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Article
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Physics
The End of Yukawa Meson Theory of Nuclear Forces
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Yukawa meson theory of nuclear force originates from the fiction of Yukawa potential function
which distorts mathematics. Its logical essence is to dismember the physical equation to fabricate
a potential function, endow the mass factor in the fictional potential function with the meaning of
meson mass, and then claim that the nucleon in the nucleus depends on the meson to exchange
force, so there is a meson theory of nuclear force. However, there has never been a meson in the
nucleus. It is obviously unreasonable to say that the interaction between nucleons is dominated by
other substances that are not in the nucleus. Here I have reviewed Yukawa’s paper on the creation
of the nuclear meson theory in detail, and strictly pointed out that the serious distortion of the
basic mathematical algorithm in the process of fabricating Yukawa potential function is irreparable,
and there is no scientific logic and experimental basis to prove that the mass factor in Yukawa
p otential function is a meson. Specifically: 1) The Yukawa meson theory introduces a delta function
without causality into the op erator equation of relativistic momentum energy relationship, thus very
covertly deleting the energy term, which essentially belongs to tearing and dismembering the operator
equation corresponding to the relativistic momentum energy relationship, seriously distorting the basic
mathematical algorithms and physical logic; 2) Yukawa’s paper tore and dismembered the relativistic
momentum energy operator equation to give a mutilated equation that meets the expectation but has
no physical significance in practice, and then introduced the unrelated Compton wavelength to knead,
interpreted the mass factor in the exponential factor of the exact solution of the defective equation
as the mass of mesons that do not exist in the atomic nucleus, and declared that mesons are the
medium of nucleon interaction, the nuclear meson theory thus created is anti-scientific logic; 3) The
stationary solution of the operator equation of the momentum energy relationship of free particles is
Bessel function, but the Yukawa meson theory deletes the energy operator in the operator equation
of the momentum energy relationship and then solves the mutilated equation, which is equivalent
to strongly twisting Bessel function into an exponential function that meets the expectation, which
is a severe distortion of the basic mathematical algorithm; 4) The calculation results of the binding
energy of the atomic nucleus show that the nucleus cannot absorb other particles such as mesons
when nucleons combine into the atomic nucleus, while Yukawa’s meson theory insists that mesons are
the medium of interaction between nucleons, which is just the hegemonic fallacy. All these facts mean
that the Yukawa meson theory, which is based on distorting the basic mathematical algorithms and
fabricating potential functions, is a pseudo-scientific theory and must be completely terminated.
Keywords: Relativistic momentum energy relation; Relativistic operator equation; Bessel function;
Yukawa function; Unitary principle; Nuclear force meson theory
PACS number(s): 02.30.Xx—Calculus of variations; 03.65.Ta—Foundations of quantum me-
chanics; measurement theory; 13.75.Cs—Nucleon–nucleon interactions; 21.30.-x—Nuclear forces;
21.30.Cb— Nuclear forces in vacuum; 42.50.Xa—Optical tests of quantum theory.
1 Introduction
The law of interaction between substances is the most
basic physical law. The discovery of physical laws comes
from the analysis of repeatable and highly accurate ex-
perimental data. The discovery of the law of universal
gravitation is not strict, which is an exception, but the
determination of the constant of universal gravitation
[1, 2]
makes up for this defect. Human cognitive ability to nat-
ural laws is not unlimited, but subject to the cognitive
scope. To study the law of interaction between ultra-fine
and ultra far substances outside the cognitive scope, we
rely more on speculation, and then test the extent to
which this speculation conforms to expectations rather
than truth through experiments.
Experimental instruments are made of substances
Citation: Dongfang, X. D. The End of Yukawa Meson Theory of Nuclear Forces. Mathematics & Nature 1, 202110 (2021).
*The core argument of this article is that Yukawa constructed a wrong differential equation through a lie distorted by mathematics. By
solving this wrong differential equation, a pseudo nuclear potential energy later called Yukawa potential function was obtained, and the
mass parameter in the pseudo nuclear potential energy function was forcibly said to be the mass of meson. In fact, this belongs to a
brainwashing rather than scientific demonstration, although mesons exist in nature. Yukawa’s paper had a lot of wrong physical concepts
and mathematical expressions, but it was later packaged as a nuclear meson theory.
2 X. D. Dongfang The End of Yukawa Meson Theory of Nuclear Forces
composed of atoms rather than nuclei. Instruments can-
not go deep into the scope of nuclei. Therefore, our un-
derstanding of atomic nuclei depends more on assump-
tions, guesses and reasoning. In 1935, Yukawa went be-
yond the logic of mathematical physics to make a special
treatment on the operator equation of the relationship
between relativistic momentum and energy
[3]
, obtained
a function containing natural exponential factor, which
was later called Yukawa potential function
[4]
, asserted
that mesons were the medium of nucleon interaction,
and thus put forward the nuclear force meson theory
[5-9]
that has not been tested by logic. However, it is well
known that there is no meson in the atomic nucleus. It
is illogical to say that mesons are the interaction medi-
um between nucleons. In the past 100 years, there has
been a strange phenomenon in physics, which can shake
the hearts of the public as long as it is claimed that
experiments or theories have made new and incredible
achievements due to the theory of relativity. The me-
son theory of nuclear force was popular soon after its
birth. However, what is the truth? Newton’s law of
motion
[10]
is the greatest founding law of physics and
cannot be surpassed. Because of this, later physicist-
s gave up scientific argumentation and opened a new
path, trying to surpass Newton’s theory by establishing
theories through assumptions. However, whether the as-
sumptions of physics are correct or not has a causal law.
Relativity
[11]
originated from the assumption that the
speed of light is constant. We have strictly proved that
the assumption that the speed of light is constant is not
tenable
[12]
. It is only an imaginary mathematical infer-
ence with incomplete thinking. It is a misunderstanding
to think that relativity and quantum mechanics surpass
Newtonian mechanics.
