Something is wrong in the state of QED

Oliver Consa

Department of Physics and Nuclear Engineering, Universitat Politècnica de
Catalunya
Campus Nord, C. Jordi Girona, 1-3, 08034 Barcelona, Spain
E-mail<EMAIL_ADDRESS>
Quantum electrodynamics (QED) is considered the most accurate theory in the
history of science. However, this precision is based on a single experimental
value: the anomalous magnetic moment of the electron (g-factor). An
examination of the history of QED reveals that this value was obtained in a
very suspicious way. These suspicions include the case of Karplus & Kroll, who
admitted to having lied in their presentation of the most relevant calculation
in the history of QED. As we will demonstrate in this paper, the Karplus &
Kroll affair was not an isolated case, but one in a long series of errors,
suspicious coincidences, mathematical inconsistencies and renormalized
infinities swept under the rug.

##  1 Introduction

After the end of World War II, American physicists organized a series of three
transcendent conferences for the development of modern physics: Shelter Island
(1947), Pocono (1948) and Oldstone (1949). These conferences were intended to
be a continuation of the mythical Solvay conferences. But, after World War II,
the world had changed.

The launch of the atomic bombs in Hiroshima and Nagasaki (1945), followed by
the immediate surrender of Japan, made the Manhattan Project scientists true
war heroes. Physicists were no longer a group of harmless intellectuals; they
became the powerful holders of the atomic bomb’s secrets. They were
militarized and their knowledge became a state secret. There was a positive
aspect: the US government created the Atomic Energy Commission (AEC) and
appointed Oppenheimer as its chief advisor. Former members of the Manhattan
Project took control of universities and research centers. They received
generous grants from the government that allowed them to invest in expensive
experimental resources, such as particle accelerators, supercomputers, or
atomic explosion tests. But no one considered the risk to the future of
science.

Former members of the Manhattan Project now enjoyed unlimited credibility.
Their hypotheses were automatically accepted and no one could refute his
theories. Their calculations and experimental data were subject to military
secrecy, and the cost of the equipment necessary to conduct the experiments
was prohibitive for the rest of the international scientific community.
Consequently, the calculations and experiments could no longer be reproduced
independently. Those who accepted the their hypotheses were rewarded with good
jobs at research centers and universities, while those who criticized their
work were rejected and ostracized. The inevitable consequences of this new
situation will soon become apparent.

##  2 Quantum Field Theory (QFT)

###  2.1 Nature is absurd

The acceptance of quantum mechanics meant the acceptance of strange
explanations, such as the wave-corpuscle duality, the uncertainty principle or
the collapse of the wave function.

With the quantization of the electromagnetic field, these strange explanations
became even more confusing, including the polarization of the quantum vacuum,
electrons and photons interacting with their own electromagnetic fields,
particles traveling back in time, the emission and reception of virtual
photons, or the continuous creation and destruction of electron-positron pairs
in a quantum vacuum.

Feynman summarized this new paradigm quite clearly: “It is whether or not the
theory gives predictions that agree with experiment. It is not a question of
whether a theory is philosophically delightful, or easy to understand, or
perfectly reasonable from the point of view of common sense. The theory of
quantum electrodynamics describes Nature as absurd from the point of view of
common sense. And it agrees fully with experiment. So I hope you can accept
Nature as She is: absurd. I’m going to have fun telling you about this
absurdity, because I find it delightful. Please don’t turn yourself off
because you can’t believe Nature is so strange.” [1]

###  2.2 The problem of infinities

After the success of the Dirac equation in 1928, quantum mechanics theorists
attempted to quantify the electromagnetic field by creating the quantum field
theory (QFT). Unfortunately, QFT was a complete failure since any attempted
calculation under this theory resulted in an infinite number.

The only solution the proponents could devise was to simply ignore these
infinities. Many methods can be used to ignore infinities, but the primary
ones are:

  * •

Substitution: replacing a divergent series with a specific finite value that
has been arbitrarily chosen (for example, the energy of an electron).

  * •

Separation: separating an infinite series into two components, one that
diverges to infinity and another that converges to a finite value. Eventually,
the infinite component is ignored and only the finite part remains.

  * •

Cut-off: focusing on an arbitrary term in the evolution of a series that
diverges to infinity and ignoring the rest of the terms of the series.

All these techniques are illegitimate from a mathematical perspective, as
demonstrated by Dirac: “I must say that I am very dissatisfied with the
situation because this so-called ’good theory’ does involve neglecting
infinities which appear in its equations, ignoring them in an arbitrary way.
This is just not sensible mathematics. Sensible mathematics involves
disregarding a quantity when it is small – not neglecting it just because it
is infinitely great and you do not want it!. ”[59]

This technique of ignoring infinities is called renormalization. Feynman also
recognized that this technique is not mathematically legitimate: “The shell
game that we play is technically called ’renormalization’. But no matter how
clever the word, it is still what I would call a dippy process! Having to
resort to such hocus-pocus has prevented us from proving that the theory of
quantum electrodynamics is mathematically self-consistent. It’s surprising
that the theory still hasn’t been proved self-consistent one way or the other
by now; I suspect that renormalization is not mathematically legitimate.” [1]

As an example of the use of these renormalization techniques we can look at
the calculation of the Casimir effect. [3] The equation of the Casimir effect
depends on the Riemann function.

$\frac{F_{c}}{A}=\frac{d}{da}\frac{<E>}{A}=-\frac{\hbar c\pi^{2}}{2a^{4}}\
\zeta(-3)=\frac{\hbar c\pi^{2}}{20a^{4}}\ \zeta(-1)$ (1)

However, the Riemann function is defined only for positive values, since for
negative values the Riemann function diverges to infinity. The Riemann
function of -1 is equal to the sum of all positive integers. Applying a
renormalization technique, the Indian mathematician Ramanujan calculated that
the sum of all positive integers is not infinity but -1/12. [2]

$\zeta(-1)=\sum_{n=1}^{\infty}{n}=1+2+3+4+5+...=\frac{-1}{12}$ (2)

And this is precisely the value that is used in the equation of the Casimir
effect.

$\frac{F_{c}}{A}=\frac{\hbar
c\pi^{2}}{20a^{4}}\left(\frac{-1}{12}\right)=-\frac{\hbar c\pi^{2}}{240a^{4}}$
(3)

##  3 Shelter Island (1947)

###  3.1 The Shelter Island conference

From June 2 to 4, 1947, the first international physics conference after World
War II was held at Shelter Island. The conference brought together 24
physicists from the Manhattan Project, including Bethe, Bohm, Breit, Feynman,
Kramers, Lamb, von Neumann, Pauling, Rabi, Schwinger, Teller, Uhlenbeck,
Weisskopf and Wheeler. Oppenheimer acted as congress master of ceremonies. The
participants were received as celebrities, and the conference made a
significant impact in the press. Despite high expectations, the conference
ended in disappointment.

Fig. 1: Shelter Island Conference participants

Two important experimental measures were presented at the Shelter Island
conference: the Lamb shift and the anomalous magnetic moment of the electron.
Lamb [4] presented an experiment that showed that the 2S1/2 and the 2P1/2
energy levels of the hydrogen atom were not identical; instead they differed
by about 1000 MHz. Rabi’s team [5] presented a 0.1% anomaly in the hyperfine
structure of hydrogen. Later, Breit [6] interpreted this anomaly as the
anomalous magnetic moment of the electron (g-factor).

