Thus all of the expressions derived in Sect. 5.10 apply to the superconducting part of the junction. The solutions to (5.84) can thus be written as ψ±ke/ h =

  u 0 ±ike/ h x e . v0

(5.86)

In the normal metal the superconducting pairing potential is zero, N (x) = 0. Then in (5.84) the wavefunction f describes an electron, but for the wavefunction g we have the time-reversed Schrödinger equation. A time-reversed electron behaves like a hole, and so f and g can be interpreted as describing electrons and holes respectively. When V(x) is constant, which is the usual approximation in the BCS model, f and g are proportional to the electron and hole occupation probabilities, u 0 and v0 [27]. Therefore, we see that (5.86) describes quasiparticle excitations in the superconductor. The one-dimensional Hamiltonian incorporating the chemical potential μ of the condensate is 2 ∇ 2 − μ(x) + V (x), (5.87) H (x) = − 2m

5.11 Andreev Reflection and the Proximity Effect

79

where the potential V (x) = hδ(x)

(5.88)

describes the contact resistance or an oxide layer between the metals at the interface x = 0, h is the height of the barrier. For simplification it is assumed that μ(x), V (x) and (x) are all constants with the values μ(x) = μ, V (x) = 0 and 

(x) =

0 for x < 0 (normal metal) ,

for x ≥ 0 (superconductor).

(5.89)

5.11.3 The Normal Metal Side In the normal metal the two equations in (5.84) decouple due to vanishing and the energy-momentum relations are E ke =

2 ke2 − εk 2m

and E kh = εk −

(5.90)

2 kh2 , 2m

(5.91)

for electrons and holes respectively, Fig. 5.13. The Fermi energy is εk = μ and the corresponding Fermi wave vector takes two values kF = ±

√ 2μm . 

(5.92)

Ek

Fig. 5.13 Degeneracy and character of the wave vector k in a normal metal

μ

0

−kF −μ

kF

k

80

5 Theory of Superconductivity

Keeping in mind the physics of the matter, below the Fermi energy only hole excitations are available since all electrons are in the ground state. However above the Fermi energy, electron excitations can occur.

5.11.4 The Interface Figure 5.14 shows the dispersion at the normal metal-superconductor interface along with the plane wave propagation modes for the BdG solutions in the normal metal ψ N (x) =

      1 ikeN x 0 ikhN x 1 −ikeN x +a +b e e e 0 1 0

(5.93)

and the superconductor ψ S (x) = c

    S u 0 ikeS x v e + d 0 e−ikh x . v0 u0

(5.94)

Using the condition of continuity to match the wavefunctions and their first derivative at the junction x = 0, ψ N (0) = ψ S (0) and ψ N (0) = ψ S (0), the coefficients a, b, c and d can be found. The main assumption of the BTK analysis is the Andreev approximation, where all wavevectors, both in the superconductor and the normal metal, are assumed to be very close to k F , keN ≈ khN ≈ keS ≈ khS ≈ k F ,

(5.95)

which considerably simplifies the analysis and results in the aforementioned coefficients of reflection in (5.83) and the following coefficients of transmission c=

iv0 Z u 0 (1 − i Z ) , d= 2 . u 20 + (u 20 − v02 )Z 2 u 0 + (u 20 − v02 )Z 2

(5.96)

Despite having clarified the concept of Andreev reflection at a normal metalsuperconductor interface, we have not explained how Cooper pairs explicitly move from the superconductor to the normal metal. One of the authors of the BTK paper, Klapwijk, stated in 2004 that the following equation called the self-consistency equation, is a necessary part of the analysis even though it had been ignored due to the simplistic geometry [35],

(r) = Vk F(r) = V



v∗ (r) u(r)[1 − 2 f (E)].

(5.97)

E>0

Here, f (E) is the Fermi distribution function, Vk is the approximate mean field pairing interaction and F(r) is the order parameter,

5.11 Andreev Reflection and the Proximity Effect

81

E

E μ a

b

Inc. keN

N kh

−keN 0

−kF

c

d

kF

S −kh

Δ 0

−kF

keS kF

k

−μ Superconductor

Normal metal

Fig. 5.14 Dispersion relation at a normal metal-superconductor interface. An incident electron from the left with energy above (Inc.) has probability of being reflected (b), retro-reflected as a hole (a), transmitted as a hole-like quasiparticle (d) or an electron-like quasiparticle (c) [30, 36]

F(r) = ψ(r↑ )ψ(r↓ ),

(5.98)

which can be said to correspond to the Cooper pair density [22]. As noted in Refs. [35, 38], even though the pairing interaction Vk vanishes at the interface, resulting in = 0, the order parameter F(r) can be non-zero in the normal metal, which is the basis of the proximity effect.

5.11.5 Optical Retro-Reflection If the energy of an incident electron (5.80) is smaller than the superconducting gap, then the wave vector obtains an imaginary component. Waves of this type are known as evanescent waves and are well known in optics. They are formed when total internal reflection occurs at a boundary. Despite having been known for a long time, only recently their measurable effect in the electromagnetic setting was theorized [39] and shortly after detected [40]. Retro-reflection and phase conjugation are considered characteristic aspects of Andreev reflection. Both properties are manifested in non-linear optics of phase conjugate mirrors [41]. A remarkable property in the optical case is that a scattering barrier between a light source and a phase conjugate mirror becomes invisible in the sense that due to retro-reflection, the scattering is retraced back along the original ray trajectory [42]. Rotational recoil keeps the chirality of the reflected wave [43]. This

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5 Theory of Superconductivity

behaviour was originally considered to be associated with time-reversal symmetry but rigorous derivation has shown that is not the case [44]. The analogy is not exact due to an additional phase factor obtained in the case of Andreev reflection and so a scattering barrier imposed between a normal metal and superconductor will not be invisible, despite retro-reflection [41].

5.12 Superfluidity There exists a completely analogous effect to superconductivity in fluids known as superfluidity. Both effects have in common the sustaining of constant currents without resistance or friction. With the same approach as in GL theory, Sect. 5.4, one can describe the superfluidic state by a wavefunction [17] ψ f (r) = ψ f0 eiθ(r) ,

(5.99)

where θ (r) is a real valued phase function. By applying the momentum operator pˆ = −i∇ on the wavefunction we uncover an expression for the superfluid velocity, v f (r) =

 ∇θ (r). m

(5.100)

Therefore, the phase of the superfluid wavefunction takes on the role of a velocity potential , m θ = . (5.101)  We are then led to the conclusion that the gauge function (5.47) and the velocity potential are analogous, which implies that the superfluid velocity v f and the magnetic vector potential have the same behaviour. This relation also holds in classic electromagnetism and fluid dynamics, as we will discuss further in Sect. 6.2. From (5.100) we see that the phase is uniform for a fluid at rest and that in the case of a constant superfluid velocity, the phase has uniform variation in the direction of the velocity. Comparable to the London equation (5.9) we have for superfluids [17] Jf = ρfvf.

(5.102)

A Navier-Stokes like equation can be formulated describing the acceleration of the superfluid using the total convective time derivative, also known as the substantial or material derivative, ∂ D = +v·∇ (5.103) Dt ∂t where v is the velocity of the fluid element itself. A similar treatment should be possible in the case of superconductors.

5.12 Superfluidity

83

Superfluids have the interesting property of having more than one mode of density waves or sound waves, known as second sound. There are essentially two more modes, the third sound having been both predicted and measured [45]. The second sound can be seen as an entropy wave, where the normal fluid part and the superfluid part of the liquid move with opposite phase resulting in zero net mass flow [17]. The phenomena was only known to occur at temperature up to a few Kelvin but in 2019 second sound was observed for the first time at temperatures above 100 K in graphene [46].