In the early days of the theory of relativity, this ab-
normal logic that goes beyond the mathematical and
physical conventions but caters to the curiosity of read-
ers was questioned. However, because the scope of peo-
ple’s query is limited to some conclusions of relativity
and fails to go deep into the level of principle, the query
lacks strict scientific demonstration, and because the rel-
ativists who have become the authority argue that ev-
ery abnormal inference is a real new discovery. Those
who doubted became the losers, and the doubt could
not continue. Since then, the experimental reports that
developed the theory of relativity and claimed to confirm
the relativistic inference, without exception, can cause a
sensation and quickly gain the recognition and praise of
authoritative organizations, but the authoritative orga-
nizations never test its authenticity. As a result, these
theories and experimental reports have become the most
dazzling content in textbooks, and generations of physi-
cists with firm beliefs have been emerging. The cycle of
life and death, be raging like a storm. Relativity will
have a long period of vitality. Academic circles exces-
sively advocate the celebrity effect, and modern physics
always despises the logical test, leading to hegemonis-
m, which seriously hinders the development of physical
theory in the right direction. If we prove that the rela-
tivistic energy equation is not tenable within the frame-
work of relativity, will physics be able to start a new
development? This problem makes us have a clear un-
derstanding of the meson theory of nuclear forces which
ostensibly relies on relativity, but this is not the fo cus of
this paper.
There is no meson in the atomic nucleus. Isn’t
Yukawa’s nuclear force meson theory indisputably and
directly denied? It can be proved that even if there are
mesons in the atomic nucleus, the experimental test of
meson theory of nuclear forces also constitutes a logical
loop of dead cycle
[13]
. The physical debate completely
confined to the principles of modern physics will never
come to an end, and will eventually end in the strange
circle of Anti Science in which truth is always strangled
by the arbitrariness of authority. So is there a new basic
principle of physics equivalent to mathematical axioms
that will make it easy for future physicists to find the
key to wrong logic, so that they will not waste time and
energy identifying the experimental data that have been
tampered with under the pretext of systematic error cor-
rection and the so-called major experimental observation
data that have even been fabricated through analog sig-
nal scaling?
Here, let us jointly reveal various principled errors in
Yukawa’s meson theory of nuclear forces, appreciate the
powerful logical power of the unitary principle original-
ly proposed in 1985 and continuously improved in its
application, and then completely terminate Yukawa me-
son theory of nuclear forces in many aspects within the
logical framework of relativity.
2 Introduction to Yukawa meson theory of
nuclear forces
Fermi treated the problem of β-disintegration on the
hypothesis of “neutrino”. According to this theory, the
neutron and the proton can interact by emitting and
absorbing a pair of neutrino and electron. But the in-
teraction energy calculated on such assumption is much
too small to account for the binding energy of neutrons
and protons in the nucleus.
Yukawa believes that it seems natural to modify the
theory of Heisenberg and Fermi in the following way.
The transition of a heavy particle from neutron state to
proton state is not always accompanied by the emission
of light particles, i.e., a neutrino and an electron, but
the energy liberated by the transition is taken up some-
times by another heavy particle, which in turn will be
transformed from proton state into neutron state. If the
probability of occurrence of the latter process is much
larger than that of the former, the interaction between
the neutron and the proton will be much larger than
in the case of Fermi, whereas the probability of emis-
Mathematics & Nature (2021) Vol. 1 3
sion of light particles is not affected essentially. The
interaction between elementary particles is usually de-
scribed by a force field, just as the interaction between
charged particles can be described by an electromagnetic
field. Yukawa cites the term of quantum mechanics that
was just emerging at that time to explain that this field
should be accompanied by a new quantum, which has an
important impact on the nuclear structure. The reason
is that it is like an electromagnetic field accompanied by
photons.
Yukawa believes that there is a p otential function Φ
that rapidly decreases with the increase of distance r in
the field describing the interaction between neutrons and
protons. This potential function satisfies the stationary
wave equation similar to the electromagnetic wave equa-
tion, and its time-free expected form is
Φ = ±g
2
e
−λr
r
(1)
Where g is the constant representing the interaction
strength and λ is the undetermined constant. The quan-
tum of the force field appears in this potential function,
and its characteristic quantity is the mass m, which is
an undetermined parameter here. Whether or how this
characteristic mass can be determined determines the
fate of the theorization of the above-mentioned idea.
Yukawa believes that the potential function should sat-
isfy the lower wave equation
∆ −
1
c
2
∂
2
∂t
2
− λ
2
Φ = 0 (2)
Then the matrix is introduced to define that there are
two opposite unit results, representing the neutron state
and the proton state respectively,
1 0
0 −1
= ±1 (3)
Then equation (2) is rewritten into a new form with ma-
trix
∆ −
1
c
2
∂
2
∂t
2
− λ
2
Φ = −4πg
˜
ψ
1
2
0 1
1 0
± i
0 −i
i 0
ψ (4)
Where ψ and
˜
ψ are a pair of conjugate wave functions
representing heavy particles. The sign in front of the
complex matrix corresponds to the proton to neutron
transition and the inverse transition respectively. A
large number of equations related to neutrons and pro-
tons without causality are omitted here. Yukawa also
lists platzwechselintegral called the Heisenberg function
J (r) = −g
2
e
−λr
r
(5)
It is claimed that the spin of the lowest energy state of
the Hamiltonian square operator H
2
is 0, which is differ-
ent from that Heisenberg takes J (r) as positive, and the
experiment requires that the spin of the lowest energy s-
tate with a negative sign above is II. The two constants g
and λ in the formula will be determined experimentally.