These two measurements contradicted the Dirac equation. Meeting participants
assumed that Dirac’s theory of the electron was incomplete and proposed that
these effects were due to the quantization of the electromagnetic field. It
was also assumed that these discrepancies could be calculated using the QFT
and that the infinities of this theory could be corrected using
renormalization techniques. This was the origin of Quantum Electrodynamics
(QED).

If Dirac had been at the Shelter Island conference, the story of the QED would
have been very different, as he said: "Renormalization is just a stop-gap
procedure. There must be some fundamental change in our ideas, probably a
change just as fundamental as the passage from Bohr’s orbit theory to quantum
mechanics. When you get a number turning out to be infinite which ought to be
finite, you should admit that there is something wrong with your equations,
and not hope that you can get a good theory just by doctoring up that number.”
[65]

###  3.2 Bethe’s calculation of the Lamb Shift

On the train trip home after the conference ended, Bethe starred in one of the
most epic moments in the history of theoretical physics. As recalled by Bethe:
"I said to myself, well, let’s try to calculate that Lamb shift. And indeed,
once the conference was over, I traveled by train to the General Electric
research lab. And on the train I figured out how much that difference might
be.” [67]

The paper published by Bethe in 1947 [7] was a short three pages where he
proposed this equation for the Lamb shift.

$W_{ns^{\prime}}=\frac{8}{3\pi}\left(\frac{e^{2}}{\hbar c}\right)^{3}Ry\
\left(\frac{Z^{4}}{n^{3}}\right)\ ln\ \frac{K}{<E_{n}-E_{m}>_{Av}}$ (4)

Where Ry is the ionization energy of the ground state of hydrogen (13.6 eV)
and $<E_{n}-E_{m}>_{Av}$ is the average excitation energy for the 2s state of
hydrogen.

In this equation, K is a series that diverges to infinity. Bethe decided to
apply renormalization by substituting this infinite value for the finite value
of the electron’s energy ($K=mc^{2}$). There is no physical justification for
making this change, except that when applying that change, the theoretical
value was in good agreement with the new experimental value.

$W_{ns^{\prime}}=136\ ln\ \left(\frac{K}{kp}\right)=136\ ln\
\left(\frac{m_{e}c^{2}}{17.8\ Ry}\right)=1040\ \mbox{Mhz}$ (5)

According to the paper, "The average excitation energy has been calculated
numerically by Dr. Stehn and Miss Steward.”. [7] They proposed a value of 17.8
Ry (242 eV), a value that Bethe considered "an amazingly high value." [7]
Informally, $ln(k_{p})$ is also known as ’Bethe logarithm’.

As you can see, Bethe’s fantastic calculation is based on data that was
calculated later, so Bethe could not have known it on his train journey. His
calculation included this value that we suspect that was entered ad hoc to
match the theoretical value with the experimental value. In the field of
physics, this trick is known as a ’fudge factor’.

###  3.3 Schwinger’s numerology

A few months after Bethe calculated the value of the Lamb shift, Schwinger
devised an even more epic calculation. He published a one-page paper [8] with
a simple theoretical value for the electron g-factor, just $\alpha/2\pi$ .

$g=1+\frac{\alpha}{2\pi}$ (6) $\begin{cases}&g\ (theor)=1.00116\\\ &g\
(exp)=1.00119\\\ \end{cases}$ (7)

This value, known as the "Schwinger factor", was in good agreement with the
experimental value published by Kush and Foley [9]. Schwinger did not explain
how he got that value because "a paper dealing with the details of this theory
and its applications is in course of preparation." [8]

The Schwinger factor had a significant impact on the scientific community due
to its simplicity and accuracy. Everyone waited expectantly for the fabulous
new theory he had used to calculate this factor. Schwinger’s theory must
signify a revolution in modern physics. But, the days passed, and Schwinger
did not publish his theory.

Why did not he publish this long-awaited theory? We suspect that Schwinger did
not publish the theory because he had no theory. How did he obtain such a
spectacular result without a theory? We suspect that he used a technique known
as numerology. Schwinger assumed that the g-factor should be directly related
to the fine structure constant ($\alpha$), which has an approximate value of
0.7%. Dividing this value by 6 provides an approximate value of 0.1%, which is
the value obtained by Rabi [5]. And $2\pi$ is about 6.

##  4 Pocono (1948)

###  4.1 The Pocono conference

The Pocono conference took place from March 30 to April 2, 1948. This
conference was attended by the same participants as the Shelter Island
conference, as well as three of the greatest physicists of the time: Bohr,
Dirac and Fermi. As with the Shelter Island conference, the expectations were
high due to recent progress from Bethe and Schwinger. And as in the Shelter
Island conference, the results were again disappointing.

The conference expectations were focused on Schwinger’s presentation. Everyone
hoped that he would finally explain the elegant way in which the Schwinger
factor had been calculated.

But Schwinger’s presentation lasted for five unbearable hours and comprised a
series of complex, totally incomprehensible equations. Oppenheimer expressed
his displeasure:  “others gave talks to show others how to do the calculation,
while Schwinger gave talks to show that only he could do it.” [68] Gradually,
the attendees left the presentation until only Bethe and Fermi remained. The
overall feeling was one of disappointment, as it was clear that Schwinger’s
theory was not based on an elegant solution.

The next day, Feynman presented his theory, explaining for the first time his
famous Feynman diagrams. However, the attendees did not respond positively to
this presentation. Feynman was convinced of the validity of his calculations
simply because they produced the correct results.

Bethe remembers the conference like this: “At Pocono, Schwinger and Feynman,
respectively presented their theories. (…) Their theories seemed to be totally
different. (…) Schwinger’s was closely connected to the known quantum
electrodynamics, so Niels Bohr, who was in the audience, immediately was
convinced this was correct. And then Feynman came with his completely new
ideas, which among other things involved positrons going backwards in time.
And Niels Bohr was shocked, that couldn’t possibly be true, and gave Feynman a
very hard time.” [66]

Feynman’s recollection of the conference is also enlightening: “This meeting
at Pocono was very exciting, because Schwinger was going to tell how he did
things and I was to explain mine. (…). We could talk back and forth, without
going into details, but nobody there understood either of us. (…) When he
tried to explain his theory, he encountered great difficulty. (…) As soon as
he would try to explain the ideas physically, the wolves would descend on him,
he had great difficulty. Also, people were getting more and more tired (…) I
didn’t have a mathematical scheme to talk about. Actually I had discovered one
mathematical expression, from which all my diagrams, rules and formulas would
come out. The only way I knew that one of my formulas worked was when I got
the right result from it. (…) I said in my talk: "This is my mathematical
formula, and I’ll show you that it produces all the results of quantum
electrodynamics." immediately I was asked: "Where does the formula come from?’
I said, "It doesn’t matter where it comes from; it works, it’s the right
formula!" "How do you know it’s the right formula?" "Because it works, it
gives the right results!" "How do you know it gives the right answers?" ’ (…)
They got bored when I tried to go into the details. (…) Then I tried to go
into the physical ideas. I got deeper and deeper into difficulties, everything
chaotic. I tried to explain the tricks I had employed. (…) I had discovered
from empirical rules that if you don’t pay attention to it, you get the right
answers anyway, and if you do pay attention to it then you have to worry about
this and that.” [69]

After the disappointing explanations of Schwinger and Feynman, the scientists
returned home, aware of the need for a new unified QED theory that could
elegantly explain Bethe’s Lamb shift results and the Schwinger factor for the
anomalous magnetic moment of the electron.

