There are physical phenomena in everyday life that are taken for granted simply because the explanation of their behavior closely matches the expectations of the observer. For some of these phenomenon, an extensive body of theoretical knowledge exists which matches the experimental observations. The electromagnetic force is one of these phenomenon. The observer can envision empty space filled with electromagnetic waves, and describe these waves and their effects on matter with mathematical precision. Devices can be constructed, based on electromagnetic theory, that confirm our belief that the electromagnetic phenomena are well understood— that is, observations are produced consistent with expectations. With further investigation new questions arise, requiring a reformulation of the theory which supports these observations.

1. A MATHEMATICAL TOUR
OF THE
ELECTROMAGNETIC FORCE
from
Michael Faraday to Quantum Field Theory
There are physical phenomena in everyday life that are taken for granted
simply because the explanation of their behavior closely matches the
expectations of the observer. For some of these phenomenon, an extensive
body of theoretical knowledge exists which matches the experimental
observations. The electromagnetic force is one of these phenomenon. The
observer can envision empty space filled with electromagnetic waves, and
describe these waves and their effects on matter with mathematical precision.
Devices can be constructed, based on electromagnetic theory, that confirm our
belief that the electromagnetic phenomena are well understood — that is,
observations are produced consistent with expectations. With further
investigation new questions arise, requiring a reformulation of the theory
which supports these observations.
The classical electromagnetic field is described by Maxwell’s equations. From
these equations much of the material world can be analyzed. At physical sizes
below molecules Maxwell’s description of nature becomes unusable. A
quantum mechanical description of the electromagnetic field is required.
Such a description is provided by Quantum Electrodynamics. Starting with
the classical description of the radiated electromagnetic field, this book makes
use of a simple human experience — the receipt of radio signals — to explore
the mathematical foundations of the electrodynamics.
Starting with the earliest experiments in electrostatics, Faraday, Maxwell and
Hertzian formulations of the radiated field are described. The theory of
antennas and electromagnetic reflection and refraction are explored. All of
this material is a prelude to the quantum mechanical description of the
electromagnetic field and its interaction with matter. In this description, the
quantized field interacts with charged particles through the exchange of a
particle which carries the electromagnetic force through free space — the
photon. The behavior of this interaction at the quantum mechanical level
provides new insight to the complexities of nature.
G. B. Alleman
Niwot, Colorado
Copyright © 2001

2. Why Study Mathematics?
I am amused, I said, at your fear of the world, which makes you guard against the
appliance of insisting upon useless studies; and I quite admit the difficulty of
believing that in every man there is an eye of the soul which, when by other
pursuits lost and dimmed, is by there purified and re–illuminated and is more
precious for then ten thousand bodily eye, for by it alone is truth seen.
— Socrates to Glaucon in Plato’s Republic Book VII
Physics ... is essentially an intuitive and concrete science. Mathematics is only a
means for expressing the laws that govern phenomena.
— Einstein to Solovine in [Solo79]

3. Table of Contents
§0. PREFACE.............................................................................................................1
§0.1 Guided Preview..........................................................................................2
§0.2 Advise to the Reader.................................................................................5
§0.3 Historical and Mathematical Endnotes................................................5
§0.4 Electrodynamics Notation.......................................................................6
§1. THE FOUR FORCES OF NATURE .....................................................................1–1
§1.1 The Everyday Force of Gravity ..............................................................1–1
§1.2 Early Astronomy — The History of Theory.........................................1–4
§1.3 The Four Forces of Nature ......................................................................1–7
§1.4 The Particle Zoo.........................................................................................1–9
§1.5 Fundamental Forces in Quantum Chromodynamics........................1–13
§1.6 Quantum Field Theory.............................................................................1–15
§1.7 Preliminaries to Modern Physics...........................................................1–16
§1.8 Unifying Principals of Nature ................................................................1–17
§2. CLASSICAL FIELD THEORY..............................................................................2–1
§2.1. Electrodynamics ........................................................................................2–2
§2.2. Electrostatics and Early Experiments..................................................2–2
§2.3. Electromagnetic Interactions.................................................................2–6
§2.4. Unifying Electricity and Magnetism.....................................................2–7
§2.4.1. Lines of Force .......................................................................................2–9
§2.4.2. Beginnings of Field Theory...............................................................2–12
§2.4.3. Removal of Action at a Distance.......................................................2–13
§2.5. Special Relativity and Electromagnetic Fields...................................2–14
§2.6. Light — Particle or Wave........................................................................2–16
§2.7. Overview of the Wave Equation.............................................................2–18
§3. MAXWELL'S EQUATIONS..................................................................................3–1
§3.1. Maxwell's 1st Equation — Coulomb's Law...........................................3–3
§3.2. Maxwell's 2nd Equation — Absence of Magnetic Monopoles...........3–5
§3.3. Ampère's Law for Steady State Fields...................................................3–5
§3.4. Maxwell's 3rd Equation — Ampere's Law............................................3–7
§3.5. Maxwell's 4th Equation — Faraday's Law of Induction ...................3–9
§3.6. Newton–Lorentz Force Equation...........................................................3–11
§3.7. Coupling Strength of the Electromagnetic Field................................3–13
§3.8. Continuity Equations...............................................................................3–16
§3.9. Summary of Maxwell’s Equations..........................................................3–17
§4. SOLUTIONS TO MAXWELL'S EQUATIONS ......................................................4–1
§4.1. Vector Algebra Solution to Maxwell’s Equations...............................4–1
§4.2. Vector Potential Solution to Maxwell’s Equations.............................4–1

4. Table of Contents
§4.3. Integral Form of Maxwell's Field Equations.......................................4–4
§4.3.1. Green's Function and the Potential Solution................................4–4
§4.3.2. Field Potential Solutions...................................................................4–6
§4.4. Traveling Waves........................................................................................4–8
§4.4.1. Displacement Current in the Field Equations..............................4–11
§4.5. Classical Explanations for Force from Fields......................................4–11
§4.6. Summary of Classical Field Theory ......................................................4–12
§5. THE RADIATED FIELD......................................................................................5–1
§5.1. Plane Waves in Free Space.....................................................................5–2
§5.1.1. Longitudinal Propagation Components.........................................5–4
§5.2. Energy in the Radiated Field .................................................................5–4
§5.3. Poynting's Theorem..................................................................................5–5
§5.4. Vector Potential Description of the Radiated Field............................5–7
§5.4.1. Quasi–Stationary Expansion............................................................5–8
§5.4.2. Multipole Expansion...........................................................................5–9
§5.4.3. Radiation Expansion..........................................................................5–9
§5.5. Polarization of the Radiated Field.........................................................5–10
§6. ANTENNAS AND RADIATED FIELDS ...............................................................6–1
§6.1. Time-Dependent Fields in Conductors .................................................6–3
§6.1.1. Wave Propagation in a Conduction Media....................................6–5
§6.2. Electromagnetic Waves Incident on a Conductor..............................6–9
§6.3. Summary of Maxwell's Classical Field Theory ...................................6–14
§7. PRINCIPLE OF RELATIVITY..............................................................................7–1
§7.1. Origins of Relativity Theory ...................................................................7–2
§7.1.1. Invariance of Newton’s Equations of Motion.................................7–3
§7.2. Velocity of Light and Absolute Motion .................................................7–4
§7.3. The Lorentz Transformation ..................................................................7–11
§7.3.1. The Components of Relativity ..........................................................7–12
§7.3.2. Relativity Principals Formally Stated............................................7–12
§7.3.3. Structure of Space–Time....................................................................7–17
§7.4. Covariant Notation...................................................................................7–19
§7.4.1. Covariant Transformations...............................................................7–20
§7.4.2. Divergence and Curl in 4 Dimensions............................................7–22
§7.5. Lorentz Transformation in Covariant Form.......................................7–22
§7.6. Maxwell's Equations in 4–Dimensions.................................................7–23
§7.7. Lorentz Transformation of Maxwell’s Equations...............................7–27
§8. HAMILTONIAN MECHANICS.............................................................................8–1
§8.1. Newton’s Equations in Lagrangian Form............................................8–1
§8.2. Variational Description of the Equations of Motion ..........................8–3

5. Table of Contents
§8.3. Calculus of Variations..............................................................................8–6
§8.4. Ordinary Maximum and Minimum Theory.........................................8–7
§8.4.1. Lagrangian Formalism and the Calculus of Variations.............8–8
§8.5. Generalized Coordinates..........................................................................8–10
§8.6. Hamiltonian Formalism...........................................................................8–11
§8.6.1. Canonical Coordinates and Poisson Brackets...............................8–14
§8.7. Standard Lagrangian of Classical Electrodynamics .........................8–16
§8.7.1. Time Independent Lagrangian........................................................8–18
§8.7.2. Lagrangian Density............................................................................8–19
§9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD .......................................9–1
§9.1. Field Energy Density................................................................................9–4
§10. A PREVIEW OF QUANTUM MECHANICS.........................................................10–1
§10.1. Domination of Modern Quantum Mechanics.......................................10–1
§10.2. Early Quantum Theory............................................................................10–2
§10.3. Experimental Necessity for the Quantum Theory of Radiation......10–3
§10.3.1. Black Body Radiation.........................................................................10–4
§10.4. States of a Mechanical System ...............................................................10–6
§10.5. Quantum Mechanics of Electromagnetic Fields.................................10–7
§10.6. Preliminaries to Quantizing the Radiation Field...............................10–8
§10.6.1. Vector Potential Expanded as a Fourier Series............................10–8
§10.6.2. Planck’s Conclusions Using the Vector Potential.........................10–10
§10.7. Radiation Field Expansion Using Canonical Variables...................10–12
§10.8. Schrödinger’s Equation............................................................................10–13
§10.8.1. Development of Schrödinger’s Equation ........................................10–14
§10.9. Formulating Schrödinger’s Wave Equation........................................10–16
§10.10.Schrödinger’s Time Dependent Equation ............................................10–17
§10.10.1. The General Solution to Schrödinger’s Equation.............10–18
§10.10.2. Semi–Classical Theory of Radiation ...................................10–18
§11. GAUGE THEORY.................................................................................................11–1
§11.1. Classical Mechanics Example of a Gauge Invariance.......................11–2
§11.2. Electromagnetic Fields and Gauge Transformations........................11–3
§11.3. Lorentz and Coulomb Transformations...............................................11–8
§11.4. Gauge Symmetries and Potential Fields..............................................11–9
§11.4.1. Gauge Invariance and the Lagrangian..........................................11–11
§11.4.2. Symmetry and Conservation............................................................11–12
§11.5. Gauge Particles and the Conveyance of Force....................................11–12
§12. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS......................12–1
§12.1. Vectors and Vector Spaces.......................................................................12–1
§12.1.1. Abstract Vector Algebra.....................................................................12–2