The following description that seems to have a subtle
relationship is that the Φ– field should be quantized ac-
cording to the general method of quantum theory, the
neutrons and protons obey Fermi statistics, the quantum
accompanying the Φ– field should obey Bose statistics,
and the quantization can be carried out according to the
method similar to the electromagnetic field. The lower
limit of the law of conservation of charge requires that
the electron charge quantum is e or −e, and the field
quantity Φ corresponds to the operators that increase
the negative charge quantum number and decrease the
positive charge quantum number 1 respectively. Then,
the momentum operator and energy operator of quan-
tum mechanics are used to define an equation between
the undetermined constants λ and m,
ˆp
2
= −~
2
∆,
ˆ
W = i~
∂
∂t
, mc = λ~ (6)
The reduced Planck constant ~ = h/2π in the original
text is written as Planck constant h. This introduction
has been modified one by one, the same below. Then
it is claimed that the wave equation satisfied by OO in
free space should be written as follows
ˆp
2
−
ˆ
W
2
c
2
+ mc
2
Φ = 0 (7)
Yukawa’s conclusion is that the quantum of the adjoint
field of neutron proton interaction has an appropriate
massm = λh/c, which actually belongs to the repeated
statement of no causality for the above definition.
Then take the potential functionΦ = Φ
′
(r) e
iωt
, where
ω = (W
N
− W
p
)/~, W
N
and W
p
are the energies of neu-
trons and protons respectively. Thus, equation (4) is
rewritten as following
∆ −
λ
2
−
ω
2
c
2
Φ
′
= −4πgνu (8)
Yukawa claims that the solution to the equation is
Φ
′
= g
1
|r − r
′
|
exp
−
λ
2
−
ω
2
c
2
|r − r
′
|
˜ν (r
′
) u (r
′
) dν
′
(9)
4 X. D. Dongfang The End of Yukawa Meson Theory of Nuclear Forces
It seems that the initial expectation, that is, the key po-
tential function of the nuclear force field, has only been
mentioned here. However, from hypothesis (1) to equa-
tion (9), Yukawa’s paper describes a large numb er of
irrelevant and non causal contents, which may be un-
precedented.
As for equation (9), only from the formal point of
view, if λ
2
> ω
2
c
2
, then
λ
2
− ω
2
c
2
1/2
is a real num-
ber, and Heisenberg function (5) has the following form
J (r) = −g
2
1
r
exp
−
λ
2
−
ω
2
c
2
r
(10)
Where
λ
2
− ω
2
c
2
is determined by the absolute val-
ue |W
N
− W
P
| of the energy difference between neu-
trons and protons. When the latter W
P
approach-
es mc
2
, it becomes smaller and smaller, which is ex-
plained that the interaction range between neutrons
and protons increases with the increase of |W
N
− W
P
|.
If λ
2
< ω
2
c
2
,
λ
2
− ω
2
c
2
becomes a pure imagi-
nary number, and the potential function Φ represents a
spherical undamped wave. Yukawa’s explanation is that
through the transition of heavy particles from the neu-
tron state to the proton state, quantum energy greater
than mc
2
can be emitted in outer space, provided that
|W
N
− W
P
| > mc
2
. The velocity of Φ−wave is greater
than the speed of light, but the group velocity is still
less than the speed of light, just like the electron wave.
The reason why the quantum with such a large mass
(if they ever existed) has not been found may be that
the mass
λ
2
− ω
2
c
2
is so large that the condition
|W
N
− W
P
| > mc
2
cannot be realized in ordinary nu-
clear conversion.
At the end of the follow-up paper on the interac-
tion of elementary particles II
[14]
published separately,
Yukawa introduced the mass m repeatedly mentioned
above. What it means is that street and Stevenson have
obtained more evidence for the existence of new parti-
cles. The mass is consistent with the result of Compton
wavelength estimation in later textbooks, which is about
130 (1 ± 25%) Mev
∼
=
268.5m
e
. If this mass is accepted
as the mass m of the meson, then it is speculated that
the range of the nuclear force is
1
λ
=
~
mc
∼
=
~
268.5m
e
c
(11)
Considering equation (10), the specific form of the ini-
tially expected potential function (1) is written as
Φ = ±g
2
1
r
e
−
mc
~
r
(12)
This seems to describe the short-range force.
The physics community believes that the mass esti-
mates listed by Yukawa are consistent with the mass
273m
e
of π
±
meson found in cosmic rays in 1947 and
the mass 264m
e
of π
0
meson found in 1950. The π
meson is thus generally accepted as a quantum of the
nuclear force field. Yukawa was rewarded by authorita-
tive organizations. His unprecedented success enhanced
the credibility of the theory, and the meson theory of
nuclear forces became popular and developed. It is now
generally believed that nucleons interact with each other
through mesons.
3 Serious problems of Yukawa meson theo-
ry of nuclear forces
The original intention of Yukawa meson theory of nu-
clear forces is very simple. In order to accurately calcu-
late the binding energy of nuclear β decay, it is assumed
that there is a centrosymmetric potential function de-
scribing the nucleon interaction, and an operator equa-
tion with the assumed potential function as its exact
solution is sought. Then, it is tried to prove through
experiments that the integral constant in the solution of
the equation is consistent with the exp ectation of the in-
tensity of nuclear force interaction, and it is determined
that the mass in the potential function is the mass of
the new particle sought.
As we know, as the most basic law of material inter-
action, it is impossible to find it through the operator
equation. We have also proved that it is impossible to
determine the parameters representing the interaction
intensity in the Yukawa function through experiments,
although the assumed Yukawa potential function has a
lower limit in theory. Yukawa’s papers and the papers
he collab orated with his students did not really solve any
problems. Even today’s scientists can not find the nucle-
ar force potential function through the operator method.
At first glance, however, Yukawa’s paper seems to be
well described, as if it had really solved a problem. Af-
ter careful reading, you will find that Yukawa’s pap er
does not list a large number of equations that appear
abstruse but have no logical relationship, and has writ-
ten a large number of hypothetical explanations of non
causal relationship. The key procedure covered by the
whitewash is to describe the expected solution of equa-
tion (12) by referring to the real solution of equation (8).
