Top page ( Quantum mechanics is false )

Particle physics, standard model is a fictional theory

Relativistic Dirac hydrogen doesn't prove "spin" (22/9/19)

*(Fig.1) QED relies on wrong math which artificially removes indefinite infinities by another indefinite infinites in the ad-hoc renormalization.*

Contrary to the media-hype, quantum electrodynamics (= QED ) is "the most **unsuccessful** quantum field theory" which can **Not** predict any meaningful values such as the tiny anomalous magnetic moment or Lamb shift ( this p.3-left ).

All calculated values in the fishy relativistic quantum field theory, QED and standard model are known to miserably **diverge** to infinities due to the unobservable infinite numbers of fictional virtual particles which always appear and have to be dealt with in QED **unphysical** calculations ( this p.3-6 ) which have nothing to do with our real world's phenomena.

The relativistic QED's fictional virtual particles must always have unreal ( imaginary ) masses and contradict Einstein relativistic energy-momentum relation ( this p.9-10 ). ← Relativistic QED contradicting the relativity is self-contradictory and wrong.

Physicists had to rely on **illegitimate** wrong math trick called "renormalization" which artificially creates other **infinite** free parameters called "( unobservable fictional ) bare charge and mass" or infinite counter terms to cancel the QED infinities for getting some finite (manipulable) physical values ( this p.4-5, this middle ).

↑ You can easily find this QED ad-hoc renormalizaiton is mathematically **wrong** with **No** power to predict any finite physical values, because we cannot "predict" such unobservable **indefinite** infinite bare charge counter parameters ( this 3rd-paragraph ). ← "Infinity" is Not the definite number which can be predicted, so the tired cliche "QED can successfully predict the anomalous magnetic moment" is a lie.

Actually, No scientists use this useless QED or its unreal virtual particles in our daily-life technology.

And the media and academia repeatedly and baselessly claiming "successful QED" **never** try to present the (dubious) **detailed methods** of how such a (wrongly) successful QED calculates the anomalous magnetic moment. ← Because the detailed QED calculation methods are terrible full of **scam**.

Thinking commonsensically, this QED unreasonable claim that some finite values can be obtained by subtracting **infinities** from other **infinities** is mathematically impossible and wrong, because the indefinite "infinities" can absorb any finite numbers (= infinity minus infinity equals indefinite infinite numbers, so QED renormalization is based on wrong logic ).

This is why even QED founders such as Paul Dirac and Feynman harshly **criticized** this QED illegitimate renormalization trick removing infinities, saying "hocus-pocus" or "there is something **wrong** with QED equations ( this p.2, this 1st-paragraph )."

On this page, we actually calculate such a dubiously successful anomalous magnetic moment (= g-2 factor ) of an electron using QED, Feynman diagram and one-loop renormalization correction of virtual particles' infinity, and show QED is just an artificial useless math trick with freely-adjustable parameters (= or freely-chosen QED methods and rules which can change the results freely ), hence, "QED is a successful theory" is proven to be a total **lie**.

*(Fig.2) Unphysical QED interaction (= ψAψ ) between mathematical Dirac electron (= ψ ) and unreal (virtual) photon inside magnetic potential (= A ) just generates meaningless infinities.*

All (unphysical) relativistic quantum field theory and QED use the very abstract unphysical interaction term (= based on wrong math ) in the unphysical Feynman diagram as the only way ( this p.6 ) to describe the fictional QED interaction between electrons and (virtual) photons ( this p.3-4, this p.5-10, this p.3 ).

↑ This abstract unphysical QED interaction math term and Feynman diagram tell us **nothing** about the detailed mechanism of how exactly each electron (= expressed as the unphysical Dirac equation ) interacts with (fictitious virtual) photons (= expressed as nonphysical math symbols allegedly hidden in fictitious magnetic potential A, this p.3, this p.2-right ) at all.

So studying QED, which **obstructs** science progress by refusing to clarify the detailed physical mechanism, is completely meaningless and waste of time.

*(Fig.3) Relativistic QED just describes electrons and photons as meaningless math symbols which clarify nothing.*

Relativistic QED combines unphysical Dirac electron (= ψ ) and mathematical photon's particle under the **fictitious interaction** Hamiltonian energy (= H_{I} or L_{int}, this p.3-6, this p.36-40 ) where electrons and photons are described just as nonphysical math symbols put side by side with No more detailed description of the concrete interaction between electrons and (imaginary virtual) photons ( this p.3-13, this p.9-10 ).

Quantum mechanics claims particles change with time obeying the exponential function of this fictitious interaction energy (= e^{iHt}, this p.32, this p.63-68, this p.51-upper (or p.55 ) ).

This mathematical exponential function can be expanded into infinite terms or infinite series which infinite terms can easily make QED calculations diverge to infinities ( this p.11-13 ).

Of course, physicists cannot calculate all these ridiculous infinite terms of QED virtual interaction allegedly contributing to the tiny tiny anomalous magnetic moment or g-2 factor.

As the degree of terms increases, the number of virtual interactions between electrons and virtual photons increases which increases the number of the loop of the internal virtual particles interactions. ← Calculation of each loop interaction diverges to **infinity** ( this p.138 )

In the calculation of electron's anomalous magnetic moment (= or g-2 factor ), physicists cannot analytically calculate even the small 4-loop virtual interactions ( this p.9 ). = They have to rely on uncertain numerical calculation, only this fact debunked the myth of the alleged "QED "predicting" g-2 factor with great precision".

Of course, in order to know the true anomalous magnetic moment's value by this ad-hoc QED method, physicists have to unrealistically calculate the infinite series, which is impossible (= hence, it's impossible to know what value QED predicts by executing the calculation of the infinite interaction terms ).

Furthermore the QED calculation of each interaction term (= each one of infinite series ) miserably diverges to infinity (= called catastrophic ultraviolet divergence, this p.6, this p.61-64 (or p.65-68 ) ), which QED infinites must be artificially removed by **freely-adjustable infinite bare charge** parameters called renormalization, which illegitimate method cannot predict any physical values ( this p.6-12, this p.56-59 (or p.68-71 ) ).

*(Fig.4) An incoming electron absorbs an external (unreal virtual) photon to become an outgoing electron via infinite numbers of internal virtual photons (with infinite energies ) and virtual electrons = successful QED anomalous magnetic moment model ? ← No way !*

In the actual calculation of "allegedly-successful" QED electron's anomalous magnetic moment (= g-2 spin factor ), physicists just "imagine" the abstract unrealistic interaction where an incoming electron (= with momentum p ) expressed as nonphysical math symbol absorbs an external virtual photon (= with momentum q ) and becomes an outgoing electron (= with momentum p' = p + q due to energy-momentum conservation, this p.18-19 ).

In all relativistic quantum field theory and QED calculations, total energy and momentum at each vertex (= interaction points between electrons and (virtual) photons ) must be always conserved ( this p.25 ).

↑ The simple calculation shows when the total energy and momentum are conserved (= energy conservation law must be obeyed ), a real electron can absorb or emit only an unreal virtual photon with imaginary mass contradicting Einstein's relativistic energy mass relation which paradoxical property is called "off-mass-shell" ( this 3rd-paragraph, this p.3 this p.16 )

In the upper three internal lines, one internal virtual electrons with momentum k (= from minus infinity to plus infinity ) absorb the external (virtual) photon's momentum (= q ) to become the second internal virtual electrons with momentum k'(= k + q ), and the remaining internal virtual photons have momentum p - k (== external incoming electron p minus internal virtual electron k ).

↑ All these unreal internal virtual electrons and photons are said to have **infinite** momentums and energies ( this p.15 ) whose internal virtual particles' energy relation clearly disobeys Einstein's energy-momentum-mass relativistic relation ( this p.33-34, this p.8 ).

These unrealistic infinite energies and momentums of the ghost-like internal virtual electrons and photons cause all QED calculations to diverge to infinities (= called ultraviolet divergence, this p.14, this p.69-70 ).