Upon his return to Princeton, Oppenheimer received a paper from a Japanese
physicist named Tomonaga with a third QED theory. Now, there were three QED
theories, and all of them were inconsistent and incompatible with one other.

###  4.2 Dyson’s series

After the Pocono meeting, the physics community searched for a unified and
covariant QED theory. The person in charge of addressing this problem was a
young English scientist named Dyson. He managed to reconcile the three QED
theories in his paper “The Radiation Theories ofTomonaga, Schwinger, and
Feynman.” [10]

Dyson proposed that the Heisenberg S-matrix could be used to calculate the
electron’s g-factor, transforming it into a series called the Dyson’s series
[11]. The Dyson’s series was an infinite series of powers of alpha, where the
first coefficient was precisely the Schwinger factor, and where each
coefficient could be calculated by solving a certain number of Feynman
diagrams.

$g=1+C_{1}\left(\frac{\alpha}{\pi}\right)+C_{2}\left(\frac{\alpha}{\pi}\right)^{2}+C_{3}\left(\frac{\alpha}{\pi}\right)^{3}+C_{4}\left(\frac{\alpha}{\pi}\right)^{4}+C_{5}\left(\frac{\alpha}{\pi}\right)^{5}...$
(8)

Dyson’s theory based on Feynman diagrams provided the solution his colleagues
were waiting for. The enthusiasm returned to the American scientific
community.

###  4.3 Internal criticism

However, not all scientists were excited about Feynman’s and Dyson’s results.
The primary critic of this new QED theory was Dirac: “How then do they manage
with these incorrect equations? These equations lead to infinities when one
tries to solve them; these infinities ought not to be there. They remove them
artificially. (…) Just because the results happen to be in agreement with
observations does not prove that one’s theory is correct.” [58]

Another critic was Oppenheimer, as Dyson relates: “When after some weeks I had
a chance to talk to Oppenheimer, I was astonished to discover that his reasons
for being uninterested in my work were quite the opposite of what I had
imagined. I had expected that he would disparage my program as merely
unoriginal, a minor adumbration ofSchwinger and Feynman. On the contrary, he
considered it to be fundamentally on the wrong track. He thought adumbrating
Schwinger and Feynman to be a wasted effort, because he did not believe that
the ideas of Schwinger and Feynman had much to do with reality. I had known
that he had never appreciated Feynman, but it came as a shock to hear him now
violently opposing Schwinger, his own student, whose work he had acclaimed so
enthusiastically six months earlier. He had somehow become convinced during
his stay in Europe that physics was in need of radically new ideas, that this
quantum electrodynamics of Schwinger and Feynman was just another misguided
attempt to patch up old ideas with fancy mathematics.” [61]

According to Dyson, Fermi also did not agree with this new way of conducting
science: “When Dyson met Fermi, he quickly put aside the graphs he was being
shown indicating agreement between theory and experiment. His verdict, as
Dyson remembered, was “There are two ways of doing calculations in theoretical
physics. One way, and this is the way I prefer, is to have a clear physical
picture of the process you are calculating. The other way is to have a precise
and self-consistent mathematical formalism. You have neither.” When a stunned
Dyson tried to counter by emphasizing the agreement between experiment and the
calculations, Fermi asked him how many free parameters he had used to obtain
the fit. Smiling after being told “Four,” Fermi remarked, “I remember my old
friend Johnny von Neumann used to say, with four parameters I can fit an
elephant, and with five I can make him wiggle his trunk.” There was little to
add.” [63]

Feynman’s response to these criticisms is well known: “Shut up and Calculate!”
[1]

##  5 Oldstone (1949)

###  5.1 The Oldstone conference

From April 11 to 14, 1949, a third conference was held at Oldstone, with the
same participants as the Shelter Island and Pocono conferences. As on the
previous occasions, the Oldstone conference began with great expectations,
this time based on Dyson’s advances. As with the previous conferences, the
results were disappointing.

The star of the Oldstone conference was Feynman, who used his immense charisma
to present Dyson’s theory as the definitive formalism of the QED theory. From
that moment on, Feynman’s diagrams became a popular tool among American
physicists, and Feynman became the leader of this new generation of
scientists.

In parallel to the QED consolidation, the conference presented important
experimental results on subatomic particles that were called pi-mesons or
pions. These particles had been discovered thanks to the new synchrocyclotron
particle accelerator at the University of Berkeley. Interest in QED rapidly
declined due to its extreme complexity and lack of practical utility, while
the pions became the primary focus. As a result, Oppenheimer decided not to
convene any further QED conferences; instead, he created the International
Conference of High Energy Physics (ICHEP).

###  5.2 Feynman diagrams in QCD

New research in high energy physics resulted in Quantum Chromodynamics (QCD),
the Electroweak Theory (EWT) and the Standard Model of particle physics. All
these developments relied heavily on the use of Feynman diagrams. However, the
Feynman diagrams are only valid when the coupling constant has a very low
value. Obviously if $\alpha>1$, the Dyson’s series would diverge.

But, in the case of fermions, the coupling constant is greater than one, so
the Dyson’s series would diverge. This means that it is not mathematically
legitimate to use Feynman diagrams for these calculations.

In 1951, Feynman himself warned Fermi of this problem: ”Don’t believe any
calculation in meson theory that uses a Feynman diagram.”[70]

###  5.3 Improved Lamb shift calculation

In 1950, Bethe [14] improved the "Bethe logarithm" from 17.8 Ry to 16.646 Ry
and incorporated the new factors of Kroll, Feynman, French and Weisskopf into
the Lamb shift calculation. With these changes, he obtained an improved value
of 1052 Mhz.

$W_{ns^{\prime}}=\frac{\alpha^{3}}{3\pi\ Ry}\left(\ ln\ \frac{mc^{2}}{16.646\
Ry}-ln\ 2+\frac{5}{6}-\frac{1}{5}\right)$ (9)

It was assumed that the difference between the theoretical and experimental
values of the Lamb shift was due to relativistic corrections. These
corrections were calculated by Baranger in 1951 [15]

$\Delta W=\alpha^{4}Ry\left(1+\frac{11}{128}-\frac{ln\ 2}{2}\right)=6.894\
\mbox{Mhz}$ (10)

Baranger obtained a theoretical Lamb shift value of 1058.3 MHz, in good
agreement with the new the experimental value of $1061\pm 2$ Mhz.

##  6 Fourth-order correction (1950-1957)

###  6.1 The Kroll & Karplus calculation

In 1949, Gardner and Purcell obtained a new experimental result for the g
factor of 1.001,146 [12]. With this new experimental value, the Schwinger
factor was no longer considered accurate. Feynman seized this new crisis as an
opportunity to demonstrate the validity of Dyson’s reformulation of the QED,
where the renormalization of infinities could be performed consistently.

In 1950, Karplus and Kroll [13] completed these complex calculations and
published a value of -2,973 for the second coefficient in the Dyson series.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-2.973\left(\dfrac{\alpha}{\pi}\right)^{2}\\\
&g\ (theor)=1.001,147\\\ &g\ (exp)=1.001,146\\\ \end{cases}$ (11)

For the second time, the new theoretical value was in good agreement with the
new experimental value.

