End of quantum electrodynamics (QED).

Top page ( correct Bohr model )
Trick of QED magnetic moment.
Special relativity is wrong !
Propagators of various fields.

Table of contents (13/1/20)

Reason why QED is wrong.

[ Quantum electrodynamics is NOT physics, but "wrong" math. ]

(Fig.1) Wrong math ?

You may hear quantum electrodymanics (QED) is most successful theory in the world, which can predict precise values such as anomalous magnetic moment (= g-factor ) and Lamb shift.
But in fact, we show QED uses wrong and "ad hoc" mathematical trick and has NO physical images at all.
( Actually, even if you study about QED for very long time, QED is completely useless in the real world. )

Though more than half a century has passed since QED was born, we do NOT know the basic reasons why QED "mathematical" method using renormalization can give precise values.
Renormalization means we artificially remove infinity to get the very tiny values.

Infinite "bare" charge = renormalization ?   Really !?

(Fig.2) Infinite bare charge and mass are real ?

According to QED, bare charge and mass of an electron are infinite ! ( Can you believe this ? )
Because infinite values are artificially packed into the charge and mass of an electron.
So there were many objections to this QED.
For example, Feynman himself referred to renormalization as a "shell game" and "hocus pocus" in his book.

Furthermore, Dirac was very unsatisfied with renormalization method in QED.
(See also "The Strangest Man" by Graham Farmelo.)
Dirac was an expert in mathematics. So he insisted that ignoring very small value is reasonable, but ignoring infinite value can not be understood.
( Though QED calculations completely depend on Dirac equation. )

Fatal paradox of relativity → QED is wrong, too.

(Fig.3) Fatal paradoxes of special relativity = QED is wrong.

And as shown on this page, fatal paradoxes shows special relativity is wrong.
( If only there is one "Paradox", it means this theory is wrong. Actually there are various paradoxes. )

Of course, if special relativity is wrong, QED is wrong, too, because QED completely depends on the relativistic energy and momentum relation through Dirac equation.
And as shown on this page, Dirac equation is special relativity itself.

In addition to the ad hoc mathematical tricks which artificially remove infinity, special relativity is wrong.
So these double blows prove QED is wrong.

Classical Lorentz force + Dirac equation = QED.

(Fig.4) Lorentz force → Feynman diagram.

As shown on this page, QED interaction among electrons and photon originates in classical Lorentz force.
In QED, a photon can be expressed as magnetic and electric potentials (= A, φ ).
If we do the replacements in Dirac's Lagrangian, we get the inteaction term, which gives Feynman diagram.

And this form of interaction is ONLY one interaction among electrons and photon in QED.
So you need to remember there are NO physical images in "mathematical" QED.

Time evolution by interaction Hamiltonian.

(Fig.5) Time evolution by interaction Hamiltonian.

As shown on this page, quantum field theory only expresses paticles such as electron (= c) and photon (= a ) as very abstract creation and annihilation operators.
Using the interaction term of Fig.4, we can get interaction Hamiltonian (= HI ), which expresses the interaction among electrons and photons.

In the quantum theory, the time evolution can be expressed using exponential function + Hamiltonian.
By expanding the exponential function, we get terms consisting of various numbers of Hamiltonians.
In this page about g factor correction, we use the term consisting of three Hamiltonians, which diverges to infinity.

Feynman diagram by three Hamiltonians forms one loop.

(Fig.6) Feynman diagram -- three Hamiltonians (= vertex ).

Fig.6 shows the case of three interaction Hamiltonians.
An "in" electron with momentum " p " and photon with momentum "q" fuse into "out " electron " p' ".
The energy and momentum are conserved, p' = p + q.
Each vertex consists of one photon (= A, brown ) and two fermions (= ψ, blue ).

Binding two identical particles ( photon-photon or electron-electron ), we get Feynman diagram.
Three internal lines of Fig.6 are called propagaters ( of photon or fermion ).

Renormalized charge "e" becomes infinite !

(Fig.7) Renomalizatoin = infinite bare charge.

As I explain later, Fig.6 includes one loop consisting of three internal lines (= propagator ).
To be Lorentz-invariant, each wavefunction must include infinite kinds of momentum and energy.
So this loop diverges to infinity, when we calculate this time evolution.