As I said, QED just **freely adjust** other infinite bare charge parameters (= unobservable, so QED validity cannot be verified ) to artificially cancel the virtual particles' infinities to obtain some finite physical values such as electron's anomalous magnetic moment or g-2 factor, which illegitimate QED math trick is useless and unable to predict any physical values at all.

*(Fig.5) ↓ Nonphysical QED math consists of unreal virtual particles' connection called propagators.*

In nonphysical QED calculation, physicists connect all internal unreal virtual particle math operators to get virtual interaction lines as shown in abstract Feynman diagram ( this p.59-69 this p.26-35 ).

Each connected internal line between (virtual) electrons and photons is called "propagator" which abstract math objects have No relation to the real world = physically meaningless QED concepts.

*(Fig.6) Three internal lines consist of unreal virtual electrons' and photon's propagators in electron's g-2 calculation. → diverge to meaningless infinity*

In this page, (-1,+1,+1,+1) version of metric tensor is used. If you want to change it into (+1,-1,-1,-1) version, see this page.

All QED calculations rely on unrealistic concepts or Feynman diagram's abstract lines called "propagators ( this p.8-11, this p.35 )" which allegedly mean "virtual interactions" among ghost-like infinite virtual electrons and photons with infinite amount of energies which always cause QED calculations to diverge to meaningless **infinity** ( this p.3-5 ).

The internal lines connecting two internal (virtual) unphysical Dirac's electrons give the fermion propagator (= ψψ, this p.10, this p.45 (or p.46 ) ) containing infinite numbers of virtual electrons and antiparticles with infinite momentums and energies disobeying Einstein relativistic energy-momentum relation ( this p.7-14, this p.24-25 (or p.14-15 ) ). ← self-contradictory relativistic QED

In one-loop Feynman diagram of these virtual propagaror interactions, one set of energy (= k^{0} ) and momentum (= k^{1}, k^{2} k^{3} ) variabes (= d^{4}k = four independent variables ) must be integrated from **minus infinity to plus infinity** ( this p.2-left ), which integration causes meaningless infinity in this unphysical QED calculation ( this p.5-6 ).

In the QED calculation of the electron's anomalous magnetic moment or g-2 factor, its equation consists of two virtual internal electrons' propagators and one virtual internal photon's propagator with three vertices (= expressed as three nonphysical γ matrices = γ^{ν}, γ^{μ}, γ^{ρ}, ) from Fig.5.

Using the formula of Eq.60, we can get the last line of Fig.6 ← This QED calculation process needs unreal zero-polarized scalar photon with negative probability expressed as g_{μν} of photon propagator ( this p.6-8 ) using Lorentz (= Feynman ) gauge ( this p.11 ), which means QED calculation is based on unrealistic concepts such as zero-polarized photon with negative probability.

*(Fig.7) ↓ QED calculation is nonphysical math trick with No reality.*

From the equation of QED anomalous magnetic moment (= g-2 factor ) of Fig.6, we can get the equation of the upper Fig.7 ( this p.1-left ). using the math trick (= we use -1,+1,+1,+1 version of metric tensor ) of nonphysical 4×4 γ matrices ( this p.5-last ).

*(Fig.8) Artificial change of equation using new variables = x,y,z = one of QED illegitimate tricks*

Next, they artificially change the equation of Fig.7 using the math trick adding new finite variables (= x,y,z = not position variables, here = integral from 0 to 1 ) called Feynman parameters ( this p.98-99 (or p.99-100 ), this p.4, this p.7 ).

The denominator (= D ) of the equation becomes the cube of [ -l^{2} - Δ ] combining all three propagators' denominators.

↑ This conversion by adding new variables (= x,y,z ) is clearly one of QED math tricks to **manipulate** the QED calculated results.

Of course, this kind of conversion itself usually should Not change or affect the calculated result of the whole equation.

But QED illegitimate renormalization separates arbitrary infinite values (= discarded artificially ) from the remaining arbitrary finite values later, which result is clearly influenced by this artificial tranformation by Feynman parameter.

*(Fig.9) the variable "k" is replaced by new variable "l" ( l = k + finite ), but the integral interval remains the same infinite = from minus infinity to plus infinity. = New "finite values" added by this illegitimate change of variables is used for a part of anomalous magnetic moment*.

After changing the original (infinite) anomalous magnetic moment equation by the artificial trick called Feynman parameters, they artificially change the variables from k to l (= l = k + some finite values or variables = yq-zp ).

↑ This artificial "change of variables" is one of important ad-hoc QED math tricks manipulating the finite calculation result of anomalous magnetic moment.

Because even after this change of variables ( k = l + finite value = l + yq - zp, this p.4-(17), this p.10, this p.16-17 (or p.17-18 ) ), the **integral interval** remains the **same** from **minus infinity to plus infinity** in both k and l variables (= Infinity can absorb any finite values ).

As shown in the upper Fig.9, after this artificial change of variables ( k → l ), the external incoming electron's momentum (p) suddenly appears in the numerator ( compare the left equation with the right one in this p.1) !

The detailed calculation from the first to last equations of the upper Fig.9 is explained in this.

*(Fig.10) Illegitimate QED wrong math trick = "change of variables" changes the result !*

Normally, "change of variables" in integrals can**not** change the entire calculated results, because "change of variables" is just a different calculating process leading to the same results irrespective of chosen different variables.

But the QED illegitimate math trick tries to **change the result itself** (= anomalous magnetic moment or g-2 factor ) by taking advantage of this **wrong** "change of variables ( this p.10 )" by exploiting the QED virtual particles' **infinity**.

In the upper Fig.10, the original anomalous magnetic moment equation contains only the **variable k** (= integral of this k from minus **infinity** to plus infinity ).

But after introducing the artificial conversion called Feynman parameter (= adding new finite variables x,y,z ), this original variable k is **replaced** by another **new variable l** ( k = l + finite parameters = l - yq + zp ).

As shown in the above Fig.10, in the original equation, the integral of "one-order k term (= linear in k, or odd function of k )" with respect to the variable k from minus infinity to plus infinity becomes **zero** (= by minus and plus infinities canceling each other ).

But after replacing this "k" by new ad-hoc valiable k = l - yq + zp, the unreasonable QED claims the integral of this "one-order k term" does **Not** become zero ! ← "Change of variables" **changes** the QED calculated **results**, which is clearly a **wrong** math.

They wrongly claim that the integral of the new "one-order l term (= linear in the new variable l )" becomes zero ( this p.18-(1.49) ), and the remaining finite parameter terms of (-yq+zp) **remain** Not-zero, which become zero or do Not appear in the original k integral !

This wrong math changing the QED result by the wrong change of variables is caused by the fact that the integral intervals both in the original "k" and new "l" variables remain the same from **minus infinity** to **plus infinity** even after the change of valiabes of k = l + finite, because the unreasonable infinity absorbs any finite parts.

↑ If the integral interval of the original variable k is from minus finite value (= -w ) from plus finite value (= +w ), after change of variables, the integral interval of this new variable l becomes **asymmetrical** from -w+finite to +w+finite, which cannot make terms linear in the new variable l zero, which gives the **wrong** anomalous magnetic moment result. ← QED exploits the **infinities** to manipulate the anomalous magnetic moment result.

So the unreasonable QED infinity can **freely change** the finite parameters (= -yq+zp part ) or the ad-hoc finite calculated results (= anomalous magnetic moment values can be freely changed ) by choosing different "change of variables (= k = l + "finite", where this "finite" part can be freely changed, which affects and **changes** the calculated result ! )"

This wrong math trick using the illegitimate "change of variables" shows the QED method is just scam and **wrong,**.

Actually, all terms of the one-order l (= terms linear in l ) disappear by this illegitimate integral with respect to l from minus infinity to plus infinity ( this p.1-right-last, this p.2-3rd-paragraph, this p.14 (or p.8 )-(2.21), this p.11-from (2.43) to (2.44) ).