5.13 Superfluid Vacuum Theory Since ancient times, the human intellect has conceived of empty space as some sort of substance of a subtle nature. The four Greek classical elements of earth, wind, water and fire were thought to have the fith element, the æther or ether, as their basis of existence. Even earlier the same concept was formulated in Vedic schools of philosophy in ancient India termed akasha [47], believed to have only the property of sound. For physicist up until the 20th century, it was assumed that the ether had a similar nature to a fluid. Maxwell’s final words in his Treatise on Electricity and Magnetism, Vol. 2 are the following [48], Hence, all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavor to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.

The textbook story today is that the idea of the luminous æther was completely abandoned following the negative result of the Michelson-Morley experiment in 1887 and Einstein’s discovery of the principle of relativity in 1905 [49]. However the great physicist Paul A. M. Dirac sent a letter to the editor of Nature magazine in November 24th 1951, titled “Is there an æther?” which contains the following remark [50], Physical knowledge has advanced much since 1905, notably by the arrival of quantum mechanics, and the situation [about the scientific plausibility of aether] has again changed. If one examines the question in the light of present-day knowledge, one finds that the aether is no longer ruled out by relativity, and good reasons can now be advanced for postulating an aether.

In the letter, Dirac refers to his 1951 paper A New Classical Theory of Electrons [51] where he put forth a new condition for the magnetic vector potential as the simplest relativistic way to remove the degeneracy of the gauge transformations Aμ Aμ = k 2 ,

(5.104)

φ , c

(5.105)

or using more standard notation |A| = k =

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5 Theory of Superconductivity

with k being a universal constant taken to be the mass/charge ratio, k = m/e.1 By a variational (Lagrangian) approach, he then derives the charge-current density or four-current, (3.36) as (5.106) Jμ = −λAμ which is a four-vector formulation of the London equation (5.5), with λ = −1 . Curiously, Dirac did not mention this connection nor a single word about superconductivity. He went on to show that the energy-momentum four-vector of the electron is (5.107) pμ = e∇μ χ , which is again a four-vector formulation of a relation from superconductivity theory, (5.46). In Dirac’s view, his theory is classical and equal to Maxwell’s equations in the absence of charges if λ = 0 [51]. Dirac published two more papers on his new classical theory of electrons [52, 53] where he related the magnetic vector potential to a charge velocity, Aμ = kvμ .

(5.108)

He found his Hamiltonian to require that even in the absence of charges, a velocity persists and stated the following [52]. One may picture vμ , as the velocity of an aether. There is no contradiction with relativity, because all the equations are Lorentz invariant. A perfect vacuum, in the theory after quantization, should be a region in which all directions for vμ are equally probable, so there is no preferred time-axis.

The Indian physicist K. P. Sinha with collaborators, recognized the similarities of Dirac’s new theory to superfluidity and proposed that the vacuum is a superfluid state of particle and antiparticle pairs [54]. Assuming that the pairs are in the ground orbital state with a zero value of the total spin in the second quantization momentum representation, the Hamiltonian describing a sea of particle-antiparticle pairs is H=



† εk ck↓ ck↓ + εk dk†↑ dk↑ −

k

 k,k

† Vk,k ck†  ↓ d−k  ↓ d−k− ↓ ck↓ . −

(5.109)

The operators c† and d † are the particle and antiparticle creation operators respectively, c and d are the corresponding annihilation operators. This model is very similar to the BCS Hamiltonian (5.60). As in BCS theory, Vk,k is an effective short range pair interaction, but here of an unknown nature. Assuming it is constant, the ground state energy of the superfluid vacuum is lowered by a gap parameter . Any object moving through the superfluid vacuum with energy below the gap parameter will not be able to exchange energy or momentum with it. The excitation energy of the vacuum will be 1 Dimensional analysis reveals that k

of c is missing.

needs to have dimension of ML/QT in (5.104). Perhaps a factor

5.13 Superfluid Vacuum Theory

85

Ek =



2 + εk2 ,

(5.110)

which is the same as (5.64). Sinha et. al further postulated that particle masses arise from interactions and that the gap energy will therefore account for the mass. By making the identification (5.111)

= mc2 and εk = pc, the relativistic energy-momentum relation is obtained, E=



(mc2 )2 + ( pc)2 .

(5.112)

The speed of light is then seen as the critical velocity of the vacuum. Let us recall Maxwell’s mechanical model of molecular vortices, Fig. 2.12. In order to have vortical flow without friction, Maxwell added “idle wheels” between the vortices which he admittedly thought was the most difficult part of his model to explain [48]. Superfluidity of the vacuum would explain the frictionless characteristic. The idea has had a lively revival in recent years in relevance to dark matter [55–57] and observational implications have been proposed [57].

5.14 Gauge Invariance in Superconductivity The question of gauge invariance becomes more pressing in superconductivity theory than in classical electrodynamics, since the observable supercurrent is directly proportional to the magnetic vector potential. By taking a closer look at the London equation (5.5), restated here for convenience Js = −

1 A, 

(5.113)

we see that the transformation A → A + ∇χ changes the supercurrent and so (5.113) is not gauge invariant. Taking the divergence of (5.113) gives ∇ · Js = −

1 ∇ · A. 

For a solenoidal supercurrent or vortex, ∇ · Js = 0 which necessitates the Coulomb gauge, also known as the London gauge, ∇ · A = 0.

(5.114)

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5 Theory of Superconductivity

From the canonical momentum for a multiply connected region, meaning a region with holes like the superconducting ring, we can rewrite (5.46) using (5.47) as vs =

e (∇χ − A) . m

(5.115)

Together with (5.2) and (5.8) we have Js =

1 (∇χ − A) . 

(5.116)

As noted by R. P. Feynman, this is the expression for the current which conserves the probability density of a Schrödinger equation for a pair of particles interacting electromagnetically. Feynman furthermore stated that the phase gradient is just as real and observable as charge density, as both are components of the supercurrent density Js [20]. For a simply connected region, meaning a region without holes, the phase is constant and London’s equation (5.113) is recovered. Let us now explore what happens to (5.114) if we allow longitudinal currents. Using the continuity equation ∂ρs , (5.117) ∇ · Js = − ∂t we obtain

∂ρs 1 = ∇ · A. ∂t 

(5.118)

It then becomes clear, that in order to have a longitudinal current component, the diverge of the magnetic vector potential has to be non-zero. Taking into account our exploration of gauge condition in earlier chapters, it is natural to have ∇ ·A=−

1 ∂φ . c2 ∂t

(5.119)

Inserting into (5.118) reveals the relation ρs = −

1 φ  c2

(5.120)

which can be seen as a complementary to London’s equation [2]. Taken together with (5.113), a four-vector formulation is straight forward with Jμ = (cρ, J), Jμ = −

1 Aμ . 

(5.121)

The problem of gauge invariance in BCS theory was found to be rooted in the assumptions that Cooper pairing consisted of zero-momentum pairs only, which corresponds to the order parameter ↑ ↓  being constant in space. A phonon like

5.14 Gauge Invariance in Superconductivity

87

mode is needed to describe fluctuations in the order parameter [58]. This mode is in essence a Goldstone mode and turns out to be the gauge function χ (5.47). According to Goldstone’s theorem, whenever a continuous symmetry is spontaneously broken massless fields of Goldstone bosons emerge [59]. Phonons in fluids are a special case of Goldstone bosons. The Goldstone boson can be incorporated as a longitudinal mode of a gauge field, but in doing so the gauge field acquires mass. This process is the now famous Higgsmechanism [60], which has its roots in superconductivity theory with the works of Anderson [61] and Nambu [62]. We can use the free energy Lagrangian in the Ginzburg-Landau model (5.26) to illustrate the process in a simplified manner [63]. The third term in (5.26) describing the gauge covariant derivative of the wavefunction can be rewritten with the form ψ = |ψ|eiθ along with (5.47), giving e∗2 2 2 (∇|ψ|) + (∇χ − A)2 |ψ|2 . 2m ∗ 2m ∗