6. Table of Contents
§12.2. Linear Functionals....................................................................................12–4
§12.2.1. Linear Operators .................................................................................12–5
§12.3. Dirac Notation and Linear Operators...................................................12–5
§12.3.1. Measurable Properties .......................................................................12–8
§12.3.2. Quantum Operators............................................................................12–9
§12.3.3. Commutators and Poisson Brackets................................................12–10
§12.3.4. Commutators and the Electromagnetic Field...............................12–12
§13. POSTULATES OF QUANTUM MECHANICS......................................................13–1
§13.1 Basic Theoretical Concepts .....................................................................13–1
§13.2 The Four Postulates of Quantum Mechanics (according to
Bohr) 13–2
§13.2.1 Postulate 1 and Postulate 2...............................................................13–2
§13.2.2 Postulate 3 ............................................................................................13–3
§13.2.3 Postulate 4 ............................................................................................13–3
§13.2.4 Postulate 5 and Schrödinger's Equation........................................13–5
§13.2.5 Lorentz Force Law from Schrödinger's Equation.........................13–7
§14. FOUNDATIONS OF QUANTUM FIELD THEORY..............................................14–1
§14.1 Problems with QFT ...................................................................................14–1
§14.2 Simple Approach to QFT..........................................................................14–2
§14.3 Mechanical Analogy .................................................................................14–3
§14.3.1 Canonical Coordinates of the String...............................................14–7
§14.3.2 Quantizing the Mechanical System ................................................14–8
§14.4 Canonical Momentum of the String......................................................14–9
§15. QUANTIZING THE CLASSICAL RADIATION FIELD.........................................15–1
§15.1 Quantizing the Schrödinger Equation .................................................15–2
§15.2 Quantizing the Radiation Field .............................................................15–3
§15.2.1 Field Commutation Modes ................................................................15–3
§15.2.2 Zero Point Energy...............................................................................15–3
§16. GAUGE THEORY AND THE CREATION OF PHOTONS....................................16–1
§16.1 Annihilation and Creation Operators...................................................16–1
§16.2 Photons States............................................................................................16–2
§16.3 Photons as Radiated Field Excitations.................................................16–3
§16.3.1 Total Hamiltonian...............................................................................16–4
§16.3.2 Photon Polarization............................................................................16–5
§17. VACUUM STATE FLUCTUATIONS....................................................................17–1
§17.1 Radiation Density of the Quantized Field............................................17–2
§17.2 Radiation Damping and Self Fields.......................................................17–4
§17.3 Open Questions about the QFT..............................................................17–7

7. Table of Contents
§18. BIBLIOGRAPHY ..................................................................................................18–1
I write to discover what I think.
— Daniel J. Boorstin, Librarian of Congress

8. Copyright 2000, 2001 1–1
§1. THE FOUR FORCES OF NATURE
We see the effects of force all around us. The force of gravity, the
electric and magnetic forces of natural and manmade objects, and the
mechanical force of machines all have well known effects. In pre–twentieth
century science, natural philosophers asked many of the same questions
that are asked here — why does nature behave in the way it does?
Although these questions have the tone of theological or philosophical
inquiries, the study of these forces and their interaction with matter is
generally the domain of physics [Alio87].
The development of the concept of a force marks the boundary
between science and pre–science [Jamm62], [Agas68], [Cajo29]. In early
history, objects were believed to have internal powers, which could account
for their movements. The motion of the planets through the night sky was
associated with gods, and supernatural powers. It was realized during the
time of Galileo that the function of a force was not to produce the
motion, but to produce a change in the motion [Whit58], [Koyr55],
[Jamm62], [Hawk87], [Roge60]. This description of force was not
significantly different from the previous occult force, since the origin of the
force was not known. However, these forces could be measured which
allowed quantitative order to be brought to nature.
One of the most significant scientific developments in the past several
centuries was the concept of a continuous field of force [d’Abro39],
[Adai87], [Hess61], [Sach73]. This discovery replaced action–at–a–distance
with action conveyed through a field. The application of this concept by
19th
century scientists lead to a new understanding of electricity and
magnetism which strongly influenced 20th
century physics [Beck74]. The
special theory of relativity exploited the concept of a continuous field to
describe the motion of objects, including electromagnetic waves, independent
of any special reference frame. The second revolution in 20th
century
physics was quantum theory, which describes matter at the atomic level in
the form of fields. With electromagnetism's fields of force, special relativity's
fields of geometry and quantum theory's fields of probability, the notion of
a field is capable of describing nearly all aspects of physical processes
[Sach73], [Agas68].

9. Forces of Nature
1–2 Copyright 2000, 2001
I do not know what I may appear to the world; but to myself I seem to
have been only like a boy, playing on the sea–shore, and diverting
myself, in now and then finding a smoother pebble or a prettier shell than
ordinary, whilst the great ocean of truth lay all undiscovered before me
— Isaac Newton [Brew55]
§1.1 THE EVERYDAY FORCE OF GRAVITY
The most familiar force in everyday life is the gravitational force, which
unifies the behavior of objects on the human scale of a few centimeters to
the galactic scale of 25
10 cm . This force holds objects to the earth, it keeps
the planets in their orbits, it maintains the path of stars in the galaxy and
it forms the glue the binds the galaxies together. The strength of the
gravitational force is proportional to the product of an object’s mass and
inversely proportional to the square of the distance between the objects.
Gravity is the only force that acts in the same manner between all types
of matter. Neutrons, protons, electrons, and the matter they form all attract
each other according to the law of gravity. Since the same law applies to
all objects, gravity can be considered the result of the geometrical properties
of space itself [Tayl66], [Hawk87]. Einstein formulated the general theory
of relativity on this basis. Unlike Newton’s inverse square law of gravity,
the strength of the gravitational force in general relativity is not a simple
inverse square relationship. [1]
Although the force of gravity dominates the
human experience, it is in fact the weakest force of nature.
1
The concept of the gravity in Newtonian mechanics implies that a test particle is
subject to an external force — the gravitational force. This force acts in a linear fashion on
the test particle as it travels through the gravitational field. In the General Theory of
Relativity, the presence of the test mass influences the behavior of the gravitational force, so
that the force felt by the test particle is non–linear. In the Newtonian view of gravity, the
force field is static and can be represented by a scalar potential, just as the electrostatic
potential can be represented by the Coulomb potential. When the electromagnetic field is
not static — it is dynamic — the addition of the vector field is required to represent the
complete system. These scalar and vector portions of the electromagnetic field can be
represented by a 4–vector potential. The consequence of this form of representation is that
electromagnetic disturbances are propagated with the speed of light. In Maxwell’s
representation, the potentials satisfy the wave equation, rather than the Poisson’s static
potential equation. In General Relativity the Poisson equation ∇ = − πκρ2
4U describing the
static gravitational potential is replaced by ∗
∇ − = πκρ2
00 4g , where ∗
ρ is the density of
mass–energy, not just mass and 00
g is the metric tensor describing the curvature of space–
time. Space–time is curved as a result of the presence of matter [Fran79], [Misn75].

10. Forces of Nature
Copyright 2000, 2001 1–3
The discovery of the gravitational force was made by Sir Isaac Newton
(1642–1726) while attempting to explain Johannes Kepler's (1571–1630)
three laws of planetary motion [Koyr55], [Holt56], [Step94]. The history of
Newton’s discovery of the laws of gravity is surrounded in popular myth.
The 17th
century laws governing the motions of celestial objects were
regarded quite differently from those governing the motions of bodies on
earth. The study of the motion of a heavenly body, particularly the planets
and the sun, was the primary subject taught in the university in the mid
1600’s. Students of natural philosophy at Cambridge in 1664 discussed these
motions in detail. In 1665, the plague broke out in England and classes at
Cambridge were suspended [Manu68], [Chri97], [Shre70]. The 23 year old
Isaac Newton student was sent home in June to Woolsthrope of that year
and did not return until March of 1666 [Manu68]. While pursuing his
B. A. Degree in the Lent Term of 1665, Newton remained home to think
about the question of planetary motion [Sedg39]. He was apparently
inspired as he saw an apple fall to earth in an orchard. [2]
It occurred to
2
Newton’s contribution to the science of physics is well documented. His formulation
of mechanics and his ideas of absolute space and time were not seriously challenged until
Albert Einstein developed the theory of special relativity nearly 250 years after Newton, in
1905. Newton also invented the fluxional calculus, conceived the idea of universal
gravitation, discovered its law, and discovered the composition of white light [Resn60].
In a biography written by Newton’s friend Dr. William Stukeley in 1752, Memoirs of Sir
Isaac Newton, Stukeley states that he was having tea with Newton in a garden under some
apple trees, when Newton said that the setting was the same as when he got the idea of
gravitation, earlier as he noticed an apple drawn to earth in his mother’s Woolsthrope garden
[Asim82], [West80], [Fren88], [Stuk36], [Manu68], [Chan95].
It was occasioned by the fall of an apple, as he sat in a contemplative
mood. Why should that apple always descend perpendicularly to the
ground, thought he to himself? Why should it not go sideways or upwards,
but constantly to the earths centre? Accordingly, the reason is that the
earth draws it.
Another account of this incident is given by Newton himself through the words of his
associate John Conduitt:
Whilst he was musing in a garden came into his thought that the power of
gravity (which brought an apple from the tree to the ground) was not limited
to a certain distance from the earth but that this power must extend much
further then was usually thought. Why not as high as the moon said he to
himself and if so that must influence her motion and perhaps retain her in
her orbit, where upon he fell to calculating that would be the effect of that
supposition. [West80]
The particular tree under which Newton was to have been siting has been identified as
a yellow–green cooking apple in the front garden of Newton’s home in Woolsthrope. When
the tree collapsed in the 18th
century, a cutting was grafted to another tree in the botanical
garden of Kew outside London.