*(Fig.11) Artificial switching between nonphysical γ matrices and the electron mass m using Dirac equation.*

Using unphysical Dirac equation of free electrons, they can get the final form of the alleged anomalous magnetic moment of one-loop virtual particles' interaction, as shown in the last line of Fig.11.

Various formulas and rules such as this are used.

What QED tries to do is change all terms of γ^{a}p_{a} (or γ^{b}p_{b} = γ^{0}p_{0} + γ^{1}p_{1} + γ^{2}p_{2} + γ^{3}p_{3} ) and γ^{a}q_{a} (or γ^{b}q_{b} = γ^{b}p'_{b} - γ^{b}p_{b} ) attached to γ^{μ} matrix in ① of the upper Fig.11 into
the terms of p'^{μ}, p^{μ} and q^{μ} **free from** (= Not attached to ) γ^{μ} by using this and this formulas, and artificially remove and renormalize only terms **attached to** γ^{μ} matrix.

But later, QED **paradoxically reverses** this process, and **returns** all terms ( such as p'^{μ}, p^{μ} ) once freed from γ^{μ} matrix
to the **original** terms attached to γ^{μ} matrix in the form of the contradictory anomalous magnetic moment of the spin = σ^{μν} matrix, which is the product of **two γ matrices** ( σ^{μν} = γ^{μ}γ^{ν}, this p.6-7-(20)-(21) ). ← This is also one of the most illegitimate tricks of QED.

The important point is that this ad-hoc QED anomalous magnetic moment calculation treats the incoming electron's momentum (= p ) is **different** from the outgoing electron's momentum (= p' ) by absorbing virtual external photon's momentum (= q ), hence p' = p + q, where
the virtual photon's momentum q is Not zero.

So in the upper calculation, the term of γ^{b}p'_{b} must move into the left side of the central γ^{μ} matrix to reach the left-end bar-u(p') (= Dirac electron ) to change into the electron's mass (= m ) through the Dirac equation's formula ( this p.65 (or p.69 )-upper, this p.1-right-middle, this p.11-middle ).

But later, they impose the additional contradictory condition of the external virtual photon's mass being zero (= q^{2} = 0 ) like a real photon. ← When the virtual photon's mass (= usually unreal imaginary mass ) becomes zero, this external virtual photon's energy and momentum must become zero (= q = 0 ), which means the incoming and outgoing electrons' momentums and energies become the **same** p' = p.

↑ When p' = p, this whole QED calculation process of getting anomalous magnetic momenty by distinguishing p' and p
is **invalid** and meaningless (= we can freely manipulate the resultant anomalous magnetic moment value, when p' = p, because we can freely choose whether each γ^{a}p_{a} or γ^{a}p'_{a} moves into the left side or right side of γ^{μ} matrix which can freely change the anomalous magnetic moment result, when p = p' )

The detailed calculation from the first to the last equations of the upper Fig.11 is explained in this.

*(Fig.12) Unphysical γ matrices spawn (unrealistic) spin magnetic moment according to QED illegitimate math trick. ↓*

How can the illegitimate QED math trick connect this meaningless virtual particles' calculation to the electron spin anomalous magnetic moment or g-2 factor ?

Physicists artificially change the single γ^{μ} matrix into two terms of ( p^{μ} + p'^{μ} ) **freed from** γ^{μ} matrix and ( p'_{ν} - p_{ν} = q_{ν} ) **attached** to γ^{μ} matrix like the upper Fig.12 using the math trick of Dirac equation and Gordon identity ( this p.1-(2), this p.12 ).

Here "q" means the virtual external photon's energy and momentum, and p' and p mean the outgoing electron's and incoming electron's energy and momentum, respectively, hence the relation of p' - p = q is satisfied due to energy and momentum conservation ( this p.18 ).

Physicists baselessly insist putting two nonphysical 4×4 γ matrices side by side may indicate the electron's spin expressed as the nonphysical 4×4 σ^{μν} matrices (= γ^{μ}γ^{ν}, this p.1-2 ), which are different from the original 2×2 Pauli matrices ( this p.3 ).

So they insist the upper eσ^{μν}/2m means Bohr magneton (= unit of magnetic moment ) or magnetic moment (= eℏ/2m, this angular momentum ℏ is replaced by nonphysical 4×4 σ matrix = spin angular momentum, they claim )

The detailed calculation of the upper Fig.12 is explained in this.

*(Fig.13) Artificially removing infinity + part of finite values to obtain the ( unphysical ) spin anomalous magnetic moment (= free parameter, so Not the QED's prediction at all ) by illegitimate QED renormalization.*

They use the Dirac equation and its Gordon identity formula to replace the blue-line term of the upper Fig.13 by two terms of γ^{μ}× 2m^{2}z(z-1) and the alleged anomalous magnetic moment part including unphysical 4×4 σ^{μν} matrices (= consisting of two 4×4 γ matrices )

And they artificially remove the terms surrounded by the black line of Fig.13 (= these terms include **infinity** represented by the infinite enegy-momentum integral of l^{2}, this p.8, this p.148-151, this p.37 ) using the illegitimate trick of renormalization to obtain the desirable anomalous magnetic moment or g-2 factor.

↑ This **artificial removal** of only unnecessary infinity + part of unnecessary finite values on the pretext of "renormalization" or regularization ( this p.12-lower, artificially removing divergent F1 part ) is **unjustifiable** and a **wrong** math trick, hence QED has **No** power to predict any physical values such as anomalous magnetic moment or g-2 factor ( this p.32-37 ).

↑ Using this QED unjustifiable renormalization (= using counter infinite bare charge renormalization, this p.3-(13) ), physicists artificially remove and "ignore" all terms (= F1 term ) except for the unphysical anomalous magnetic moment term including
σ^{μν} (= F2 term ), as shown in this p.2-2nd-last-paragraph, this p.65 (or p.69)-(7.36), this p.7(or p.13)-(2.10), this p.11-(2.47)

*(Fig.14) ↓ Obtaining anomalous magnetic momenty by illegitimate QED math trick, which is useless with No power to predict actual physical values.*

After artificially removing only unnecessary infinity and unnecessary finite values (= F1 ), they insist they could obtain the extremely tiny finite anomalous magnetic moment (= F2 ) close to experimengal values. = One loop virtual particle's correction gives the simple value of α/2π where α is fine structure constant (= α = 1/137 ), which QED one-loop calculation value ( α/2π = 0.00116.. ) is slightly differnet from the actual experimental value (= 0.001159.., this p.2-3, this p.42 (or p.48), this p.9 (or p.15), this p.5 ).

Here, they use various math formulas ( such as Wick rotation, this p.4, this p.35-37, this p.2 ) and the artificial condition of changing the virtual photon into real photon with zero mass (= q^{2} = 0 ), which means all these (virtual) photon's energy and momentum become zero (= because an electron cannot emit or absorb a real massless photon ), and the energy and momentum of the incoming and outgoing electrons become the same (= p'=p ), which makes the above QED calculation meaningless (= anomalous magnetic moment value can be freely changeable, when p'=p or virtual photon has zero mass instead of the original imaginary mass ).

The final calculation of the above anomalous magnetic moment integral is in this.

*(Fig.15) QED one-loop anomalous magnetic moment (= g-2 = 0.00116.. ) contradicts the original Dirac magnetic moment (= g = 2 ) calculation*

The serious problem is that this artificial QED math trick for anomalous magnetic moment is based on the paradoxical **inconsistent** rule, so wrong.

First, they obtained the necessary term (= F2 ) of ( p'^{μ} + p^{μ} ) mz(z-1) **freed from** γ^{μ} matrix by getting as many terms as possible out of γ^{μ}(..) terms using this formula.

↑ They imposed the **artificial rule** where only this unnecessary terms (= F1 ) **attached to** γ^{μ} matrix = γ^{μ}(..) should be artificially **ignored** and removed by (infinite bare) charge renormalization ( this p.2-2nd-last-paragraph, this p.65 (or p.69)-(7.36), this p.7-(22) ).