(5.122)

We can further rewrite this term by defining the field

and obtain

 A = A − ∇χ

(5.123)

e∗2 2 2 2 (∇|ψ|)2 + A |ψ| . ∗ 2m 2m ∗

(5.124)

The second term is a mass term, as quadratic terms are interpreted in field theory, analogous to classical kinetic energy 1 p2 = mv2 . 2m 2

(5.125)

By making the transformation (5.123), the Goldstone mode χ has been incorporated into the potential  A and in the process, made the potential become the only mass term. In quantum field theory it is often said that the massless gauge potential A “eats” the Goldstone boson χ and becomes the massive gauge potential  A, thus breaking gauge symmetry. This narrative has been argued as being misleading [64], as it is U(1) rotational phase symmetry that is broken, rather than gauge symmetry. In field theory, the transformation (5.123) is called the unitary or unitarity gauge [65], yet in a recent paper by N. R. Poniatowski, it is emphasised that (5.123) is not a gauge transformation since the field  A is itself gauge invariant and therefore observable [63]. Since the Anderson-Higgs mechanism lies at the intersection of condensed matter and particle physics, mismatches in phrasing are perhaps not surprising. In the spirit of unification, one hopes that an underlying general principle can be clarified and established. Interestingly, we will find hints of this phenomena by exploring the analogy of electromagnetism to hydrodynamics in Sect. 6.2.

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5.15 Hole Superconductivity and the H-Index In recent years, a bibliometric index, the h-index [66], has become very popular— possibly a standard in the evaluation of academic researchers. The man behind the h-index is J. E. Hirch and his reasons for inventing the index, are related to his unconventional theory of hole superconductivity [67]. Neither the h-index nor the theory of hole superconductivity is without criticism [68, 69]. Hirsch himself has even criticised the h-index [70], using the value of his own h-index as an example, for failing to highlight the main body of his research. For the last 30 years he has worked extensively on a theory of hole superconductivity and published well over 100 papers on the subject, that have been sparsely cited, apart from self-citations. Even though he has made controversial statements [71], he has a point in that pairing of holes could play a significant role in unconventional high temperature superconductors. Furthermore, he notes that the Meissner effect might be understood in terms of Alvén’s theorem from magnetohydrodynamics. BCS theory like all theories has its limitations and is not the ultimate theory of all superconductors. Whether the theory of hole superconductivity is significant or not, only time will tell. One thing is for sure, there are many things yet to be discovered in the field of superconductivity.

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Chapter 6

Applications and Analogies

Abstract In this chapter, proposed applications of the magnetic vector potential in telecommunications are reviewed with reference to existing patents. The analogy of electromagnetism with hydrodynamics is then examined in detail and a one to one correspondence between the two field theories is uncovered. The electromagnetic gauge conditions are found to correspond to fluid compressibility, with implications for pressure waves. Inspired by the use of the material derivative in hydrodynamics, we put forth a generalized induction law for the magnetic vector potential. As an interesting example of the analogy at play, we explore the population inversion of two dimensional vortices and nuclear spins.

6.1 Application in Telecommunications In light of the versatile role taken on by the magnetic vector potential in physical theory we now turn to applications based on the magnetic vector potential and its properties. Since its conceptual emergence, the vector potential has mainly been applied as a theoretical device in order to simplify calculations of magnetic field intensity. Having been seen as a non-physical concept by most scientists in the last century, the lack of physical applications of the vector potential is not surprising. In the same way that electromagnetism was first applied to the field of telecommunications, the main application of the magnetic vector potential when looking into published patents is in communication systems. There exist over ten patents for vector potential communication systems, held by at least five unrelated inventors and companies. Curiously, one of the inventors is electrical engineer Harold E. Puthoff who is mainly known for his affiliation with the Central Intelligence Agency (CIA) in Project StarGate, a very unorthodox program researching the possibilities of using psychic phenomena for military purposes, now declassified [1]. According to Ref. [2] the vector potential receiver is a Josephson junction, Fig. 6.1, consisting of two type-I superconductors on either side of a thin electric insulator. The junction is situated in a proper cryogenic environment, shielded both electrically and magnetically with an aluminum/copper shield superposed with mu-metal, a material of very high magnetic permeability. In that way only the potentials are measured. The © Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_6

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Fig. 6.1 Diagrams of a Josephson junction receiver (left) and dual toroidal coil radiator (right) [2]

radiator or transmitter consists of two toroidal wound coils transmitting the vector potential signal along with two conductive plates to cancel out the corresponding electric field, to obtain a pure signal of potentials only. The inventor claims that by proper tuning the electric fields can be cancelled out. With the toroidal winding of a coil, which is a standard solenoid closing in on itself in a loop, the magnetic field is situated only within the toroid having zero intensity outside of it. Due to this fact, toroidal wound coils have been widely applied in electronic circuitry in order to minimize leakage of the magnetic field. Figure 6.2 depicts the magnetic vector potential field for a toroidal wound coil. Even though the form of the field is the same as the equipotential lines in Fig. 4.7, the vector structures are orthogonal, following the current. The claimed radiator does indeed transmit a near-field vector potential signal and by modulating the input current, information can be transmitted via the magnetic vector potential. Since the receiver in this case has to be cooled to temperatures well below −100 ◦ C to invoke a superconducting state, the practicality of this system can be questioned. Two well established researchers, Natalia K. Nikolova, electrical engineer and professor at McMaster University in Canada, along with the late Robert K. Zimmerman, physicist and radio engineer, together hold a patent for a magnetic vector potential communications system. Their method of communication, referring to Fig. 6.3, is as follows. A source signal (13) is modulated to a transmission signal current in an conventional RF transmitter (14), it is then amplified (15) by a current driving amplifier which is matched with a low impedance wave-potential radiator (16). The radiator consists of radial electrical conductors emitting a time varying longitudinally polarized magnetic vector potential field, resulting in zero far field electric and magnetic field strengths. The receiver is a biased fluorescent plasma tube (17) which functions as a DC-RF converter, with the output power being controlled by the received 4-potential wave signal. The output signal is then processed by a conventional RF receiver (18).

6.1 Application in Telecommunications

93

Fig. 6.2 Sketch of the vector potential field (A) of a toroidal current (J), arrows indicate direction of circulation. The magnetic field (B) points into the page inside the right cross section of the toroid and out of the page on the left

Fig. 6.3 Block diagram of the system by Nikolova and Zimmerman [3]

According to the inventors, the main difference and advantage of this system over the conventional one is that little power is needed for transmission, since the power for detection can be provided locally at the receiving end. Also, the strength falls in relation to the inverse of the distance instead of the distance squared as with electric and magnetic fields. One reason for transmitting information with potential signals is secrecy, since it would be harder to tap into the potential signal. Inventor Raymond C. Gelinas has nine registered patents in Europe for detection and modulation of the magnetic vector potential field. The patents are all owned by the American conglomerate company Honeywell International Inc. which has been active for 125 years in various industries such as alarm systems, avionics, aerospace, engineering, military and natural gas. Another Japanese company, Sumitomo Metal Industries, which is currently the world’s third largest steel manufacturer and holds 107 patents in Europe, one of which is a magnetic vector potential based communications system. In their patent, it is

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acknowledged that the received signal is an electric field generated by the propagation of the magnetic vector potential, in accordance with (2.5). Being a mining company one could guess their interest in the system stems from the need for subterranean communications. The author is in no way affirming the functionality of the devices described nor dismissing it but simply reporting on patented applications of the magnetic vector potential. Obtaining a patent is no simple matter and costly as well. Nevertheless many physically impossible devices have been patented. If the magnetic vector potential can be used for communications as proposed, we may be looking at the next breakthrough in the fundamentals of communication systems.