In QED, we artificially pack that infinity into very small charge "e" of an electron.
As a result, the very small electron charge "e" becomes infinite !
The form of this interaction is kept by this ad hoc manipulation, so they are satisfied with this method.
( How do you think about this ? )

QED g-factor depends on "wrong" math.

(Fig.8) Change of variables = QED trick.

As shown on this page, calculating anomalous magnetic moment by QED is based on very artificial tricks.

The upper equation in Fig.8 includes only k2 term, and does NOT include the information of g-factor.
When we integrate it with respect to the variable "k", it diverges to infinity.
Because the interval of integration is from -∞ to +∞.

When we change the variables of integration as k → l - β, new β term appears and this β includes the information of g-factor.
Single " l " term becomes zero by the integration, becuase it is an odd function.
l2 term diverges to infinity, but this term is removed artificially by the renormalization.
As a result, we can get the g-factor value.

∞ = ∞ + finite value ?   ← Trick !

(Fig.9) Infinity remains infinity ( NOT finite value ) !

Of course, the method of Fig.8, which is used in calculating g-factor, is wrong.
The original integration of k2 is equal to the whole equation of ( l2 -2βl + β2 ), so, ∞ = ∞ is satisfied.
Finite β2 term must be neglected, because the whole equation itself remains infinite.
We are NOT allowed to pick up only the finite β terms after the change of variable, according to the usual mathematics.
So this method is wrong math.

Using this wrong methematics, we can easily add the g-factor value to the terms, which originally have NO g-factor.

Odd function:   ∞ - ∞ = 0 ?

(Fig.10) QED takes advantage of infinity.

In QED calculation, we can delete odd functions of " l " by integrating them from minus infinity to plus infinity.
If the original interval of integration is finite ( from -α to + α ), we cannot delete terms including single " l ".
Because the change of variables cannot give odd functions of " l ".

Of course, if we don't delete these terms, we cannot get the g-factgor.
This means QED makes good use of infinity, and later removes them artificially.

QED wrong math. part II.

(Fig.10') This math is correct ?

In Fig.10' upper, integration of odd function "k" becomes zero.
After the change of variables ( k = l - β ), The part of " l " becomes zero due to odd function.
As a result, only "-β" is left, which includes the information of g-factor.
This math is indispensable for getting g-factor in QED.
Do you think the math of Fig.10' is correct ?

Change of variable,   ∞ → ∞ + "finite".   Trick !

(Fig.10'') Right math.

If we use a right mathematical method, the part of "-β" becomes zero, too.
Because after integration, " l2 " term diverges much faster than " βl " term.
As a result, only " l2 " term is left, and becomes zero.
So QED mathematical method of Fig10' violates basic math.
About the actual example, see this page.

Virtual photon satisfies Einstein formula = QED trick.

[ Virtual photons always appear in QED. ]

In this section, we show the photon "q" of Fig.6 is a virtual photon, which disobeys special relativity.
These tachyonic virtual photons are peculiar to QED, gauge theory, and standard model, though they are relativistic theories !
Suppose one electron (= p ) at rest absorbs one photon (= q ) and becomes p' electron.
(Fig.11)

The energy of an electron at rest (= p ) is only rest mass energy ( E/c = mc ).
And an electron (= p' ) after absorbing photon has the momentums.
The electrons ( before and after the interaction with photon ) need to satisfy Einstein's energy-momentum relation,
(Eq.1)

From Fig.11 and the conservation of energy and momentum, the photon's energy and momentum is
(Eq.2)

As you know, photon's mass is zero, but from Eq.1 and Eq.2,
(Eq.3)

Eq.3 means this photon's mass is imaginary number !
Because energy-momentum equation of Eq.3 must be zero or less than zero for real particles.
(Eq.4)

So Eq.3 shows this photon is tachyon, which disobeys special relativity.
As shown in Fig.6, this tachyonic photon "q" plays the leading role in calculating g-factor.
( So relativistic QED includes self-contradiction from the beginning. )

Einstein's mass formula of Dirac equation.

In this section, we shows this virtual photon "q" becomes zero in g-factor.
( Though ordinary QED textbooks do NOT say this important fact clearly. )
And if this virtual photon is zero ( q = 0 ), "out" electron's momentum " p' " becomes just equal to "in" electron's " p ".
( p' = p + q, ← q = 0. )

As a result, the g-factor ( anomalous magnetic moment ) can take any values.
This means our nature does NOT determine only one g-factor through QED.
( So QED is NOT a true theory. )

(Eq.5) Dirac's relation = Einstein's mass formula.

