Conceptual Origins of Maxwell Equations
and
of Gauge Theory of Interactions
It is usually said that Coulomb, Gauss, Ampere and Faraday
discovered 4 laws
experimentally, and Maxwell wrote them into equations by adding the displacement
current.
That is not entirely wrong, but obscures the subtle interplay between geometrical and
physical intuitions that were essential in the creation of
field theory.
19th Century
19.1
The first big step in the study of electricity was the invention in 1800 by Volta (1745-1827) of the Voltaic Pile, a simple device of
zinc and copper plates dipped in seawater brine.
19.2
Then in 1820 Oersted
(1777-1851) discovered that an electric current would
always cause magnetic
needles in its neighbor-
hood to move.
Oersted’s experiment electrified
the whole of Europe, leading to
such devices as the solenoid,
and to the exact mathematical
laws of Ampere.
Ampere (1775-1836) was
learned in mathematics. He worked out in 1827 the exact magnetic forces in the
neighborhood of a current,
as “action at a distance”.
Faraday (1791-1867) was also greatly excited by Oersted’s discovery. But he lacked
Ampère’s mathematical training.
In a letter Faraday wrote to
Ampère we read:
“I am unfortunate in a want to
mathematical knowledge and the power of entering with facility any abstract reasoning. I am obliged to feel my way by facts placed closely together.”
Without mathematical training, and rejecting Ampere’s action at a distance, Faraday used his geometric intuition to “feel his way” in understanding his
experiments.
• In 1831 he began to compile his
<Experimental Researches>, recording eventually 23 years of research (1831-1854). It is
noteworthy that there was not a single formula in this whole
monumental compilation.
19.3
Then in 1831
Faraday discovered
electric induction!
Faraday discovered how to convert kinetic energy to electric energy, thereby how to make electric
generators.
• This was of course very very important.
• But more important perhaps was his vague geometric
conception of
• the electro-tonic state
“a state of tension, or a state of vibration, or perhaps some other
state analogous to the electric current, to which the magnetic forces are so
intimately related.”
<ER> vol. III, p.443
This concept first appeared early, in Section 60, vol. I of
<ER>, but without any
precise definition.
(Sec. 66) All metals take on the peculiar state
(Sec. 68) The state appears to be instantly assumed
(Sec. 71) State of tension
Faraday seemed to be impressed and perplexed by 2 facts:
• that the magnet must be
moved to produce induction.
• that induction often produce effects perpendicular to the cause.
• Faraday was “feeling his way”
in trying to penetrate electromagnetism.
• Today, reading his
<Experimental Researches>, we have to “feel our way” in trying to penetrate his geometric
Faraday seemed to have 2 basic geometric intuitions:
• magnetic lines of force, and
• electrotonic state
The first was easily experimentally seen through sprinkling iron
filings in the field. It is now called H, the magnetic field.
The latter, the electro-tonic state, remained Faraday’s
elusive geometrical intuition when he ceased his
compilation of <ER> in 1854.
He was 63 years old.
• That same year, Maxwell
graduated from Cambridge
University. He was 23 years old.
• In his own words, he
“wish to attack Electricity”.
Amazingly 2 years later Maxwell published the first of his 3 great
papers which founded
Electromagnetic Theory
as a Field Theory.
19.4
• Maxwell had learned from reading Thomson’s mathematical papers the usefulness of
• Studying carefully Faraday’s voluminous <ER> he finally
A
H
• He realized that what Faraday had described in so many
words was the equation:
• Taking the curl of both sides, we get
This last equation is Faraday’s law in differential form. Faraday himself had stated it in words, which tranlates into:
E dl dt d H d
Comment 1 Maxwell used Stokes’
Theorem, which had not yet appeared in the literature. But in the 1854 Smith’s
Prize Exam, which Maxwell had taken as a student, to prove Stokes’ theorem was question #8. So Maxwell knew the
theorem.
Comment 2 Maxwell was well
aware of the importance of his paper 1. To avoid possible controversy with Thomson about the origin of equation
A
H
With respect to the history of the present theory, I may state that the recognition of certain
mathematical functions as expressing the
“electrotonic state" of Faraday, and the use of them in determining electrodynamic potentials and electromotive forces is, as far as I am
aware, original; but the distinct conception of the possibility of the mathematical expressions arose in my mind from the perusal of Prof. W.
5 years later,
1861 paper 2, part I 1861 paper 2, part II 1862 paper 2, part III 1862 paper 2, part IV
19.5
The displacement current first appeared in Part III:
“Prop XIV – To correct Eq. (9) (of Part I) of electric currents for the effect
due to the elasticity of the medium.”
I.e. He added the displacement
Maxwell arrived at this
correction, according to his paper, through the
study of a network of
vortices.
Maxwell took this model seriously and devoted 11 pages to arrive at the
correction.
I made several attempts
to understand these 11
pages. But failed.
With the correction, Maxwell happily arrived at the
momentous conclusion:
“we can scarcely avoid the
inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”
Comment Maxwell was a religious person. I wonder after this momentous discovery, did he in his prayers ask for God’s forgiveness for revealing one of His Greatest Secrets.
