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# Something is rotten in the state of QED

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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 QED history reveals that this value was obtained using illegitimate mathematical traps, manipulations and tricks. These traps included the fraud of Kroll & Karplus, who acknowledged that they lied in their presentation of the most relevant calculation in QED history. As we will demonstrate in this paper, the Kroll & Karplus scandal was not a unique event. Instead, the scandal represented the fraudulent manner in which physics has been conducted from the creation of QED through today.
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February 2020
Something is rotten in the state of QED
Oliver Consa
Independent Researcher,
Barcelona, Spain
Email: oliver.consa@gmail.com
Quantum electrodynamics (QED) is considered the most accurate theory in the his-
tory of science. However, this precision is based on a single experimental value: the
anomalous magnetic moment of the electron (g-factor). An examination of QED history
reveals that this value was obtained using illegitimate mathematical traps, manipula-
tions and tricks. These traps included the fraud of Kroll & Karplus, who acknowledged
that they lied in their presentation of the most relevant calculation in QED history.
As we will demonstrate in this paper, the Kroll & Karplus scandal was not a unique
event. Instead, the scandal represented the fraudulent manner in which physics has been
conducted from the creation of QED through today.
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 confer-
ences. 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 had become the powerful holders of the secrets of the
atomic bomb. The members of the Manhattan Project had
been militarized, and their knowledge had become a state se-
cret. There was a positive aspect to this development: the US
government created the Atomic Energy Commission (AEC)
and appointed Oppenheimer as its chief advisor. The former
members of the Manhattan Project took control of universi-
ties, and American research centers received generous gov-
ernment grants. With these grants, the research centers were
able to invest in expensive experimental resources, such as
atomic explosion tests, particle accelerators and supercom-
puters.
Although this situation provided the American scientists
with unlimited resources to conduct their research, no one
had considered the great risk it posed to the future of science.
The former members of the Manhattan Project now enjoyed
unlimited credibility. Their hypotheses were automatically
accepted, and no one could refute their theories. Their cal-
culations and experimental data were subject to military se-
crecy, and the cost of the equipment necessary to perform
the experiments was prohibitive for the rest of the interna-
tional scientiﬁc community. Consequently, calculations and
experiments could no longer be reproduced. Modern physics
had become “not falsiﬁable,” according to the criteria of the
philosopher Popper. It was no longer possible to dierenti-
ate scientiﬁc theories from dogmas. Those who accepted as
truth the hypotheses of the former members of the Manhat-
tan Project were rewarded with positions of responsibility in
research centers, while those who criticized their work were
separated and ostracized. The devil’s seed had been planted
in the scientiﬁc community, and its inevitable consequences
would soon grow and ﬂourish.
2 Shelter Island (1947)
2.1 The problem of inﬁnities
After the success of the Dirac equation in 1928, quantum me-
chanics theorists attempted to quantify the electromagnetic
ﬁeld by creating the quantum ﬁeld theory (QFT). Unfortu-
nately, QFT was a complete failure since any attempted cal-
culation under this theory resulted in an inﬁnite number.
The only solution the proponents could devise was to sim-
ply ignore these inﬁnities. Many methods can be used to ig-
nore inﬁnities, but the primary ones are
Substitution: replacing a divergent series with a spe-
ciﬁc ﬁnite value that has been arbitrarily chosen (for
example, the energy of an electron).
Separation: separating an inﬁnite series into two com-
ponents, one that diverges to inﬁnity and another that
converges to a ﬁnite value. Eventually, the inﬁnite com-
ponent is ignored and only the ﬁnite part remains.
Cut-o: 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 mathemati-
cal 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 ap-
pear 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 ne-
glecting it just because it is inﬁnitely great and you do not
want it!.” [33]
Oliver Consa. Something is rotten in the state of QED 1
February 2020
This technique of ignoring inﬁnities is called renormal-
ization. Feynman also recognized that this technique was
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 mathe-
matically legitimate.” [1]
The extent of the magic used by the QFT mathematicians
is exempliﬁed by the sum of the whole numbers. According
to an illegitimate mathematical demonstration conducted by
the Indian mathematician Ramanujan, the result of the sum of
all positive integers is not inﬁnite, but -1/12 [2]:
X
n=1
n=1+2+3+4+5+... =1
12 (1)
Despite the absurdity of these renormalization techniques,
they have been accepted in modern physics, and they are con-
stantly used in current theories.
2.2 Nature is absurd
The acceptance of quantum mechanics meant the acceptance
of seemingly illogical physical explanations, such as the
wave-corpuscle duality, the uncertainty principle and the col-
lapse of the wave function. With the quantization of the elec-
tromagnetic ﬁeld, these physical explanations became even
more confusing. The long list of meaningless explanations
includes 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, and the continuous creation and destruction
of electron-positron pairs in a quantum vacuum.
Feynman summarized this quite clearly: “The theory of
quantum electrodynamics describes nature as absurd from
the point of view of common sense. And it fully agrees with
experiment. So I hope you can accept nature as She is —
Absurd.” [1]
Accepting QED means giving up on understanding
Nature. After all, “Nobody understands quantum mechan-
ics.