As shown on this page, Dirac equation originates in relativistic energy and momentum relation.
( In this page, (-1,1,1,1) version metric tensor is used. See this page. )

So the meaning of Dirac's relations of Eq.5 is just equal to Einstein's relation of p2 - E2/c2 + m2c2 = 0.
Like Eq.1, "in" and "out" electrons satisfy Einstein's energy and momentum relation, when we use Eq.5.
As a result, the photon "q" becomes virtual photon. ( ← p' = p + q. )

(Eq.6) QED g-factor calculation.

As I explain later, one loop propagators give the form like Eq.6.
In Eq.6, p' is at the left of γμ, and p is at the right.
Using the mass relations of Eq.5, we can change γapa at the right into - m, and γap'a at the left can be changed into -m.

Moving p or p' into another side of γμ causes g-factor.

So, p at the left of γμ cannot be changed into -m.
Because p is different from p'.
In the same way, p' at the right of γμ cannot be changed into -m.

(Eq.7) Difference between p and p'.

Of course, the differece between p and p' is due to the existence of virtual photon q.
So the rule of Eq.6 is effective, ONLY when this virtual photon q is NOT zero.
( You should remember this fact. )
(Eq.8)

In g-factor calculation, we often use the relation of Eq.8 to transfer p or p' from one side of γμ to another side.
Due to this manipulation, new term, which doesn't include γμ, appears.
This new term represents g-factor according to their theory.

Of course, these are ONLY rules they introduced, when they first calculate QED g factor.
So there are NO reasonable reasons why we cannot renormalize "p" at the left as it is.

γ matrices obstructs free transfer of "p".

(Eq.9) Transfer "p" of left side to right side.

When the virtual photon q is NOT zero, p is not equal to p' ( p' = p + q ).
So, the left side "p" cannot be changed into "-m" using Eq.5.
In this case, they use the relation of Eq.8, and transfer "p" to the right side and change it to -m.
In this process, one of the g-factor value is generated (= -2pμ ).

(Eq.10) Transfer p' of right side to left side.

In Eq.10, p' is at the right, so it cannot be changed into -m using Eq.5.
So they transfer p' to the left side and change it to -m.
Also in this process, one of the g-factor is generated.

When virtual photon is zero, "g-factor" can be "arbitrary". ← Meaningless.

(Eq.11) Virtual photon q = 0 → g-factor can be manipulated freely.

When the virtual photon q is zero, the difference between p and p' is gone.
As a result, p' becomes p, and we don't need to transfer p (= p' ) into another side to make them "-m".
Of course, we can freely transfer them into another side to generate g-factor.
This means when virtual photon q is zero, QED g-factor can be any values !

Surprisingly, the present QED methods make this virtual photon zero to get the g-factor.
So if we try to ask about the real mechanism of this QED precise calculation, this method is wrong in determining a single value.
And our law of nature does NOT adopt this QED to determine g-factor. ( See also g-factor can be manipulated. )

Virtual photon needs to be zero to get g-factor value.

(Eq.12) q2 = 0 is indispensable for getting g-factor.

As I explain later, to get the value of g-factor, the condition of q2 = 0 is indispensable.
The notation of q2 represents Einstein's energy and momentum relation, as follows,

(Eq.13) q2 = Einstein's relation.

In this page, (-1,1,1,1) version metric tensor is used ( see this page ).
Photon's mass is zero, so Einstein's relation of Eq.13 seemes to be true.
But you need to recall that this photon q is a virtual photon, which disobeys special relativity.

When Virtual photon satisfies special relaivity → It vanishes !

(Fig.12) Virtual photon + relativity = energy zero.

When this virtual photon obeys special relativity, its energy and momentum are completely zero.
Next, we prove this fact.

(Fig.13) p' = p + q.

"in" and "out" electrons of Fig.13 satisfy Einstein's formula,
(Eq.14)

So,
(Eq.15)

From Eq.15, if we suppose q2 = 0 (= photon mass is zero ),
(Eq.16)

where "pq" also means Einstein's formula.

In some direction, the energies and momentums of "in" electron and photon are
(Eq.17)

where zero component means energy, and other components are momentums.

From Eq.16 and Eq.17, we can get q=0, as follows,
(Eq.18)

Due to electron's "mass" energy, p0 is always greater than p1.