As indicated in the paper, “The details of two independent calculations which
were performed so as to provide some check of the final result are available
from the authors.” [13] The calculations had been performed independently by
two teams of mathematicians who had obtained the same result. Therefore, it
was impossible that there were any errors in the calculations. Nor was it
possible to imagine that a theoretical result that was identical to the
experimental result could have been achieved by chance.

This was the definitive test. QED had triumphed. Feynman’s prestige
dramatically increased and he began to be mentioned as a candidate for the
Nobel Prize.

###  6.2 Dyson’s divergence

In 1952, two years after this great success, Dyson published a paper entitled
“Divergence of Perturbation Theory in Quantum Electrodynamics” where he
states: “An argument is presented which leads tentatively to the conclusion
that all the power-series expansions currently in use in quantum
electrodynamics are divergent after the renormalization of mass and charge.”
[16]

The creator of the QED theory stated that his Dyson’s series was divergent. As
Dyson admitted: "That was of course a terrible blow to all my hopes. I really
meant that this whole program made no sense." [62] After the publication of
this paper, Dyson moved to England, abandoned this line of research and
dedicated the rest of his career to other areas of physics.

Surprisingly, Dyson’s claim that the series was divergent did not diminish
QED’s credibility.

###  6.3 The Witch-Hunt

In 1949, the USSR had obtained the atomic bomb thanks to information provided
by Fuchs, a Manhattan Project researcher with communist sympathies. Senator
McCarthy began a witch hunt where espionage accusations became widespread
among the American scientific community. Oppenheimer was accused of espionage
and treason. He was tried and found not guilty, but his position as chief
advisor at the AEC was removed.

Now the scientists’ own lives were in jeopardy. Definitely, 1952 was a bad
year to admit the complete failure of the main theoretical research program in
modern physics.

The witch hunt ended in 1957, when the Russians sent the Sputnik satellite
into space and the US government realized that it needed scientists to create
NASA and win the space race.

###  6.4 The infamous paper

In 1956, Franken and Liebes [17] published new, more precise experimental data
that provided a very different g-factor value (1.001,165). This value was
higher than the Schwinger factor, so the value of the second coefficient that
had been calculated by Kroll and Karplus not only did not improve the
Schwinger factor; it made the calculation worse.

With the new experimental data, the value of the second coefficient of the
series should have been +0.7 instead of -2.973. The difference between these
values was huge and unjustifiable. The probative force of QED was upended. In
addition, there was no explanation for why Kroll and Karplus’s calculation
provided the exact expected experimental value when that value was incorrect.
It was evident that the QED calculations had matched the experimental data
because they were manipulated. It was a fraud, a scandal.

Karplus and Kroll confessed that they had not independently reached the same
result; instead, they had reached a consensus result. Therefore, it was
possible that there were errors in the calculation.

According to Kroll: “Karplus and I carried out the first major application of
that program, to calculate the fourth order magnetic moment, which calculation
subsequently turned out to have some errors in it, which has been a perpetual
source of embarrassment to me, but nevertheless the paper I believe was quite
influential. (…) The errors were arithmetic (…) We had some internal checks
but not nearly enough. (…) it was refereed and published and was a famous
paper and now it’s an infamous paper.”[64]

Feynman’s version of these events does not fully correspond to reality: “It
took two ‘independent’ groups of physicists two years to calculate this next
term, and then another year to find out there was a mistake - experimenters
had measured the value to be slightly different, and it looked for a while
that the theory didn’t agree with experiment for the first time, but no: it
was a mistake in arithmetic. How could two groups make the same mistake? It
turns out that near the end of the calculation the two groups compared notes
and ironed out the differences between their calculations, so they were not
really independent”. [1]

###  6.5 Petermann’s solution

Petermann [20] detected an error in the Kroll and Karplus calculations (one
that no one had detected in the seven years since the article was published).
He made the correct calculation and obtained a result of -0.328, which was
almost ten times lower than the previous calculation.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}\\\
&g\ (theor)=1.001,159,6\\\ &g\ (exp)=1.001,165\\\ \end{cases}$ (12)

The same error was independently detected by Sommerfield [21]. Once again, two
independent calculations provided the same theoretical value.

For the third time, the new theoretical value was in good agreement with the
new experimental value.

##  7 The Unpublished Feynman diagram IIc

###  7.1 Karplus and Kroll’s paper

At this point, we have doubts about everything that was reported, so we
reviewed the original article published by Kroll and Karplus as well as the
corrections of Petermann and Sommerfield published seven years after.

Fig. 2: Feynman diagrams for the fourth-order corrections

Karplus and Kroll’s paper consists of 14 pages full of complex mathematical
calculations[13]. On the second page of the document, the authors indicate
that to obtain the coefficient it is necessary to calculate 18 Feynman
diagrams grouped in five groups (I, II, III, IV and V). However, on pages 3
and 4, they argue that groups III, IV and V are not necessary.

Therefore, it is only necessary to calculate seven Feynman diagrams,
identified as I, IIa, IIb, IIc, IId, IIe, IIf. A lot of calculations are done
between pages 4 and 11 that only serve to show that diagrams IIb and IIf are
not necessary either. Therefore, it is only necessary to calculate five
Feynman diagrams (I, IIa, IIc, IId, IIe).

The calculation of diagrams IIe (0.016) and IId (-0.090) are performed on
pages 11 and 12 respectively. It follows that: “The expressions for I, IIa and
IIc become successively more complicated and very much more tedious to
evaluate and cannot be given in detail here.” [13] In other words, the
complete calculation of three of the five diagrams was never published. On
page 13, the results of the three remaining diagrams are shown (I = -0.499,
IIa = 0.778 and IIc = -3.178). Finally, page 14 of the paper presents the
"Summary of Results" with the results of each of the five diagrams.

$C_{2}=I+IIa+IIc+IId+IIe=-2,973$ (13)

I | IIa | IIc | IId | IIe | Total
---|---|---|---|---|---
-0.499 | 0.778 | -3.178 | -0.090 | 0.016 | -2.973
Table 1:  Values of the five Feynman diagrams.

Diagrams IId and IIe were the only diagrams whose calculations are included in
the document; however, its values were completely irrelevant. Diagrams I and
IIa practically cancel each other out. IIc was the dominant diagram and
consisted of four components:

$II_{c}=-\frac{323}{24}+\frac{31}{9}\pi^{2}-\frac{49}{6}\pi^{2}ln(2)+\frac{107}{4}\zeta(3)=-3,178$
(14)

Constant | $\pi^{2}$ | $\pi^{2}ln2$ | $\zeta(3)$ | Total
---|---|---|---|---
-13,458 | 33,995 | -55,868 | 32,153 | -3,178
Table 2:  Value of the four components of Feynman diagram IIc.

The four components of IIc have abnormally high values
(-13, 34, -55 and 32) which surprisingly compensate for each other, resulting
in -3,178, an order of magnitude lower. It is not possible to say anything
more about the calculation of diagram IIc because the complete calculation was
never published.