This is very much like a kind of deceptive anti science
behavior that deliberately confuses right and wrong. In
fact, equation (9) is only the formal solution of equation
(8) rather than the real solution, and the explanation
of formal solution (9) in the case of λ
2
< ω
2
c
2
is com-
pletely right and wrong. In Yukawa’s paper, the reduced
Planck constant ~ = h/2π is written as Planck constant
h, and it is obvious that the error of term m
2
c
2
with
momentum square dimension in equation (7) is written
as term mc
2
with energy dimension. These are by no
means clerical mistakes! The writing error of the former
exposes that the author is very unfamiliar with quantum
Mathematics & Nature (2021) Vol. 1 5
mechanics. In particular, the writing error of the latter
exposes that the author is not familiar with the solu-
tion of the wave equation and does not really solve the
corresponding wave equation, because the clerical error
of the equation can always be found and corrected in
time in the process of solving the equation. All the de-
scriptions and conclusions are more likely to come from
the fuzzy memory of relevant literature, of course, the
play of which is very random. Because of this, the real
solution of the equation is ignored, and the difference be-
tween the formal solution and the real solution and their
different physical meanings has not been paid attention
to, but there have been random explanations and wrong
conclusions about the virtual exponential solution.
Yukawa meson theory of nuclear forces, qualitatively
speaking: 1) it has not really proved that the medium
of nuclear force is meson; 2) An operator equation is
not constructed successfully, which shows that the so-
lution of the equation is the desired function; 3) It is
impossible to prove that the mass in the potential func-
tion is the mass of the meson; 4) The expected bind-
ing energy of β decay cannot be calculated; 5) Ignoring
the hypothesis that there are mesons in the nucleus will
completely negate the serious consequences of the nu-
clear mass measurement results in history; 6) It is full
of false or non causal explanations. It is puzzling why
such a pap er equivalent to machine generation has been
widely recognized and developed in the physics field for
nearly a century, and is still developing today. Yukawa
meson theory of nuclear force being simply summarized
to that, the logical construction process is to knead the
mathematical expressions and mathematical equations
that have no causal relationship together, list the con-
cepts and equations of quantum mechanics and particle
physics without really understanding their significance,
introduce the wave equation of quantum mechanics and
cannot correctly solve the wave equation, especially the
calculation of Dirac quantum force matrix, More than
80% of the space has nothing to do with the construction
of potential function. However, the behavior catered to
the trend of Dirac quantum theory and particle physic-
s at that time, giving readers the illusion of profound
knowledge. To put it bluntly, Yukawa’s paper is simply
the dream language of insane patients. Regardless of
mathematical equations or language descriptions, they
deliberately go around the Bush and put together the
cutting-edge physics vocabulary, physical equations and
the names of celebrities of the era that has no logical rela-
tionship. From equation (1) to equation (13), including
the Hamiltonian expressed by the matrix in the middle,
few points can be proved. Yukawa meson theory of nu-
clear forces is very angry to read! The only thing worth
thinking about is the quantitative relationship between
relativistic operator equation and potential function ex-
pectation (1), which needs to be discussed in detail and
given a scientific answer.
The electric field strength and potential satisfy the
operator equations. The solutions of these operator e-
quations of course contain the electric field strength or
potential, but usually there are other solutions. For un-
known interaction forces, the solution of an operator e-
quation may not be the potential function describing
the interaction. Construction of wave equation requires
a physical model. Equation (2) is a wave equation con-
structed according to the expected function (1). Equa-
tion (7) is an operator equation of relativistic momen-
tum energy relationship. Its solution is not Yukawa’s
expected potential function, but Bessel function.
Yukawa assumes that there is a function with param-
eters to be determined in the nucleon interaction, and
considers that it satisfies the wave equation (1), and
then assumes that the parameters to be satisfied with
the relationship, mc = λh, and then compares the wave
equation with the relativistic operator equation, suggest-
ing that the assumption is correct. Here, with good in-
tentions, and the middle description goes around a lot,
which distracts the reader’s attention. It is very cunning.
The real motive is covered up. However, the assumption
belongs to the assumption after all. The so-called po-
tential function (1) must have
1
c
2
∂
2
Φ
∂t
2
= 0 (13)
This belongs to the steady-state meaning of the real
wave equation
[15]
∇
2
ψ +
4π
2
σc
2
(W − U) ψ −
1
c
2
∂
2
ψ
∂t
2
= 0 (14)
with total energy W and potential energy U introduced
by Dongfang. Where σis a determinable constant and c
is the speed of light. It must b e pointed out that if the
steady-state meaning of the Dongfang real number wave
equation is selected, the real number wave equation (14)
must be used. However, with this choice, even if U in
the real wave equation is arbitrarily removed and the
wave function ψ is replaced by the potential function
Φ, the potential function (1) cannot be derived. The
steady-state meaning of the imaginary wave equation in
quantum mechanics is
1
c
2
∂
2
ψ
∂t
2
= −
W
2
~
2
c
2
ψ (15)
Referring to the op erator definition (6), equation (2) and
equation (7) are regressed as
∆ +
W
2
~
2
c
2
− λ
2
Φ = 0
∆ +
W
2
~
2
c
2
−
m
2
c
2
~
2
Φ = 0
(16)
These two equations can only repeat the assumption
λ~ = mc of the last term of the definition equation
(6). Therefore, it is impossible to derive the potential
function (1) by choosing the steady-state meaning of the
imaginary wave equation in quantum mechanics.
6 X. D. Dongfang The End of Yukawa Meson Theory of Nuclear Forces
There is no meson in the nucleus. Otherwise the mass
of the nucleus or atom will not be expressed by the mass
number in the periodic table of elements. Perhaps the
most thought-provoking question is, how can the inter-
action between nucleons in atomic nuclei be dominated
by mesons and derive a huge meson theory of nuclear
forces?