11. Forces of Nature
1–4 Copyright 2000, 2001
Newton that the same force that attracts the apple to the earth could also
attract the moon to the earth. Newton postulated that the centripetal
acceleration of the moon in its orbit and the downward acceleration of a
body on the earth might have the same origin. The idea that celestial
motions and terrestrial motions followed similar laws was a major break in
the tradition of 17th
century science [d’Arbo27], [d’Arbo39]
Newton postulated that a universal attractive force between two bodies
could explain the motions of the moon around the earth as well as the
motions of the planets [3]
. Before the time of Galileo, most natural
philosophers thought that some external influence or force was needed to
keep a body moving. They thought that a body was in its natural state
when it was at rest. In order for a body to move in a straight line at
constant speed, they believed that some external agent had to continually
propel it along — otherwise the body in motion would naturally stop
[Resn60].
3
Newton wrote down his laws of motion in Philosophiae Naturalius Principia
Mathematica, between 1684 and 1687. In this text, Newton collected his previous incomplete
studies in mechanics and mathematics. The writing of Principia arose from a discussion at
the Royal Society in 1684 between astronomer Edmond Halley (1656–1742) the architect Sir
Christopher Wren (1632–1723) and Newton’s archival Robert Hooke (1635–1703) [Manu68],
[Rona69], [Chri97], [Hall32]. The discussion revolved around the conjecture by Wren that the
inverse square law implies that elliptical orbits of the planets must be produced. Hooke
claimed that he had a proof of this theory, but could not actually produce the mathematics.
Halley went to ask Newton the same question. Newton claimed he could prove this
conjecture, but he also did not have the mathematics to back up his claim. Using Kepler’s
observations, Newton produced, in April of 1685, a nine page paper (in Latin) De Motu
Corporum (On the motion of bodies in Orbit), which described the elliptical paths of the
planets in terms of the Laws of Gravitation and the Laws of Motion [Manu68]. This paper
laid the foundation for the mathematical description of the laws of classical mechanics
described in Principia, first published in 1687.
Newton reasoned that the forces between bodies must be the consequences of a force
between particles, which make up the bodies. 22 years after the Lent Term, Newton
consolidated his ideas in Principia. Newton wrote the Principia in three parts, using the
methods of Euclidean geometry to derive his results. The first part describes the motion of a
body from the forces acting on it. The second part describes the forces encountered in
nature and the third examines the solar system and the motion of planets under the force of
gravity. All of these subjects are developed through axioms, lemmas and theorems in the
same manner as a Greek mathematical exposition. The result is a text that is very difficult
to read, even by today’s standards, because of the geometric language. The differential and
integral calculus that was invented for describing motion was not included in Principia.

12. Forces of Nature
Copyright 2000, 2001 1–5
§1.2 EARLY ASTRONOMY — THE HISTORY OF THEORY
Early astronomy provides a clear example of the growth and use of
theory in the development of a deeper understanding of nature. Astronomy
is almost as old as mankind. When early civilization ventured outside their
known world, trade routes were formed. These routes required navigation
aides in order to be reliably traveled. The compass, clock and calendar
became essential components of modern civilizations. Astronomy provided
all three.
The relative individualism of the history of science as opposed to
general history, is also due to the fact that if it is not altogether easy to
analyze and to estimate a man’s contributions in the field of science, at
least it is a good deal easier than is any other field, except art.
— George Sarton [Sart31]
The earliest attempts to describe the solar system were made by the
Greeks in the 4th
century BC. Aristarchus of Samos (310–230 B.C.)
proposed a heliocentric system [Heat13], [Clag55], [Cole60], [Neug52].
Archimedes (287–212 B.C.) assumed the earth moved in an orbit whose
radius, when compared to the fixed stars, was the same ratio of the center
of the earth to its surface. A detailed description of the conclusions of the
Greek astronomers was published in the 2nd
century by (Claudius
Ptolemaeus) Ptolemy [4]
and described a geocentric system in which the
earth is stationary at the center of the universe. The sun, planets and all
stars revolve around the earth in complex orbits. This theory had great
influence on the philosophy and literature for fifteen centuries. Since the
theory was computationally complex, it could not be used to quantitatively
4
The Egyptian astronomer Ptolemy (90–168 AD) recorded his astronomical observations
in The Almagest (Arabic for The Greatest). The exact birth, death and publication dates of
Ptolemy are not reliably known. He lived during the reigns of emperors Trajan, Hadrian,
Antonious Pius and Marcus Aurelius from around 100 AD to 178 AD. He worked in or
near Alexandria Egypt. Ptolemy drew on the works of Hipparchus of Nicaea (180–125 BC)
[Sart31], [Clag55], [Farr49], [Ging93], who was a well-respected Greek astronomer.
Hipparchus’ observations led to the development of trigonometry using theorems of similar
triangles. From these theorems, the concepts of sine, cosine and tangent were defined.
Ptolemy also wrote Geography, which summarizes all Greek knowledge on the subject
of maps, including various methods of projecting the surface of the earth onto flat maps.
Ptolemy’s book was lost during the Dark Ages and cartography became a lost science.
Ptolemy remained one of the greatest astronomers until Copernicus, Tycho and Brahe
[Farr49], [Whit58], [Adle60].

13. Forces of Nature
1–6 Copyright 2000, 2001
account for the increasing number of accurate observations of the motions
of the stars and planets. In 1514 Nicolus Copernicus (1473–1543) suggested
that a simpler description of the motions of the planets could be
developed, by placing the sun at the center of the universe, with the earth,
planets orbiting this center [Rose71], [Kuhn56], [Kuhn57], [Ging93],
[Armi57], [Banv76].
Copernicus agreed that for certain phenomena, which were used to
justify the evidence for the stationary theory of the earth, this evidence
would not be altered if the earth moved and the sun was stationary.
The centre of the earth is not the centre of the universe. We
revolve around the sun like any other planet. The earth’s
unmobility (is) due to an appearance [Rose59].
The controversy over the heliocentric theory prompted astronomers to
gather more accurate data about the motions of celestial objects. The
observations made by Tycho Brahe (1546–1601), [5]
recorded in Astronomiae
Instauratae Mechanica, were analyzed and interpreted by Kepler, who had
been Brahe’s assistant [Holt56], [Banv81].
Using the precise observations of Tycho Brahe, including error
measurements, Kepler found regularities in the motion of the planets and
formulated his three laws of planetary motion. [6]
Kepler’s laws reinforced
5
Tycho Brahe observed a supernova in 1572, which bears his name. Tycho’s visual
observations were made with great care and were sufficiently accurate to deduce the rate of
decrease of the brightest supernova of the time. He was able to make these accurate and
systematic measurements with the help of instruments constructed with funds provided by
King Frederick II of Denmark. He made several advances in measuring celestial objects. He
derived methods for measuring the flex in the instruments. He corrected for the effects of
refraction when stars were observed at different elevations above the horizon. He included
the error values for his observations. These techniques are recorded in Epistolarum
Astronomicarum. This information was vital for the proper interpretation of Brahe’s
observation by Kepler 20 years later.
6
Kepler assumed circular orbits, but the closest he could come to describing the
plant’s motion had an error of 6–8 arc minutes. This error was outside the error band of
Tycho Brahe’s observations. The 6–8 arc minutes is equivalent to the width of a wooden
pencil when viewed from a distance of ten feet.
From these eight minutes, we will construct a new theory that will explain
the motions of the planets – Kepler.
Kepler’s next attempt used ovoid orbits and an inverse square law of the force driving
the planet’s motion. Kepler attempted to use this law to describe the velocity of the planets
in their orbits. After some difficulty, he intuitively adopted the idea of equal areas swept out
in equal time – the aeral law.

14. Forces of Nature
Copyright 2000, 2001 1–7
the Copernican theory and showed the simplicity with which planetary
motions could be described when the sun was placed at the center of the
orbital system. These laws described the motions of the planets using
empirical data, without any theoretical interpretation. However, Kepler had
no concept of the force that caused the planets to move with regularity. It
was Newton’s great triumph that the laws of motion, using gravity as the
force, could be derived from Kepler’s laws of planetary motion. Newton
could account for the motion of the planets in the solar system and for
the motion of the bodies falling near the earth with the same concept. He
unified, in one theory, the theory of terrestrial mechanics and celestial
mechanics. [7]
Kepler’s first two laws were published in Astronomia Nova (The New Astronomy: Based
on Causes or Celestial Physics) (1609) [Kepl09] and the third in Harmonice Mundi (Harmony of
the World) (1619) [Kepl16]. Kepler's three laws are: (i) each planet moves in an elliptical
orbit, with the Sun at one focus of the ellipse; (ii) the focal radius from the Sun to a planet
sweeps equal areas of space in equal intervals of time; (iii) the square of the sidereal periods
of the planets are proportional to the cube of their mean distance to the Sun. This third law
can be stated as =3 2
A kT where T is the period of the planet and A is the semimajor axis
of its elliptical orbit and k can be given in terms of Newton's gravitational constant
[Emch84].
Kepler’s first law expresses the constancy of the observed orbits and the total angular
momentum of the plant–sun system. This observation which was seen as...
... a marvelous manifestation of the harmony of Nature. [Banv81]
This observation revels itself today as a consequence of the laws of dynamics. In fact,
Kepler’s Law is incorrect because it is not the angular momentum of the plant–sun system
that remains constant, but the angular momentum of the entire solar system. The angular
momentum vector for the entire system is perpendicular to the invariable plane of Laplace.
Fortunately for Newton, Kepler’s error has negligible impact because of the weak interaction
between the plants compared to the interaction between the Sun and the planets [Doug90].
7
Newton was the first to state that his work was the culmination of the work of
others. In a letter to Robert Hooke...
If I have seen further (than you) it is by standing upon the shoulders of
Giants.
Although the quote of Newton has been popularized into a comment regarding the
substantial works of previous scientist, it is more complex. The reference to the shoulders of
giants is taken from John of Salisbury’s The Metalogicon [Spey94], [Thor90]…
Bernard of Chatres used to compare us to (puny) dwarfs perched on the
shoulders of giants. He pointed out that we see more and further than our
predecessors, not because we have keener vision or greater height, but
because we are lifted up and borne aloft on their gigantic stature [Sals55].
The quote can actually be taken as backhanded slap at Robert Hooke (1635–1703), who
Newton carried on a life time rivalry. Newton used this quote in a letter responding to
Hooke’s claim that Newton stole the hypothesis on light from Hooke’s Micrographia