6.2 Hydrodynamics Analogy With the hope of gaining a deeper understanding of the electromagnetic fields and potentials, we now turn to the analogy with hydrodynamics.

6.2.1 Vorticity, Acceleration and Induction In the spirit of Maxwell’s model of molecular vortices, Fig. 2.12, the first step in seeking our analogy is pairing the magnetic field B with the vorticity of a fluid ω, B = ∇ × A, ω = ∇ × v.

(6.1) (6.2)

We will see later on as we dive into the analogy that this is the appropriate pairing of concepts, with the use of dimensional analysis. At the end of this chapter is Table 6.1 which summarizes the analogy. The careful reader may want to refer to it right away. From the analogy of vorticity to the magnetic field we see that the magnetic vector potential, A, can be compared to the fluid velocity, v. That can be justified by the fact that velocity can be understood as momentum per mass whereas the magnetic vector potential is momentum per charge. A natural next step would be to seek an hydrodynamic analogue of the electric field. Recall that the time derivative of the magnetic vector potential gives the electric field, from the potential formulation of Faraday’s law of induction (2.5), E=−

dA . dt

(6.3)

Since the magnetic vector potential is analogous to the fluid velocity, then the electric field is analogous to the acceleration of the fluid. In general, acceleration can occur

6.2 Hydrodynamics Analogy

95

not only with temporal changes of velocity but also when the velocity changes with respect to position. From a vector version of the chain rule we obtain the convective derivative Dv ∂v = + v · ∇v, (6.4) Dt ∂t with the latter term being the non-linear convective acceleration. We can use the convective derivative operator, also known as the total time derivative D ∂ = +v·∇ Dt ∂t

(6.5)

and apply it to Faraday’s law (6.3) obtaining E=−

∂A − v · ∇A. ∂t

(6.6)

As noted in Ref. [4], Maxwell wrote Faraday’s law in this manner in order to account for the motion of the circuit through the surrounding magnetic vector potential field. The convective term v · ∇A can be rewritten by combining the following two gradient vector calculus identities ∇ (A · v) = (A · ∇) v + (v · ∇) A + A × (∇ × v) + v × (∇ × A)

(6.7)

∇ × (v × A) = v (∇ · A) − A (∇ · v) + (A · ∇) v − (v · ∇) A

(6.8)

and

resulting in v · ∇A =

1 ∇(v · A) − A × (∇ × v) − v × (∇ × A) 2  − ∇ × (v × A) + v(∇ · A) − A(∇ · v) .

(6.9) (6.10)

We can then rewrite (6.6) as a force equation of six components, since the electric field is force per charge E = F/q, giving ∂A 1  − q ∇(v · A) ∂t 2 − A × (∇ × v) − v × (∇ × A)

(6.11b)

− ∇ × (v × A)

(6.11c)

F = −q

 + v(∇ · A) − A(∇ · v) .

(6.11a)

(6.11d)

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Only one force term involving the velocity is familiar, from the Lorenz force law (3.13), qv × (∇ × A) = qv × B. (6.12) Furthermore it is the only term containing the magnetic field. Other terms seem to be suggesting different ways for induction, using the magnetic vector potential rather than the magnetic field. Let us analyse each term, beginning with (6.11a). First we have the time derivative of the vector potential times the charge, analogous to Newtons second law ∂v (6.13) F=m ∂t with the vector potential corresponding to velocity. The latter term on the right hand side of (6.11a) is telling us that if a charge or a circuit element move parallel to a vector potential, where either the velocity or vector potential change in magnitude, a force is experienced. Interestingly there exists a controversial generator/motor design known as the Marinov motor or Siberian Coliu which was explained with the force term ∇(qv · A) by Stefan Marinov. Theoretical physicist Wesley [5] agreed with the analysis and noted that the Hooper-Monstein effect, another example of induction with a zero magnetic field, could be explained by it as well. Moving on to the next term, (6.11b), we have qA × (∇ × v) = qA × ω, (6.14) suggesting that a rotation about an axis perpendicular to the magnetic vector potential field induces a force perpendicular to both. As an experiment to realize this behavior, the first thing that comes to mind would be to rotate a conductor about an axis above a toroidal wound coil, with the plane of the coil parallel to the axis of rotation. The force term in (6.11c) is not as easily interpreted, being a rotation of the vector resulting from the cross product of the velocity and vector potential, given their orthogonality. Let us define a new vector c such that c = v × A.

(6.15)

Starting out with a curl free vector potential field and moving orthogonal to it with a constant speed will give us a uniform vector field of c. For the force term to kick in we would need a rotation in the field of c so that ∇ × c = 0. This will be the case if either the velocity or the vector potential have varying magnitude with the movement, Fig. 6.4. For the final two terms in (6.11d) the divergence of both the vector potential and velocity come into play, however with different signs, which make their corresponding force terms anti-parallel. First we have the velocity times the divergence of the vector potential, v(∇ · A). We recognize the divergence of the vector potential to be the gauge condition. Therefore this term is zero in the Coloumb gauge ∇ · A = 0. In

6.2 Hydrodynamics Analogy

97

c(x, y)

Fig. 6.4 Rotation in the field of c with a spatially varying magnitude. The force term − q2 ∇ × c would point into the plane of the paper, parallel to either the velocity of the magnetic vector potential, depending on initial conditions

∇×c c(x + dx, y + dy )

the Lorenz gauge, which we have argued to be the natural gauge to give a physical interpretation to, we have have the force term −

q ∂φ q v(∇ · A) = μ0 ε0 v . 2 2 ∂t

(6.16)

Since the divergence of a vector field is a scalar, this force term would be in the direction of the velocity. Finally, we have the magnetic vector potential times the divergence of the velocity, A(∇ · v). If we were to have an expanding loop conductor in a magnetic vector potential field, resulting in a divergent velocity, then according to the latter term of (6.11d), we would get a force in the direction of the vector potential field itself. The truthfulness of the above statements will have to be verified with experiment. New ways of induction in the absence of magnetic fields would open doors in generator and motor designs. This matter is thus quite significant. Strictly speaking, the analogy with hydrodynamics was not necessary to arrive at the expansion of the convective term for the magnetic vector potential into (6.11a) to (6.11d). We solely applied the convective time derivative, routinely used in the mathematical description of fluid flow. In the following sections we explore the analogy in greater detail.

6.2.2 The Navier-Stokes Equation The convective derivative for a velocity field, as commonly used in hydrodynamics, simplifies to 1  ∂v Dv = − v × (∇ × v) + ∇ v2 . (6.17) Dt ∂t 2 The second term on the left hand side is known as the Lamb vector, named after the applied mathematician Horace Lamb, defined as the cross product of the velocity and vorticity L = v × (∇ × v) = v × ω. (6.18)

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Rewriting the magnetic term of the Lorenz force (3.13), again keeping in mind that E = F/q, we have E = v × (∇ × A) = v × B. (6.19) Thus the Lamb vector corresponds to the electric field, which we found to represent the fluid acceleration. Dimensional analysis is in agreement as can be seen in Table 6.1. The main equation of motion in hydrodynamics is the Navier-Stokes equation 1 Dv = ηf ∇ 2 v − ∇ p (6.20) Dt ρ where η f is the kinematic fluid viscosity, which gives rise to diffusion of momentum and vorticity [6], ρ is the density and p is the pressure. The diffusion term should not have an analog in vacuum electromagnetism, since there is no diffusion within empty space. We could thus expect that an equation describing electromagnetic fields in materials is needed to find an analog to fluid viscosity. This is indeed the case as we will see shortly. Written with the value of the convective derivative in (6.4) inserted, the Navier-Stokes equation reads 1 ∂v + v · ∇v = ηf ∇ 2 v − ∇ p. ∂t ρ