In conclusion, three conditions "in" and "out" electrons (= p and p' ) satisfy Einstein mass formula, energy conservation ( p' = p + q ), and virtual photon (= q ) satisfies Einstein mass formula ( q2 = 0 ), mean the virtual photon energy zero ( q = 0 ).
( So every component 0-3 of virtual photon becomes zero. )
In this very easy condition, QED calculation can give any g-factor values.

Various "artificial" rules are one of QED tricks.

(Fig.14) Various rules.

In the three-vertex g-factor correction, we use ONLY the pattern of Fig.14 A.
And we ignore the pattern of Fig.14 B, though this also has three vertices and one loop.
If we consider Fig.14B, too, the g-factor becomes almost twice as original values, of course, this case is wrong.
So, ignoring Fig.14B pattern is one of the artificial rules in QED.

In higher-order corrections, we can manipulate more variables of integration and other values of ( p, p' q ).
And other patterns of self-energy appear and we can choose some Feynman diagrams to get the g-factor.
Also in Lamb shift calculation, various other "ad-hoc" rules were introduced to get the experimental values.
But QED Lamb shift includes vertex correction above as one of them, which means QED Lamb shift cannot be determined as a single value, when virtual photon is zero.

Quantum electrodynamics (QED) is really correct ??

(Fig.15) QED = infinity.

It is said that the quantum electrodynamics (QED) gives extremely correct values such as electron g-factor (anomalous magnetic moment) and Lamb shift.
But it gives infinite values, which are far away from the correct values.
So, according to QED, bare charge and mass of an electron must be infinite.
"Nature" really obeys QED "infinite" rule ??

Here we explain some "mathematical" tricks and unreasonable interpretations used by QED in calculating electron g factor ( g-2 ).
The recent interesting experiments show that the fine structure constant alpha varies.
( J. K. Webb, et.al. Phys. Rev. Lett. 107, 191101 (2011) )
And recent interesting NIST's experiment using trapped Ti ion is inconsistent with QED.
( C. T. Chantler, et.al. Phys. Rev. Lett. 109, 153001 (2012) )
If these results are true, the reliability of QED will be gone.

Quantum electrodynamics and Feynman rules.

[ Feynman propagators in various fields. ( scalar and Dirac. ) ]

(Fig.16) Internal lines = propagator.

Propagator is a main tool in calculating various values in quantum electrodynamics.
So in this section, we explain feynman propagators in various fields ( scalar, Dirac, and electromagnatic fields.)
To understand propagators, we need to understand the basic quantum field theory first.
So first read this page (if you have not read ).

Propagators of Dirac and photon's fields.

(Eq.19) Dirac propagator (= electron ).

(Eq.20) Photon propagator.

Eq.19 is Dirac propagator ( electron - electron ), and Eq.20 is photon propagator.
Propagators are internal lines of Feynman diagram of Fig.16, and the time order is expressed using step function θ.
About the detailed calculation, see scalar, Dirac, photon propagators.

Time evolution by exponential function of interactive Hamiltonian.

As shown on this page, interaction Hamiltonian density between electron and electromagnetic field is
(Eq.21)

where we have to sum up all components of Aμ γμ (μ = 0, 1, 2, 3) according to the basic QFT rule.

Hamiltonian (= HI ) is gotten by the space integral of Hamiltonian density,
(Eq.22)

We have to expand the exponential part of Eq.22 in an Taylor series, as follows,
(Eq.23)

Using Eq.23, Eq.22 can be expanded, as follows,
(Eq.24)

where c(p) and a(q) mean external electron and ( virtual ) photon entering interaction vertex.
So these are expressed as creation operators.
And c(p') means external electron which leaves vertex.
( p, q, and p' mean momentum and energy of each particle.)
Here p + q → p'.
About the time evolution, Heisenberg and interaction pictures in QED, see this page.

One interactive Hamiltonian causes NO loop.

First we calculate the first-order of Hamiltonian density in Eq.24. (Fig.17.)
( First order is the second term of Eq.24. )

(Fig.17) One Hamiltonian.

Fig.17 doesn't contain internal lines (= propagators ) and loops.
Feynman diagram of Fig.17 means
(Eq.25)

As shown in Eq.25, two identical particles ( electron - electron, photon - photon ) are linked in operators.
As shown on this page, Dirac's solutions are,
(Eq.26)

where c satisfies anticommutation of
(Eq.27)

Using Eq.26 and Eq.27, the external line 1 (electron-vertex) of Fig.17 and Eq.25 becomes
(Eq.28)

where kp means the wave number = p / ħ of electron 1.
" cc " is annihilated by the vacuum of both ends.