2.3 The Shelter Island conference
From June 2 to 4, 1947, the ﬁrst international physics confer-
ence after World War II was held at Shelter Island. The con-
ference brought together 24 physicists from the Manhattan
Project, including Bethe, Bohm, Breit, Feynman, Kramers,
Lamb, von Neumann, Pauling, Rabi, Schwinger, Teller, Uh-
lenbeck, Weisskopf and Wheeler. Oppenheimer acted as the
master of ceremonies for the congress. The participants were
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 anoma-
lous magnetic moment of the electron. Lamb [3] presented
an experiment that showed that the 2S1/2 and the 2P1/2 en-
ergy levels of the hydrogen atom were not identical; instead
they diered by about 1000 MHz. Rabi’s team [4] presented
a 0.1% anomaly in the hyperﬁne structure of hydrogen. Later,
Breit [5] interpreted this anomaly as the anomalous magnetic
moment of the electron (g-factor). These two measurements
worked correctly for 20 years without exception.
Questioning the validity of the Dirac equation
meant questioning the validity of quantum mechanics, which
had generated many controversies in its beginning stages.
Questioning quantum mechanics also meant questioning the
legitimacy and status of the Manhattan Project heroes.
The meeting’s participants discussed how to handle this
crisis. However, all the options that were considered involved
accepting errors in their theories without oering alternative
solutions. They devised a compromise solution to manage
this dilemma by deﬁning QED as the renormalized perturba-
tion theory of the electromagnetic quantum vacuum. Under
this hypothesis, it was assumed that the Dirac equation of the
electron was absolutely correct and that the small measure-
ment discrepancies were due to disturbances caused by the
polarization of the quantum vacuum. It was also assumed
that these perturbations could be calculated using the QFT
and that the inﬁnities of this theory could be corrected using
renormalization techniques.
For this compromise solution to work, it was necessary to
use the QED equations to calculate the experimental values of
the Lamb shift and the g-factor—something that no one knew
how to do.
3 Pocono (1948)
3.1 Bethe’s fudge factor
On the train trip home after the conference ended, Bethe
starred in one of the most epic moments in the history of the-
oretical physics. Using only a pencil and paper, he solved
2 Oliver Consa. Something is rotten in the state of QED
February 2020
the Lamb shift electron equation, obtaining a result of 1040
MHz.
This is the story, as recalled by Bethe: “The combina-
tion of these two talks that by Kramers and that by Lamb,
stimulated me greatly and 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 dierence
might be. I had to remember the interaction of electromag-
netic quanta with electrons. And I wasn’t sure about the
factor of two. So if I remembered correctly, I seemed to get
just about the right energy separation of 1000MHz. But I
might be wrong by a factor of two. So the ﬁrst thing I did
when I came to the library at General Electric was to look
up Heitler’s book on Radiation Theory, and I found that,
indeed, I had remembered the number correctly and that,
indeed, I’d got a 1000MHz.” [40]
three pages. In the article, Bethe proposed this equation for
the Lamb shift.
Wns0=8
3π e2
~c!3
Ry Z4
n3!ln K
<EnEm>Av
(2)
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=
mc2).
Wns0=136 ln K
k p !=136 ln mec2
17.8Ry!=1040 Mhz (3)
All the values in Bethe’s equation were known physical
constants, except for the value of 17.8 Ry. The origin of this
value is unknown, but it is essential to obtain the desired re-
sult. According to the document, “The average excitation
energy kp =<EnEm>Aufor the 2s state of hydrogen has
been calculated numerically by Dr. Stehn and Miss Steward
and found to be 17.8 Ry, an amazingly high value.That is,
Bethe’s fantastic calculation is based on data that was calcu-
lated later, data that Bethe could not have known on his train
journey. It is a value that was entered ad hoc to match the
theoretical value with the experimental value. In the ﬁeld of
physics, this illegitimate trick is known as a fudge factor.
The value of 1 Ry is deﬁned as the ionization energy of
the hydrogen atom (13.6 eV). Ionization energy is the maxi-
mum energy that an electron can receive in an atom, so the
electrons’ excitation values must be much lower than this
amount of energy. By deﬁnition, the average excitation en-
ergy must be less than the maximum excitation value, that
is, less than 1 Ry. However, in this case, Bethe proposed 17.8
Ry (242 eV), an absurd value. Bethe, himself, considered this
fudge factor "an amazingly high value."
3.2 Schwinger’s numerology
A few months after Bethe calculated the value of the Lamb
shift, Schwinger devised an even more epic calculation for
the g-factor of the electron. This value was known as the
Schwinger factor [7].
g. f actor =1+α
2π=1.001162 (4)
Kush and Foley [8] [9] had obtained an experimental g-
factor value of 1.00119. Schwinger published his result in
February 1948 in a short article, comprising just one sheet,
that said, “the detailed application of the theory shows that
netic moment associated with the electron spin of magni-
tude δµ/µ =α/2π=0.001162.” [7] At no time was it ex-
said that "a paper dealing with the details of this theory and
its applications is in course of preparation."