(6.21)

Using (6.17) it can also be written as 1  ∂v 1 − v × (∇ × v) + ∇ v2 = ηf ∇ 2 v − ∇ p. ∂t 2 ρ

(6.22)

We can formulate a Navier-Stokes like equation for the magnetic vector potential, using Ohm’s law J = σE (6.23) where σ is conductivity. However we have then restricted the analysis to Ohmic materials. Adding to (6.6) the gradient of the electric scalar potential for generality, we have DA − ∇φ. (6.24) E=− Dt Combining (6.23) with (3.20) and (6.24) we obtain −

1 DA − ∇φ. ∇2A = − σ μ0 Dt

(6.25)

A well known parameter from magneto-hydrodynamics [7], the magnetic diffusivity, happens to be defined as 1 ηm = . (6.26) σ μ0

6.2 Hydrodynamics Analogy

99

Thus by rearranging, (6.25) becomes DA = ηm ∇ 2 A − ∇φ, Dt

(6.27)

which is anagolous to (6.20). We see that the electric scalar potential is equal to the pressure divided by the density, a parameter known as kinematic pressure, the dimension of which fit within the framework of our analogy, see Table 6.1. By expanding the convective derivative as in (6.21) we have the same form ∂A + v · ∇A = ηm ∇ 2 A − ∇φ. ∂t

(6.28)

One could then use some of the immense literature on tricks for solving the NavierStokes equations to solve (6.27) for the magnetic vector potential. On the other hand the Navier-Stokes equation can be seen as a hydrodynamics version of Ohms law (6.23), with Ohmic materials corresponding to Newtonian fluids in which the stresses are linearly proportional to the strain rate. The theoretical techniques of nonNewtonian fluids, called Rheology, may therefore aid in the study of non-Ohmic materials.

6.2.3 The Continuity Equation and Incompressible Flow From mass conservation we obtain the continuity equation for fluids ∂ρf + ∇ · jm = 0, ∂t

(6.29)

where ρf is the fluid density and jm is the mass flux, jm = ρf v.

(6.30)

The continuity equation in electromagnetism is a statement of the conservation of charge, ∂ρq + ∇ · J = 0, (6.31) ∂t denoting charge density with ρq and the current density with J. As noted by Swinger, a particular example of electric current density is the charge flux vector [8] jq = ρq v.

(6.32)

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6 Applications and Analogies

Now, the continuity equation for fluids can be written in the following equivalent forms ∂ρf Dρf ∂ρf + ∇ · (ρf v) = + v · ∇ρf + ρf ∇ · v = + ρf ∇ · v. ∂t ∂t Dt

(6.33)

If the fluid density ρ f is constant, then the fluid is incompressible and the continuity equation reduces to ∇ · v = 0, (6.34) which is known as the incompressibility condition. An interesting aspect of fluids is that in order for the propagation of sound waves to be possible, the fluid has to be compressible. In our analogy the incompressibility condition would correspond to the Coulomb gauge ∇ · A = 0. (6.35) Using the first approximation to the relationship between the fluid density and pressure ∂ρf p, (6.36) ρf = ∂p along with the equation for the speed of sound in a fluid at a constant entropy cs2 =

∂p , ∂ρf

(6.37)

we can rewrite the continuity equation for fluids as 1 ∂( p/ρf ) 1 + v · 2 ∇( p/ρf ) + ∇ · v = 0. 2 cs ∂t cs

(6.38)

The analogous electromagnetic equation would be 1 1 ∂φ + A · 2 ∇φ + ∇ · A = 0. c2 ∂t c

(6.39)

For homogeneous fluids, ∇( p/ρf ) = 0. In that case (6.38) describes a slightly compressible fluid and is analogous to the Lorenz gauge.

6.2.4 Irrotational Flow and the Velocity Potential For irrotational flow we have ∇ × v = 0.

(6.40)

6.2 Hydrodynamics Analogy

101

According to the curl theorem (see the Appendix) we then have 

 ∇ × v · dS = S

∂S

v · d = 0

(6.41)

and so the velocity can be written as the gradient of a scalar function v = ∇ .

(6.42)

The scalar function is known as the scalar velocity potential. In our analogy it corresponds to the electromagnetic gauge function χ . Irrotational flow corresponds to a magnetic field free configuration, ∇ ×A=0

(6.43)

A = ∇χ .

(6.44)

and so

In the special case of incompressible flow, the velocity potential satisfies Laplace’s equation as seen by inserting (6.42) into (6.34). Pressure is related to the velocity potential as the time derivative of the velocity potential equals the kinematic pressure [9] ∂ p (6.45) = ρf ∂t which is analogous to the electrostatic potential being the time derivative of the phase function ∂χ . (6.46) φ= ∂t Lamb in Ref. [9] interprets the velocity potential as impulsive pressure, not unlike a perturbation in the state of the fluid. The velocity potential satisfies the wave equation which describes spherical sound waves in a compressible fluid ∇2 −

1 ∂ 2 =0 cs2 ∂t 2

(6.47)

where the speed of sound, at constant entropy, is given by  cs =

dp . dρ

(6.48)

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6 Applications and Analogies

The electromagnetic gauge function satisfies the wave equation as well, given the Lorenz gauge (3.22), which is furthermore invariant under gauge transformations if the wave equation for χ holds [10] ∇2χ −

1 ∂ 2χ = 0. c2 ∂t 2

(6.49)

This is the hint of Goldstone modes we referred to in Sect. 5.14.

6.2.5 Table of Comparison In the comparison of concepts we can see the dimension of the electromagnetic concepts is equal to those in hydrodynamics by a factor of the ratio of mass to charge, which is Dirac’s constant from (5.105). This holds for the first five rows of the table. However when it comes to mass itself in the hydrodynamical concepts containing mass, one has to divide by the ratio in order to obtain the charge. The author feels that the analogy may shed some light on the relationship of mass to charge in general. This is put forth merely as a notion of intuition. Having seen how fluid vorticity and the magnetic field have analogous mathematical and even physical behavior, both stemming from angular momentum, we now move on to explore an interesting theoretical problem in the theory of turbulence, Table 6.1.

Table 6.1 Analogous concepts in hydrodynamics and electromagnetism. Key: Concept [symbol: Dimension in SI]. The dimension is presented in terms of the basic quantities mass (M), length (L), time (T) and charge (Q) Hydrodynamics Electromagnetism Velocity [v: L/T] Vorticity [ω: 1/T] Acceleration [L: L/T2 ] Kinematic pressure [P =

Magnetic vector potential [A: ML/TQ] Magnetic field [B: M/TQ] Electric field [E: ML/T2 Q] Electric scalar potential [φ: ML2 /T2 Q]

Velocity potential [ : Mass [m: M] Fluid density [ρf : M/L3 ] Mass flux [jm : M/TL2 ]

Phase function [χ: ML2 /TQ] Charge [q: Q] Charge density [ρq : Q/L3 ] Current density [J: Q/TL2 ]

p 2 2 ρ : L /T ] 2 L /T]

6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain

103

6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain In order to explain the stability of large scale vortices, such as the giant red spot on Jupiter and in weather systems here on earth, physicist Lars Onsager turned to a controversial concept in statistical mechanics, that of statistically defined negative absolute temperatures. Let us shortly introduce the thermodynamics involved.

6.3.1 Entropy In the early days of thermodynamics, the concept of entropy surfaced as a means of quantifying the degradation of available energy of a system due to frictional forces. Rather confusingly, the word was initially also used to refer to the energy that could be converted into mechanical work [11]. The former definition became the standard with the change of entropy of a closed system being defined as dS =

dQ T

(6.50)

where d Q is a small amount of heat entering the system at a temperature T . Our modern day version comes from Stefan Boltzmann, who put forth the statistical definition of absolute entropy of an ensemble of microstates in the late 19th century [12] S = kB ln(W ) (6.51) where kB is the Boltzmann constant and W is the number of microstates available to the system at a certain temperature, if all states are equally probable. If we denote the probability of state j with p j then (6.51) takes the form S = −kB



p j ln( p j ).