And the conjugate transpose (× γ0) of Dirac wavefunction ( Eq.26 ) is
(Eq.29)

In the same way, using Eq.27 and Eq.29, the external line 2 (electron which leaves vertex ) of Fig.17 and Eq.25 becomes,
(Eq.30)

Vector potential A = photon.

As shown on this page, the solution of Aμ (x), which satisfies Maxwell equation is
(Eq.31)

And creation and annihilation operators of each component satisfy.
(Eq.32)

And using Eq.31 and Eq.32, the external line 3 (photon-vertex) of Fig.17 and Eq.25 becomes
(Eq.33)

This means " μ component " of the external photon.
When ν is not equal to μ, the metric tensor gνμ is zero.

Evergy and momentum are conserved in QED calculation.

And integrating Eq.28, Eq.30, and Eq.33 with resprct to d4 x, as shown in Eq.25, the exponential parts become
(Eq.34)

Eq.34 means the conservation of momentum and energy.
( pμ + qμ = p'μ )

There is NO divergence in the one vertex function, because it contains no loop in Fig.17.
And due to gμμ, γ matrices of vertex (Eq.25, Eq.33) changes into γμ

Two vertex (= Hamiltonians ) function G1 becomes zero.

Next we try two vertices in Eq.24, which contains two Hemiltonian densities and two integrations.

(Eq.35)

But in this case, as shown in Eq.35, there is one A(y) operator left, which can not be a pair.
This A(y) contains creation and annihilation operators which belong to different terms.
So due to the vacuum at both ends, the equation of Eq.35 becomes zero.
As a result, we need not consider the two vertex function.

Divergent one loop and three vertex function

Next we try three vertex function of Eq.24.
This contains three Hamiltonian densities and three integrations , as shown in Fig.18 and Eq.37.

(Fig.18)

where wave number and momentum of each particle is
(Eq.36)

(Eq.37)

And as shown in Eq.37, all operators form pairs, so there is no single operator left.
As a result, Eq.37 doesn't become zero.
The coefficient of the expanded exponential function 1/3! can be cancelled out, because there are six patterns ( 6 = 3 × 2 ) in arranging three vertices in Eq.37.
So we need not consider the coefficient 1/3!.

Dimensionless fine structure constant α is included in one loop.

Comparing Eq.37 (Fig.18) and Eq.25 (Fig.17), G2 function has two more vertices (integrations) than G0,
(Eq.38)

So here we aim to calculate renormalized vertex γ matrix of
(Eq.39)

Eq.38 and Eq.39 are only rules, which were defined only for getting g-factor first.
Basically, ikx and iωt of the exponential parts are "dimensionless", as follows,
(Eq.40)

And the fine structure constant α also is dimensionless, as follows,
(Eq.41)

Eq.38 includes two charge e in the two Hamiltonian densities.
As a result Eq.38 includes fine structure constant α, which is explained in detail later.
( Caution; Fig.18 contains three vertices, but one of them "originally" exists in Fig.17. )

Calculation of three vertex function.

Three external lines are the same as Eq.28, Eq.30, and Eq.33 in G0 function.
So the external electron ( ex 1 ) entering the vertex is
(Eq.42)

and the external electron ( ex 2 ), which leaves the vertex, is
(Eq.43)

and the external photon ( ex 3 ) entering the vertex is
(Eq.44)

where vertex γ changes into γμ.
This is important.

Using Eq.19 and this page, the propagator 1 of fermions (pro1) is
(Eq.45)

where
(Eq.46)

pk means momentum, and ωk means angular frequency.

As shown in Eq.45, propagator ( internal line ) contains infinite kinds of momentum and energy.
This is very difficult to imagine.
But accoeding to QED, even in very weak electromagnetic interaction, there are infinite particles (and antiparticles) created (and annihilated ) in the process.
(So QED is not a real one, as I said many times.)

In the same way, the propagator 2 of the fermions (pro2) is
(Eq.47)

As shown in Fig.18, pk' = pk + q, which means the conservation of momentum and energy.