The Schwinger factor had a signiﬁcant impact on the sci-
entiﬁc community due to its simplicity and accuracy. Every-
one 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 he not publish
this long-awaited theory?
We suspect that Schwinger did not publish the theory be-
cause 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 (α),
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 [4]. And 2πis about 6.
3.3 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 great-
est physicists of the time: Bohr, Dirac and Fermi. As with the
Shelter Island conference, the expectations were high due to
the recent progress of Bethe and Schwinger. 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 ﬁ-
nally explain the elegant way in which the Schwinger factor
had been calculated. Schwinger’s presentation lasted for ﬁve
unbearable hours and comprised a series of complex, totally
incomprehensible equations. Oppenheimer expressed his dis-
pleasure: “others gave talks to show others how to do the
calculation, while Schwinger gave talks to show that only he
could do it. [41]” Gradually, the attendees left the presenta-
tion 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.
Oliver Consa. Something is rotten in the state of QED 3
February 2020
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 accuracy of his calculations
simply because they produced the correct results. He did not
bother to defend the mathematical basis for the equations or
to present any physical hypotheses. For Feynman, the result
was everything.
Bethe remembers the conference like this: “At Pocono,
Schwinger and Feynman, respectively presented their the-
ories. (...) Their theories seemed to be totally dierent.
(...) Schwinger’s was closely connected to the known quan-
tum electrodynamics, so Niels Bohr, who was in the au-
dience, 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. [39]”
Feynman’s recollection of the conference is also enlight-
ening: “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, with-
out going into details, but nobody there understood either
of us. (...) When he tried to explain his theory, he en-
countered great diculty. (...) As soon as he would try to
explain the ideas physically, the wolves would descend on
him, he had great diculty. Also, people were getting more
and more tired (...) I didn’t have a mathematical scheme to
pression, from which all my diagrams, rules and formulas
would come out. The only way I knew that one of my for-
mulas 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 electro-
dynamics." immediately I was asked: "Where does the for-
mula 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 diculties, everything chaotic. I tried to explain
pirical 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
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 re-
turn to Princeton, Oppenheimer received a third QED theory
from a Japanese physicist named Tomonaga. Now, there were
three QED theories, and all of them were inconsistent and in-
compatible with one other.
4 Oldstone (1949)
4.1 Dyson’s series
After the Pocono meeting, the physics community searched
for a uniﬁed, covariant QED theory. The person in charge
of addressing this problem was a young 26-year-old English
scientist named Dyson, who managed to reconcile the three
QED theories in his article “The Radiation Theories of
Tomonaga, Schwinger, and Feynman.” [11] The article ab-
stract indicated that “The chief results obtained are (a) a
demonstration of the equivalence of the Feynman and
Schwinger theories, and (b) a considerable simpliﬁcation of
the procedure involved in applying the Schwinger theory.
In reality, Dyson had created his own QED theory based on
the ideas of Tomonaga, Schwinger and Feynman. Dyson’s
theory was subsequently published in an article titled “The
Dyson proposed that the Heisenberg S-matrix could be used
to calculate the electron’s g-factor, transforming it into a se-
ries called the Dyson series. The Dyson series was an inﬁnite
series of powers of alpha, where the ﬁrst coecient was pre-
cisely the Schwinger factor, and where each coecient could
be calculated by solving a certain number of Feynman dia-
grams.
a=C1α
π+C2α
π2
+C3α
π3
+C4α
π4
+C5α
π5
... (5)
Dyson’s theory, based on Feynman’s diagrams, appeared
to provide the deﬁnitive solution for which his peers had been
waiting. Enthusiasm returned to the American scientiﬁc com-
munity.
4.2 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 hap-
pen to be in agreement with observations does not prove that
one’s theory is correct.” [32].
Another critic was Oppenheimer, as Dyson relates:
“When after some weeks I had a chance to talk to Oppen-
heimer, I was astonished to discover that his reasons for be-
ing uninterested in my work were quite the opposite of what
my program as merely unoriginal, a minor adumbration of
Schwinger and Feynman. On the contrary, he considered
it to be fundamentally on the wrong track. He thought ad-
umbrating Schwinger and Feynman to be a wasted eort,
because he did not believe that the ideas of Schwinger and
Feynman had much to do with reality. I had known that he
4 Oliver Consa. Something is rotten in the state of QED
February 2020
had never appreciated Feynman, but it came as a shock to
hear him now violently opposing Schwinger, his own stu-
dent, 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. [35]”
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 cal-
culations 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
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
Feynman’s response to these criticisms is well known:
“Shut up and Calculate!” [1]
4.3 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 confer-
ences, 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 Amer-
ican physicists, and Feynman became the leader of this new
generation of scientists.
In parallel to the QED consolidation, the conference pre-
sented important experimental results on subatomic particles
that were called pi-mesons or pions. These particles had been
discovered thanks to the new synchrocyclotron particle accel-
erator 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 con-
ferences; instead, he created the International Conference of
High Energy Physics (ICHEP).