4 Distorted logic of tearing and dismember-
ing operator equations
Ignoring for the time being the large-scale imitative
mathematical explanation without causality in Yukawa
meson theory of nuclear forces, Yukawa’s theory is equiv-
alent to directly putting forward two assumptions: 1) the
potential function of nucleon interaction is equation (1),
and it is the solution of the op erator equation (7) of rela-
tivistic momentum energy relationship; 2) Mesons exist
in the nucleus, and the mass in the potential function is
the mass of mesons.
Trying to prove these hypotheses, Yukawa meson the-
ory seriously distorts the mathematics, which is mani-
fested in many aspects. The main ones are: a) the so-
called equation representing the calculated values of the
neutron state and proton state matrix (3) is not tenable;
b) The solution (9) of equation (8) listed is only a situa-
tion solution, and the explanation of the physical mean-
ing of the formal solution in the case of λ
2
< ω
2
c
2
com-
pletely reverses right and wrong; c) The introduction of
Heisenberg function (1) used as an indirect proof of the
expected function (12) does not constitute a causal re-
lationship with the expected potential function (12); d)
Introducing δ function to rewrite relativistic momentum
energy operator equation is tearing and dismembering
the existing equation, which is an anti logical behavior;
e) The sp ecific form of the expected potential function
(12) is the pseudo solution of the relativistic operator
equation. The last two aspects are discussed in detail
here.
In order to disguise the expected potential function (1)
as a logical inference, Yukawa meson theory of nuclear
forces rewrites the operator equation (7) of relativistic
momentum and energy relationship into equation (8).
However, the specific form (12) of the expected potential
function listed at the end is obtained from the mutilat-
ed equation after secretly deleting the energy term (i.e.
W = 0) in equation (8), i.e. equation (16), and introduc-
ing irrelevant equations in the middle is to cover up the
anti logic characteristics of its operation. Later text-
books pushed Yukawa meson theory of nuclear forces,
adding fuel to the flames by introducing the δ function
to write the mutilated equation into a fashionable and
scholarly form
∆ −
m
2
c
2
~
2
Φ = −4πgδ (r) =
−4πg (r = 0)
0 (r ̸= 0)
(17)
The only solution of this mutilated equation is the so-
called Yukawa potential function
Φ = g
1
r
exp
−
mc
~
r
(18)
Where g is an integral constant, which needs to be deter-
mined by the definite solution conditions, but cannot be
determined in practice. Changing the operator equation
(8) or (16) into the mutilated equation (17) essential-
ly belongs to tearing and dismembering the relativistic
momentum energy equation, integrating the mutilated
equation, and then expressing the undetermined inte-
gral constant with g, claiming that this undetermined
integral constant represents the strength of the nuclear
force, and the result of the integration is the potential
function of the nuclear force, which is a complete blas-
phemy to mathematics and physical logic. The expected
solution (12) or (18) is only the pseudo solution of the
operator equation (2) or the relativistic operator equa-
tion (7) originally set by Yukawa meson theory.
Although we can prove that the equation of the re-
lationship between relativistic momentum and energy is
not correct, the relativistic equation does not lose it-
s physical meaning. Yukawa’s tearing and dismember-
ing of the relativistic momentum energy equation over-
turned the most basic physical meaning. The second
equation of equation (7) or (16) is the relativistic mo-
mentum energy operator equation of free particles. The
definition of the relativistic energy W of a free particle
comes from the relativistic kinetic energy theorem
v
0
F · dr =
v
0
d
dt
mv
1 − v
2
c
2
· dr
=
mc
2
1 − v
2
c
2
− mc
2
= E
k
− 0
(19)
Where the kinetic energy E
k
> 0, and the relativistic
energy
W =
mc
2
1 − v
2
c
2
= E
K
+ mc
2
> mc
2
(20)
The key trick of Yukawa’s nuclear force meson theory
is to make W = 0 in disguised form to give a mutilated
equation (17) that is not only of no physical significance
but also completely wrong, and then use the mutilated
equation (17) to piece up the desired solution (18), giv-
ing people the illusion that the function (18) is from the
relativistic wave equation. This kind of operation is very
cunning and unprecedented. It is not an unintentional
error in mathematical knowledge, but a deliberate dis-
tortion of mathematical rules, which is a blasphemy to
scientific logic.
Mathematicians and theoretical physicists should
know that the general solution of the second equation of
Mathematics & Nature (2021) Vol. 1 7
equation (7) or (16) is the product of angular spherical
harmonics and radial spherical Bessel functions
[16]
with
oscillatory properties. Therefore, capturing the light of
the relativistic wave equation, which was popular for a
while, can not actually derive the so-called Yukawa po-
tential function (18). In order to highlight the singular
point of the general solution and make a trade-off, here
I use the convergent series to express the solution of e-
quation (7) or (16). The specific form is
Φ =
C
l+1
√
W
2
− m
2
c
4
~c
l+1
r
l
exp
i
√
W
2
− m
2
c
4
~c
r
∞
n=0
b
n
√
W
2
− m
2
c
4
~c
r
n
Y (θ, φ)
C
−l
√
W
2
− m
2
c
4
~c
−l
1
r
l+1
exp
i
√
W
2
− m
2
c
4
~c
r
∞
n=0
d
n
√
W
2
− m
2
c
4
~c
r
n
Y (θ, φ)
(21)
It can be called the Dongfang function. Perhaps we
should all solve the equations by ourselves without com-
pletely relying on the conclusions of the textb ooks.
Otherwise we will ignore the different physical mean-
ings of the solutions in different situations. Where
C
l+1
and C
−l
are undetermined integral constants, l =
0, ±1, ±2, ···, and the coefficients of the series satisfy
the recursive relationship
b
n+2
= −
4 (n + l + 2) (n + l + 1)
(n + 2l + 3) (n + 2l + 2) (n + 2) (n + 1)
b
n
d
n+2
= −
4 (n − l + 1) (n − l)
(n + 2) (n + 1) (n + 1 − 2l) (n − 2l)
d
n
(22)
By substituting (21) into equation (17), it can be verified
that the recursive relation (22) is valid. If Φ is a wave
function in the meaning of Born’s statistical interpre-
tation, the viewpoint of traditional quantum mechanics
is to normalize the wave function to determine the in-
tegral constant; If Φ is all kinds of potential functions
with point radiation properties, including point charge
potential, the integral constant is determined by exper-
iment.