But after getting this necessary term (= for anomalous magnetic moment ? ) which are allegedly **freed from** γ^{μ} matrix ( this p.1-right-last ), they paradoxically **return** this term (= (p'^{μ} + p^{μ}) with No γ^{μ} attached ) to the original γ^{μ}(..) terms **attached to** γ^{μ} matrix, using Dirac equation or Gordon identity in the upper Fig.15 by reversing this formula ( this p.2-(a.4)-lower, this p.11-(2.44)-(2.46), this p.35 (or p.36 )-1st-paragraph ).

This reversed result is shown in the allegedly-necessary anomalous magnetic moment term containing unphysical **σ ^{μν} matrix which consists of** two

This QED math trick deliberately choosing which terms attached to γ^{μ} matrix should be removed is clearly paradoxical and **inconsistent**, hence wrong.

The point is that they try to remove only unnecessary terms attached to γ^{μ} matrix, but this "unnecessary γ^{μ} matrix" is a part of spin matrix (= σ^{μν} = γ^{μ}γ^{ν}, this p.1, this p.65(or p.69)-(7.35) ) of the necessary anomalous magnetic moment.

This tiny anomalous magnetic moment part (= 0.00116.. ) must be added to the Dirac spin-g factor (= 2 ), hence, the original Dirac spin-g-factor = spin magnetic moment of "2" not only tiny QED anomalous magnetic moment is indispensable.

This necessary Dirac spin g-factor (= 2 ) is obtained by replacing one γ^{μ} matrix with two terms of anomalous magnetic moment part σ^{μν} (= **including** two γ matrices ! ) and the (p'^{μ} + p^{μ}) term **freed from** γ^{μ}, using Gordon identity ( this p.3-right, this p.18 ).

↑ So the equation of the original Dirac spin-g-factor = 2 must **contain** the term of (p'^{μ} + p^{μ}) **freed from** γ^{μ} ( this p.7-(23),(24)-1st-term of p^{μ} ), which term is artificially removed or replaced by γ^{μ} term in the QED anomalous magnetic moment ad-hoc calculation.

This is the most serious **contradiction** between the original Dirac equations of "spin g-factor = 2" and the QED tiny anomalous magnetic moment change of "0.00116.." with respect to whether the (p'^{μ} + p^{μ}) term freed from γ^{μ} matrix remains or not in the final form of equations.

If QED wants to observe the **common rule** with the original no-loop (= containing only one vertex or one γ^{μ} with No other values or correction ), they have to change the whole γ^{μ}(...) = γ^{μ}(1/2l^{2}..+(1-2z-z^{2})m^{2}) terms **attached** to γ^{μ} in the upper figure F1 (= infinite + unnecessary finite terms diverging to infinity, which give No meaningful finite physical values ) into (p'^{μ}+p^{μ}) term and the alleged anomalous magnetic moment term attached to σ^{μν} matrix (= this case adopting the **common** rule gives completely **different** divergent anomalous magnetic moment value ) by obeying the same rule and the same Gordon equation (= changing γ^{μ} terms into the two terms of (p'^{μ}+p^{μ}) term and σ^{μν} magnetic moment term ) as the no-loop's original spin magnetic moment (= 2 ), instead of stupidly returning (p'^{μ}+p^{μ}) term with no γ^{μ} to the beginning terms attached to γ^{μ} (= like Fig.11-upper equation ) and σ^{μν} term which **also contains** γ^{μ} matrices.

↑ If physicists want to remove all the terms attached to γ^{μ} matrix, they also have to remove the anomalous magnetic moment term containing the original γ^{μ} matrix inside σ^{μν} = γ^{μ}γ^{ν}. ← paradox !

The original equation of this QED anomalous magnetic moment (= upper equation of Fig.11 ) tacitly includes the anomalous magnetic moment terms as the form of " γ^{μ}γ^{a}q_{a} (= not removed by renormalization, though this spin anomalous magnetic moment term is clearly attached to γ^{μ}, hence, this remormalization rule is **inconsistent** )", which is one of the most important QED illegitimate scams.

*(Fig.16) QED artificially separates unnecessary terms from necessary terms also inside γ ^{μ}(...) terms, which artificial rule is different from the original one-loop calculation ! ← QED artificial rule for which terms should be renormalized or removed is illegitimately changeable.*

If QED ad-hoc calculation of the electron's anomalous magnetic moment of g-2 factor was right, QED has to use the **same consistent common** rule for determining which terms must be artificially removed or renormalized in one-loop, two-loop and all other higher loop corrections.

But in fact, QED uses the completely **different inconsistent** methods between one-loop and two-loop of anomalous magnetic moment calculations, so QED renormalization method is inconsistent and **wrong**.

In the one-loop calculation of anomalous magnetic moment, QED artificially removes or **renormalizes all terms attached to** γ^{μ} matrix = all γ^{μ}(....) terms are artificially removed.

But in the two-loop calculation of anomalous magnetic moment, they artificially **change the rule**, and pick up only a part of γ^{μ}(....) terms to remove or renormalize, while all other terms **attached to** γ^{μ} remain **un**renormalized, which is inconsistent and illegitimate, as shown in this paper's p.2 (= IIa two loop in Feynman diagram ) p.8-(30-32) p.9-(33)-(36b) p.10-(41) where only the terms of γ^{μ}[ k^{2} - 4m^{2}(1-u-1/2u^{2}) ] is artificially removed and renormalized while all other γ^{μ}(....) terms attached to γ^{μ} matrix remain **without** being removed or renormalized ! ← **inconsistent** with one-loop renormalization.

↑ This Kroll and Karplus calculation of two-loop anomalous magnetic moment agreed with the later official calculation in this Feynman diagram of two-vetex-loop correction (= IIa, this p.8-M-IIa, p.14-u-IIa = 0.778 α^{2}/π^{2} equal to Petermann's later result of this p.2 )

So QED two loop correction officially adopted the different **inconsistent** rules compared with the one-loop correction in anomalous magnetic moment calculation.

And this two-loop correction of anomalous magnetic moment uses various **illegitimate** manipulations.

For example, in the upper Fig.16, they try to generate (p'-p'')_{ν} terms in addition to the main (p'-p'')_{μ} by exchanging the γ^{a}p_{a} with the irrelevant γ_{ν} on the edge insteand of exchanging it with the original central γ_{μ}, and artificially removed some terms exchanged with the irrelevant γ_{ν} ( this p.8-(31) ).

↑ So they artificially create and impose **different rules** for which terms should be artificially removed or renormalized in one-loop and two-loop corrections of anomalous magntic momenet, which inconsistent QED method is illegitimate and **unable** to predict any physical values.

*(Fig.17) If an electron interacts with a real photon with zero mass satisfying relativity (= q ^{2} = 0 ), this real photon's energy must be zero (= q = p' - p = 0 ), and an electron's energy and momentum do Not change (= p' = p, if q^{2} = 0 ) ← In this condition ( p' = p ), the calculated anomalous magnetic moment value can be freely changed with No QED prediction.*

In the QED anomalous magnetic moment calculation, they try to change the original external virtual photon with momentum and energy of q^{μ} into a real massless photon (= relativistic energy momentum equation = q^{2} = -q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2} = - m^{2} = 0 ) in the last process ( this p.3-(13), this p.2-right-lower, this p.9(or p.15)-(2.23) ).

QED demands that the total energy and momentum between electrons and (virtual) photon should be **conserved** ( this p.16, this p.3 ) where an incoming electron (= with momentum p ) can absorb only an unreal virtual photon (= with momentum q ) with imaginary mass to become an outgoing electron (= with momentum p' ), if relativistic relation of energy (= E = cp^{0} ) and momentum (= p_{j}, j = 1,2,3 ) is satisfied.

So if we impose the condition of an unreal virtual photon's mass equal to zero (= q^{2} = 0, this p.3-top ), it means this virtual photon becomes a **contadictory** real massless photon, hence this massless virtual photon's energy and momentum must become zero (~ q = 0, or q^{0} = 0, q^{0} = 0, q^{1} = 0, q^{2} = 0 ). ← QED assumption of the interaction between electron and a (virtual) photon, which becomes zero, is invalid and meaningless.