New research in high energy physics resulted in quan-
tum chromodynamics (QCD), the electroweak theory and the
standard theory of particle physics. All these developments
relied heavily on the use of Feynman diagrams. However,
the Feynman diagrams are only valid when the coupling con-
stant has a very low value. In the case of electromagnetism,
the coupling constant alpha is much smaller than one. But,
in the case of fermions, the coupling constant is greater than
one, so it is not mathematically legitimate to use the Feynman
diagrams for these calculations. In 1951, Feynman himself
warned Fermi of this problem: ”Don’t believe any calcula-
tion in meson theory that uses a Feynman diagram.” [43]
4.4 More fudge factors for Bethe
In 1950, Bethe [15] published a new calculation of the Lamb
shift that adjusted the fudge factor value from 17.8 Ry to
16.646 Ry. This value has not been modiﬁed since this
change. Other researchers, such as Kroll, Feynman, French
and Weisskopf, have expanded Bethe’s original equation with
new fudge factors, resulting in a value of 1052 Mhz with the
following equation:
Wns0=α3
3πRy ln mc2
16.646 Ry
ln 2+5
61
5!(6)
This strategy of adding to existing equations new factors of
diverse origin with the objective of matching the theoretical
and experimental values has been widely used in QED. This
strategy is known as perturbation theory, and it is often used
recursively. That is, each of these factors is, in turn, formed
by another series of factors. For example, it was assumed
that the dierence between the theoretical and experimental
values of the Lamb shift was due to relativistic corrections.
These corrections were calculated by Baranger in 1951 [16],
who obtained a theoretical Lamb shift value of 1058.3 MHz,
while the experimental value was 1061±2Mhz. As expected,
this new factor was, in turn, composed of three other factors
of diverse origin.
W=α4Ry 1+11
128 ln 2
2!=6.894 Mhz (7)
5 The Kroll &Karplus Scandal (1950-1957)
5.1 Fourth Order Correction
In 1949, Gardner and Purcell obtained a new experimental
result for the g-factor of 1.001,146 [13]. With this new ex-
perimental value, Schwinger’s factor was no longer consid-
ered accurate. Feynman used this new crisis as an opportu-
nity to demonstrate the validity of his theory. Using the QED
reformulation with Dyson’s S-matrix, the renormalization of
inﬁnities could be performed in a consistent manner. Accord-
ing to this theory, Schwinger’s factor was only the ﬁrst coe-
cient of the Dyson series. The calculation of each coecient
in the series required the resolution of an exponential num-
ber of extremely complex equations. The calculation of the
Oliver Consa. Something is rotten in the state of QED 5
February 2020
next factor in the series (the fourth order correction) required
seven Feynman diagrams. Kroll and Karplus, two of Feyn-
man’s assistants, performed these calculations. In 1950, they
published their results [14].
The second coecient of the Dyson series, as calculated
by Kroll and Karplus, was -2.973. Consequently, the new
theoretical value of the g-factor was 1.001,147, which was
almost the same as the experimental result that had been re-
ported by Gardner and Purcell.
g. f actor =1+α
2π
2.973 α
2π2
=1.001,147 (8)
As indicated in the paper, “The details of two indepen-
dent calculations which were performed so as to provide
some check of the ﬁnal result are available from the au-
thors.” [14] That is, the calculations had been performed in-
dependently by two teams of mathematicians who had ob-
tained 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 ex-
perimental result could have been achieved by chance. This
was the deﬁnitive test. QED had triumphed. Along the way,
been dispensed. These faults did not matter; the theoreti-
cal calculations coincided with the experimental data with
great precision. There was nothing more to discuss. Feyn-
man’s prestige dramatically increased, and he began to be
mentioned as a candidate for the Nobel Prize.
5.2 Dyson’s betrayal
In 1952, two years after this great success, Dyson published
an article entitled “Divergence of Perturbation Theory in
Quantum Electrodynamics,” [17] which said that “An ar-
gument is presented which leads tentatively to the conclu-
sion that all the power-series expansions currently in use in
quantum electrodynamics are divergent after the renormal-
ization of mass and charge. The creator of the QED the-
ory had questioned its theoretical validity by stating that his
Dyson series was divergent.
Dyson had hinted at this fact in his previous works. The
abstract of the article “The Radiation Theories of Tomonaga,
Schwinger, and Feynman” [11] indicated that “the theory of
these higher order processes is a program rather than a
deﬁnitive theory, since no general proof of the convergence
of these eects is attempted. The abstract of the article “The
S-Matrix in Quantum Electrodynamics” [12] also indicated
that “Not considered in this paper to prove the convergence
of the theory as the order of perturbation itself tends to in-
ﬁnity.After this dicult confession, Dyson moved to Eng-
land, abandoned this line of research and dedicated the rest of
his career to other areas of physics. Perhaps this confession
is the reason he never received the Nobel Prize.
5.3 The scandal
Surprisingly, Dyson’s claim that the series was divergent did
not diminish QED’s credibility. However, in 1956, Franken
and Liebes [18] published new, more precise experimental
data that provided a very dierent g-factor value (1.001,165).