15. Forces of Nature
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The force Newton postulated would be proportional to the product of
the masses of the two bodies and inversely proportional to their separation.
Newton then developed the laws of motion that govern the path taken by
a body in the presence of this gravitational force [Cajo62], [Fren88],
[West80]. [8]
The most well known of these laws is Newton's second law of
motion, ma=F ,[9]
which states that a force F produces an acceleration on
a body proportional to the mass of body — given the same force, light
bodies are accelerated faster than heavy bodies. [10]
This law describes the
acceleration, force and mass properties of material bodies [Wein61].
[Hook61]. Newton was familiar with Micrographia and claimed that Hooke took much of
that work from Descartes who — claimed Newton — took his work from Marcantonio de
Dominis and Ariotto [Hall62], [Manu68], [Koyr65].
8
There is evidence in Newton’s student notebooks that he had learned of Kepler’s first
and third laws from, Astronomia Carolina written in 1661 by Street [Robi90].
9
Unlike computer languages, which we are familiar with, mathematical notation is read
left to right. As the above sentence says, a force produces an acceleration on a mass. In
Einstein's General Theory of Relativity, accelerations produce force, so that Newton's
Second Law has reciprocity. In general though equations do not exhibit such behaviors.
Although the point seems trivial, the mathematics of physics, unlike the mathematics of
computing (some would argue this), is a language in which physical phenomenon are
described in a self contained manner. The language of mathematics is capable of describing
the behaviors of nature that can be visualized — in addition, mathematics is capable of
describing unobservable behaviors as well. It is possible to invent a mathematical model for
a process of nature that has no equivalent visualization. There are several models of nature
that can not be visualized.
The behavior of objects whose size is small compared to molecules can be described
by quantum mechanics. The laws of physics at the quantum level may have no equivalent
visualization in the classical world [Polk85]. The situation of increasing abstraction was
predicated by Joseph Lamor…
There has been of late a growing trend of opinion, promoted in part by
general philosophical views in the direction that the theoretical
constructions of physical science are largely factitious, that instead of
presenting a valid image of the relation of things on which further progress
can be based, they are still little better than a mirage [Lam05].
10
Newton's three laws of motion are formally given in Philosophiae Naturalis Principia
Mathematica (Mathematical Principals of Natural Philosophy) [Cajo62], [Andr56], [Asim82],
[Motz89], [Cohe78], [Heri65] as:
Lex I (in editions of 1687 and 1713) – Corpus omne perseverare in statu
suo movendi uniformiter in directum, nisi quatenus illud a viribus impressis
cogitur statum suum mutare.
Lex I (in edition of 1726) – Corpus omne perseverare in statu suo

16. Forces of Nature
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§1.3 THE FOUR FORCES OF NATURE
Although it is the electromagnetic force that is of interest here, three
other forces exist in nature, the gravitational force, the nuclear or strong
force and the weak force. These four forces are the source of all the
variety in the universe [Neem86]. Without them attraction and repulsion of
physical bodies would not be possible and interaction between matter
would not take place. Bodies would simply pass through each other with
no effect.
Gravity or the gravitational force was first identified by Isaac Newton
in the 1680's. Although the gravitational force acts on all matter, its
strength is the weakest of the four forces. As humans, we are conscious of
quiescendi vel movendi uniformiter in directum, nisi quantenus illud a
viribus impressis cogitur statum suum mutare. (Every body continues in its
state of rest, or of uniform motion in a right line, unless it is compelled to
change that state by forces impressed upon it.)
A Body at rest remains at rest and a body in a state of uniform linear motion
continues its uniform motion in a straight line unless acted on by an unbalanced force. This
law is often called the law of inertia. This means that the state of motion in a straight line
remains at rest of continues its uniform motion unless acted on by an unbalanced force. The
presence of the unbalanced force is indicated by changes in the state of motion of a body.
Lex II – Mutationem motis proportionalem esse vi motrici impressae, et
fieri secundum lineam qua vis illa imprimitur. (The change of motion is
proportional to the motive force impressed; and is made in the direction of
the right line in which that force is impressed).
An unbalanced force, F, applied to a body gives it an acceleration, a, in the direction
of the force such that the magnitude of the force divided by the magnitude of the
acceleration is a constant, m, independent of the applied force. This constant, m, is identified
with the inertial mass of the body. The inertial mass is a derived rather than basic quantity.
Newton's equations of motion establish a procedure for measuring this mass. This is done
by applying a known force to a body and measuring its acceleration. The result of this
measure is the mass of the body. There is an additional interpretation of the second law of
motion. If a body is observed to be accelerating than a force must be acting on it, but if no
force is known to be physically applied to the body, Newton concluded that this force must
act–at–a–distance.
Lex III – Actioni contrariam semper et aequalem esse reactionem: sive
corporum duorum actiones is se mutuo semper esse aequales et in partes
contrarias dirigi. (To every action there is always opposed an equal
reaction; or, the mutual actions of two bodies upon each other are always
equal, and directed to contrary parts.)
If a body exerts a force of any kind on another body, the latter exerts an exactly equal
and opposite force on the former. This law introduces a symmetry that does not appear in
the first two laws. It states that forces appear in equal and opposite pairs.

17. Forces of Nature
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the force of gravity only because of the immense mass of the earth and
celestial objects.
The weak and strong forces are not detectable at human scales, not
because of their relative strength but because of their short range. The
weak force has a range of 17
10 m−
to 18
10 m−
. At distances small compared
to the range of these forces, both the strong and the weak force obey the
inverse square law the same as the gravitational and electromagnetic forces
[Hugh91]. Although unfelt by humans, the weak force plays a critical role
in the generation of energy in the sun and the building of heavy elements
through nuclear synthesis [Kane93]. The weak force is also responsible for
the instability of neutrons. Although neutrons are stable within the nucleus
of an atom, under the influence of the weak force a neutron placed in
isolation will split into a proton, an electron and an antielectron neutrino
within fifteen minutes. [11]
This instability is called Beta Decay.
In 1958 Robert E. Marshak (1916–1992) and E. C. G. Sudarshan
(1931 – ) observed that the weak force appeared to involve an action
between two currents similar to the attraction or repulsion between two
current carrying wires [Neem86], [Mars92]. In the 1960's and 1970's a
theory emerged which unified the weak force with the electromagnetic force
— the electroweak force [Rent90], [Mars92].
The strong force which acts between protons and neutrons (nucleons)
is effective only when the nucleons are within 15
10 m−
of each other. The
strong force is responsible for the interactions between nucleons, nucleons
and mesons and a number of other particles. The nucleus contains both
protons and neutrons and the electrostatic repulsive force of protons must
be overcome by an attractive force in order to maintain the stability of the
nucleus. Since the 1930’s, some form of nuclear force has been postulated.
In modern particle physics, it is believed the quarks are the particles that
undergo strong nuclear interactions and are described by the theory of
Quantum Chromodynamics (QCD).
A century after the discovery of the gravitational force, Charles A.
Coulomb (1736–1806) measured the electrostatic force acting between two
11
One of the first experimental confirmations of the neutron’s instability was
performed by Enrico Fermi (1901–1954). Using the atomic pile at the University of Chicago,
Fermi placed an evacuated spherical container inside the reactor. After some time some of
the fission neutrons while passing through the container would decay into a proton an
electron and an antielectron neutrino. The electron and the proton would become trapped in
the container and be combine to form hydrogen gas. The rate at which the gas formed
could be used to estimate the neutron’s mean half–life of approximately 14 minutes
[Gamo65].

18. Forces of Nature
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charged bodies. Like Newton's inverse square law for gravity, the
electromagnetic force obeys an inverse square law — Coulomb's Law.
Instead of being proportional to the masses of the bodies, the electric force
is proportional to the product of the bodies’ electric charge. Since electric
charges can be positive as well as negative, the electric force can attract as
well as repel bodies.
In the late 19th
century, the effects of magnetism were carefully
measured and it was determined that magnetism was a force created by the
current produced by the motion of electrically charged objects. The
electromagnetic force was first thought to be two unrelated forces,
electricity and magnetism. Experiments showed that they were connected
and are a single force.
Although the electrostatic force acts only between charged bodies, the
electromagnetic force can effect uncharged bodies as well. The neutral
charged neutron has a non–zero magnetic moment and is influenced by
magnetic fields. The photon, which has no charge or magnetic moment, is
effected by the electromagnetic force during its absorption and remission by
atoms.
§1.4 THE PARTICLE ZOO
The study of the universe can be described as the search for the basic
constitutes of matter, the forces that effect this matter and the calculation
of the motions of this matter given these forces [Kane92]. Starting with the
Greeks and Chinese, there have been theories that describe the behavior of
matter. The Greeks thought all matter was made up of four elements —
air, fire, water and earth. This atomistic theory originated with the Greek
philosopher Leucippus, the probable founder of the School of Abdera in
Thrace, 5th
century B.C. [12]
This school of thought claims that both empty
space and the matter composed of atoms that filled the space are real. The
changing world was described in terms of the isolation of groups of atoms,
which was in direct conflict with the views put forth by the teachings of
the Eleatic School of Parmenides of Elea (515–450 B.C.), which stated that
everything that had existed had always done so and could never change.
12
Little is known of the life of Leucippus. He was probably a contemporary of
Empedocles (490–435 B. C.) and Anaxagoras (499–428 B. C.) [Gres64] and possibly a pupil of
Zeno of Elea (~462 B.C.). Leucippus assumed the existence of empty space as well as
matter and held that all things are composed of atoms. Space is infinite in extent and atoms
are infinite in number and are indivisible. The atoms are always engaged in activity and the
worlds produced by them have various shapes and weights [Sedg39].