In this case of a massless virtual photon, an electron cannot absorb or emit this paradoxical massless virtual photon, and the energy and momentum of an incoming and outgoing electrons must be the same (= p' = p, this p.66 (or p.70)-top ).

QED anomalous magnetic moment calculation depends on the assumption that that the energies and momentums between an incoming electron (= p ) and an outgoing electron (= p' ) are **different** and these electrons must be differently moved inside the anomalous magnetic moment equation.

So if the (virtual) photon becomes massless, the energy and momentum of an incoming and outgoing electrons become the same (= p = p' ), and physicists can freely manipulate and change the calculated results of the anomalous magnetic moment, which makes QED prediction meaningless.

*(Fig.18) σ ^{μν}q means (unphysical) spin-magnetic field interaction ? ← It's impossible and unrealistic. *

QED tries to misinterpret the unphysical notation of σ^{μν}q_{ν} as the (virtual) interaction between electron's spin (= σ^{μν} ) and magnatic field (= q, this p.7, this p.3, this p.36 ).

↑ This is impossible and **inconsistent** with the actual magnetic field, because this unphysical matrices =
σ^{μν} = product of two γ matrices (= γ^{μ}γ^{ν} ) includes the paradoxical matrices which are Not Pauli spin matrices when μ = ν or μ, ν = 0 (= there is No zero component spin )

Furthermore, when an electron is moving around in the static magnetic field, the moving electron must be always absorbing varying (virtual) photons with constantly changing momentums (= q is always changing, this p.8 (or p.14)-(2.14)-(2.17) ) for an electron to constantly change its direction under the magnetic field.

But the static external magnetic field (= B = ∇×A, where A is magnetic potential, QED virtual photon's momentum q corresponds to the derivative of ∇ ) means the value of static B = qA must be always constant and unchanged, which **contradicts** the constantly-changing electron's momentum (= which means the constantly-changing virtual photon's momentum q ) in the static magnetic field.

↑ This means the QED ad-hoc far-fetched interpretation of the (virtual) spin-magnetic field interaction is paradoxical, hence the QED spin anomalous magnetic moment is **wrong**.

*(Fig.19) The initial rest electron (= p ) absorbs a photon (= q ).*

In fact, an electron can absorb or emit only an **unreal** virtual photon with imaginary mass violating Einstein relativity, when the total energy and momentum are conserved ( this p.16 ).

We can check this fact by a simple calculation.

An initial electron at rest with rest mass energy (= mc^{2} = cp^{0} ) and zero momentum (= p^{i} = 0, when i = 1,2,3 or x,y,z ) is supposed to absorb a (virtual) photon with the energy (= q^{0} ) and momentum (= q^{i}, i = 1,2,3), and become a resultant electron with the energy E' (= cp'^{0} ) and momentum (= p'^{i} ).

↑ So the resultant electron's energy (= E' = p'c ) after absorbing the (virtual) photon's energy (= q ) becomes greater than the initial electron (= p ) at rest ( E' > mc^{2} = rest mass energy ).

*(Eq.1) Einstein relativistic energy (= E ) momentum (= p ) relation.*

Relativistic energy (= E ) and momentum (= p ) are known to always satisfy the relativistic mass relation in any reference frames seen by any observers moving at different speeds.

So we can pick up one reference frame of Fig.19 (= initial electrons is at rest ) to explain any other frames in the relativity.

The total energy (= E ) and momentum (= p ) must be always conserved between an electron and a (virtual) photon ( = an initial electron p absorbs a photon q to become p', q = p' - p ), p' > p E' > mc^{2}

*(Eq.2)*

So by inserting Eq.2 into the relativistic relation, the (virtual) photon's mass from its energy and momentum becomes

*(Eq.3)*

Eq.3 means this (virtual) photon's mass is an unreal imaginary number !

*(Eq.4)*

Eq.3 and 4 shows the square of this virtual photon's mass becomes negative (= m^{2} < 0 ), which means this virtual photon has an unreal imaginary mass !

Only when the resultant electron's energy E' is equal to the initial electron's rest mass energy ( E' = mc^{2}, or p' = p ), this photon's mass becomes zero, but in this case, all the energy and momentum of this photon become **zero** in Eq.3 (= q = 0, p = p', when q^{0} = 0).

*(Eq.5) When γ ^{a}p_{a} meets the right-side Dirac electron (= u(p), = Not the left-side u(p') ), it can change into -mc. When γ^{a}p'_{a} meets the left-side Dirac electron (= u(p'), = Not the right-side u(p) ), it can change into -mc. = QED basic unphysical trick distinguishing p and p' *

QED calculation uses the simple relativistic Dirac equation of free electrons.

QED basically treats the energies and momentums of incoming electron (= p) and the outgoing electron (= p' ) after absorbing the external (virtual) photon's energy (= q ) as different values ( p is Not equal to p', hence, photon's energy q is Not zero = this virtual photon's mass becomes imaginary, Not zero, q^{2} = 0 is Not satisfied contrary to the QED last manipulation ! ). ← QED calculation of anomalous magnetic moment is **contradictory** and wrong.

Only when the incoming electron's math operator = γ^{a}p_{a} directly touches the right-side Dirac's electron spinor operator (= u(p) ), this γ^{a}p_{a} operator can change into "-mc (= constant rest mass, this p.18, this p.10-11, this p.33-(3.31) )".

On this page, we use (-1,+1,+1,+1) version of metric tensor

QED considers this p to be different from p', so when the incoming electron's γ^{a}p_{a} touches the left-side Dirac outgoing electron operator = u(p'), this γ^{a}p_{a} cannot change into -mc. ← This is also one of many **artificial rules** created for unphysical QED.

*(Eq.6) QED anomalous magnetic moment calculation.*

In Eq.6, **p'** (= outgoing electron ) is at the **left** of γ^{μ}, and **p** (= incoming ) is at the **right**.

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

At first, they treat p as one different from p', so even if γ^{a}p_{a} touches the left-side Dirac external electron spinor (= u(p') ), or γ^{a}p'_{a} touches the right-side Dirac external electron spinor (= u(p) ), they can**not** change into the rest mass of "-m"

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

Of course, the **difference** between p and p' is due to the **existence** of virtual photon q.

So the artificial rule of Eq.6 is effective, **ONLY** when this virtual photon q is **NOT zero**.

But QED inconsistent rule for calculating anomalous magnetic moment imposes the condition of the (virtual) photon mass being zero in the last process (= QED changes the rule between the initial and last calculation process ! ), hence, after all, this virtual photon's energy and momentum become zero, and the energy and momentum of an incoming and outgoing electrons become the same (= p = p' ), which makes us freely manipulate the anomalous magnetic moment value ignoring the rule, invalidating QED prediction.

*(Eq.8)*

QED uses nonphysical γ matrices which math objects can magically change into completely different things such as spins, electric charges and interaction vertex.

Matrix means we cannot exchange these two γ matrices freely.

As shown in Eq.6. the incoming electron's math operator γ^{a}p_{a} has to move into the right side of the central γ^{μ} to meet the right-side Dirac equation's incoming electron spinor (= u(p) ) and change into the constant rest mass (= -m or -mc ).

Because this γ^{a}p_{a} cannot change into the constant -m, even if it touches the left-side Dirac external outgoing electron spinor (= u(p') ). ← This is just one of artificial rules which have nothing to do with our real world.

When the matrices γ^{a}p_{a} come across the central γ^{μ}, the upper Eq.8 formula must be used where γ^{μ} changes its sign and produces the new term of -2p^{μ} (= without γ^{μ} attached ) which newly-emerging term by this exchange ( this p.3-lower ) is used as a part of anomalous magnetic moment according to this QED artificial rule which is eventually violated.

*(Eq.9) Transfer of "γ ^{a}p_{a}" from the left side to right side of γ^{μ} .*

*(Eq.10) Transfer of γ ^{a}p'_{a} from the right side to left side of γ^{μ}.*

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

When the external virtual photon's energy and momentum is zero which happens when this virtual photon becomes a real photon with zero mass (= q^{2} = 0 ), the energies and momentums of an incoming and an outgoing electrons become the same (= p = p ' ).