And using Eq.20, the propagator 3 of the photons (pro3) are
(Eq.48)

where kp = p / ħ

Fig.18 contains three integrations. First we integrate with respect to d4x.
Eq.42, Eq.45, and Eq.48 includes eix, so
(Eq.49)

We use (2π)4 which belongs to one of three propagators 1 - 3.

Integrating Eq.49 with respect to d4kp-k, which is included in the propagator 3 of Eq.48,
(Eq.50)

Eq.50 means external electron (ex 1) is divided into propagators 1 and 3, as shown in Fig.18.

Eq.44, Eq.45, and Eq.47 includes eiy, so integrating it with respect to d4y,
(Eq.51)

This is wrong and just an artificial mathematical trick to manipulate calculation results to get convenient g-factor.

You cannot get the right answer without this artificial manipulation, which means "QED is the successful theory" is a total lie.

Next integrating Eq.51 with respect to d4k', which is included in the propagator 2 of Eq.47,
(Eq.52)

Eq.52 means that k and photon q are united into k'. (See Fig.18.)

Eq.43, Eq.47 and Eq.48 includes eiz, so integrating it with respect to d4z,
(Eq.53)

Eq.53 can use (2π)4, which is included in the propagators, because Eq.53 is an "excessive" part, as I said.
(So only one (2π)4 of three propagators is left.)

Substituting Eq.50 and Eq.52 into Eq.53,
(Eq.54)

Eq.54 means the conservation of
(Eq.55)

As you notice, in Eq.54, k is cancelled out, so we need NOT integrate it with respect to d4k.
So the integration of d4k in the propagator 1 (Eq.45) is left.
Of course, this integration is from -∞ to +∞, so it diverges to " infinity ".

Summary of one loop correction.

(Eq.56) Summary in one loop correction.

As a result, the coefficient of Eq.56 (= three propagators and two vertices ) becomes
(Eq.57)

where
(Eq.58)

Eq.57 is also dimensionless, which is similar to the fine structure constant α of

Coefficient of Eq.57 and remaning parts in three propagators of Eq.45, Eq.47, and Eq.48 and two vertices (= γ ), we get
(Eq.59)

In the second line of Eq.59, we use
(Eq.60)

This is very important.
"Unreal" zero component photon is indispensable for caluculation of propagators.
( gνρ of Eq.60 originates in these photons. )

Formula of γ matrices.

Next we calculate the numerator of Eq.59.
Using the formula of γ matrices of
(Eq.61)

where we use
(Eq.62)

One term included in the numerator of Eq.59 becomes
(Eq.63)

And using the formula of
(Eq.64)

Another term included in the numerator of Eq.59 becomes
(Eq.65)

And using the formula of
(Eq.66)

The terms left in the numerator of Eq.59 become
(Eq.67)

Final result of one loop Feynman diagram.

Using the results of Eq.63, Eq.65, and Eq.67, Eq.59 becomes
(Eq.68)

If we use ( +1, -1, -1, -1 ) version of metric tensor, taking the conversions of
(Eq.69)

( See also appendix. )

Eq.68 can be expressed as
(Eq.70)

From here we simplify the expression of Eq.68 using the definition of
(Eq.71)

Using Eq.71, Eq.68 is expressed as
(Eq.72)

Due to ħ = 1, we can change the notation, as follows,
(For example, we can use the momentum p instead of kp.)
(Eq.73)

The notation of "red" line is used from here.

Various "mathematical" tricks in QED.

[ Feynman parameters. ]

Next we squeeze the three denominator factors of Eq.72 into single quadratic polynomial in k, raised to the third power.
Here we use the identity of
(Eq.74)

First we prove the identity of Eq.74.
Integrating the right side of Eq.74 with respect to z
(Eq.75)

Integrating Eq.75 with respect to x,
(Eq.76)

And lastly integrating Eq.76 with respect to y,
(Eq.77)

We can prove Eq.74.

Manipulating "arbitrary" constant shift is the trick of QED.

Using Eq.74, the denominator of Eq.72 can be expressed as D3, which D means
(Eq.78)

Here we use the relations of x + y + z = 1 and k' = k + q ( see Eq.52 ).
And the notations in Eq.78 are
(Eq.79)

Here we introduce new variable l,
(Eq.80)

Eq.80 is a very important trick in QED.
l and k can be from -∞ to +∞.
This is constant shift. ( " l " replaces " k " in the integral. )
And when we remove infinity of l2 artificially by renormalization, important "p" and "q" values are left, which we want.