This value was higher than the Schwinger factor, so the value
of the second coecient that had been calculated by Kroll
and Karplus not only did not improve the Schwinger factor;
With the new experimental data from Franken and Liebes,
the value of the second coecient of the series should have
been +0.7 instead of -2.973. The dierence between these
values is huge and unjustiﬁable. The probative force of QED
was upended. QED must necessarily be an incorrect the-
ory. In addition, there was no explanation for why Kroll and
Karplus’s calculation provided the exact expected experimen-
tal 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.
But the creators of QED refused to accept defeat. QED
could not be an incorrect theory because that placed them in
an indefensible situation. All the developments in the ﬁeld of
based on this theory. All the privileges they had obtained af-
ter the success of the Manhattan Project were at stake. Even
worse, now the scientists’ own lives were in jeopardy. In
1949, the USSR had obtained the atomic bomb thanks to in-
formation provided by Fucks, a Manhattan Project researcher
with communist sympathies. From then on, espionage ac-
cusations became widespread among the American scientiﬁc
community. Senator McCarthy began a witch hunt in which
Oppenheimer was accused of treason and had to submit to
trial. The witch hunt ended in 1957, when the Russians sent
the Sputnik satellite into space and the US government re-
alized that it needed scientists to create NASA and win the
space race.
In response to this scandal, surprising facts were revealed.
First, Kroll and Karplus confessed that they had not indepen-
consensus result. Therefore, it was possible that there were
errors in the calculation. Next, Petermann [19] detected an er-
ror in the Kroll and Karplus calculations (one that no one had
detected in the seven years since the article was published).
Petermann made the correct calculation and obtained a result
of -0.328, which was almost 10 times lower than the previ-
ous calculation of -2.973. This correct calculation resulted
in a new theoretical value of the g-factor (1.001,159,6) that
was within the margin of error of the new experimental value
(1,001,165). The same error was independently detected by
Sommerﬁeld [21]. Once again, two independent calculations
provided the same theoretical value. Miraculously, QED had
been saved.
For the third time in 10 years, experimental data had con-
6 Oliver Consa. Something is rotten in the state of QED
February 2020
tradicted theoretical calculations (Kursh, Gardner and
Franken), and, for the third time in 10 years, a theoretical cor-
rection had allowed the reconciliation of the theoretical data
with the experimental data (Schwinger, Kroll and Karplus,
and Petermann). This was all very suspicious. After the Kroll
and Karplus scandal, two facts had become clear: no one was
reviewing the theoretical calculations that were being pub-
lished, and the researchers had lied to indicate that the cal-
culations had been performed independently by two dierent
groups.
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 per-
petual 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 fa-
mous paper and now it’s an infamous paper.” [38]
5.4 The analysis
At this point, we have doubts about everything that was re-
Karplus in 1950 [14], as well as the corrections of Peter-
mann [19] and Sommerﬁeld [21] that were published in 1957.
Kroll and Karplus’s article consists of 14 pages and is full
of complex mathematical calculations. The document indi-
cates that to obtain the coecient, it is necessary to calculate
the 18 Feynman diagrams that are presented in Fig. 2. These
diagrams are grouped into ﬁve groups (I, II, III, IV and V).
Fig. 2: Feynman diagrams for the fourth-order corrections
Subsequently, it is argued that groups III, IV and V are
not necessary, so only seven Feynman diagrams (I, IIa, IIb,
IIc, IId, IIe and IIf) need to be calculated. A large number
of calculations are then performed to demonstrate that dia-
grams IIb and IIf are also not necessary. This occurs on page
11 of the article. The values for diagrams IId and IIe, which
appear the simplest, are quickly calculated, while indicating
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.Consequently, the cal-
culations for three of the ﬁve diagrams were never published.
Finally, the results summary shows the results of each of
the ﬁve Feynman diagrams. The sum of the ﬁve diagrams
provides a result of -2.973.
Petermann’s paper consists of only two pages, and the cal-
culations are not shown; only the results are presented. Peter-
mann indicated that he only found relevant errors in the cal-
culations for diagram IIc. The calculations for diagrams IIa,
IId and IIe were correct, while diagram I had a very small
error. The following table demonstrates the contribution of
each diagram to the ﬁnal coecient:
I IIa IIc IId IIe Total
K&K -0.499 0.78 -3.178 -0.09 0.02 -2.973
Pet. -0.467 0.78 -0.564 -0.09 0.02 -0.328
Di. 6% 0% 82% 0% 0% 89%
Table 1: Feynman diagrams values.
The error in diagram I is small (6%), but the error in di-
agram IIc is huge (82%). Petermann focused on diagram IIc
which is the dominant diagram. Suspiciously, this is the dia-
gram that included the main errors. The results of the other
four diagrams are not relevant and practically cancel each
other out.