In this situation (= p = p' ), we can freely choose whether each γ^{a}p_{a} should move into the right side of γ^{μ} to touch u(p) and change into the constant -m (= and produce a new term of p^{μ} used as a part of anomalous magnetic moment ), or the γ^{a}p_{a} stays in the left-side of γ^{μ} and touches the left-side u(p') and changes into -mc (= when p = p', γ^{a}p_{a} can change into the -m, also when it touches the left-side u(p') ) **without** producing p^{μ}.

↑ So when p = p', we can freely choose whether new p^{μ} or p'^{μ} terms (= without γ^{μ} ) are produced or not, which means we can **freely choose** and manipulate the calculated value of anomalous magnetic moment. ← QED prediction is meaningless.

↑ This meaningless state is what the QED calculation is.

In the QED calculation, they impose the condition of a virtual photon becoming a massless real photon in the last process, which means the virtual photon's energy and momentum are equal to zero, and p is just equal to p'

*(Eq.12) virtual photon's mass is zero, q ^{2} = 0 is indispensable for getting g-2 factor or anomalous magnetic moment.*

QED calculation of anomalous magnetic moment makes a **paradoxical** claim that the mass of the unreal virtual photon with imaginary mass should be zero in the last stage of the calculation ( this p.7 (or p.13 )-(2.10) this p.4 ).

↑ It means the energies and momentums of an incoming and outgoing electron become the same ( p = p' ), and the QED ad-hoc manipulation of moving each γ^{a}p_{a} (or γ^{a}p'_{a} ) into the right (or left ) side of
γ^{μ} becomes **meaningless**, and this QED calculation can freely manipulate the calculated result of anomalous magnetic moment which fact **invalidates** QED itself.

*(Eq.13) q ^{2} = Einstein's energy-momentum relation.*

In this page, (-1,1,1,1) version of metric tensor is used ( see this ).

So the notation of q^{2} = 0 means the photon's mass is zero like q^{2} = - q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2} = -m^{2}c^{2} = 0

*(Fig.20) an outgoing electron's momentum = p' = p (= an incoming electron ) + q (= virtual photon ).*

Both an incoming electron (= p ) and an outgoing electron (= p' ) after absorbing a virtual photon (= q ) must satisfy Einstein relativistic energy-momentum relation ( this p.18 ) ( in this version of notation ).

*(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 (= relativistic energy and momentum relation must be unchanged as seen by any observers moving at different speeds in any directions, so we can choose one particular direction or momentum to justify all situations in relativity ), the energies and momentums of an incoming electron (= p ) and an absorbed virtual photon (= q ) are

*(Eq.17)*

where zero component means energy (= q^{0} ), and other components are momentums (= q^{j}, j = 1,2,3 ).

From Eq.16 and Eq.17, we can get q=0 which means any energy and momentum of a massless virtual photon become **zero**, as follows,

*(Eq.18)*

Due to electron's "rest mass" energy, the electron's relativistic energy = p^{0} is always **greater** than the electron's momentum p^{1}.

In conclusion, when an incoming and outgoing " electrons (= p and p' ) satisfy Einstein mass formula, their energy and momentum are conserved ( p' = p + q ), and virtual photon (= q ) satisfies Einstein mass formula ( q^{2} = 0 = massless virtual photon ), the virtual photon's energy and momentum must always become zero ( **q = 0** ).

In this case of the virtual photon's zero energy and mass (= q^{0} ), the incoming and outgoing electrons' energies and momentums are the same ( p = p ' ), and this QED calculation can freely control the anomalous magnetic moment value. ← QED calculation of anomalous magnetic moment is meaningless.

*(Fig.21) QED calculated results freely change depending on what artificial rules are created and applied, which is irrelevant to the real physical principle.*

QED calculation of anomalous magnetic moment is **illegitimate** and unable to predict any "finite" physical values.

QED calculated results completely depend on "which finite values" are artificially left by imposing many **different inconsistent rules** in different-loop interactions.

For example, they artificially remove and renormalize different parts by different inconsistent rules in the one-loop and two-loop anomalous magnetic moment calculation.

And the two-loop anomalous magnetic moment contains many different-types of Feynman diagram or virtual interactions where there is much more room to adopt different artificial rules.

In the IIe type of two-virtual-loop anomalous magnetic moment calculation ( this p.2-right IIe two-loop diagram ), they use the completely different-type of aritifical "change of varibles" trick, and remove (= renormalize ) the different part of infinite + a part of finite values ( this p.4-right-lower and footnote, p.5-left-artificial separation of renormalized part as seen in p.9-M-IIe equation ) than the one-loop case ( two-loop "change of variable" artificially makes the equation more complicated by artificialy creating the complicated finite part incorporated into anomalous magnetic moment, this p.9-(1.88) ).

↑ Completely different inconsistent rules are created to artificially remove the chosen infinite and finite values means QED calculation of anomalous magnetic moment is Not the result of QED prediction at all.

*(Fig.22) QED = infinity is removed by another infiniy = infinite bare charge ?*

As I said, all calculated results of QED miserably diverge to infinites.

QED just artificially manipulate and **choose** another infinity called (infinite) bare charge and mass parameters ( this p.100 (or p.101 ), this p.56-57 (or p.68-69) ) to cancel the virtual particles' infinity to obtain the finite anomalous magnetic moment (= bare charge renormalization in one-loop, this p.7-2nd-paragraph ).

↑ This QED math trick of subtracting another infinity from infinity to obtain **arbitrary** finite physical value is illegitimate and cannot be called "prediction", because the indefinite infinite values such as infinite bare charge and mass cannot be determined (= which means the resultant finite values cannot be determined, either ) as the predicted values at all

*(Fig.23) Internal lines = propagator of virtual particles.*

In the QED virtual interaction between an electrons and a (virtual) photon expressed in Feynman diagram, three internal lines in the upper one-loop interaction are called propagators of virtual electrons (= fermion ) and virtual photons whose masses disobey Enstein energy-momentum relation ( this p.11-14, this p.3-7 ). ← Relativistic QED is self-contradictory and wrong.

*(Eq.19) Dirac propagator (= virtual electrons with infinite energies and momentums ).*

*(Eq.20) Photon propagator (= virtual photons including unreal zero-component scalar photon ).*

Eq.19 is Dirac fermion propagator ( internal virtual electron - virtual electron ), and Eq.20 is photon propagator.

Propagators are nonphysical interaction between virtual particles with unreal masses ( this p.2-3, this p.9-10 ) and infinite kinds of energies and momentums expressed as internal lines of Feynman diagram.

About the detailed calculation, see this.

As shown on this, the interaction Hamiltonian energy between electron (= ψ ) and electromagnetic field (= A ) in the nonphysical quantum field theory is

*(Eq.21)*

All these nonphysical photon field (= A ) and electron (or antiparticle ) fields (= ψ ) must always contain unrealistic virtual photons or virtual particles having all kinds of momentums (= k ) and energies (= integral of infinite virtual particles from minus infinite energies to plus infinite energys, -∞ < k < ∞, this p.6-10 ) to satisfy (unrealistic) Einstein relativity (= infinite energies of virtual particles look the same infinite as seen by any observers moving at different speeds = fantasy Einstein relativistic condition in QED ), which always causes the unrealistic infinity in any QED calculations.

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 obtained by the space integral of Hamiltonian density ( this p.65-70 ),

*(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 particles are expressed as nonphysical creation operators.

And c(p') means external electron which

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

Here

First we calculate the **first-order ** of interacting Hamiltonian energy in Eq.24. (Fig.24.)

( First order is the second term of Eq.24. )

*(Fig.24) One interacting Hamiltonian energy.*

Fig.24 doesn't contain internal lines (= propagators ) and loops (= so Fig.24 includes only an external vitual photon = q as unreal particles ).

