Top page ( correct Bohr model )

Trick of QED magnetic moment.

Special relativity is wrong !

Propagators of various fields.

- Reason why QED is wrong.
- Virtual photon satisfies Einstein formula = QED trick.
- Quantum electrodynamics and Feynman rules.
- Various "mathematical" tricks in QED.
- Anomalous magnetic moment and contradiction of magnetic field.

*(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.

*(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. )

*(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.

*(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.

*(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 (= H_{I} ), 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.

*(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 ).

*(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 ? )

*(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 k^{2} 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.

**l ^{2}** term diverges to infinity, but this term is

As a result, we can get the g-factor value.

*(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 k^{2} is equal to the **whole** equation of ( l^{2} -2βl + β^{2} ), so, ∞ = ∞ is satisfied.

Finite β^{2} 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.

*(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.

*(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, " l^{2} " term diverges **much faster** than " βl " term.

As a result, only " l^{2} " term is left, and becomes zero.

So QED mathematical method of Fig10' **violates basic math**.

About the actual example, see this page.

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. )

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 **p ^{2} - E^{2}/c^{2} + m^{2}c^{2} = 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

*(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 γ^{μ}, and **p** is at the **right**.

Using the mass relations of Eq.5, we can change **γ ^{a}p_{a}** at the

So, p at the left of γ^{μ} **cannot** be changed into -m.

Because p is **different** from p'.

In the same way, p' at the right of γ^{μ} 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 γ^{μ} to another side.

Due to this manipulation, new term, which doesn't include γ^{μ}, 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.

*(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 (= -2p^{μ} ).

*(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.

*(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. )

*(Eq.12) q ^{2} = 0 is indispensable for getting g-factor.*

As I explain later, to get the value of g-factor, the condition of **q ^{2} = 0** is

The notation of q

*(Eq.13) q ^{2} = 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.

*(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 q^{2} = 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, p^{0} is always **greater** than p^{1}.

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 ( q^{2} = 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.

*(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.

*(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.

*(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 ).

*(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.

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 A_{μ} γ^{μ} (μ = 0, 1, 2, 3) according to the basic QFT rule.

Hamiltonian (= H_{I} ) 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

So these are expressed as creation operators.

And c(p') means external electron which

( p, q, and p' mean momentum and energy of each particle.)

Here

About the time evolution, Heisenberg and interaction pictures in QED, see this page.

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 **k _{p}** means the wave number = p / ħ of electron 1.

" c

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)*

As shown on this page, the solution of A_{μ} (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 g_{νμ} is zero.

And **integrating** Eq.28, Eq.30, and Eq.33 with resprct to **d ^{4} x**, as shown in Eq.25, the exponential parts become

Eq.34 means the **conservation** of momentum and energy.

( *p ^{μ} + q^{μ} = p'^{μ}* )

There is NO divergence in the one vertex function, because it contains no loop in Fig.17.

And due to g_{μμ}, γ matrices of vertex (Eq.25, Eq.33) changes into γ^{μ}

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.

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!.

Comparing Eq.37 (Fig.18) and Eq.25 (Fig.17), G_{2} function has **two more vertices** (integrations) than G_{0},

*(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. )

Three external lines are the **same** as Eq.28, Eq.30, and Eq.33 in G_{0} 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 **γ ^{μ}**.

This is important.

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

*(Eq.45)*

where

*(Eq.46)*

p_{k} means *momentum*, and ω_{k} 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, *p _{k'} = p_{k} + q*, which means the conservation of momentum and energy.

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

*(Eq.48)*

where k_{p} = p / ħ

Fig.18 contains three integrations. First we **integrate with respect to d ^{4}x**.

Eq.42, Eq.45, and Eq.48 includes e

We use

Integrating Eq.49 with respect to **d ^{4}k_{p-k}**, which is included in the propagator 3 of Eq.48,

Eq.50 means external electron (ex 1) is

Eq.44, Eq.45, and Eq.47 includes e^{iy}, so **integrating it with respect to d ^{4}y**,

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 **d ^{4}k'**, which is included in the propagator 2 of Eq.47,

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

Eq.43, Eq.47 and Eq.48 includes e^{iz}, so **integrating it with respect to d ^{4}z**,

Eq.53 can use (2π)

(So

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 d ^{4}k**.

So the integration of d

Of course, this integration is

*(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.

( g_{νρ} of Eq.60 originates in these photons. )

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)*

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 k_{p}.)

*(Eq.73)*

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

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.

Using Eq.74, the denominator of Eq.72 can be expressed as D^{3}, 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 l^{2} 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 D ^{3}**. )

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.

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 l^{2} temrs are removed **artificially** by the renormalization, which is one of QED "mathematical" tricks.

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, " l^{2} " term diverges **much faster** than " βl " term.

As a result, only " l^{2} " 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**.

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, "k^{2}" 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.

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.

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 l^{2}.

To remove the minus sign of l^{0}, we consider the contour of the integration in the l^{0} plane.

The locations of the poles, and the fact that the integrand falls off sufficiently rapidly at large | l^{0} |, allow us to rotate the contour counterclockwise by 90 degrees.

We then define a Euclidean 4-momentum variable l_{E}, as follows,

*(Eq.95)*

Using Eq.95,

*(Eq.96)*

By simply changing variables to l_{E}, we can now evaluate the integral in **four-dimensional spherical coordinates**.

For example, 1/D^{3} 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 q^{2} is supposed to be **zero**.

As I said in the first section, the instant q^{2} 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 q^{2} = 0 and z=1, as follows,

*(Eq.101)*

This is called "**infrared divergence**".

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 q^{2} = 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 q^{2} = 0 is **indispensable** for getting the g-factor.

As I said, the instant virtual photon satisfies Einstein's mass shell condition ( q^{2} = 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**.)

2013/1/20 updated. Feel free to link to this site.