To prove Eq.80 is equal to Eq.78, we have to use the Klein-Gordon relations (mass shell) of
(Eq.81)

From Eq.81, we obtain
(Eq.82)

Using Eq.81, Eq.82 and x+y+z =1, we can prove Eq.80 is equal to Eq.78.
( As a result, the denominator of Eq.72 is D3. )

From Eq.80,
(Eq.83)

Using Eq.81 and x+y+z=1 the last two terms of Eq.78 become
(Eq.84)

Eq.83 is equal to Eq.84.
So Eq.78 is equal to Eq.80.

Originally, there is NO external electron spin in one loop function.

Next we try to prove that the numerator of Eq.72 can be expressed as
(Eq.85)

The detailed calculation method of Eq.85 is explained in appendix.

If you see the first line of Eq.85, you will notice that it doesn't include "external electrons" ( p and p' ) at all.
k' = k + q. k and k' are virtual particles with infinite kinds of momentums.
Without initial information of "p", these infinite virtual particles have NO relation with electron and g factor.

(Eq.86) "p" appears by constant shift.

But in the second line of Eq.85 (Eq.86), the information of external electron (p) is included.
This trick exists in the constant shift in the integral.

(Eq.87) Manipulating the "integration variable".

As shown in Eq.87, the new variable l is made by shifting k by a constant.
This constant includes the information of the external electron (= p ).
And l2 temrs are removed artificially by the renormalization, which is one of QED "mathematical" tricks.

Wrong math II

Single "l" is also removed by odd function, but this math is wrong, as shown in Fig.10' and Eq.10''.

(Fig.10') This math is correct ?

In Fig.10' upper, integration of odd function "k" becomes zero.
After the change of variables ( k = l - β ), The part of " l " becomes zero due to odd function.

As a result, only "-β" is left, which includes the information of g-factor.
This math is indispensable for getting g-factor in QED.
Do you think the math of Fig.10' is correct ?

(Fig.10'') Right math.

If we use a right mathematical method, the part of "-β" becomes zero, too.
Because after integration, " l2 " term diverges much faster than " βl " term.

As a result, only " l2 " term is left, and becomes zero.
( According to the right math, "β" part is absorbed into "l" term. )
So QED mathematical method of Fig10' violates basic math.

Application of wrong math.

Here we explain a "wrong math" example, which is actually used in calculating g-factor.
Of course, if "wrong math" is actually used, QED g-factor is wrong.
Because, it's just a mistake in calculation.
(Eq.W-1)

As shown in Eq.80, we replace the original variable " k " by " l ".
And the odd function " l " of the numerator becomes zero by integration.
( The denominator is an even function of l. )

As a result, only the part of g-factor (= p ) is left, they insist.
Eq.W-1 seems to be correct, but in fact this math is wrong, and QED g-factor is wrong, too.

(Eq.W-2)

We can use simpler example like Eq.W-2 in the integration of rational function.
After replacing " k " by " l ", the odd function becomes zero by integration.
As a result, only g-factor part is left.
(Eq.W-3)

Here we express the integral of the denominator of Eq.W-2 as G(l).
And we do integration by parts, as shown in Eq.W-3.
So we have.
(Eq.W-4)

If we change ( l - β ) to only l, infinity ( or minus infinity ) remains infinity ( minus infinity ).
(Eq.W-5)

Using Eq.W-5, the result of Eq.W-4 becomes zero, too.
(Eq.W-6)

So, the mathematical trick of picking up only g-factor is wrong math.
(Eq.W-7)

Due to the infinity, "k2" term of the denominator in Eq.W-7 becomes dominant.
So the difference between k and l is gone.
As a result, Eq.W-7 becomes zero, too.

This means QED g-factor calculation depends on wrong math.
Of course, the instatnt we know they are wrong math, QED cannot describe truth.

Like Eq.W-3, if we define
(Eq.W-8)

Eq.W-7 is
(Eq.W-9)

The result of Eq.W-9 is just equal to Eq.W-6.
Two approches can get the same result, so this result is right.
As a result, g-factor cannot be picked up from the original equation like Eq.W-2.

(Eq.W-10)

In Eq.W-4, we should not calculate after dividing the equation into two terms.
Because in higher degree equations, we get the wrong result from ∞ + C - ∞ = C.
A finite value "C" must be neglected in the infinity, and this result is inconsistent with Eq.W-9.