19. Forces of Nature
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The Chinese on the other hand thought there were five elements rather
than four — metal, wood, water, fire and earth and named five planets
accordingly.
Along with Newton’s work in celestial mechanics, he laid the
groundwork for particle physics. Newton’s reasoning is considered
traditional and theologically based in the times.
All these Things being considered, it seems probable to me
that God in the beginning formed Matter in solid, massy, hard,
impenetrable, moveable Particles, of such Sizes and Figures,
and with other Properties, and in such Proportion to Space, as
most conduced to the End for which He formed them; and that
these primitive Particles being Solids are incomparably harder
than any porous Bodies compounded of them; even so very
hard, as never to wear or break in Pieces; no ordinary Power
being able to divide what God himself made in the first
Creation ... And therefore that Nature may be lasting, the
Changes of corporeal Things are to be placed only in the
various Separations and new Associations and Motions of
these permanent Particles. [Cohe52]
With the beginning of chemistry in the early 17th
century, John Dalton
(1776–1844) proposed there was an elementary component within each
element, which itself was unalterable, called an atom. [13]
In the middle of
the 19th
century, Dmitry Ivanovich Mendeleyev (1834–1907) discovered that
the chemical elements could be classified into a table that had a periodic
structure.
In 1897 the electron was discovered by Joseph John (J. J.) Thompson
(1856–1940) [Thom99] followed by the discovery of the nucleus of the
atom by Ernest Rutherford (1871–1937) in 1911 [Ruth11]. These discoveries
resulted in a model of the atom based on the planetary like motion of
electrons orbiting the nucleus. The nucleus of an element can have the
same numbers of electrons but have a different mass and still have identical
chemical properties. These elements are called isotopes. The study of
chemical isotopes suggested that it is the number electrons in the element
that is responsible for its chemical properties. With the discovery of the
neutron by Sir James Chadwick (1891–1974) in 1932, the behavior of
13
In the Principia, page 6, Newton laid the foundation of the atomic theory ...
Because the hardness of the whole arises from the hardness of the parts,
we ... justly infer the hardness of the individual particles not only from the
bodies we fell but of all others.

20. Forces of Nature
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isotopes was explained [Chad32]. Since the electrically neutral neutron
resides in the nucleus it has no effect on the chemistry of the element, but
changes its atomic weight. With the additional discovery of the proton, the
description of the nucleons making up the nucleus of the atom was
complete. Isotopes are now understood to be elements with different
numbers of neutrons, but the same numbers of proton and electrons.
With the detection of cosmic rays in the 1930’s, other particles were
discovered to exist. Using accelerators, still other constituents of matter
were produced through the collisions between particles. The existence of
these particles lead to the discovery of the nuclear force and the
classification of particles that are subject to the nuclear force — hadrons.
Protons and neutrons are hadrons that are held together in the nucleus of
the atom by the nuclear force. Electrons are not hadrons since they are
held in the atom by the electromagnetic force. By 1939 the fundamental
constituents of matter were composed of the proton (p), the neutron (n),
the electron (e) and the neutrino (ν ), plus their anti–particles [Mars93].
After World War II, the number of particles exploded. Using accelerators
hundreds of particles were created adding to the complexity of the
underlying structure of nature.
This situation can be simplified if there is some order given to the vast
zoo of particles. The first approach is to classify a particle by how it
behaves in the presence of an identical particle. This can be done by
describing the statistics of its interaction between large numbers of identical
particles. Two types of particles exist using this description — fermions and
bosons. If a number of identical fermions are placed in a confined area,
they will statistically tend to avoid each other. If a number of identical
bosons are placed in a confined area together, they will statistically tend to
stay together [Kim91].
A second method of classifying particles is by describing their
interaction with the forces of nature. By sorting through the remnants of
the particle collisions, it was discovered that there are two nuclear forces at
work, the weak nuclear force and the strong nuclear force. Particles subject
to the weak nuclear force and the electromagnetic force are a class of
fermions called leptons. These leptons are — the electron (e), muon (µ),
tau (τ ) and the neutrino (ν ). The particles that are subject to the strong
nuclear force remained hadrons. The hadrons can be further classified into
mesons and baryons. [Clos87], [Clos86], [Dodd84], [Frau74], [Schw92].
As early as 1964, Murray Gell–Mann (1929– ) and George Zweig
(1937– ) independently produced a theory that would explain the growing
complexity of the hadrons and their interaction. Their original theory

21. Forces of Nature
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described a universe made up of three types of elementary particles: (i)
quarks, which come in two flavors, up and down, (ii) the electron and (iii)
electron neutrino. In this theory, forces including gravity are carried by
other particles — gauge bosons [Bloo82].
In 1970, there was no theory capable of describing the strong force. A
nuclear force had been postulated in the 1930’s since the nucleus contains
several protons that must be held together while the electrical force
attempts to pull them part. It is the quarks that participate in the strong
force [Ishi82], [Clos79]. Quarks carry color charge and combine to make
color neutral hadrons just as electrically charged electrons combine with
charged nuclei to form electrically neutral atoms [Chew64].
The material of the universe can be described as being made up of
leptons and quarks, which are held together by the force carrying bosons.
The force carriers are the photon for the electromagnetic force, gluons for
the strong force and the 0
, andW W Z+ −
for the weak force [Garv93].
The theory of quarks and their interaction with each other and other
matter is called Quantum Chromodynamics (QCD). The simplifying theory
of quarks quickly became complex as more behaviors of hadron interaction
were discovered at higher collision energies.
The original up and down quarks were joined by four other quarks
named charm, strange, top, and bottom [Namb76]. These six quarks and
their related leptons can be classified into three generations. The first
generation makes up the matter we see in everyday life. The constituents
of the second and third generation are unstable at normal energies and are
only produced in accelerators — or during the formation of the universe.
The six quarks can be arranged into three groups or doublets
u c t
d s b
 
 
 
. The top rows of quarks have charge 2 3 and the bottom row
have charge 1 3− . The six leptons can also be arranged in three doublets
e
e
µ τν ν ν 
 µ τ
 
where e is an electron, µ is a muon, τ is a tau all of
which have charge –1, while each particles' neutrino has no charge
[Namb76], [Neem86], [Okun85]. [14]
14
Neutrinos are massless (or nearly massless) particles with no charge. The neutrino
was invented by Wolfgang Pauli (1900–1958) in 1930 to account for the missing energy
created during Beta decay [Rein79]. Beta decay was discovered in 1896 when it was observed
that certain atoms decay into other atoms [Frit83]. Early theories of Beta decay predicted

22. Forces of Nature
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that the neutron in the nucleus of an atom would be changed into a proton and a free
electron [Lipk62]. It was also predicted that the products of the decay would conserve
charge, energy and momentum with a fixed value. In an experiment performed in 1927 it
was found that the free electron produced by the Beta decay had a continuous spectrum of
energy values, contrary to the theory [Sutt92], [Brow78].
Pauli's new particle was needed to carry off the momentum and energy, preserving the
conservation laws that were violated by the earlier naive models [Sutt92]. This particle was
named the neutrino after Enrico Fermi developed the theory of Beta decay and was quoted
as saying...
It is a little neutron, it is a neutrino [Rein79].
Pauli’s original particle was named the neutron since today’s chargeless particle called
the neutron had yet to be discovered. Pauli’s neutron name was not copyrighted since it only
appeared in private correspondence and never in print. In 1932 James Chadwick (1891–1974)
presented evidence of a neutral charge particle with nearly the same mass as the proton he
called the neutron. When Enrico Fermi (1901–1954) reported Chadwick’s discovery, a
member of the audience asked if Chadwick’s neutron was the same as Pauli’s neutron,
Fermi answered...
No, the neutrons of Chadwick are large and heavy, Pauli’s neutrons are
small and light, they have to be called Neutrinos [Gamo65], [Ferm54],
[Segr70].
The neutrino has an extremely low interaction rate with other forms of matter. In a
cubic centimeter of water there are approximately × 22
7 10 free protons available in the
nuclei of hydrogen. The protons in the nuclei of oxygen are bound and unavailable for any
interaction. A neutrino passing through this cubic centimeter of water has one chance in
44
10 of being captured by any one of the 22
10 protons. The result is one chance in 22
10 of
any proton capturing the neutrino — very low odds. Converting this probability to a human
scale it would require 22 3
10 cm of water to capture a single neutrino. This length is 1000
light years or 63,000 times the distance between the sun and the earth [Sutt92].
Free neutrinos were first observed in 1956 by Fred Reines (1918 – ) and Cylde L.
Cowan (1919 – ) using a liquid scintillator placed in a neutrino beam generated by a nuclear
reactor [Cowa56], [Rein56], [Rein5]. Their first proposal was to place the scintillator 40 m
from ground zero during the test of the first atomic bomb. After 100 days of operation over
a period of a year, on June 14th, 1956 Reines and Cowan captured the poltergeist particle
[Rein79], [Rein79a], [Rein94], [Krop94].
This discussion of neutrinos may seem far removed from the goal of the book, but it
does have several connections. The speculation of the existence of the neutrino by Pauli and
its subsequent theoretical prediction by Fermi lead to the theory of Beta decay. Fermi's
theory was built on a quantum field theory in which particles need not preexist but can be
created from a vacuum [Bern89]. No theorist was saying the neutrinos preexist inside the
nucleus and are ejected during Beta decay. They are rather created during the Beta decay
process, then ejected [Brow78]. The concept of the creation and subsequent annihilation of
particles will be used later in the quantum field description of the electromagnetic field.
The second connection is between quantum field theory and observational astronomy.
On the night of February 23, 1987 a star named Sanduleak (SK) –69° 202, cataloged by
Nicholas Sanduleak in 1969, located in the region of the Tarantula Nebula, on the edge of
the Large Magellanic Cloud became the first supernova to occur in our own galaxy in four