The so-called Yukawa meson theory of nuclear forces
obviously does not specifically solve equation (16) ac-
cording to the requirements of mathematical norms, but
only writes the solution of equation (8) or (16) in the
form of (9) according to the impression of Green’s func-
tion
Φ
′
= g
1
|r − r
′
|
exp
−
λ
2
−
ω
2
c
2
|r − r
′
|
˜ν (r
′
) u (r
′
) dν
′
Then reference the Heisenberg function
J (r) = −g
2
1
r
exp
−
λ
2
−
ω
2
c
2
r
Distract readers’ attention, pretend to discuss real num-
ber solutions and imaginary number solutions, use irrel-
evant energy mc
2
= ~ω sophistry to reverse the Bessel
function of the solution of the equation, explain the oscil-
lation function of equation (16) as an exponential func-
tion solution, and finally write the expected function
(18) to achieve personal goals.
Listing too many specious equations for unnecessary
explanation, and then interpreting the oscillation func-
tion solution of the equation as an exponential function
solution; or tear and dismember the known operator e-
quation and use the solution of the mutilated equation
to perjure the expectation function. The essence of be-
havior is to resort to mathematical equations for fraud.
Its absurd logic fools physics scholars all over the world
and steals fame. Its nature is very bad and should be
widely condemned.
5 Uncertainty of undetermined mass in
Yukawa function
Relativity is a theory that can be systematically
proved to be incorrect. However, it is much easier to
prove the error of relativity than to expose Yukawa’s
chaotic meson theory of nuclear forces. This is because
the theory of relativity has its independent mathemati-
cal logic, while the so-called Yukawa nuclear force meson
theory is purely mathematical tearing and patchwork.
So, is it possible that the distortion or dismemberment
of the wrong theory will produce a correct theory, so the
so-called Yukawa potential function pieced together with
the meson mass can be preserved? Although this prob-
ability is very low, in order not to omit some possible
results that can arouse people’s interest, we still add to
prove that there is no trace connection between mesons
and Yukawa function, or even if there is a reasonable
equation derived potential function (18), it is impossi-
ble to determine that the mass in this potential function
corresponds to mesons.
Here again, the universal logic test principle called
Dongfang unity principle is introduced: The result of the
8 X. D. Dongfang The End of Yukawa Meson Theory of Nuclear Forces
transformation of the mathematical form of the natural
law in different metrics to the same metric is unique.
The united principle is described in detail as follows:
There is a definite transformation relationship between
different metrics describing the natural law, and the nat-
ural law itself does not change due to the selection of d-
ifferent metrics. When the mathematical expression of
natural laws under different metrics is transformed into
one metric, the result must be the same as the inherent
form under this metric, 1=1, meaning the transforma-
tion is unitary
[12, 17, 18]
. The United principle is not only
an efficient tool for logic testing, but also an efficient
tool for establishing new theories. It can help us test
whether a theoretical reasoning conforms to logic from
different angles. A theory that conforms to the unitary
principle is not necessarily a scientific theory or a cor-
rect theory, and it must be wrong to violate the correc-
tion principle. Therefore, the establishment of physical
theory must consider whether it conforms to the unity
principle. The error nature of theories and experimental
reports that violate the unitary principle is usually very
serious. In particular, if the experimental report violates
the unitary principle, it will expose lies.
Let us now prove the unreliability of constructing op-
erator equations with respect to mass. According to
Maxwell’s electromagnetic theory, the electrostatic po-
tential ϕ of a point charge with electric quantity q in a
medium with dielectric constant ε satisfies the operator
equation
∆ϕ =
q
ε
δ (r) =
q
ε
(r = 0)
0 (r ̸= 0)
(23)
Assuming that there is an unproven logic that allows the
introduction of δ function to directly write the operator
equation about mass (17), then by selecting the grav-
itational field metric, electromagnetic field metric and
nuclear force field metric at the same time, the operator
equation satisfied by the potential functions of the three
fields can be written at the same time
∆
G
−
m
2
G
c
2
~
2
Φ = −4πg
G
δ (r)
∆
α
−
m
2
α
c
2
~
2
Φ = −4πg
α
δ (r)
∆
σ
−
m
2
σ
c
2
~
2
Φ = −4πg
σ
δ (r)
(24)
Where m
G
, m
α
and m
σ
respectively represent the mass-
es of the medium quantum of the gravitational field, the
medium quantum of the electromagnetic field and the
medium quantum of the nuclear force field. So there are
three potential functions.
Φ
G
= ±g
2
G
1
r
e
−
m
G
c
~
r
Φ
α
= ±g
2
α
1
r
e
−
m
α
c
~
r
Φ
σ
= ±g
2
σ
1
r
e
−
m
σ
c
~
r
(25)
So, what is the reason why m
G
= 0, m
α
= 0 and
m
σ
= m
π
are inevitable conclusions? Obviously, there
is no answer. Just because experimental physicists have
observed π mesons, arbitrarily claiming that m
σ
= m
π
is not in line with scientific logic, and considering that
function (18) is a potential function of nuclear force in-
teraction dominated by meson media is obviously a fluff.
It can be seen that the later textbooks introduced δ func-
tion into Yukawa meson theory of nuclear forces, which
did not increase the logic power, but brought more trou-
ble. The logic of Yukawa nucleon meson theory to de-
termine the meson mass violates the unitary principle,
and the conclusion is only an anti scientific assumption.
To sum up, we draw a conclusion of universal signifi-
cance: it is impossible to construct an operator equation
about unknown mass, so that the solution of the equa-
tion becomes a potential function describing the action
of unknown force without logical contradiction, so as to
accidentally discover a new basic law of material inter-
action.