19.6
Paper 3 was published in 1865. It had the title: A Dynamical Theory of the Electromagnetic Field. In it we find the formula for energy density:
.
8
1
2 2H E
Its Section (74) we read a very clear exposition of the basic philosophy of Field Theory:
“In speaking of the Energy of the field,
however, I wish to be understood literally.
All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any
other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in the electrified
bodies, conducting circuits, and magnets, in the form of an unknown quality called
“On our theory it resides in the
electromagnetic field, in the space surrounding the electrified and
magnetic bodies, as well as in those bodies themselves, and is in two
different forms, which may be
described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis as the motion
That was historically
The f
irst clear formulation of the fundamental principle ofField Theory
But Maxwell still believed there had to be an
“aethereal medium”:
Comment Throughout his life time, M.
always wrote his equations with the vector potential A playing a key role.
After his death, Heaviside and Hertz gleefully eliminated A.
• But with QM we know now that A has physical meaning. It cannot be eliminated (E.g. A-B effect).
20th Century
Comment Thomson and Maxwell had both discussed what we now call the
gauge freedom in
It was in the 20th century, with the development of QM, that this freedom acquired additional meaning in physics and mathematics, as we shall discuss
A
H
20.1
The first important development in the 20th century in physicists’
understanding of interactions was Einstein’s 1905 special
relativity, according to which:
There is no aethereal medium.
20.2
The next important development was the 1930-1932 discovery of the positron, which led to
Dirac’s sea of negative energy particles, to
QED
QED was very successful in the 1930s in low order
calculations, but was reset
with infinities in higher order
calculations.
20.3
1947-1950 Renormalization
a=(g-2)/2
Accuracy one pair in 10
9!
1950-1970
• Efforts to extend filed theory.
• Efforts to find alternatives of field theory.
• Return to field theory, to
20.4
1919 H Weyl:
“…the fundamental conception on
which the development of Riemann’s geometry must be based if it is to be in agreement with nature, is that of the infinitesimal parallel
displacement of a vector. ...”
If in
infinitesimal displacement of a vector, its direction keepchanging then:
“Warum nicht auch seine
Länge?“ (Why not also its length?)
Based on this idea Weyl
introduced a Streckenfacktor or Proportionalitätsfacktor,
where γ is real
/
exp eA dx
Then in 1925-1926 Fock and
London independently pointed out that in QM
I.e. Weyl’s γ should be imaginary in QM
A
becomes A ,
ie i
e p
20.5
In 1929 Weyl published an important paper,
accepting that γ should be imaginary, arriving at:
(a) A precise definition in QM of gauge
transformation both for EM field, and for wave function of charged particles.
(b) Maxwell equations are invariant consider this combined gauge transformation.
Weyl’s gauge invariance
produced no new experimental results. So for more than 20
years, it was regarded as an elegant formalism but not
essential.
After WWII many new strange particles were found.
How do they interact with
each other?
20.6
This question led to a
generalization of Weyl’s gauge invariance, to a possible new theory of interactions beyond
EM. Thus was born non-Abelian gauge theory.
Motivation for this generalization was concisely stated in a 1954 abstract:
…the electric charge serves as a source of
electromagnetic field; an important concept in this case is gauge invariance which is closely connected with (1) the equation of motion of the electromagnetic field, (2) the existence of a
current density, and (3) the possible
interactions between a charged field and the
Non-Abelian gauge theory was very beautiful, but was not
embraced by the physics community for many years
because it seemed to require the existence of massless
20.7
Starting in the 1960s the concept of spontaneous symmetry breaking was introduced which led to a series of
major advances, finally to a
U(1) x SU(2) x SU(3) gauge theory of electroweak interactions and strong
interactions called the Standard Model.
In the forty some years since 1970 the
international theoretical and experimental
physics community working in “particles and fields” combined their efforts in the
development and verification of this model, with spectacular success, climaxing in the discovery of the “Higgs Boson” in 2012 by
two large experimental groups at CERN, each
Comment Despite its spectacular success, most physicists believe the
standard model is not the final story. One of its chief ingredients, the symmetry breaking mechanism, is a phenomenological
construct which in many respects is similar to the four ψ interaction in Fermi’s beta
decay theory. That theory was also very successful for almost 40 years after 1933.
Entirely independent of developments in
physics there emerged, during the first half of the 20th century, a mathematical theory
called fiber bundle theory, which had diverse conceptual origins: differential forms
(Cartan), statistics (Hotelling), topology (Whitney), global differential geometry
(Chern), connection theory (Ehresmann), etc..
The great diversity of its conceptual origin
20.8
It came as a great shock to both physicists and mathematicians when it became clear in the
1970s that the mathematics of gauge theory, both Abelian and non-Abelian, is exactly the
same as that of fiber bundle theory. But it was a welcome shock as it served to bring back the close relationship between the two disciplines which had been interrupted through the
increasingly abstract nature of mathematics
Comment: In 1975 after learning
the rudiments of fiber bundle theory from my mathematician colleague
Simons, I showed him the 1931 paper by Dirac on the magnetic
monopole. He exclaimed "Dirac had discovered trivial and nontrivial