###  7.2 Petermann’s numerical calculation

Petermann was the first person to identify an error in the original
calculation of Karplus and Kroll. He performed a numerical analysis of the
five Feynman diagrams. He found that the solution of diagram IIc was clearly
wrong, since its value was outside the limits. The rest of the diagrams were
within limits: “The numerical results for the terms I, IIa, IIc, IId, IIe in
the work by Karplus and Kroll have been checked by rigorous upper and lower
bounds. Whereas every other term fell well between these bounds, agreement
could not be obtained for diagram IIc. (…) The numerical value for this term
has been found to satisfy IIc = -1.02 +/- 0.53.” [18]

Petermann published a second paper where he adjusted his calculations: ”the
diagram IIc is found to satisfy IIc = -0.60 +/- 0.11 in contradiction with the
value -3.18 given by the previous authors.” [19]

Between the publication of these two papers, Petermann communicated privately
to Sommerfield the result of another calculation: "Petermann has placed upper
and lower bounds on the separate terms of Karplus and Kroll. He finds that
their value for IIc does not lie within the appropriate bounds. Assuming the
other terms to be correct, he concludes that the result is -0.53 +/- 0.37.”
[21]

Petermann worked for three months following a numerical methodology that
allowed him to narrow the margin of error in diagram IIc. Surprisingly,
fourteen days after his third numerical calculation, he made an unexpected
change in his methodology and published the exact analytical calculation, with
no margins of error.

The articles published by Petermann on the calculation of the Feynman diagram
IIc are summarized in the following table:

Date | IIc | Method |  Publication
---|---|---|---
28/5 | -1.02 +/- 0.53 | Numerical |  Nuclear Phys. 3
1/7 | \- 0.53 +/- 0.37 | Numerical |  Phys. Rev. 107, Note added in proof.
3/8 | -0.60 +/- 0.11 | Numerical |  Nuclear Phys. 5
17/8 | -0,564 | Analytical |  Helvetica Physica Acta 30
Table 3:  Petermann’s publications.

###  7.3 Sommerfield and the Green’s functions

After the publication of the new experimental value by Franken and Liebes
[17], Schwinger commissioned a 22-year-old student named Sommerfield to redo
the Kroll and Karplus calculations. Schwinger proposed using his own method
based on Green’s functions instead of using Feynman diagrams. According to
Sommerfield: "Julian assigned us three problems, one of which involved the
anomalous magnetic moment (…). At my meeting with him, he suggested that I
continue the calculation of the anomalous magnetic moment to the next fourth
order (…). Schwinger wanted me to use the other method, while respecting gauge
invariance at every step. Many years later Roy Glauber told me that the
faculty was not entirely happy that a graduate student had been given such a
problem." [23]

In May 1957, Sommerfield sent a two-page paper to the Physical Review Journal
where he published his results:

$C_{2}=\frac{197}{144}+\frac{1}{12}\pi^{2}-\frac{1}{2}\pi^{2}ln(2)+\frac{3}{4}\zeta(3)=-0,328$
(15)

Sommerfield’s paper does not include the calculations performed, but the
author states that: “The present calculation has been checked several times
and all of the auxiliary integrals have been done in at least two different
ways.” [21] As a guarantee that the calculations were correct.

He also states that: “The discrepancy has been traced to the term I y IIc of
Karplus and Kroll.” [21] But this statement about the origin of the error
cannot be deduced from Sommerfield’s calculations, since he used Green’s
functions instead of Feynman diagrams.

In 1958 Sommerfield published the full calculation of the g-factor
calculations in an extensive 32-page paper as part of his doctoral thesis
[22]. He used Green’s functions instead of Feynman diagrams. Then, the
calculation of the enigmatic Feynman diagram IIc does not appear in his paper.

As Schwinger states: “Interestingly enough, although Feynman-Dyson methods
were applied early [by Karplus and Kroll], the first correct higher order
calculation was done by Sommerfield using [my] methods.”.[66] In the third
volume of "Particles, Sources, and Fields" published in 1989 [24], Schwinger
devoted more than 60 pages to a detailed calculation of the second coefficient
of Dyson series getting exactly the same result. But, once again, using
Green’s functions instead of Feynman diagrams.

###  7.4 Petermann’s definitive correction

The definitive solution to the problem was presented in 1957 by Petermann in a
paper published in the Swiss journal Helvetica Physica Acta [20].

The article was signed by a single author due to an internal conflict between
the researchers. As Sommerfied recalls: "In the meantime Schwingerian Paul
Martin had gone to the Niels Bohr Institute in Copenhagen and had spoken to
Andre Petermann, a postdoc with the Swedish theoretician Gunnar Kallen. Martin
told Petermann about my work (…) In the end, however, after both of our
calculations were completely finished they were in agreement with each other
but not with Karplus and Kroll. We agreed to cite each other’s work when
published. However, Schwinger and Kallen had had a somewhat acrimonious
discussion (…) and Kallen had forbidden Petermann to mention my work.
Petermann’s apology to me was profuse.". [23] Although the paper was signed by
a single author, Petermann acknowledges that the result was obtained by
consensus: “The new fourth order correction given here is in agreement with:
(a) The upper and lower bounds given by the author. (b) A calculation using a
different method, performed by C. Sommerfield. (c) A recalculation done by N.
M. Kroll and collaborators.”. [20] Petermann’s final result was identical to
the Sommerfield’s result published three months earlier.

$C_{2}=I+IIa+IIc+IId+IIe=-0,328$ (16)

I | IIa | IIc | IId | IIe | Total
---|---|---|---|---|---
-0.467 | 0.778 | -0.564 | -0.090 | 0.016 | -0.328
Table 4:  Corrected values of the five Feynman diagrams.

$II_{c}=-\frac{67}{24}+\frac{1}{18}\pi^{2}+\frac{1}{3}\pi^{2}ln(2)-\frac{1}{2}\zeta(3)=-0,564$
(17)

The following table compares the calculations of the four components of the
Feynman IIc diagram made by Karplus and Kroll (Equation 14) with the
calculations made by Petermann (Equation 17).

| Const. | $\pi^{2}$ | $\pi^{2}\ ln(2)$ | $\zeta(3)$ | Total
---|---|---|---|---|---
K&K | -13,458 | 33,995 | -55,868 | 32,153 | -3,178
P&S | -2,791 | 0,548 | 2,280 | -0,601 | -0,564
Diff. | 10,667 | -33,447 | 58,148 | -32,754 | 2,614
Table 5:  Comparative components of Feynman diagram IIc.

The corrections are huge, one or two orders of magnitude for each component of
diagram IIc. We cannot know the origin of these discrepancies because the
correction calculations were also not published.

###  7.5 Smrz & Uleha

In 1960, Smrz & Uleha published a short paper of two pages where the situation
generated in 1957 by Petermann’s correction is explained. The authors states
that they performed an independent calculation of the Feynman IIc diagram and
obtained exactly the same result as Petermann. “Since the considerable
difference between the original value of the magnetic moment (Karplus & Kroll
[13]) and the values calculated later (Petermann [20]) originates in the
calculation of the contribution from the third diagram, only the value of this
contribution was determined by the standard technique and the above
regularization in the infra-red region. The contribution from the third
diagram (-0.564) is in complete agreement with Petermann’s value.” [26]

Unfortunately when looking for the reference of the work with the
calculations, it has not been published either: "Smrz P.: Diploma thesis,
Faculty of Tech. and Nucl. Physics, Prague 1960, unpublished." [26]

###  7.6 Terentiev

In 1962, Terentiev published a long paper of 50 pages [27]. The paper is only
in Russian and there is no English translation. We identify the equation 60 of
the paper as the second coefficient of the Dyson’s series, with the same
expression and value obtained by Petermann. Analyzing the document, we
interpret that this equation is the result of the sum of nine other equations
identified as equations 22, 24, 27, 31, 33, 47, 51, 58 and 59. There are nine
equations instead of the five Feynman diagrams of Karplus and Kroll and none
of the these equations correspond to the Feynman Diagram IIc.