In the article summary, Petermann showed the result of
the original calculation of diagram IIc, the result of the cor-
rected calculation and the dierence between the
calculations:
[Karplus & Kroll]
IIc=323
24 +31
9π249
6π2ln(2) +107
4ζ(3) (9)
[Petermann]
IIc=67
24 +1
18 π2+1
3π2ln(2) 1
2ζ(3) (10)
[Dierence]
IIc=32
361
18 π2+17
2π2ln(2) 109
4ζ(3) (11)
If we analyze the calculations for diagram IIc (Table 2),
we observe that the four components of IIc have abnormally
high values (-13, 34, -55 and 32). When added together, sur-
prisingly result in -3.18, a ﬁgure that is an order of magnitude
lower. In contrast, Petermann’s corrections were enormous,
Oliver Consa. Something is rotten in the state of QED 7
February 2020
the size of one or two orders of magnitude for each compo-
nent of diagram IIc. It is dicult to believe that Kroll and
Karplus made so many large mistakes. These circumstances
cast doubt on Kroll’s assertion that the discrepancies were due
to “simple arithmetic errors.
Const. π2π2ln(2) ζ(3) Total
K&K -13,46 34,00 -55,87 32,15 -3,18
Pet. -2,79 0,55 2,28 -0,60 -0,56
Di. 10,67 -33,45 58,15 -32,75 2,61
Table 2: Components of IIc Feynman diagram.
We analyzed another paper by Petermann that was pub-
lished a few months earlier in the journal of Nuclear Physics
[20]. In this paper, Petermann indicated that “some of the big
contributions have been evaluated analytically, the others
estimated by analytic upper and lower bounds. The numer-
ical value for this term has been found to satisfy IIc =-1.02
+/- 0.53“. The types of calculations and the obtained results
clearly indicate that the problem was not due to simple arith-
metic errors; the issue was related to discrepancies about how
and where renormalization techniques should be applied to
eliminate inﬁnities.
The Sommerfeld paper [21] is a press release that con-
ﬁrms Petermann’s results without providing any further in-
formation. The press release merely indicates that: “The
present calculation has been checked several times and all
of the auxiliary integrals have been done in at least two dif-
ferent ways,without oering any substantive proof.
Since the original calculations for diagrams I and IIc were
not published, Petermann and Sommerfeld must have had ac-
cess to private data to ﬁnd the errors. Since they also did not
publish their corrective calculations, we cannot know what
the errors were or if the corrections were accurate.
The version of the facts conveyed by Feynman does not
correlate with reality and completely ignores the seriousness
of what occurred: “It took two ‘independent’ groups of
physicists two years to calculate this next term, and then an-
other year to ﬁnd out there was a mistake — experimenters
had measured the value to be slightly dierent, 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 dierences between their calcula-
tions, so they were not really independent”. [1]
6 The Nobel Prize (1965)
6.1 Direct g-factor measurement
In 1953, a research team from the University of Michigan [22]
proposed a new experiment to calculate the magnetic moment
of the electron directly from the precession of the free elec-
tron spin. This new technique provided more precise exper-
imental values than the previous techniques that were based
on atomic levels. The Michigan experiment only presented
a proof of concept, demonstrating that the idea was viable
while obtaining irrelevant results.
A few years later, the Michigan team obtained the nec-
essary resources to conduct the real experiment. In 1961,
Schupp, Pidd and Crane [23] published their results with an
experimental value of 1.0011609 (24). 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 stan-
dard error which is great enough to include the weighted
average of Table IV, namely ±0.0000020. Finally, we com-
bine with this the estimated systematic standard errors (...).
This results in a ﬁnal value of 0.0011609 ±0.0000024.
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 pub-
lished, the result would have been 0.0011609 ±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 av-
eraged the measurements in Table IV, generating a result of
0.0011627 ±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. Al-
though they published what they considered to be the correct
result (0.0011609), 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 basis; this was the only way
to keep Petermann’s theoretical value within the margin of
error.
8 Oliver Consa. Something is rotten in the state of QED
February 2020
6.2 The experimenter’s bias
At that time the situation was dramatic again. Predictably,
subsequent experiments would discredit the g-factor theoret-
ical value. And after the Kroll and Karplus scandal, the theo-
retical 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 [24]
published a third improved version of the experiment. In the
report of the results of this third experiment, 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 eort of
the laboratory on the g factor of the free negative electron.
Just when QED seemed doomed to disaster, the miracle
happened again. This time, the new experimental value was
1,001,159,622 ±0.000,000,027, nearly the same as Peter-
mann’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 experi-
ment, and it was conducted at the same University, with the
same teams, only two years later. The margin of error could
not have improved so much from one experiment to another,
and 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.
It is evident that the measuring devices were calibrated to
obtain the expected theoretical value. This type of error is
known as experimenter bias.
In this case, the error does not seem to be involuntary;
instead, it appears to be a conscious manipulation of the ex-
perimental data with the sole objective of, once again, sav-
ing QED. The trap worked perfectly. After this experiment,
all doubts about QED were cleared, and, in 1965, Feynman,
Schwinger and Tomonaga were awarded the Nobel Prize in
physics.