(6.52)

j

Shannon put forth an analogous formula for the information content of a message [13]. Let p j be the probability of the message m j from the message space M. Then the average information, H , is given by H =−



p j log2 ( p j ).

(6.53)

j

For a set of equally probable elements that can carry a certain message or information, the uncertainty in the message is given by H = log2 ()

(6.54)

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6 Applications and Analogies

where  is the number of permutations of the elements m i , given by the multinomial coefficient or the binomial for a two-level system. Base 2 logarithm is used since information is conventionally measured in bits. Nonetheless the natural logarithm as in (6.51) could be used, the unit of information would then be nat. Equations (6.51) and (6.54) have the exact same structure. The Boltzmann entropy can thus be seen as information expressed in nats of energy at a given temperature. Shannon proposed that the concepts were equal since with increased entropy, due to more available states, the information content would be greater. Let’s take a trivial example to illustrate this line of thought. Imagine a book containing only the letter “X”. Each character has only one state available, therefore the entropy is zero. Given a single character we can say for sure what the next one will be. The information content of the book is zero as well, we cannot code any message in a book containing only one equally spaced character. In this way Shannon argued that thermodynamical entropy can be understood as information. The second law of thermodynamics states that the entropy of any given irreversible process in a closed system will increase, including the universe as a whole, given it is a closed system. In light of information theory, this means that both the information content and possible states of the universe are constantly increasing. This might shed light on the increasing complexity of life and even the technology of man. Another viewpoint, from Jaynes, was that both Shannon’s information and Boltzmann entropy can be understood as uncertainty [14]. With more possibilities, the more uncertain is the unfolding.

6.3.2 Negative Absolute Temperature Before we can appreciate Onsager’s approach to long lived vortices we have to see how (6.51) allows for a statistical definition of negative absolute temperatures. Let us theorize a four particle system, each with two possible states where one state is of higher energy. Let us symbolize the system at rest with four white dots and the system fully excited with four black dots. For the macrostate of one particle being excited, we have four available microstates • ◦ ◦◦, ◦ • ◦◦, ◦ ◦ •◦, ◦ ◦ ◦ •. The entropy of the system would then be S1 = kB ln(4). In the same way for two excited particles we would have six available states; • • ◦◦, • ◦ •◦ and ◦ • •◦ along with the inverses, as calculated by the binomial coefficient, with entropy increasing to S2 = kb ln(6), see Table 6.2. When we move on to three excited particles of four total then we have more particles in a state of higher energy than in a state of low energy. By this step entropy has decreased even though we increased the energy of the system, Fig. 6.5. This happens only because the number of particles (states) are limited.

6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain

105

Table 6.2 Number of excited particles in a two level system (N), their corresponding available states (W) and entropy (S) there of N W S 0 1 2 3 4

1 4 6 4 1

S0 S1 S2 S3 S4

= kB ln(1) = 0 = kB ln(4) = kB ln(6) = kB ln(4) = kB ln(1) = 0

In our daily life, energy can always migrate out into the surrounding space having unlimited degrees of freedom. Temperature is defined in terms of the change of entropy with respect to energy dS 1 = =β (6.55) T dE where E is the energy and the inverse of temperature is β which will be the slope of the energy-entropy graph, Fig. 6.5. The slope of the energy-entropy graph is zero when we reach the state of maximum entropy at an equal distribution of energy states over the energy levels. The corresponding energy-temperature graph has a vertical asymptote at E = 0, for arbitrary units. Coming to the point of maximum entropy, the temperature therefore runs from +∞ to −∞ and so we have a negative absolute temperature of the system. In the literature this is called population inversion. In spite of how exotic the notion of negative absolute temperature seems, the concept has been extensively applied in Lasers (Light Amplification by Stimulated Emission of Radiation) for more than half a century. For stimulated emission to occur, there has to be a population inversion of electron states in the active medium of the laser [15].

Entropy

Temperature

Smax

Emin

Energy

Emax Emin

Energy

Emax

Fig. 6.5 Energy-entropy graph of a limited system and corresponding energy-temperature graph

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6 Applications and Analogies

6.3.3 Onsager’s System of Parallel Vortices in a Plane Large long-lived vortices are rather common phenomena in planetary atmospheres, Fig. 6.6. The giant red spot on Jupiter is considered to be a storm cyclone that has existed at least since the 17th century when it was first observed. Normally, vortices break down into smaller ones which is termed forward cascade of energy. In the formation and persistence of large isolated vortices, inverse-cascade is displayed as smaller vortices cluster together into a single large vortex. Onsager took on this problem in turbulence, continuing the work of the Soviet mathematician A. Kolmogorov and published a paper on statistical hydrodynamics [18], shortly after Shannon’s work of relating entropy and information. He presented a model consisting of n parallel vortices of intensities or circulations κ1 , ..., κn defined as  κ = v · d, (6.56) in a frictionless and incompressible fluid. The system has limited n-degrees of freedom and the equations of motion may be written as κi d xi /dt = ∂ H/∂ yi , κi dyi /dt = −∂ H/∂ xi ,

(6.57)

where H is the fluid energy Hamiltonian H =− ri2j

1 κi κ j ln(ri j ), 2π i> j

(6.58)

= (x j − xi ) + (yi − y j ) . 2

2

Onsager refers to a series of papers by C.C. Lin where the Hamiltonian for vortex motion in a multiply connected region is found in terms of a Green’s function for the Laplacian operator [18]. Now if the said finite system of vortices is confined to

Fig. 6.6 Two examples of long-lived large scale vortices: A Storm maturing off the east coast of the US [16], lifetime in days and the giant red spot on Jupiter as seen by Voyager [17], lifetime in centuries

6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain

107

Fig. 6.7 Clustering of vortices after the point of population inversion (dashed vertical line) with an emerging dipolar structure. Reprinted from Yuichi Yatsuyanagi, Journal of Plasma and Fusion Research Series, Vol. 8, p. 932, 2009, with permission from JSPF

an area A, the phase-space is finite. Furthermore the phase-space is identical to the configuration space since the coordinates x and y are interchangeable. The behavior of the system is analogous to our hypothetical two-level system since the vortices have either clockwise or anti-clockwise spin. The energy-entropy and energy-temperature relations are therefore described by the graphs in Fig. 6.5 [19]. Above an equal energy distribution over all possible states, the system has a negative statistical temperature and the entropy (the number of possible states) will decrease with added energy. This results in the clustering of vortices of the same orientation. The result has been observed in numerical simulations [19]. In a system of vortices with both spin orientations, a dipolar structure forms from the ensemble of vortices, Fig. 6.7. Onsager was well aware that his model was highly idealized and stated that “it must obviously be taken with a grain of salt” [18]. Though still controversial to this day, statistical negative absolute temperatures have been experimentally realized in various settings, one of which supports the hydrodynamics analogy with electromagnetism covered in the preceding section. We close this section by briefly introducing a few experiments for the readers referral.