23. Forces of Nature
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Why matter is composed of leptons and quarks and why these leptons
and quarks should be arranged in families with specific masses is not
known. The search for the answer to this question is the quest of the
current generation of physicist [Wein93], [Lede93]. Using the quark model,
material objects can be built from these particles. Protons, neutrons, pions,
etc. are built from quarks. Since these hadrons are constructed from quarks,
they are not considered elementary [Robe79]. [15]
§1.5 FUNDAMENTAL FORCES IN QUANTUM CHROMODYNAMICS
In Quantum Chromodynamics, the quarks that compose hadrons are
bound together by gluons. The residual force of the gluon, when seen
outside a hadron becomes the nuclear force that binds hadrons into stable
nuclei. The electrically stable nuclei and the only electrically stable lepton
— the electron — are bound into atoms by the electromagnetic force. The
residual electromagnetic force outside the atom binds atoms into molecules.
Since these molecules form the basis of life, the study of particle physics
can be considered of primary interest to mankind [Geor81], [Wein93],
[Lede93], [Geog80], [Frit83], [Barr91].
centuries. Other than the observation of this very rare event SK 69° 202, a.k.a. SN 1987A
was important for what was not seen by the astronomers. A burst of approximately × 58
1 10
neutrinos, lasting nearly 6 seconds were emitted from SN 1987A. Nearly 30 million billion
of the neutrinos then passed through a detector located 2000 feet deep in a salt mine in
Painesville, Ohio. Out of these particles × 15
30 10 neutrinos, 8 interactions occurred. Nearly
three hours later the visible photons from SN 1987A arrived at the telescopes in the
Southern Hemisphere. The energy necessary to produce × 58
1 10 neutrinos is approximately
× 58
3 10 ergs sec or × 20
1 10 times the total energy production of the sun.
15
Neutrons are composed of 2 down–quarks and an up–quark whose charges are
summed produced a neutral particle ( )( )( ) ( )− − = =1 3 1 3 2 3 0 0 0 while a proton is
composed of two up–quarks and one down–quark whose charge is ( )( )( )− =2 3 2 3 1 3 1 .
Other hadrons are composed of three different quarks and are called baryons, while other
hadrons are composed of a quark and an anti–quark and are called mesons.
There is symmetry between quarks and leptons in that leptons have integral units of
charge while the electrical units of quarks are multiples of 1/3. This factor of 3 is actually
accompanied and compensated by another factor 3; each quark comes in three invented labels
called color, which is simply a quantum number for the behavior of quarks, not an actual
color as we know green or red.
All of these behaviors are described through a theory based on group symmetries called
SU(3) or the Eightfold Way [Gell64], [Dodd84], [Clos83]. In this theory the nucleons belong
to a multiplet of eight and the pions and kaons, which are mesons with quark contents
, , ,su sd us ds, belong to a separate multiplet which also has a multiplicity of eight.

24. Forces of Nature
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In addition to quarks and bosons there is one more particle needed to
complete the theory in a consistent manner — the Higgs Boson (Peter
Ware Higgs (1929– )) [Velt86]. The theory of the electroweak interaction
and the large masses of the 0
, andW W Z+ −
particles requires one electrically
neutral Higgs Boson [Clin82], [Clin74]. The field produced by the Higgs
Boson is a background field pervading all space, ever present, even in the
vacuum state [Guid91]. The presence of this field produces an energy
density in the vacuum which would curve space–time through the
gravitational interaction. At this time, there is no experimental evidence for
the Higgs Boson, but the search continues [Roln94].
The electromagnetic and gravitational forces have long range, the weak
and nuclear forces have short ranges, all four forces obey an inverse square
law. Why are their ranges different? Why do all these forces obey the
inverse square law? What generates these four forces? How are the forces
conveyed?
Although the four forces of nature all appear to follow the inverse
square law, the force binding quarks together — the chromostatic force —
behaves differently. The electrostatic force described by ( ) 2
4V e= − πr r is
replaced by ( )V = κr r , when r is very large. This chromostatic force
behaves Coulombic when r is small, but the potential increases linearly for
large r. The result is that this force permanently confines the quarks inside
their host [Adle81].
A theory of the electromagnetic force must not only explain observable
effects of electromagnetic fields, but also must explain the source of the
forces and the mechanism that conveys these forces. The search for the
answer to these questions is similar to the mathematically undecidable
question in Gödel's Theorem (Kurt Gödel (1906–1978)) [Gode62], [Hofs79],
[Nage58], [Penr89b] — that in order to describe sufficiently one set of
axioms, an external (super) set of axioms (theories) is needed, which in
turn requires another external set of axioms. [16]
Gödel summarized this
16
Gödel's Theorem appears as Proposition VI in his 1931 paper, "On Formally
Undecidable Propositions in Principia Mathematica and Related Systems I". It states: "To
every ω –consistent recursive class κ of formulae there correspond recursive class–signs r,
such that neither ~Gen r nor Neg(~Gen r) belongs to κ( )Flg (where v is a free variable
of r).
In layman's terms this says: All consistent axiomatic formulations of number theory
include undecidable propositions — arithmetic is not completely formalizable. Gödel
observed that a statement about number theory could be about a statement of number
theory (possibly even itself), if only the numbers could somehow stand for statements.
Gödel's work was part of a long attempt to define what proofs are. Proofs are

25. Forces of Nature
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dilemma in the development of the understanding of nature, using the tools
of mathematics as…
The human mind is incapable of formulating (or mechanizing)
all its mathematical intuitions, i.e. if it has succeeded in
formulating some of them, this very fact yields new intuitive
knowledge, e.g. the consistency of this formalism. This fact
may be called the incompletability of mathematics. On the
other hand, on the basis of what has been proved so far, it
remains possible that there may exist (and even be empirically
discoverable) a theorem–proving machine which in fact is
equivalent to mathematical intuition, but cannot be proved to
be so, nor even be proved to yield only correct theorems…
[Gode51].
demonstrations within fixed systems of propositions. Gödel was saying that the system of
Principia Mathematica [Whit27] is incomplete — there are true statements of number theory
that its methods of proof are too weak to demonstrate. (The Principia Mathematica is a
monumental work consisting of 4 volumes that attempted to build the foundation of
mathematics upon a paradox–free set of logical axioms.)
The concept that mathematics is nothing but symbols in some formal mathematical
system is the definition of formalism in which mathematics becomes a meaningless game.
Gödel dealt formalism a devastating blow with his theorem and restored meaning to the
symbols. In the world of physics it is the meaning of the symbols that provides the means
for describing nature through mathematics.
In the early 1920's the logician Alfred Tarski (1901 – 1983) took Gödel's argument
further to show that logical systems are also semantically incomplete as well [Tars56]. He
showed that if a mathematical system is consistent then the notion of truth is not definable.
The result of this discovery is that logical and mathematical systems are logically
incomplete in that there is no formal system in which the truth of all mathematical
statements could be decided or in which all mathematical concepts could be defined
[Barr92]. The totality of mathematics cannot be brought to complete order on the basis of
any system of axioms. Just as Heisenberg had done for the physical sciences, Gödel ended
the search for certainty in mathematics [Asim82b].
In the proof of Gödel’s theorem there are two important notions that must be dealt the
simultaneously: (i) the notation that mathematics is simply the manipulation of symbols and
(ii) the concept that a mathematical proof can be substituted for the concept of the truth.
Together these concepts provide for the translation of a verbally stated logical paradox into
an arithmetic statement. One example of such a paradox is the Liar’s Paradox: This sentence
is false.
All of this background would be of no interest if not for the logical paradoxes that
will be encounted when quantum mechanics is applied to the measurement and behavior of
photons as they interact with matter [Bohm51], [Bohm62], [Stew92], [Cast96].

26. Forces of Nature
Copyright 2000, 2001 1–19
In the realm of classical physics, these questions are unanswerable and
perhaps meaningless — the force is just there. In the quantum world
however new answers may be found. [17]
§1.6 QUANTUM FIELD THEORY
In order to address the questions raised in classical physics, a new
theory has been developed — Quantum Field Theory [Gros93], [Guid91],
[Hari72], [Itzy80], [Kaku93], [Mand84], [Brow90], [Chai84], [Chen83],
[Quig83], [Roma69], [Visc69]. When quantum mechanics and special
relativity are combined, the resulting description of nature is based on the
interaction of quantized fields and their associated force carrying particles —
gauge bosons. Like classical field theories, quantum field theory describes a
force created when one particle acts on another particle, after an
appropriate delay due to the finite propagation speed of light. When
quantum theory is used, the energy carriers of the force can only assume
discrete values. It is these quanta of the field energy that are identified with
the particles that transmit force. In quantum field theory, the interaction of
elementary particles is interpreted as the exchange of force carrying particles
among the material particles.
It will be shown later that these interactions obey specific symmetry
rules and the force of the interaction is proportional to a charge of some
kind. In such theories, the interaction of objects takes place locally in the
form of the creation and annihilation of particles. Forces are transmitted by
the propagation of particles known as exchange particles. These exchange
particles have properties of mass, spin and charge, like the material
particles. The four forces of nature differ because the exchange particles
differ. In quantum field theory, the electromagnetic force is conveyed by an
exchange particle — the photon.
17
Although it may be meaningless to ask the question how is the force conveyed using
the vocabulary of classical mechanics, quantum mechanics produces a similar set of
meaningless questions. Heisenberg's uncertainty principle restricts the description of the
words position and velocity to any accuracy exceeding the uncertainty relation [Heis30].
Heisenberg cautioned that
... one should be particularly careful to remember that the human language
permits the construction of sentences which do not involve any
consequences and which therefore have no content ...
The use of the words reality often leads to a picture of a physical process that can
neither be proved nor disproved. The description of the physical process of the
electromagnetic force will become increasingly abstract as this book progresses. The
conceptualization of modern physical theory will be the most difficult hurdle to overcome.