6 The formation of nuclei cannot accommo-
date mesons
If Yukawa meson theory of nuclear forces discards the
twisted logic of tearing and dismembering relativistic op-
erator equations to piece up the so-called potential func-
tion, and only assumes that mesons are the medium of
interaction between nucleons, is this assumption possi-
ble?
In fact, according to the well-known common sense,
the mass difference between neutrons and protons is
much smaller than that of mesons, indicating that neu-
trons cannot be constructed by protons and mesons;
Considering the binding energy of the atomic nucleus,
neutrons and protons combine to form the atomic nucle-
us to lose mass, and it is impossible to absorb new parti-
cles to make them become the medium quantum for the
nuclear force interaction between Nucleons to take ef-
fect. Therefore, middle school students who understand
the basic knowledge of atomic nuclei can also prove that
there is no meson in atomic nuclei, and Yukawa’s nuclear
force meson theory is not tenable! In order to popularize
this basic knowledge to middle school students in the fu-
ture, so that more physics readers will no longer blindly
listen to the lies of well-known physicists, these simple
calculation equations and results are listed below. Ac-
cording to the recent evaluation of atomic mass
[19, 20]
, the
Mathematics & Nature (2021) Vol. 1 9
internationally recommended basic physical constants
[21]
may be more reliable. The masses of electrons, protons
and neutrons are as following,
m
e
= 9.109383 56(11) × 10
−31
kg
m
p
= 1836.15267389(17)m
e
m
n
= 1838.68366158(90)m
e
(26)
And the mass of π meson is ab out
m
π
∼
=
268.5m
e
(27)
The mass of neutrons is slightly larger than that of pro-
tons, and the mass difference between the two is
m
n
− m
p
= 2.53098769(73)m
e
(28)
The result is only 2.53/268.5
∼
=
0.0094 times the mass of
mesons, which is far less than the mass of mesons. This
indicates that neutrons cannot be composed of protons
and mesons, and also indirectly indicates that the inter-
action medium between neutrons and protons cannot be
π mesons.
On the other hand, taking deuteron as an example,
the calculation of its binding energy can evaluate that
the medium of interaction between neutrons and proton-
s is not mesons. A deuteron consists of a proton and a
neutron. The mass of a deuteron is
m
d
= 3670.48296785(10.13)m
e
(29)
The binding energy of deuteron is expressed in the unit
of electron mass, and the result is
∆M
d
= m
p
+ m
n
− m
d
= 4.35336762(94)m
e
(30)
This means that the deuteron formed by the combina-
tion of a neutron and a proton loses the equivalent of
4.4 electron masses, rather than absorbing a π meson to
make the nuclear gravity between neutrons and protons
in the deuteron take effect. Therefore, the composition
of deuterons cannot be related to π meson and other
particles with mass. The calculation methods of oth-
er nuclei are similar, and the results are consistent: it
is impossible to absorb any other leptons, mesons and
baryons when the nucleus is formed.
The simple calculation results that the above middle
school students are good at are enough to prove the com-
mon sense conclusion: there is no meson in the atomic
nucleus at all, and the nucleons of the atomic nucleus
cannot rely on the control of mesons that do not exist in
the atomic nucleus to generate nuclear force! Yukawa’s
hypothetical meson theory of nuclear forces violates the
unity principle. It is a lie and sophistry that random-
ly accumulates a large number of irrelevant and wrong
mathematical formulas to whitewash itself.
7 Conclusions and comments
The chaotic logic of Yukawa meson theory of nuclear
forces without causality is very angry. Trying to correct
its distorted calculations or explanations cost years of
time and a lot of energy but ended in failure.
Far from being able to compare with the theory of
relativity which can also prove that it is not tenable,
Yukawa’s meson theory of nuclear forces makes us un-
able to have any respect for it. It’s too much low-level
logical errors and cunning sophistries make people angry.
1) Yukawa knows only a little about the basic theories of
quantum mechanics and particle physics, and there are
principled errors in the writing of mathematical expres-
sions from beginning to end; 2) Listing a large number
of non causal modern physical concepts, nouns, equa-
tions and mainstream achievements, reversing right and
wrong, concocting untenable logic, whitewashing aca-
demic foundation, catering to celebrities and controlling
readers’ thinking; 3) Do not understand the significance
of the basic equations of relativity, tear apart and dis-
member the operator equations of the relationship be-
tween momentum and energy, and disguise assumptions
as mathematical inferences; 4) The meaning of wave e-
quation and the analytical theory of wave equation are
not really grasped, and the result of Bessel function is
distorted into a simple expected potential function, mis-
leading physics readers; 5) Demonize the mesons that do
not exist in the atomic nucleus into the medium of nu-
cleon interaction, and fool the physical world. Because
there are a large number of serious uncorrectable math-
ematical errors and untenable physical logic, Yukawa’s
meson theory of nuclear forces is only anti scientific logic.
Up to now, the new theories about the development
of relativity and the experimental reports about relativ-
ity have always been able to excite the hearts of millions
of people and shake the authoritative organizations of
physics. However, assuming that the theory of relativi-
ty is completely correct, and then establishing a theory
with the theory of relativity, we should first consider
that the inference of relativity is different from that of
Newtonian mechanics only when describing high-speed
motion and it does not fundamentally change the law of
material interaction. This means that the existence of
the law of interaction of matter can never take relativi-
ty as a sufficient and necessary condition. Therefore, it
is obviously self deception to claim that the most basic
law of material interaction has been discovered through
relativity. It seems reasonable to use the experimental
data to construct a nuclear force potential function. The
reason why it can not be realized may be that a large
number of experimental data on high-energy physics are
far from enough to be used to construct the nuclear force
potential function. However, the nuclear force potential
function, which is the most basic law of matter interac-
tion, can be easily obtained by tearing and dismembering
the relativistic equation of momentum and energy, which
10 X. D. Dongfang The End of Yukawa Meson Theory of Nuclear Forces
obviously exposes the fatal defect of thinking. From this
point of view, experimental scientists of particle physic-
s should first question and oppose the meson theory of
nuclear force. However, if, as experimental scientists in
particle physics, we have no confidence or courage to
question and oppose the meson theory of nuclear force,
does it prove that we have never had any confidence in
our own experimental data and reports? Not to mention
that outsiders do not always believe our experimental re-
ports, which are often immersed in error analysis. The
experimental report can also test its reliability with the
principle of logical test.