6.3.4 Experiments The predicted phenomena of vortex clustering in a bounded domain was recently observed experimentally in the turbulence of an atomic Bose-Einstein condensate, stirred with a focused repulsive Gaussian laser beam, seventy years after Onsager’s paper [20]. A few years before the experiment, Onsager’s model was extended for a realistic superfluid and experiments proposed for atomic Bose-Einstein condensate [21]. In light of the analogy of the magnetic field to the vorticity, we may ask if a system similar to Onsager’s two dimensional vortex model exist within the realm of electromagnetism. Indeed the system of nuclear spins in a plane, known as the Ising

108

6 Applications and Analogies

model, provides a near exact analogy. For an ensemble of unpaired electron spins in a magnetic field we obtain a two-state system where the magnetic moment of the electron can either be aligned or anti-aligned to the surrounding magnetic field. Furthermore, since the number of particles is finite, the phase space and number of states is finite as well, giving the possibility of both population inversion and statistical negative absolute temperatures, Fig. 6.8. Interestingly, the exact solution of the two-dimensional Ising model was obtained by none other than Onsager, five years before his paper on statistical hydrodynamics [22]. Curiously, there is little written on the possibility of a population inversion of the spin states in that paper. However, shortly after the publication, result were published from an experiment with a paramagnetic LiF crystal, whereby the external magnetic field is reversed in a time lesser than the spin relaxation time, creating a population inversion in the states of the nuclear spins [23]. It has also been reported that silver shows different magnetic behaviour depending on spin temperature, being anti-ferromagnetic for positive spin temperatures but ferromagnetic for negative spin temperatures [24]. A new cooling method has been proposed with the application of negative effective spin temperatures of isolated spin distributions [25]. It has been theorized that a negative absolute temperature can be obtained for motional degrees of freedom of atomic gases by reversing the confining potential of cold atoms in optical lattices [27, 28]. This was experimentally realized in 2003 along

Fig. 6.8 Population inversion of nuclear spins in a external magnetic field (B) leading to negative absolute spin temperatures [26]

6.3 Population Inversion of Two Dimensional Vortices in a Finite Domain

109

with the fact that negative absolute temperatures lead to negative pressure and may therefore be applicable in the description of dark energy in cosmology [29]. In spite of these experiments and the theoretical background supporting them, the concept of statistical negative absolute temperatures is still being met with heavy criticism. Recently the mathematics of the matter was covered rigorously and arguments against the validity of the concept refuted [30].

6.4 Future Directions We close this final chapter with a few remarks about the future outlook for the concepts presented in the totality of this treatise. First and foremost, as has been argued, electromagnetic theory should be presented, taught and applied in terms of the electric scalar potential and the magnetic vector potential, with the electric and magnetic fields emerging as consequences of temporal or spatial variations in the potentials. Their natural conceptual interpretation as the momentum and energy of charges along with the simplification of the geometry of interactions should suffice as support for this view. On top of that, the potentials also lay down a nice pathway to introduce more advanced concepts of modern physics such as quantization, gauge fields and symmetry in the course of physics education. By the strong analogy with hydrodynamics, the mathematical and physical structure of both fields can be clarified. Experiments in electromagnetism could be undertaken with the analogue hydrodynamical setup. The analogy can also be extended to seismic waves in the curriculum for geophysics. Regarding the mathematical description of electromagnetism in general, it is well worthwhile to get acquainted with the Clifford geometric calculus formulation, see for example Ref. [31], as it is a more revealing mathematical language than traditional vector calculus. Another interesting approach emerging is the application of topology to anomalous electromagnetic effects [32], explained by the magnetic vector potential acting in setups with higher group symmetries. The powerful notion of population inversion applied to nuclear spins might open doors to new types of thermodynamic work cycles. Some authors [29] claim that negative absolute temperatures lead to over-unity Carnot cycles, however this theoretical implementation has subtle specifics [30] and the law of conservation of energy of course holds as before. No matter the means of energy usage, an energy source is needed. Our best bets still seems to be the sun’s nuclear fusion or the earth’s nuclear fission and resulting processes. The quest for new frontiers in energy science would however most certainly be aided by understanding the nature of the electromagnetic vacuum. In the authors view, the most significant part of the work is the factorization of the convective derivative of the magnetic vector potential resulting in the force terms in (6.11a) to (6.11d). These could in theory lead to new techniques of current induction without magnetic fields, by the movement of particles within the magnetic vector potential field, possibly practical in motor and generator design.

110

6 Applications and Analogies

References 1. H. Puthoff, R. Targ, Central Intelligence Agency (CIA) (1974), https://www.cia.gov/library/ readingroom/document/cia-rdp96-00787r000100220005-4 2. H.E. Puthoff, Communication method and apparatus with signals comprising scalar and vector potentials without electromagnetic fields (1998) 3. N.K. Nikolova, R.K. Zimmerman, Electromagnetic wave-potential communication system (2012) 4. J.D. Jackson, L. Okun, Rev. Modern Phys. 73(3), 663–680 (2001) 5. J.P. Wesley, Apeiron 5(3–4), 219–226 (1998) 6. J.R. Holton, G.J. Hakim, Dynamic Meteorology, 5th edn. (Academic Press, Cambridge, MA, 2013) 7. W. Baumjohann, R.A. Treumann, Basic Space Plasma Physics (Imperial College Press, London, 1997) 8. J. Schwinger, L.L. Deraad, K. Milton, W. Yang Tsai, J. Norton, Classical Electrodynamics (Perseus Books, MA, 1998) 9. H. Lamb, Hydrodynamics (C.J. Clay and Sons, London, 1895) 10. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999) 11. J.C. Maxwell, Theory of Heat (Longmans, Green and Co., London, 1871) 12. J.S. Dugdale, Entropy and its Physical Meaning (Taylor & Francis, London, 1996) 13. C.E. Shannon, Bell Sys. Tech. J. 27(3), 379–423 (1948) 14. E.T. Jaynes, Phys. Rev. 106(4), 620–630 (1957) 15. R.W. Waynant, M.N. Ediger, Electro-Optics Handbook (McGraw-Hill, 2000) 16. NOAA, GOES (2018), https://www.nasa.gov/image-feature/geocolor-image-from-noaasgoes-16-satellite-of-powerful-east-coast-storm 17. K.M. Gill, JPL-Caltech, NASA (2018), https://www.flickr.com/photos/kevinmgill/ 26008230898/ 18. L. Onsager, Nuovo Cimento 6(2), 279–287 (1949) 19. Y. Yatsuyanagi, J. Plasma Fusion Res. Ser. 8(7), 931–935 (2009) 20. S.W. Seo, B. Ko, J.H. Kim, Y. Shin, Scienti. Rep. 7, 4587 (2017) 21. T.P. Billam, M.T. Reeves, B.P. Anderson, A.S. Bradley, Phys. Rev. Lett. 112(14), 145301 (2014) 22. L. Onsager, Phys. Rev. 65(3–4), 117–149 (1944) 23. E.M. Purcell, R.V. Pound, Phys. Rev. 81(2), 279–280 (1951) 24. P.J. Hakonen, K.K. Nummila, R.T. Vuorinen, O.V. Lounasmaa, Phys. Rev. Lett. 68(3), 365–368 (1992) 25. P. Medley, D.M. Weld, H. Miyake, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 106(19), 195301 (2011) 26. S. Vrtnik. Spin Temperature (Seminar). University of Ljubljana, Faculty of Mathematics and Physics (2005), http://mafija.fmf.uni-lj.si/seminar/files/2004_2005/Spin_Temperature.pdf 27. A.P. Mosk, Phys. Rev. Lett. 95(4), 040403 (2005) 28. A. Rapp, S. Mandt, A. Rosch, Phys. Rev. Lett. 105(22), 220405 (2010) 29. S. Braun, J.P. Ronzheimer, M. Schreiber, S.S. Hodgman, T. Rom, I. Bloch, U. Schneider, Science 339(6115), 52–55 (2013) 30. E. Abraham, O. Penrose, Phys. Rev. E 95(1), 012125 (2017) 31. D. Hestenes, Space-Time Algebra (Springer International Publishing, Birkhäuser Basel, Switzerland, 2015) 32. T.W. Barrett, Topological Foundations of Electromagnetism (World Scientific, Singapore, 2008)