This experimental manipulation is the most serious aspect
of this story. We suspect that some corrections to systemic
errors that were fraudulently added to this experiment, are
still maintained in current experiments. As an indirect con-
sequence of this manipulation, no alternative theory to QED
can oer better results, since the theoretical results must be
compared with manipulated experimental data.
7 To Inﬁnity and Beyond
7.1 The Penning trap
The story did not end here, as the cycle was repeated a ﬁfth
time. Between 1977 and 1987, Van Dyck and Dehmelt of
the University of Washington published experimental results
using 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 indi-
vidual electrons. These experiments improved the previous
results by three orders of magnitude, and, again, the new re-
sult excluded Petermann’s theoretical value (0.001,159,615).
[1977] : 0.001,159,652,410(200) [25]
[1981] : 0.001,159,652,222(50) [26]
[1984] : 0.001,159,652,193(4) [27]
[1987] : 0.001,159,652,188,4(43) [28]
To resolve this discrepancy, the theoretical physicists de-
cided to calculate another coecient of the Dyson series
(sixth-order correction). This new coecient required solv-
ing 72 Feynman diagrams. In 1965, Drell and Pagels [29] had
published a ﬁrst approximate theoretical value of 0.15 for this
coecient, the precision of which was now insucient. In-
creasingly precise numerical calculations were presented for
30 years until Laporta and Remiddi [45] published their ﬁnal
analytical calculation in 1996. Their result was 1.181241 (10
times higher than the initial calculation), leaving the theoreti-
cal value of the g-factor as 0.001,159,652,201.2(271), within
the range of experimental error. Once again, QED had been
miraculously saved.
What happened to Dyson’s controversial statement about
the divergence of his series? Dyson’s argument is correct, as
his series does diverge. How is it possible that 72 Feynman
diagrams resulted in a value close to 1? For some unknown
reason, the results of the Feynman diagrams tend to cancel
each other out, leading to a total result on the order of 1, re-
gardless of the number of diagrams that are calculated. Just
another Christmas miracle.
7.2 Harvard experiment
After 20 years of tranquility, history repeated for a sixth time.
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
Oliver Consa. Something is rotten in the state of QED 9
February 2020
University of Washington. These new data also excluded the
theoretical value of the g-factor.
[2006] : 1.001,159,652,180,85(76) [30]
[2008] : 1.001,159,652,180,73(28) [31]
Fig. 5: Harvard vs Washington errors
To resolve this new discrepancy, the theoretical physicists
decided to calculate two new coecients of the Dyson series
(eighth-order and tenth-order corrections). These new coef-
ﬁcients involved the calculation of 891 and 12,672 Feynman
diagrams respectively.
Fig. 6: electron g-factor errors
Unfortunately these two new coecients were not enough
to ﬁt the experimental value. As in the case of Bethe’s fudge
The ﬁrst coecient was derived from the interaction of
the electron with leptons, the second coecient was derived
from the electroweak interaction and the third coecient was
derived from the electron’s interaction with hadrons.
Despite these diculties, the QED theorists returned to
work the miracle. A team led by the Japanese physicist
Kinoshita managed to perform the necessary calculations,
thanks to the use of supercomputers, and obtained a g-factor
of 1,001,159,652,182,032(720), within the margins of exper-
imental error [48] .
7.3 Supercomputer calculations
In 2017, after 36 years of calculation reﬁnements, Laporta
[46] published a ﬁnal calculation of the fourth coecient of
the Dyson series, which required solving 841 Feynman dia-
grams. As Kinoshita [48] indicates, “It took 36 years since
the preliminary value A8 =-0.8 was reported. For the pur-
pose of this article it is sucient to list the ﬁrst ten digits of
Laporta’s result: A8 =-1.912 245 764.
This implies a ﬁnal result more than double the initial esti-
mation. Furthermore, Laporta’s published value had an accu-
racy of 1100 digits (sic), about 100 times the necessary pre-
cision. Presenting a theoretical value with this unnecessary
level of precision leads us to suspect the results even more
(“Excusatio non petita, accusatio manifesta.”).
Fig. 7: First 1100 digits of A8
In 2018, the Kinoshita team published [48] a new theo-
retical value for the ﬁfth coecient of the series. As usual,
this most recent calculation included a review of the previous
calculation that was published in 2014. The paper indicated,
“During this work, 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. The
correction of this error causes a shift of -1.25 to the Set V
contribution, and hence to the tenth-order universal term.
The value of the ﬁfth coecient that was calculated in 2014
was 7,795; while the value that was calculated in 2018 was
6,675, which means admitting an error of 15%.
Given the serious errors committed by Karplus and Kroll
in 1950 in the calculation of a single Feynman diagram (IIc),
it seems ridiculous to propose that 12,672 Feynman diagrams
could be calculated without errors
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. That errors continuously appear in theoretical cal-
culations is no longer a surprise. Recall that the calculation
of each Feynman diagram implies the resolution of multi-
ple factors, and that each of these factors diverges to inﬁnity.
Therefore, renormalization techniques must be arbitrarily ap-
plied 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 indepen-
dently validate them. None of this matters, so long as you
can arm without blushing that “QED is the most accurate
theory man has produced”.