###  7.7 Barbieri & Remiddi

In 1972, Barberi & Remeddi published a long 93-page paper where they performed
a recalculation of the Feynman diagrams corresponding to the fourth-order
coefficient and confirmed the results obtained by Petermann. The authors give
the descriptive name of "Corner Graphs" to the the Feynman IIc. On the
specific calculations, the authors state the following: "Integrations by
parts, differentiations and so on, was done by computer, using the program
SCHOONSCHIP of Veltman.” [25] That is, they used a computer program to perform
the mathematical calculations, but they did not publish the source code used,
so, again, it is not possible to replicate the calculations.

On the first page of Barbieri & Remiddi paper there is a reference to
Terentiev’s paper, where the authors claim that Terentiev’s results were
incorrect and manipulated: “Dispersion relations are used in the Terentiev
work only to write down suitable multiple integral representations, which are
in general manipulated to get the final result, without explicitly evaluating
the discontinuities. The problem of infra-red divergences has been further
overlooked, and many of the intermediate results are wrong, even if somewhat
ad hoc compensations make the final result correct.” [25]

###  7.8 Summary of the situation

The history of this calculation is surrounded by big errors and inexplicable
coincidences.

  * •

The original calculation of the Feynman diagram IIc published in 1950 was
completely wrong.

  * •

Karplus and Kroll stated that the calculation had been performed by two teams
independently. This statement was made to give guarantees about the validity
of the calculations, and yet it turned out to be false.

  * •

Despite having published a completely wrong result, the prestige of Karplus
and Kroll was not affected at all. On the contrary, both enjoyed brilliant
careers full of awards and recognition for their professional achievements.

  * •

Karplus and Kroll’s miscalculation was consistent with the experimental value
previously published by Gardner and Purcell, even though that experimental
value was also wrong.

  * •

The error in the calculation was not reported until seven years after its
publication.

  * •

The error in the calculation was detected just when a new experimental value
was published by Franken and Liebes. The corrected theoretical value also
coincided with the new experimental value.

  * •

Neither the original calculation of the Feynman diagram IIc nor its subsequent
correction has been published to date.

  * •

The recalculations of the Feynman diagram IIc of Smrz & Uleha and Barbieri &
Remiddi have not been published either.

  * •

The recalculation of Terentiev were revisited ten years later by Remmidi, who
claims that these calculations were wrong and manipulated with ad-hoc
compensations to make the final result correct.

Year | Author |  Status of calculations
---|---|---
1950 | Karplus & Kroll |  Wrong and Unpublished
1957 | Petermann |  Right but Unpublished
1957 | Sommerfield |  Right but Unpublished
1958 | Sommerfield |  Right but using Green’s Functions instead of Feynman Diagrams
1960 | Smrz & Uleha |  Right but Unpublished
1962 | Terentiev |  Wrong intermediate results with ad hoc compensations to make the final result correct
1972 | Remiddi |  Right but Unpublished Computer calculation
1989 | Schwinger |  Right but using Green’s Functions
Table 6:  fourth-order coefficient calculation.

##  8 The Nobel Prize (1965)

###  8.1 Direct g-factor measurement

In 1953, a research team from the University of Michigan [28] proposed a new
type of experiment to calculate the magnetic moment of the electron directly
from the precession of the free electron spin. This new technique provided
more precise experimental values than the previous techniques that were based
on atomic levels.

In 1961, Schupp, Pidd and Crane carried on the experiment and published a new
experimental value of 1.0011609. The experiment was revolutionary because of
the measured precision, however, the authors were cautious with their results,
presenting large margins of error. The explanation for this strange decision
is found in the paper: “In deciding upon a single value for a to give as the
result of the experiment, our judgement is that we should recognize the trend
of the points (…). The value a=0.0011609, obtained in this way, may be
compared with a simple weighted average of the data of Table IV, which is
0.0011627. We adopt the value 0.0011609 but assign a standard error which is
great enough to include the weighted average of Table IV, namely
$\boldsymbol{\pm 0.0000020}$. Finally, we combine with this the estimated
systematic standard errors (…). This results in a final value of
$\boldsymbol{0.0011609\pm 0.0000024}$”. [29]

Fig. 3: Table IV, the g-factor anomaly calculated for the various electron
energies

According to this explanation, the estimated systematic standard error was
0.0000004. If this error had been published, the result would have been
$1.0011609\pm 0.0000004$, leaving Petermann’s theoretical value outside the
margin of error and creating a new crisis in the development of QED.

The authors proposed another possible approach: they averaged the measurements
in Table IV, generating a result of $1.0011627\pm 0.0000024$. But this
alternative result also left out of the margin of error the Petermann’s
theoretical value.

Finally, the authors published a meaningless result. Although they published
what they considered to be the correct result (1.0011609), they added a margin
of error of +0.0000024 to include the average of the actual results. They also
added a negative symmetrical margin of error of -0.0000024, without any
logical; this was the only way to keep Petermann’s theoretical value within
the margin of error.

###  8.2 The experimenter’s bias

At that time the situation was dramatic again. Predictably, subsequent
experiments would discredit the g-factor theoretical value. And after the
Kroll and Karplus scandal, the theoretical calculations could not be modified
again to adapt them to the experimental data without completely distorting the
QED.

And the moment come in 1963, Wilkinson and Crane published a improved version
of the experiment. In the report of the results all the previous cautionary
language disappeared. The accuracy of this result was presented as 100 times
higher than that of the previous experiment, and the tone of the paper was
blunt: “mainly for experimental reasons, we here conclude the 10-year effort
of the laboratory on the g factor of the free negative electron.” [30]

Just when QED seemed doomed to disaster, the miracle happened again. This
time, the new experimental value was $1,001,159,622\pm 0.000,000,027$, nearly
the same as Petermann’s theoretical value (1,001,159,615).

Fig. 4: Experimental values

This experimental result is incredibly suspicious. It was obtained after a
simple improvement of the previous experiment, and it was conducted at the
same University, with the same team, only two years later. It is extremely
strange that all the measurements from the previous experiment were outside
the range of the new experimental value. Even stranger, the theoretical value
fit perfectly within the experimental value. Most disturbing, this value is
not correct, as was demonstrated in later experiments.

###  8.3 The Nobel Prize

In 1965, Drell and Pagels [31] published the first calculation of the third
coefficient of the Dyson’s series (sixth order correction), that implied
solving 72 Feynman diagrams. The result was 0.15, which allowed to slightly
improve the theoretical value of the electron g-factor.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+0.15\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&g\ (theor)=1,001,159,617\\\ &g\ (exp)=1,001,159,622\\\ \end{cases}$ (18)

For the fourth time, the new theoretical value was in good agreement with the
new experimental value. At this point all doubts about QED were cleared and
Feynman, Schwinger and Tomonaga were awarded the Nobel Prize in physics in
1965.

##  9 Sixth-order Correction (1968-1972)

###  9.1 Wesley and Rich, preliminary

In 1968, Rich [34] reevaluated the Wilkinson’s experiment and obtained a
different result. Other researchers found more errors in the experiment which
allowed different interpretations of the results. It became clear that the
Wilkinson’s experiment had to be repeated to fix these discrepancies.