27. Forces of Nature
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According to the quantum theory of Maxwell's electrodynamics —
quantum electrodynamics (QED) — electromagnetic forces between two
charged particles are generated by the transmission of a massless gauge
boson — the photon — between them. [18]
This photon brings with it a
command from one particle to another. The receiving particle obeys the
command of the arriving gauge particle with the result interpreted as the
conveyance of the force. In this theory, the photon passes a force message
to the receiving particle, rather than imparting some physical force to the
receiving particle. In the classical electrodynamics, during the transmission of
radio waves, external energy is available to communicate commands. Such
energy is not always available in the quantum world of charged particles.
According to the uncertainty principle of quantum mechanics, this energy
can be borrowed for a short time to enable the command to be carried by
the exchange particle, thus allowing force to be conveyed while obeying the
conservation of energy.
§1.7 PRELIMINARIES TO MODERN PHYSICS
Modern physics is so strongly based on quantum models of the
microworld it is difficult to bridge the gap between classical electrodynamics
and quantum theories of Maxwell's equations and their interaction with
matter. A course of instruction at a university covers classical
electromagnetic theory, developing an understanding of electrostatics,
magentics and Maxwell's equations. This knowledge is then used in the
formulation of electromagnetic radiation and its practical applications,
usually in antenna theory, wave guides and electro–optics. In parallel, the
student develops an understanding of quantum mechanics. This material
describes the duality of particles and waves, which is the basis of the
descriptions of atomic and subatomic phenomena. What is missing from
these parallel courses of study is the description of the electromagnetic field
as a Quantum Field Theory. Although the development of the quantum
description of Maxwell's radiation field presented prior to proceeding with
the Quantum Field descriptions of subatomic particles, the details of the
18
The massless nature of the photon is a consequence of the gauge (phase) symmetry
of the electromagnetic field. The photon’s masslessness also guarantees that the electric
charge is conserved since the symmetry responsible for the masslessness is a result of the
invariance of Maxwell’s equations to arbitrary phase changes in the quantum fields
associated with the electron and the photon. The arbitrary phase changes in the electron
field can be compensated for by the redefinition of the photon field which leaves the form
and structure of the equations of motion for the electromagnetic interactions unchanged
[West93].

28. Forces of Nature
Copyright 2000, 2001 1–21
Quantum Radiation Field of Maxwell's equations is not fully developed at
the undergraduate level.
There will be an attempt to bring some understanding to the
phenomenon of the electromagnetic force, through the various theories
beginning with Classical Electrostatics, Electrodynamics, Quantum Mechanics
and concluding with Quantum Field Theory. Because of the compressed
form of this book, mathematical expressions are brief and usually given in
non–rigorous form and in some cases simply stated without supporting
derivations. Whenever possible the derivation or expanded material is given
in an endnote.
The electromagnetic force is well understood from the view of classical
physics. Its characteristics include: an infinite range, allowing macroscopic
phenomena to be observed and a reasonably strong force, allowing
microscopic phenomena to be observed. It is the interaction of the
electromagnetic force with microscopic matter that leads to the formulation
of Quantum Field Theory (QFT) — the goal of this thesis.
The path taken from classical electrodynamics to quantum field theory
will often take a diversion to cover background material needed to
illuminate the primary subject. This approach is necessary, since the reader
may require an additional understanding of the mathematical
developments. [19]
The unification of the concepts found in the literature
and texts as well the diverse notation has been a significant effort. I
apologize for any over simplifications that may have entered the text.
Finally, none of the material presented here should be considered original,
but is a compendium of ideas found in the literature given in the
bibliography. Direct quotations have been kept to a minimum, but there is
material which is best conveyed verbatim from the source. In such cases,
references to the original text are given within the sentence. All other
references provide the reader with a rich set of source materials to continue
the search for understanding of the primary forces of nature.
19
Much of the difficulty in learning physics comes from understanding the background
material, including the mathematical notation.
In any branch of science the terminology becomes so cumbersome in the
process of its progress that it is very difficult to put it in a simple way for a
reader who encounters all these complicated notations for the first time
[Gamo66], pp. 68.
A second problem is the notation used in the description of electromagnetic phenomena
can be somewhat cryptic when compared to ordinary notation of calculus and differential
equations. When the notation is beyond the norm, footnotes will be used to provide
additional information on the subject matter.

29. Forces of Nature
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§1.8 UNIFYING PRINCIPALS OF NATURE
One of the great achievements of 20th
century physics has been the
attempted unification of the four forces of nature. Although Einstein first
proposed the grand unified theory, it wasn’t until the 1970’s that any
serious progress was made. The first unification took place at the end of
the 19th
century, when Maxwell formulated the theory of electro–
magnetism. In modern terms – Quantum Electrodynamics (QED) – all
phenomena can be understood in terms of the force–carrying particles
exchanged between electrically charged matter and the photon [West93].
In the last 20 years, the four forces of nature, electromagnetic, weak,
strong and gravitational are being described by a small number of unifying
principals [West93]. Four of these principals will be developed further in
this book.
The first is the Principal of Relativity, which restricts the kinematic
description of the motion of particles and fields, both classical and
quantum. This principal states that the laws of physics are independent of
spatial location and the passage of time, while being observed in a
uniformly moving reference frame.
The second is the Principal of Stationary Action which describes the
motion of both classical and quantum mechanical systems in terms of the
minimization of action.
The third is the Gauge Principal, which describes the rules governing
the interaction of fields of force with material particles. This principal is
formulated as a classical description of nature, but it has inherent quantum
mechanical properties. It may also be formulated as a relativity principal
which allows internal and external properties of a physical system to be
described independent of the observers reference frame.
The fourth principal is the Quantum Principal, which states that all
physical systems in nature are inherently quantizable. This principal is
founded on extensive experimental experience.
Using these four simple principals, it is surprising that the basic
properties of various forms of matter and the interaction between matter
can be described in detail. One question raised in the late 20th
century is
whether there is a Theory of Everything [Davi88], [Wein77], [Wein77],
[Wein93], [Lede93]. A theory in which the forces and matter of nature are
described with a very small number of principals, which would reduce the
current gap between the mathematical model of the world and physical
reality.

30. Forces of Nature
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The cornerstone of all physical sciences is experiment. The
understanding of the basic forces of nature is advanced through the
interplay of physical experiment and theoretical ideas. The largest
impediment in the formulation of the Theory of Everything is the
increasing energies needed to probe deeper into nature. With this increased
energy, comes increased cost. At the energy level required for complete
unification of natures forces — the Planck energy 19
1 10 GeV≈ × — the direct
observation of this unification may be beyond our ability to fund the
experiments [Doug90].

31. Forces of Nature
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I wish we could derive the rest of the phenomena of nature by the
same kind of reasoning as for mechanical principals. For I am induced
by many reasons to suspect that they may all depend on certain forces.
— Preface to the 1
st
Edition (1686) of Newton’s Principia

32. Copyright 2000, 2001 2–1
§2. CLASSICAL FIELD THEORY
Newtonian mechanics and the theory of gravity dominated much of the
physical sciences into the middle of the 19th century. When applied to
astronomy, the mathematics of Newton's equations of motion yielded
dramatic results. Problems such as the rise and fall of tides, the flow of
rivers, the orbits of projectiles and the motion of machinery were well
understood. Compared to the ease of these mechanical problems, the
discovery of light and the study of the structure of matter were
significantly more difficult. Much of the work on electromagnetism and
physical chemistry took place by trial and error. Unlike Newtonian
mechanics, these branches of science had no mathematical basis from
which to develop. [1]
In the physical sciences, the word field describes a continuous
distribution of some type of condition, which pervades a continuum
[d’Abr39]. The nature and magnitude of this condition can take on many
forms. If the condition can be described by a single valued function for
each point in space, then a scalar field is said to exist. A temperature
distribution in a volume of gas is an example of a scalar field. In many
cases, the condition at each point in space has a direction as well as a
magnitude. In this case, a vector field is said to exist. A field of velocities of
a fluid in motion is an example of a vector field. The distribution of stress
in an elastic medium can be described by a tensor field, as can the
gravitational field using a 4–dimensional space–time coordinate system
[Sope75].
Before the development of modern electrodynamics, all field theories
1
The description of nature using the language of mathematics was introduced by
Pythogorus. The Greeks before him envisioned the world in terms of matter. Pythogorus
envisioned the world through a mathematical description of form. This form can be
described by the fact that matter exists under definite structured conditions and develops or
moves according to definite laws, which can be described mathematically.
Mathematics is the way of describing the relationships of these movements. The
discovering (by the Greeks) that nature moves according to the laws of mathematics is
profound insight into the basic order of the universe. Although Kant has stated that
mathematics is merely a …category of our thinking, every phenomenon of Nature can be
placed in some form of mathematical language [Kant87], [Kant88]. Mathematics is the
vocabulary by which the human mind preserves the intrinsic order of nature. All of this
classical order will be challenged in later chapters by the mathematics of quantum theory.
For now however, chaos does not rule the realm of electrodynamics and classical field
theory.