Using the unitary principle, we can widely test the
principled errors of modern physics represented by rel-
ativity, and find traces of tampering with and fabricat-
ing observation data in experimental reports. The so-
called achievements of relativistic quantum mechanics,
meson theory of nuclear forces and gravitational wave
of spiral binary stars as relativistic applications seem
to be supported by experiments. If the inferences of
erroneous theories and anti scientific theories are wide-
ly supported by experiments, is it not in itself the evi-
dence of lies? What it usually exposes is the behavior
of simulating observation data, computer-generated ob-
servation data and manual tampering with observation
data, or the inference is exaggerated without substantive
breakthrough. The unitary principle has a powerful log-
ic checking function. 1) Using the unitary principle, the
complete space-time transformation is found
[12]
, which
shows that the Lorentz transformation is not a complete
transformation between the time coordinate parameter-
s in each inertial system. Therefore, it is proved that
the Einstein assumption of constant speed of light is not
tenable within the framework of relativity, which means
that relativity is not a scientific theory; 2) The applica-
tion of the unitary principle has discovered the morbid
equation of quantum numbers
[22]
hidden in quantum me-
chanics, which is one of the fatal logical contradictions of
quantum mechanics. It shows that the work done by de-
veloping quantum mechanics in the past without reveal-
ing the essence of quantum mechanics may have very lit-
tle real scientific significance except for the positive im-
pact of physical thought; 3) Using the unitary principle,
it is found that the Dirac energy level formula written
by the Dirac equation of relativistic quantum mechanic-
s applied to the hydrogen atom is a pseudo solution of
the equation
[23]
, so the challenge solution of the Dirac
equation of the hydrogen atom is proposed; 4) Using the
unitary principle, it is also found that the frequency dis-
tribution of LIGO signal not only does not conform to
the relativistic frequency equation
[22]
, but also complete-
ly negates the relativistic frequency equation. LIGO’s
spiral binary black holes failed to merge
[24, 25]
. Further
verification of the LIGO experiment report by the uni-
tary principle will prove that the LIGO signal belongs
to the ground signal. The unitary principle is also used
here to draw a totally negative conclusion from the test
of Yukawa meson theory of nuclear forces. The so-called
gravitational wave that detects the merging of ancient
spiral double black holes has exactly the same proper-
ties as the meson theory that experimentally confirms
the nuclear force. The meson theory of nuclear force
only tears and dismembers the relativistic momentum
square operator equation, and describes the irrelevan-
t meson as a sophistry of the medium quantum of the
nuclear force field. Detecting the so-called gravitational
wave of the fusion of spiral double black holes is not even
sophistry, but a lie.
We also discussed an exceptional question: is it pos-
sible that the distortion or dismemberment of a wrong
theory can produce a correct theory by accident? Tak-
ing the meson theory of nuclear force as an example,
for a stationary atomic nucleus, there is only a formal
difference between the relationship between relativistic
momentum and energy and the relationship between mo-
mentum and energy in Newtonian mechanics. However,
Yukawa function can not be obtained by using the op-
erator equation of the relationship between momentum
and energy in Newtonian mechanics. This obviously vi-
olates the unity principle. The theory that violates the
unity principle cannot be correct. Mesons exist, but
there are no mesons in the nucleus. Even the philos-
ophy of meson theory of nuclear forces has no merit.
Quietly reviewing the origin of Yukawa meson theory of
nuclear forces, comparing the operator equation of rel-
ativistic momentum and energy relationship with that
of Newtonian mechanics, we can find and falsify how
the so-called Yukawa function distorts mathematics and
violates the unity principle. Yukawa meson theory of
nuclear forces is an epitome of the eagerness for quick
success and instant benefit in mo dern physical theory
and experimental research: it does not discuss the in-
evitable causal relationship, emphasizes personal views
and explanations, makes false conclusions, and can al-
ways list a large number of plausible physical concepts,
mathematical equations and references without causal
relationship, refusing to accept the questions of reader-
s. How long will this anti scientific trend, described as
scientific research, dominate the theoretical physics com-
munity? Can the lies of physics really cover the sky and
the sun forever?
To sum up, Yukawa’s meson theory of nuclear forces
is not a scientific theory and must be completely aban-
doned due to its serious distortion of mathematical and
physical principles, random tearing and dismembering
of known equations, illogical patchwork of personal ex-
pectation functions, arbitrary interpretation of events
without cause, and whitewashing of illusory structures
that do not exist in fact.
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Latest findings
The End of Klein-Gordon Equation for Coulomb Field
The unitary principle is used to test the mathematical procedures and conclusions of the standard theory of the Klein-Gordon equation in a Coulomb field, and it is revealed that the exact solution of the Klein-Gordon equation in a Coulomb field hides the inexorable wave function divergence and virtual energy difficulty. The divergence was not found in the past because it was concealed by an unnecessary function transformation introduced in the process of solving differential equations. Since the expected solution of the Klein-Gordon equation for a Coulomb field does not meet the boundary conditions, but is only a pseudo solution, and there is no other exact solution of the Klein-Gordon equation for the Coulomb field that meets the expectation of energy quantization, which declares the end of the Klein-Gordon equation for the Coulomb field. This conclusion is irreversible. The Klein-Gordon equation is constructed by the evolution of the operator of relativistic momentum energy equation. It is the foundation equation of relativistic quantum force. The end of the desired solution indicates that relativistic quantum mechanics and even relativistic mechanics are facing severe challenges.