Feynman diagram of Fig.24 means

*(Eq.25)*

As shown in Eq.25, two identical particles ( electron - electron, photon - photon ) are linked in operators.

As shown on this, Dirac's solutions are,

*(Eq.26)*

The nonphysical electron's math creation operator (= c ) satisfies the unphysical rule called anticommutation of

*(Eq.27)*

Using Eq.26 and Eq.27, the external line 1 (*electron-vertex*) of Fig.24 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.24 and Eq.25 becomes,

*(Eq.30)*

As shown on this, the solution of the magnetic (or electric ) vector potential A_{μ} (x), which satisfies broken Maxwell equation is

*(Eq.31)*

And creation and annihilation operators of each component satisfy.

*(Eq.32)*

where the g_{00} component is the unrealistic zero-polarized scalar photon with negative probability. ← QED photon is **unreal**.

And using Eq.31 and Eq.32, the external line 3 (*photon-vertex*) of Fig.24 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 Dirac delta function like (= four space time dimensions )

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

( *p ^{μ} + q^{μ} = p'^{μ}*, this p.78 )

There is no divergence (= no ultraviolet divergence caused by infinite internal virtual particles' propagators ) in the one vertex function (= the upper delta function becomes finite by integral with k ), because it contains no loop in Fig.24.

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 Hamiltonian 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**. ← just nonphysical artificial rules which have **nothing** to do with the actual physical principles.

As a result, we need not consider the two vertex function.

Next we try three-vertex function of Eq.24.

This contains *three interacting Hamiltonian densities and three integrations *, as shown in Fig.24 and Eq.37.

*(Fig.25) QED anomalous magnetic moment by one virtual loop correction.*

where wave number (= k ) and momentum (= p = ℏk ) 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!.

This one virtual loop of virtual internal electrons and photons causes a serious (ultraviolet) divergence to infinity which must be artificially removed to obtain the tiny finite anomalous magnetic moment in the illegitimate QED renormalization.

Comparing Eq.37 (Fig.25) and Eq.25 (Fig.24), 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 just artificial QED rules with No physical meaning, which were defined **only** for getting the anomalous magnetic moment or g-2 factor.

Basically, the interacting energy and momentum = **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 (← This artificial choice of the number of interacting Hamitonian included in the anomalous magentic moment is one of QED **tricks** to manipualte the calculated anomalous magnetic moment value ).

As a result Eq.38 includes fine structure constant α, which is explained in detail later.

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 (virtual) photon ( ex 3 ) entering the vertex is

*(Eq.44)*

where vertex γ changes into **γ ^{μ}**.

This is important.

Using Eq.19 and this, the **propagator 1** of fermions 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 momentums and energies which make all QED calculation results to diverge to meaningless **infinities** ( this p.8-9 ).

This unrealistic QED world is very difficult to imagine.

But according 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 is

*(Eq.47)*

As shown in Fig.25, *p _{k'} = p_{k} + q*, which means the conservation of momentum and energy.

And using Eq.20, the virtual photon propagator is

*(Eq.48)*

where k_{p} = p / ħ

Fig.25 contains three integrals. 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 virtual photon 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**,

↑ Comparing this Eq.51 integral with Eq.49 integral, only this Eq.51 don't use 1/(2π^{4}), which 1/(2π^{4}) value is necessary to obtain the ad-hoc anommalous magnetic moment of one virtual loop correction. ← This is one of QED artificial tricks.

Next integrating Eq.51 with respect to **d ^{4}k'**, which integral is included in the virtual electron's propagator 2 of Eq.47,

Eq.52 means that k and photon q are united into k'. (See Fig.25.)

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 energy and momentum conservation

*(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 integral of d

This result shows each one-virtual-loop contains this unphysical integral of virtual energies and momentums *from -∞ to +∞*, which always diverges to " **infinity** ".

*(Eq.56) Unphysical electron's anomalous megnatic moment calculation by one-loop virtual particles' renormalization of infinity.*

Combining all Fig.25, Eq.37, Eq.45, Eq.47, Eq.48. Eq.55, we can obtain the (unphysical) QED equation (= above Eq.56 ) of electron's anomalous magnetic moment or g-2 factor based on one-loop meaningless **infinity** ( this p.2-left ) caused by unreal virtual particle (= disobeying Einstein relativity ) propagators ( this p.5-6, this p.7-11 ).

The coefficient of Eq.56 comes from combining the three virtual propagators and two vertices, like

*(Eq.57)*

*(Eq.58)*

Eq.57 is also dimensionless, which is similar to the fine structure constant α (= about 1/137 ) of

Putting together the coefficient of Eq.57 and remaning parts in three propagators of Eq.45, Eq.47, and Eq.48 and two vertices (= γ ), we have the nonphysical QED anomalous magnetic moment one-loop equation of

*(Eq.59)*

In the second line of Eq.59, we use the relation ( this p.40 (or p.46) ) of

*(Eq.60)*

This is very important.

"**Unreal**" zero component photon with paradoxical negative probability is **indispensable** for calculation of virtual photon propagators.

( g_{νρ} of Eq.60 originates in these unreal four-polarized photons, one of which is unreal zero-component scalar photon. )

Next we calculate the numerator of Eq.59 (= anomalous magnetic moment one-loop equation ).

In this page, we use (-1,+1,+1,+1) version of metric tensor.

Using the formula of nonphysical γ matrices ( this p.100 (or p.118), this p.2 ) of

*(Eq.61)*

Here 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, the result of Eq.59 becomes

*(Eq.68)*

If we replace this page's (-1,+1,+1,+1) version of metric tensor by another ( +1, -1, -1, -1 ) version of metric tensor, using the conversions of

*(Eq.69)*

Eq.68 can be changed into (+1,-1,-1,-1) version of anomalous magnetic equation of

*(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.

They try to artificially change and squeeze the three denominator factors of Eq.72 into single quadratic polynomial in k, raised to the third power even by introducing several new variables (= x,y,z ).

This artificial conversion **manipulating** the anomalous magnetic moment's calculated result is called "Feynman parameter" ( this p.1-(a.3), this p.98 (or p.99)-last, this p,17 (or p.18 ). this p.40-41 (or p.46-47) ), which is like

*(Eq.74)*

First we prove the identity of the upper 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 Eq.83-84 ).

And the notation in Eq.78 is

*(Eq.79)*

Here we introduce new variable l,

*(Eq.80)*

Eq.80 (= artificial change of variables, k = l - yq+zp ) is one of the most crucial **illegitimate** tricks in QED.

Uncertain variables l and k can take values from -∞ to +∞ (→ diverge to QED infinity ).

This artificial change of variables in Eq.80 is the shift of some **finite** values. ( " l " replaces " k " in the integral. )

And when we remove infinity of l^{2} artificially by the ad-hoc renormalization, important "p" and "q" values are magically left only after the artificial change of variables ( this p.10 ).

↑ Originally the "change of variables" must Not change the **whole** calculated results themselves.

But in this illegitimate QED math trick, they artificially choose and remove only infinite + unnecessary finite values on the pretext of "renormalization", which means the "change of variables" can freely change the remaining finite values which are used as the anomalous magnetic moment. ← This wrong math trick can Not be called the prediction of QED at all

To prove Eq.80 is equal to Eq.78, we have to use the relativistic energy and mass relation (= external electrons p and p' are on mass shell, this p.18, this p.64 (or p.68 )-upper ) of

*(Eq.81)*

where an incoming electron (= p ) absorbs an external virtual photon (= q ) to become an outgoing electron (= p' ), so p' = p + q.

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 blue line of the 2nd-last equation 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 (= numerator of equation ), 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 (= an incoming external electron )", these infinite virtual particles have NO relation with electron's anomalous magnetic moment (= terms including "p" is necessary ) and g-2 factor, which means the QED artificial "change of variables" artificially add the terms of anomalous magnetic moment, which is Not the result of QED prediction but just the result of the artificial **parameters** slipped into "change of variables" !