In 1970, Rich and Wesley [33] repeated Wilkinson’s experiment, fixing the
detected ambiguities and obtaining a result higher that the previous one.

The same year, Brodky & Drell [32] recalculated the sixth-order factor and
obtained a value of +0.55, three times higher that the previous value.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+0.55\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&g\ (theor)=1,001,159,644\\\ &g\ (exp)=1,001,159,644\\\ \end{cases}$ (19)

For the fifth time, the new theoretical value was in good agreement with the
new experimental value.

Brodky and Drell summarized the situation: “Quantum electrodynamics has never
been more successful in its confrontation with experiment than it is now.
There is really no outstanding discrepancy despite our pursuing the limits of
the theory to higher accuracy (…) however, and despite its phenomenal success,
the fundamental problems of renormalization in local field theory and the
nature of the exact solutions of quantum electrodynamics are still to be
resolved.”[32]

Starting in the 1970s, all the mathematical calculations necessary to obtain
the coefficients of the Dyson series were performed by computer. No source
code for these calculations has been published, so it is not possible to
reproduce any calculations independently.

###  9.2 Wesley and Rich, definitive

The results published by Wesley and Rich were preliminary. In 1971 they
published the final results of the experiment that turned out to be higher
than expected.[35]

The same year, Levine & Wright [37] recalculated the sixth-order factor and
obtained a value of +1.49, three times higher that the previous value.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.49\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&g\ (theor)=1,001,159,655\\\ &g\ (exp)=1,001,159,657\\\ \end{cases}$ (20)

For the sixth time, the new theoretical value was in good agreement with the
new experimental value.

Wesley and Rich summarized the situation: "The agreement between the
experimental measurements and QED predictions of the electron g-factor at a
level of four parts per billion represents five most accurate comparison
between theory and experiment in physics. In spite of the unsatisfactory
nature of the renormalization aspects of the theory, there has been no
significant evidence for a breakdown of QED”. [36]

A few months later, Kinoshita & Cvitanovic [39] published a new calculation
for the sixth-order factor, five times more accurate than Levine & Wright’s
previous one. However, this new theoretical value (+1.29) slightly worsened
the agreement with the experimental value.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.29\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&g\ (theor)=1,001,159,653\\\ &g\ (exp)=1,001,159,657\\\ \end{cases}$ (21)

###  9.3 Cvitanovic’s conjecture

According to Dyson, the value of the coefficients of the series should grow
exponentially, but this did not happen with the coefficients calculated up to
that moment. Cvitanovic observed that calculating the coefficients of the
Dyson’s series produced many more cancellations than expected. In 1977 he
proposed a conjecture [40] according to which the calculation of the
coefficients would remain at values close to one. So far, Cvitanovic’s
conjecture has not been proven nor has Dyson’s argument been shown to be
wrong.

On the other hand, in 1978, the Nobel laureate ’t Hooft presented another
argument, completely different from Dyson’s argument, but with the same
consequences. According to’ t Hooft: “We understand how to renormalize the
theory to any finite order in the perturbation expansion, but it is expected
that this expansion will diverge badly, for any value of the coupling
constant." [41]

###  9.4 Landau pole

The Landau pole (or the Moscow zero, or the Landau ghost) is the momentum
scale at which the coupling constant of a quantum field theory becomes
infinite. Such a possibility was pointed out by Landau in 1955. [42]

The asymptotic freedom of QCD was discovered in 1973 by Gross and Wilczek, but
asymptotic freedom is not applicable to QED. Landau poles appear in theories
that are not asymptotically free, such as QED. In these theories, the
renormalized coupling constant grows with energy. In a theory purporting to be
complete, this could be considered a mathematical inconsistency.

##  10 Eight-order Correction (1977-2008)

###  10.1 The Penning trap

In 1977, Van Dyck and Dehmelt of the University of Washington used a new
technique known as free electron spin resonance. These measurements were based
on a device called a Penning trap, which allowed measurements to be obtained
from individual electrons. These experiments improved the previous results by
three orders of magnitude, and, again, the new result excluded previous
theoretical value. [43]

To resolve this new discrepancy, the theoretical physicists needed to
calculate the the fourth coefficient (eighth-order correction), which involved
solving 891 new Feynman diagrams. In 1981, Kinoshita & Lindquist [52]
published the first calculation of the eight-order coefficient with a value of
-0.8.

In 1982, Levine [38] published a new calculation of the sixth-order
coefficient with a value of +1.176, lower that the previous one.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.176\left(\dfrac{\alpha}{\pi}\right)^{3}-0.8\left(\dfrac{\alpha}{\pi}\right)^{4}\\\
&g\ (theor)=1.001,159,652,460\\\ &g\ (exp)=1.001,159,652,410\\\ \end{cases}$
(22)

For the seventh time, the new theoretical value was in good agreement with the
new experimental value.

###  10.2 Feynman’s Book

In 1981, Van Dyck and Dehmelt [44] published a second experimental value,
lower that the previous one.

$\begin{cases}&g\ (theor)=1.001,159,652,460\\\ &g\ (exp)=1.001,159,652,222\\\
\end{cases}$ (23)

Just these numbers were used in 1985 by Feynman in his famous book titled
"QED: The Strange Theory of Light and Matter". On page 7 of the book, Feynman
shows this data and says: "At the present time I can proudly say that there is
no significant difference between experiment and theory! (…) To give you a
feeling for the accuracy of these numbers, it comes out something like this:
If you were to measure the distance from Los Angeles to New York to this
accuracy, it would be exact to the thickness of a human hair." [1]

However, at the time of publication of Feynamn’s book there was a real
discrepancy between the theoretical and the experimental value that remained
for a decade. Van Dyck and Dehmelt [45] published another two new experimental
values that increased that discrepancy:

  * •

$[1984]:1.001,159,652,193$ [45]

  * •

$[1987]:1.001,159,652,188,4$ [46]

###  10.3 Analytical sixth-order calculation

In 1995, Kinoshita’s team published a new value of the eighth-order correction
(-1.557), the double of his first calculation. [53]. And in 1996, Laporta and
Remiddi [47] published the analytical calculation of sixth-order (+1.181).
This definitive value was eight times higher than the initial calculation of
Drell (+0.15).

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.181\left(\dfrac{\alpha}{\pi}\right)^{3}-1.557\left(\dfrac{\alpha}{\pi}\right)^{4}\\\
&g\ (theor)=1.001,159,652,201,2\\\ &g\ (exp)=1.001,159,652,188,4\\\
\end{cases}$ (24)

For the eighth time, the new theoretical value was in good agreement with the
new experimental value.

###  10.4 Muon’s g-factor

QED was also used to calculate the anomalous magnetic moment of the muon. The
muon is an unstable subatomic particle with a mean lifetime of $2.2\ \mu s$,
making high resolution measurements extremely complex. From 1961 to 1976, CERN
made the first measurements of the muon g-factor. The following experiment was
carried out at Brookhaven National Laboratory (BNL). The experiment was named
E821 and the results were published from 1997 to 2001. Unfortunately, the
theoretical value did not match the new experimental value.