10 Oliver Consa. Something is rotten in the state of QED
February 2020
7.4 The muon anomaly
QED was also used to calculate the anomalous magnetic mo-
ment of the muon. To date, the most accurate experimental
value was obtained in 2004 in the E821 experiment that was
conducted at the Brookhaven National Laboratory (BNL).
The experimental result was 0.001,165,920.9(6) [47]. Unfor-
tunately, the theoretical value did not match the experimental
value and had an error that was greater than 3 sigmas. Despite
enormous eort in recent decades, this discrepancy could not
be eliminated. Currently, the theoretical value of the muon
g-factor is 0.001,165,918.04(51).
Fig. 8: Muon anomaly
ancy, as it is perhaps the most palpable evidence that the stan-
dard model is incomplete.
In 2011, the E989 experiment was devised to improve the
accuracy of the E821 experiment. This extremely complex
experiment will be performed at the Fermilab’s Tevatron. Be-
fore 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 oper-
ation 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
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. It is expected that the E989
experiment can be concluded in 2020 and that a new experi-
mental value will be presented for the muon g-factor.
Given the scientiﬁc precedents, we are convinced that,
one way or the other, the discrepancy will be resolved, and
the myth of QED’s precision will be preserved.
8 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 this statement is nothing more than a false myth.
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 math-
ematical 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 struc-
ture still stands.” [44]
QED should be the quantized version of Maxwell’s laws,
but it is not that at all. QED is a simple addition to quantum
mechanics that attempts to justify two experimental discrep-
ancies in the Dirac equation: the Lamb shift and the anoma-
lous magnetic moment of the electron.
The reality is that QED is a bunch of fudge factors, nu-
merology, ignored inﬁnities, hocus-pocus, manipulated cal-
culations, illegitimate mathematics, incomprehensible theo-
ries, hidden data, biased experiments, miscalculations, suspi-
cious coincidences, lies, arbitrary substitutions of inﬁnite val-
ues and budgets of 600 million dollars to continue the game.
Maybe it is time to consider alternative proposals. Winter
is coming.
1 February 2020
Oliver Consa. Something is rotten in the state of QED 11
February 2020
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12 Oliver Consa. Something is rotten in the state of QED
... Therefore, I find such reasoning to be absurd and not worthy of further discussion just for the sake of destructive criticism. And besides that, as far as the credibility of quantum electrodynamics is concerned, the foundations are still very embarrassingly questionable[81]. ...
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... worthy of further discussion just for the sake of destructive criticism. And besides that, as far as the credibility of quantum electrodynamics is concerned, the foundations are still very embarrassingly questionable [69]. 9 Necessary premise for the definition of electric field Although I have briefly mentioned the necessary assumption to have a definition of electric field, let me now write it clearly as a postulate 38 , followed by some crucial comments. ...
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Physical dimensions are {\it not numbers}, but used as {\it numbers} to perform dimensional analysis by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like $r\to 0$'', $q\to 0$'', used by the physicist, are inconsistent on dimensional grounds because $r$'', $q$'' represent {\it quantities} with physical dimensions of $[L], [Q]$ respectively and $0$'' represents just a number devoid of physical dimension. Consequently, the involvement of the statement $\lim_{q\to 0}$, where $q$ is the test charge'' in the definition of electric field leads to either circular reasoning or a contradiction regarding the experimental verification of the smallest charge in the Millikan-Fletcher oil drop experiment. Considering such issues as problematic, by choice, I make an inquiry regarding the basic language in terms of which physics is written, with an aim of exploring how truthfully the verbal statements can be converted to the corresponding physico-mathematical expressions (involving physical dimensions). Necessarily I demonstrate the meaning of the terms self inquiry'', middle way'', dependent origination'', emptiness/relational existence'', which signify the basic tenets of Buddhism. In light of this, I explain my view of definition''; the relation among quantity, physical dimension and number; the logico-linguistic fallacy of zero quantity''; difference between unit and unity. I present a critical analysis of the definitions of electric field due to Maxwell and Jackson. A postulate regarding the existence of charges becomes necessary which, however, resolves the contradiction regarding the oil drop experiment. This work is an appeal for the dissociation of the categorical disciplines of logic and physics and on the large, a fruitful merger of Eastern philosophy and Western science.
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This is the first biography ever written on the distinguished physicist Julian Schwinger. Schwinger was one of the most important and influential scientists of the twentieth century. The list of his contributions is staggering, from his early work leading to the Schwinger action principle, Euclidean quantum field theory, and the genesis of the standard model, to later valuable work on magnetic charge and the Casimir effect. He also shared the 1965 Nobel Prize in Physics with Richard Feynman. However, even among physicists, understanding and recognition of his work remains limited. This book by Mehra and Milton, both of whom were personally acquainted with Schwinger, presents a unique portrait that sheds light on both his personality and his work through discussion of his lasting influence on science. Anyone who wishes to gain a deeper understanding of one of the great physicists of this century needs to read this book.
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