33. Classical Field Theory
2–2 Copyright 2000, 2001
represented conditions of space, usually mechanical categories such as
force, velocity or stress. The aether of the electromagnetic field was first
described using the notation of a mechanical field theory. After the
withdrawal of the aether, the field theory describing electromagnetics still
referred to the aether as merely another name for empty space. It was a
field of space, which became an active agent instead of a passive void. [2]
§2.1. ELECTRODYNAMICS
The force controlling the interaction of particles, whether charged or
neutral, macro or micro in size can be described with the concept of a field
of force. Instead of saying that one particle acts on another through direct
contact or through some physical medium, as was believed in Newtonian
physics, it can be said that the particle creates a field and a certain force
resulting from the field, unique to the particle class, then acts on every
other particle of the same or similar class, located in this field. In classical
physics, the concept of a field is merely a mathematical description of the
physical phenomenon—the interaction of particles. [3]
Here and elsewhere, we shall not obtain the best insights into
things until we actually see them growing from the beginning.
— Aristotle
§2.2. ELECTROSTATICS AND EARLY EXPERIMENTS
When a piece of amber is rubbed, it attracts small pieces of material.
This discovery is attributed to Thales of Miletus (640–548 B. C.)
[Jean25]. [4]
A second discovery by Titus Lucretius Carus (~ 99/94–55/51
2
The development of field theories took place in the early 19th century only after the
development of the theory of partial differential equations [d’Abr39]. Even after the
mathematics was available, field theories were not immediately constructed. The
mechanical models of stress in elastic media, with their mechanical magnitudes were the
basis of much of the work. Only when Maxwell formulated the non–mechanical description
of the electromagnetic field did field theories take hold. Maxwell’s formulation though
depended on the experimental and theoretical results of the pioneers of electrical research,
who awaited Maxwell to make the necessary connections before the theory can be turned
into practice.
3
The use of the term particle does not imply subatomic particles, but rather a body of
matter that is physically very small, so as to approach a mathematical point.
4
Thales was considered one of the Seven Sages of Ancient Greece. He was the father of
Greek, and consequently of European philosophy and science. His speculations embraced a
wide range of topics relating to political and celestial matters.

34. Classical Field Theory
Copyright 2000, 2001 2–3
B.C.) and described in De Rerum Natura (On the Nature of the Universe)
[Lucr52] was that a mineral ore — lodestone — possessed the ability to
attract iron. [5] Plato (428–348/7 B.C.) refers to the attributes of amber in
his dialog Timaues [Bury29]. By the Middle Ages, the properties of a
compressed form of coal called jet had been described by Venerable Bede
(673–735). [6]
In the 13th century, Pierre de Maricourt demonstrated the
existence of two poles in a magnet by tracing the direction of a needle,
which was laid onto a neutral magnetized material. His publication
described the first observation connected with one of the modern laws of
electromagnetics — Gauss’s Law for the absence of magnetic charges.
The first use of the word electricity has been attributed to William
Gilbert (1544–1603) in 1600. [7]
He used electricity to characterize a
Thales studied astronomy in Egypt which allow him to construct accurate tables
forecasting the flooding of the River Nile. He first became widely known by anticipating an
eclipse of the sun in May of 585 B.C., which coincided with the final battle in the war
between the Lydians and the Persians. He used tables drawn by the Babylonian
astronomers, but did not succeed in forecasting the exact day (May 28th) or the hour of the
event.
He believed that certain substances, like lodestone (magnetic rock) and the resin
amber, possessed psyche (a soul). Many centuries lapsed before Thales’ soul was identified
as static electricity and magnetism. William Gilbert (1544–1603) who had read about the
unexplained observation of Thales, also became interested in the intangible property, and
decided to call it electricity, from the classical Greek word for amber, which is electron.
5
Little is known of Lucretius since he lived by the motto of the Epicureans Live in
Obscurity [Lucr65]. The only contemporary he mentions in his writings was a work of
Memmius, who was a politician and praetor in the 58 B.C. De Rerum Natura, is a didactic
poem on the subject of Nature, creation, and the universe. The basic concepts developed in
the poem are that nothing can be created from nothing, nothing can be reduced, the
elemental form of matter is tiny particles which are invisible and indivisible. These
particles were called seeds, first bodies, and first–beginnings. In modern physics of course,
these particles are called atoms. Lucretius also conjectured that there is also empty space
or void. All things in the universe consist of a mixture of particles and this void.
6
Bede was an English monk who also studied the tides, calculated the dates of Easter
for centuries to come and wrote one of the world’s great works of history, The Ecclesiastical
History of English. In this book Bede describes the material jet as..
... like amber, when it is warmed by friction, it clings to whatever is applied
to it [Bede55].
The cause of this attraction is not well founded, since Bede confuses friction and the
warmth produced by friction.
7
Gilbert was a London physician and president of the Royal College of Surgeons and
court physician to Elizabeth I and James I. Gilbert studied the effects of lodestone
(magnetite) and introduced the term electric, in his 1600 Latin text De magnate,

35. Classical Field Theory
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quantity that many substances share with amber when they are rubbed,
including glass, sulfur, wax and certain gems. These observations of
electrical attraction led to the idea that electricity was not an intrinsic
property of the material, but rather a substance unto itself, which was
produced or transferred when the material was rubbed.
Stephen Gray (1670–1736) showed in 1729 that static electricity can be
moved between bodies by some substances, among them metals, which are
now called conductors [Gray31]. Early laboratory experiments by
Benjamin Franklin (1706–90) [Fran51] and Charles–François de
Cisternay Du Fay (1698–1739) [Dfay33] established that electrically
charged bodies both attract and repel each other, resulting from the
presence of negative or positive charges. [8]
Du Fay concluded that there are...
...two electricities, very different from each other; one of these I
call vitreous electricity; the other resinous electricity [Dfay33].
magneticisque corporibus, et de magno telluse; physiologia nova, plurimis & argumentis, &
experimentis demonstrata (On the Magnet: Magnetic Bodies also, and On the Great
Magnet the Earth; A New Physiology Demonstrated by Many Arguments and
Experiments) after the Greek word elektron (ηλεκτρου) for amber. This book was an
attempt to explain the nature of the lodestone and to account for the five movements
connected with magnetic phenomena [Gill77]. Gilbert's work is one of the oldest
publications on the theory of magnetism. An English translation of Gilbert's work On the
Magnet, [Pric58] is a facsimile edition of a previous translation [Thom00], which itself is a
replica of Gilbert's original Latin edition published in London in 1600.
In his book, Gilbert stated that many substances besides amber could be electrified
when rubbed and they would attract light objects. Although Gilbert denied the existence of
electrical repulsion, several other researchers observed repulsion later in the 17th century.
Niccolò Cabeo (1596–1650) has often been credited with first observing electrical repulsion,
but he regarded it as a mere mechanical rebounding of the attracted objects from amber
[Home92b]. Gilbert made the distinction between the behavior of dissimilar electrified
objects and the behavior of the properties of the lodestone [Heil79]. This separation of
electricity from magnetism would be rejoined two centuries later by Øersted in 1813.
Gilbert went on to speculate that magnetism was responsible for holding the planets in
their place around the Sun. His improper explanation of orbital mechanics did lay the
groundwork for the concept of action–at–a–distance which paved the way for the future
concept of universal gravity in the 1680's [Benn80].
8
It has been known since early times that a piece of amber, when rubbed with fur,
will acquire the power to attract ... feathers, straws, sticks and other small things [Bari68].
The amber acquires a negative charge when rubbed by the fur, while glass rubbed with
silk acquires a positive charge. The exact mechanism involved in transferring a charge
(positive or negative) from the surface of one material to another is still not well understood
[Wein90], [Moor73].

36. Classical Field Theory
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The vitreous electricity, from the Latin vitreus for glassy, is produced
when glass or crystal is rubbed with silk. The resinous electricity is
produced in amber or copal, when they are rubbed with fur. Both of these
types of electricity were observed to attract ordinary matter. Vitreous
electricity was assumed at attract resinous electricity, but materials
containing vitreous electricity were assumed to repel each other and
likewise for materials containing resinous electricity.
The strength of this attraction or repulsion is given by an inverse
square law. The natural philosophers of the 18th century were disposed to
the idea of an inverse square law for electrostatics, following the success of
Newton's inverse square law for gravity. It was Charles Augustin de
Coulomb's (1736–1806) careful experiments in 1785, using a very
sensitive torsion balance, which gave direct quantitative verification of the
inverse square law, know as Coulomb's Law [Heav50], [Elli66], [Whit52],
[Shan59]. Coulomb stated that
... the repulsive forces between two small spheres charged
with the same kind of electricity is in inverse ratio to the square
of the distance between the center of the two spheres... (and)
the law of inverse square was found to hold also the case of
attraction.
The inverse square law was found to hold accurately for various
charges and separations. [9]
This law states that the force between two
point charges, 1q and 2q — whose dimensions are small compared to their
separation — exerted on one another has the direction of the line joining
the charges and is inversely proportional to the square of their separation
9
The inverse force law for electrostatic charges was first found by Joseph Priestly
(1733–1804), who also was the discoverer of oxygen in 1767 [Segr84]. He formulated the
inverse square law in a unique manner, which was more convincing that any direct
measurement that had been performed at the time. By the same reasoning, Cavendish
derived a similar law [Crow62]. After Coulomb’s Law had been established, the science of
electrostatics became mathematical rather than subjective. The most important problem
faced by the scientist of the time was,
Given the total quantity of electricity on conducting bodies, calculate the
distribution of charges on them under the action of their mutual influence
and also the forces due to these charges [Born24].
The solution to this problem, usually called the theory of potentials, does not represent
the true theory of contiguous action since the differential equations (or difference equations
used by Coulomb) describe the charge density as a function of position. They do not describe
the transmission of the electrostatic force as a function time. Therefore, they still represent
an instantaneous action at a distance.