Chapter 7

Final Words

In this treatise we have covered the historical, physical and mathematical role of the magnetic vector potential. Starting out as the electrotonic state, an elusive state of matter envisioned by Faraday in order to explain electromagnetic induction, to being understood as a vector potential for the non-conservative magnetic field by Maxwell. Then forgotten and dismissed as unphysical due to gauge invariance, only to be rediscovered in the theoretical development of quantum electrodynamics. Regaining status as a physical phenomena, though controversially, with the Aharanov-Bohm experiment and at the same time evolving to being understood as a gauge field. In analogy with hydrodynamics and by dimension analysis, the magnetic vector potential takes on the sober role of momentum per charge. The magnetic field then arises from circulation of charge, as Faraday and Maxwell had argued. By plotting the field for both a linear and loop conductor, we have seen the natural relation of the magnetic vector potential to the current and how the equipotential surfaces of the field give rise to the structure of the magnetic field. The magnetic vector potential is truly a rich concept with a colorful history, perhaps still yet to be fully comprehended. To the reader who feels a subtle sense of intuition having been stirred up, I encourage you to investigate further. Our intuition can be said to be paradoxical, since it is telling us that “we might know more than we think we know”. Even though the development of science is most often put forth as having been steady linear stepby-step progression, original ideas often come by conceptual leaps of intuition. In order to be able to learn, we must admit that we do not know nor understand. The one who knows and understands everything has nothing to learn. All to often in the history of science and other fields, have humans carried invalid assumptions between generations. Until something is discovered or invented, it remains hidden and might as well be non-existent. Though the path to discovery may be complex and difficult, the gained knowledge can turn out to be obvious in hindsight, leaving one to wonder why no one thought of it before. On the other hand, highly original work is most often with a complete lack of understanding and said to be ahead of its time, since it can take half a century or more for the community to catch up. Progress is thus also © Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3_7

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7 Final Words

a matter of timing, with the right ideas being put forth and appreciated at the right time. With the great accessibility to both education and scientific literature in modern day and age, along with rapid progress in many areas of science, it looks as if the time is ripe for a new paradigm. The author’s hope has been that by thinking in terms of the magnetic vector potential, a shift in focus might come by, leading to a fruitful approach. Mankind is now in desperate need of new solutions in transportation and energy in order to become independent of fossil fuels. Even though there are many who have financial interest in the way this branch of science will develop, research will still, first and foremost, be driven by the spark of curiosity igniting the wonderful guiding force of inspiration.

Appendix

The following are special cases of what is known in the literature as Stokes’ Theorem, f denotes a continuous function defined on the interval (a,b), F denotes a vector field while ∂ S and ∂ V denote the boundary of the surface S and volume V respectively. Fundamental theorem of calculus ˆ b f  (x) d x = f (b) − f (a) a

Gradient theorem ˆ

b

∇F · d = F(b) − F(a)

a

Curl theorem ˛

¨ ∇ × F · dS =

∂S

S

F · d

Divergence theorem ‹

˚ ∇ · F dV = V

∂V

F · dS

These theorems can be combined and generalized to n-dimensions using differential forms1 along with the exterior derivative. Let ω be a differential form defined on the manifold M of dimension k with dω being its exterior derivative of dimension k + 1. 1 For a thorough treatment of differential forms, Clifford algebra and geometric calculus refer to the

book Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics by David Hestenes and Garret Sobczyk. © Springer Nature Switzerland AG 2020 K. Ó. Klausen, A Treatise on the Magnetic Vector Potential, https://doi.org/10.1007/978-3-030-52222-3

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Appendix

Generalized Stokes theorem ˛

ˆ dω = M

∂M

ω

This generalization is still only half of the story. In order to see why we need two fundamental concepts of generalized vector calculus.

Geometric algebra and calculus In geometric algebra, also known as Clifford algebra, one can formulate the complete geometric product ab = a · b + a ∧ b where the center dot denotes the symmetric inner product and the operator ∧ named wedge denotes the anti-symmetric outer product. The inner product is dimension-reducing in the sense that for vectors a and b their inner product is a scalar of zero dimensions. The magnitude of the inner product is based on the familiar cosine of the angle between the two vectors |a · b| = |a||b| cos(θ ) while the magnitude of the outer product rests on the sine of the angle, being interpreted as the area of the parallelogram spanned by a and b, |a ∧ b| = |a||b| sin(θ ). The outer product is a generalization of the cross product to n-dimensions yet still different in three dimensions. For two orthogonal vectors a and b the cross product a×b=c is a third vector, with magnitude and direction, orthogonal to both a and b while the wedge product a ∧ b = cˇ is a two dimensional bivector lying in the plane spanned by a and b, having magnitude and area, Fig. A.1. The watchful reader should be getting a hunch on how this subtle difference influences geometrical interpretation. As a sidenote, the complex number i can be understood as a unit bivector through the eyes of geometric algebra. The interested reader can refer to the paper Imaginary Numbers are not Real—the Geometric Algebra of Spacetime by Stephen Gull, Anthony Lasenby and Chris Doran. With the outer product one can then generate higher dimensional geometric forms such as three dimensional trivectors d˜ = a ∧ b ∧ c

Appendix

115

Fig. A.1 The vectors a and b span the plane of the bivector cˇ . The vector c is orthogonal to the plane and dual to the bivector

c=a×b

a b

cˇ = a ∧ b

and in general n-vectors (known as blades in Clifford algebra) which are the building blocks of geometric algebra along with linear combinations thereof named multivectors. Multivectors can be said to generalize the concept of number to higher dimensions. For a differentiable multivector field F defined on a set M in Rn with an orthonormal basis ei we can define the gradient operator such that ∇ F = ei ∂i F = e1 ∂1 F + e2 ∂2 F + · · · + en ∂n F , using the shorthand notation ∂i = ∂∂i . The operator ∇ = ei ∂i behaves as a vector algebraically, therefore the gradient above can be factored into two components using the geometric product since aB = a · B + a ∧ B also holds for a vector a and a multivector B [1]. Thus ∇F = ∇ · F + ∇ ∧ F where ∇ · F is the divergence of the field F and ∇ ∧ F is the generalized curl or rotation of the field. In the same way as noted before the generalized curl is different from the one in standard literature (which exists only in three dimensions) as here it is a bivector; signifying the plane that the standard curl vector is orthogonal to.

The boundary theorem Let M be a smooth n-dimensional oriented manifold with a piece-wise smooth boundary ∂M of dimension n − 1 where n is any positive integer greater or equal to one. A directed element of the manifold M is a differential n-vector dxn and similarly dxn−1 for the boundary. Then ˛

ˆ M

dxn ∇ F =

∂M

dxn−1 F ,

meaning that a change in the field F within M will effect the field on the boundary and vice versa. At its root the boundary theorem is stating conservation of information.

116

Appendix

With the geometric product we can split the boundary theorem into two separate components ˛ ˆ M

and

dxn · (∇ ∧ F) =

(A.1)

dxn−1 ∧ F

(A.2)

˛

ˆ M

∂M

dxn−1 · F

dxn ∧ (∇ · F) =

∂M

The first part is equivalent in information content to the generalized Stokes theorem for differential forms [2]. The second part is the orthogonal counterpart which completes the story. The theorem strongly reminds one of the so called AdS/CFT correspondence in physics [3] with its holographic duality.

References 1. A. Macdonald, Vector and Geometric Calculus (CreateSpace Independent Publishing Platform, 2012) 2. D. Hestenes, in Clifford Algebras and Their Applications in Mathematical Physics (NATO ASI series) (Springer, Switzerland, 1986) 3. H. N˘astase, Introduction to the AdS/CFT Correspondence (Cambridge University Press, Cambridge, 2015)