New factors were added to adjust the theoretical final result. These new
factors came from empirical data obtained from the Standard Model of particle
physics. The first coefficient was derived from the interaction of the
electron with leptons, the second coefficient was derived from the electroweak
interaction and the third coefficient was derived from the electron’s
interaction with hadrons.

$g=g\ (QED)+g\ (noQED)$ (25) $g\ (noQED)=g\ (\mu,\tau)+g\ (weak)+g\ (hadron)$
(26)

The inclusion of these new coefficients in the g-factor calculation improved
the theoretical muon g-factor value but worsened the theoretical electron
g-factor value. This change created a new discrepancy between the theoretical
and experimental value of the electron g-factor.

###  10.5 Harvard experiment

In 2006 a team from Harvard University led by Gabrielse improved the
experimental results of Van Dyck and Dehmelt by two orders of magnitude. The
Harvard University data were not compatible with previous experimental data
provided by the University of Washington. These new data also excluded the
theoretical value of the g-factor.

  * •

$[2006]:1.001,159,652,180,85(76)$ [48]

  * •

$[2008]:1.001,159,652,180,73(28)$ [49]

Fig. 5: Harvard vs Washington errors Fig. 6: electron g-factor errors

In 2007, Kinoshita’s team detected an error in his previous calculation of the
eighth-order: "Comparing the contributions of individual diagrams of old and
new calculations, we found an inconsistency in the old treatment of infrared
subtraction terms in two diagrams. Correcting this error leads to the revised
value -1.914 for the eight-order term." [54]

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.181\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&-1.914\left(\dfrac{\alpha}{\pi}\right)^{4}+g\ (noQED)\\\ &g\
(theor)=1.001,159,652,182.79\\\ &g\ (exp)=1.001,159,652,180,73\\\ \end{cases}$
(27)

For the ninth time, the new theoretical value was in good agreement with the
new experimental value.

##  11 Tenth-order correction (1997-2021)

###  11.1 Tenth-order correction

Every new calculations of $g\ (noQED)$ allowed to reduce the discrepancy of
the muon g-factor but worsened the discrepancy of the electron g-factor. To
resolve this new discrepancy, the theoretical physicists needed to resolve the
12,672 Feynman diagrams of the tenth-order corrections.

In 2012, Kinoshita’s team published the first tenth-order coefficient with a
value of +9.16, and in the same paper, they published an improved value of
-1.909 for the eighth-order contribution. [55]

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.181\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&-1.909\left(\dfrac{\alpha}{\pi}\right)^{4}+9.16\left(\dfrac{\alpha}{\pi}\right)^{5}+g\
(noQED)\\\ &g\ (theor)=1.001,159,652,181,78\\\ &g\
(exp)=1.001,159,652,180,73\\\ \end{cases}$ (28)

For the tenth time, the new theoretical value was in good agreement with the
new experimental value.

In 2015, Kinoshita’s team published an improved value of the tenth-order
coefficient with a value of +7.795, and in the same paper, they published an
improved value of -1.912 for the eighth-order contribution. [56]

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.181\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&-1.912\left(\dfrac{\alpha}{\pi}\right)^{4}+7.795\left(\dfrac{\alpha}{\pi}\right)^{5}+g\
(noQED)\\\ &g\ (theor)=1.001,159,652,181,64\\\ &g\
(exp)=1.001,159,652,180,73\\\ \end{cases}$ (29)

For the eleventh time, the new theoretical value was in good agreement with
the new experimental value.

In 2017, Laporta [50] published his final calculation of the eighth-order
coefficient of the Dyson’s series with a value of -1.912, the double of the
initial estimation (-0.8). The published value had an unnecessary accuracy of
1100 digits, as a proof that this numerical calculation can be considered the
definitive result.

Fig. 7: First 1100 digits of A8

In 2018, Kinoshita’s team detected an error in his previous calculation of the
tenth-order: “we found that one of the integrals, called X024, was given a
wrong value in the previous calculation due to an incorrect assignment of
integration variables.” [57] The new value was +6.675.

$\begin{cases}&g=1+\dfrac{\alpha}{2\pi}-0.328\left(\dfrac{\alpha}{\pi}\right)^{2}+1.181\left(\dfrac{\alpha}{\pi}\right)^{3}\\\
&-1.912\left(\dfrac{\alpha}{\pi}\right)^{4}+6.675\left(\dfrac{\alpha}{\pi}\right)^{5}+g\
(noQED)\\\ &g\ (theor)=1.001,159,652,182,032(720)\\\ &g\
(exp)=1.001,159,652,180,73(28)\\\ \end{cases}$ (30)

For the twelfth time, the new theoretical value was in good agreement with the
new experimental value.

###  11.2 The muon anomaly

Despite the enormous effort made in recent times, the discrepancy between the
theoretical value and the experimental value of the muon g-factor could not be
eliminated, maintaining an error greater than 3 sigmas. Theoretical physicists
are concerned about this discrepancy, as it is perhaps the most palpable
evidence that the Standard Model is incomplete.

$\begin{cases}&g\ (theor)=1.001,165,918.04(51).\\\ &g\
(exp)=1.001,165,920.9(6)\\\ \end{cases}$ (31)

Fig. 8: Muon anomaly

In 2011, the E989 experiment was devised to improve the accuracy of the E821
experiment. This extremely complex experiment was performed at the Fermilab’s
Tevatron. Before the experiment could be conducted, a gigantic magnet (15
meters in diameter and 600 tons in weight) had to be moved 1300 km, from BNL
to Fermilab. This delicate operation was successfully performed in June 2013.
The magnet transfer lasted 35 days and cost 3 million dollars.

Fig. 9: Transportation of the 600 ton magnet to Fermilab

In addition, the Fermilab particle accelerator had to be enlarged. The related
investment plan, the PIP-II Reference Design Report, had an estimated cost of
600 million dollars and was approved in July 2018. The results of E989
experiment were published in 2021, but they confirmed the anomaly once again.

##  12 Summary

According Feynman: “We have found nothing wrong with the theory of quantum
electrodynamics. It is, therefore, I would say, the jewel of physics; our
proudest possession.” [1] But the reality of the QED is better reflected by
Dyson’s description in a letter to Gabrielse in 2006 “As one of the inventors
of QED, I remember that we thought of QED in 1949 as a temporary and jerry-
built structure, with mathematical inconsistencies and renormalized infinities
swept under the rug. We did not expect it to last more than 10 years before
some more solidly built theory would replace it. Now, 57 years have gone by
and that ramshackle structure still stands.”[71]

All calculations performed in QED always result in an infinite value.
Renormalization techniques must be used to obtain finite results. These
Renormalization techniques are not mathematically legitimate. Despite this
fact, they have continued to be used because they provide results that fit
perfectly with the experimental results. This has provided a kind of
"empirical legitimacy" to renormalization.

But for this "empirical legitimacy" to be acceptable, there can be no doubt
about the mathematical calculations used. Nevertheless, the lack of critical
review of the theoretical results that have been published is evident. The
theoretical results are only scrutinized when they do not match the
experimental values. It is no longer a surprise that errors continually appear
in theoretical calculations. Recall that the calculation of each Feynman
diagram implies the resolution of multiple factors, and that each of these
factors diverges to infinity. Therefore, renormalization techniques must be
arbitrarily applied to eliminate these infinities and to obtain finite
results. Moreover, these calculations are extremely complex and are not
published in their entirety, so it is impossible to independently validate
them.

1 October 2021

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