Next we try to prove Eq.85 (= "1" of Eq.88 ) is equal to the following equation,
(Eq.88)

About the detailed calculation methods, see this appendix.

Anomalous magnetic moment and contradiction of magnetic field.

[ Contradiction of the external magnetic field. ]

As shown on this page, we can get the equation of
(Eq.89)

where
(Eq.90)

This "q" is ( virtual ) photon = external magnetic field.
According to this definition, when the electron's momentum (= velocity ) is constant ( p'-p = 0 ), external magnetic field is zero, even if it exists.
This is strange.

(Fig.19) External magnetic field B = 0 ?

In Fig.19, the electron is moving in the z direction.
And the external magnetic field "B" is also in the z direction.

Lorentz force is
(Eq.91)

( In Fig.19, E is supposed to be zero. )

In Fig.19, the spin - magnetic field interaction exists.
Because the "constant" electron spin (= 1/2) always exists, and the external magnetic field is supposed to exist, too.
But according to the interpretation of Eq.89, the external magnetic field becomes zero.
Because the electron momentum doesn't change by the Lorentz force ( p' - p = 0 ), when the magnetic field is parallel to the electron's motion.

Renormalization of spin g factor is "artificial" rule.

From Eq.89,
(Eq.92)

Substituting Eq.92 into Eq.88-2, the spin-magnetic field interaction appears.
(Eq.93)

Due to the antisymmetric arrangement of (x-y), the last term of Eq.93 vanishes, when we integrate it by x and y.
So using Eq.72, Eq.74 and Eq.80 (= D ), Eq.93 can be expressed as
(Eq.94)

where
(Eq.80)

Here we use "Wick rotation" in the integration of l2.
To remove the minus sign of l0, we consider the contour of the integration in the l0 plane.
The locations of the poles, and the fact that the integrand falls off sufficiently rapidly at large | l0 |, allow us to rotate the contour counterclockwise by 90 degrees.
We then define a Euclidean 4-momentum variable lE, as follows,
(Eq.95)

Using Eq.95,
(Eq.96)

By simply changing variables to lE, we can now evaluate the integral in four-dimensional spherical coordinates.
For example, 1/D3 is, (using Eq.96)
(Eq.97)

where the surface of a four-dimensional unit sphere is used

But in the following case, the integral diverges to infinity, as follows,
(Eq.98)

This is called "ultraviolet divergence".

Using Eq.97 and Eq.98, Eq.94 becomes
(Eq.99)

where q2 is supposed to be zero.
As I said in the first section, the instant q2 becomes 0, virtual photon's energy becomes zero ( q = 0 ).
When q = 0, g-factor cannot be fixed at a single value, which means QED g-factor is NOT true.

Using the fine structure constatnt α (Eq.41), the coefficient of Eq.99 can be expressed as
(Eq.100)

See also Eq.68.

And further, the second term of Eq.99 becomes divergent when q2 = 0 and z=1, as follows,
(Eq.101)

This is called "infrared divergence".

Infinity must be removed by force !

So we artificially remove all these troublesome values of Eq.99 by renormalization, as follows,
(Eq.102)

"counter" term is gotten when the particles satisfy "mass shell" condition.
So Eq.102 itself is "counter term", and cancels itself out.
And these tricks cause strange infinite bare charge and mass of an electron.
Of course, this is an "ad hoc" manipulation.

The remaining part of Eq.99 is the last term.
And the coefficient of the last term (= Bohr magneton ) is
(Eq.103)

This is called "form factor 2".

When q2 = 0, Eq.103 becomes,
(Eq.104)

Fortunately, Eq.104 converges, as follows,
(Eq.105)

As a result, Eq.89 becomes
(Eq.106)

And the correction to the g factor of the electron is
(Eq.107)

The experimental value of Eq.107 is 0.0011597.
(This is called "anomalous magnetic moment".)
So only the one-loop correction gives almost same value as the experimental value.

And as shown in Eq.103 and Eq.104, the condition of q2 = 0 is indispensable for getting the g-factor.
As I said, the instant virtual photon satisfies Einstein's mass shell condition ( q2 = 0 ), its energy and momentum become all zero.
In these very easy restrictions, QED can give any g-factor as shown on this page.
This means law of "nature" does NOT obey QED to determine anomalous magnetic moment including renormalization.

And we can not know the results after calculating infinite kinds of loop patterns.
(This calculation is impossible.)

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