*(Eq.86) An incoming electron's value = "p" suddenly appears in nemerator only after the artificial change of variables of k = l - yq + zp. ← QED illegitimate trick*

As shown in the upper Eq.86 (= numerator of equation ), the incoming electron's parameter (= p ) suddenly appears only after conducting the artificial change of variables, and this newly-emerging electron's parameter "p" will be used as a part of anomalous magnetic moment of QED.

↑ "Change of variables", which should Not change the calculated results, changes the QED calculation result of anomalous magnetic moment or g-2 factor (= together with unjustifiable renormalization ), which **wrong** way of using "change of variables" is illegitimate, and QED is false.

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

As shown in the above Eq.87, the new variable l is made by *shifting k by a finite value*.

This finite part includes the information of the external electron (= p ), which will be eventually used as a part of anomalous magnetic moment.

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

*(Fig.26) This QED calculation is just a wrong math, so false.*

In the upper Fig.26, in the original anomalous magnetic moment equation, the integral of the odd-function term of k from minus infinity to plus infinity becomes zero.

But after imposing the artificial "change of variables ( k = l - β )", this original term splits into two terms.

And physicists claim only the first term linear in "l (= odd function )" becomes zero by integration from minus infinity to plus infinity (= integral interval remains the same as the original variable k, because infinity + finite remains the same infinity, this p.140-(5.77), this p.1-footnote ).

And the remaining finite 2nd-term (= -β ) remains as a non-zero value, which will be used as a part of the QED anomalous magnetic moment or g-2 factor.

↑ It means if we change the finite parameter = β inside the change of variables, we can **freely manipulate** the anomalous magnetic moment value, which makes QED method **invalid**.

The trick of this QED illegitimate calculation exploits the fact that even if they change variables from k to l + finite, the intergral interval of the original variable k from minus **infinity** to plus infinity remains unchanged also in the new variable l whose integral interval is also from minus infinity to plus infinity **absorbing** finite part.

↑ Infinity absorbs the finite shift in the integral interval, which is the trick of this illegitimate change of variables in QED.

*(Fig.27) Right math. = the whole equation remains the same zero regardless of change of variables*

If we use a **right** mathematical method, the finite 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 (= infinity minus inifnity ), and becomes zero.

( According to the right math, the finite "β" part is absorbed into the infinite "l" term. )

So QED mathematical method of Fig.26 **violates basic math**, and QED anomalous magnetic moment calculation is wrong.

*(Eq.W-1) ↓ QED change of variables changes the result ! = wrong math*

As shown in Eq.87, 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 finite part which will be used as part of anomalous magnetic moment or g-2 factor (= -yq+zp ) is left, they insist.

"Change of variable" is just a **means** of calculation which must **Not** change the result itself.

But QED **distorts** this original meaning of change of variables to artificially obtain the anomalous magnetic moment value, which is clearly **wrong** math.

*(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 (= anomalous magnetic moment ) part is left, they baselessly claim.

*(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)*

In both "(l-β)" and "l", the integration of them from minus infinity to plus inifinity becomes zero (= finite part of β is **absorbed** into infiity, so β is meaningless )

*(Eq.W-5)*

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

*(Eq.W-6)*

So, the QED mathematical trick of picking up only g-factor (= anomalous magnetic moment ) part 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, and "QED is successful theory to predict anomalous magnetic moment !" is a total lie (= actually, those who baselessly claim "successful QED" **never** present the actual ad-hoc QED calculation methods to us )

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-2 factor or anomalous magnetic moment 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 ( this p.1-right-last, this p.11 ).

As shown on this, QED anomalous magnetic moment calculation depends on the artificial misinterpretation of Dirac γ matrix as electron spin ( this p.3, this p.3-(12) ), and external magnetic field using nonphysical Gordon identity ( this p.1-(2), this p.7(or p.13)-lower )

*(Eq.89) Unphysical QED definition of spin and magnetic field*

The difference between an incoming electron's momentum (= p ) and an outgoing electron's momentum (= p' ) is an absorbed virtual photon's momentum (= q )

*(Eq.90)*

They try to treat this "q (= p' - p )", which is a virtual photon absorbed into an incoming electron, as a (fictitious) **external magnetic field**, which causes a **paradoxical** magnetic field (= B = ∇×A (= magnetic potential A ), QED tries to **misinterpret** the absorbed virtual photon q as the irrelevant space drivative = ∇ used as a part of magnetic field, this p.7 (or p.13)-lower ), so wrong.

According to this paradoxical definition of magnetic field acting on anomalous magnetic moment, when the electron's momentum (= velocity ) is **constant** ( p'-p = 0 = an incoming electron doesn't absorb a virtual photon's momentum ), external magnetic field is **zero** (= q = p'-p is used as a part of external magnetic field in QED illegitimate definition ), even if the external magnetic field exists.

This is strange.

*(Fig.28) External magnetic field B = 0, even if B is not 0 in QED ?*

In Fig.28, the electron is moving in the *z direction*.

And the static external magnetic field "B" is applied in the *z direction*, too.

Lorentz magnetic force is known as

*(Eq.91)*

( In Fig.28, E is supposed to be zero. )

In Fig.28, the **spin - magnetic field** interaction exists.

Because the "constant" electron spin (= 1/2) magnetic moment is supposed to *always exist*, and the external magnetic field is supposed to exist, too.

But according to the QED (**mis**)interpretation of magnetic field in Eq.89, when an electron is flying in the magnetic field direction at a constant speed, the external magnetic field must be **zero**, though the nonzero external magnetic field is applied in QED !

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.

This paradoxical artificial definition of QED external magnetic field shows QED calculation of spin anomalous magnetic moment interacting with this fictitious paradoxical magnetic field is **illegitimate**.

From Eq.89 derived from the original nonphysical Dirac γ matrices and Gordon identity ( this p.1-right, this p.12 ), we have

*(Eq.92)*

Substituting Eq.92 into Eq.88-2, the spin-magnetic field interaction appears. they claim ( this p.4-(19), this p.2-lower )

*(Eq.93)*

Due to the antisymmetric arrangement of (x-y), the last term of Eq.93 vanishes, when we integrate it with respect to 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**.

Using this math formula, the denominator (= 1/D^{3} ) of the anomalous magnetic moment equation becomes ( this p.35-36, this p.4, this p.2, this p.5, this p.9 )

*(Eq.97)*

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

But in the following case, the integral diverges to **infinity** ( this p.148-151, this p.6, this p.2-right-1st-sentence, this p.141 ), 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, the instant q^{2} becomes 0 (= virtual photon becomes a real massless photon ), virtual photon's energy becomes **zero** ( q = 0 ).

When q = 0 (= virtual photon becomes a real massless photon, which means the virtual photon's energy and momentum becomes zero, q = 0 ), the anomalous magnetic moment of g-2 factor value **cannot** be fixed at a single value, which means QED-calculated anomalous magnetic moment value can be an **arbitrary free** paraemter, and QED prediction itself is **invalid** and meaningless.

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

*(Eq.100)*

See also Eq.68.

And 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 QED has to **artificially** remove all these troublesome values (= infinity + **unnecessary finite** values ) attached to γ^{μ} matrix (= the first term of Eq.99 ) by the ad-hoc **renormalization**, as follows,

*(Eq.102)*

↑ This **unscientific** removal (= or intentional ignoring ) of the unnecessary infinite + part of finite value (= they call it "form factor F1" ) by QED illegitimate (infinite bare) charge renormalization is just a wrong math which is Not based on any real physical principles ( this p.4, this p.12-2nd-last paragraph, this p.65 (or p.69)-(7.36), this p.7-(22), this p.3-4 ).

Relying on this artificial ad-hoc renormalization method means QED has **NO** power to predict any physical values.

"Counter" term is obtained when the particles satisfy "mass shell" condition.

So Eq.102 itself is "counter term", and cancels itself out. ← Convenient artificial rule.

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 (= fictitious spin (= nonphysical σ^{μν} matrices )-magnetic interaction using Bohr magneton ) is

*(Eq.103)*

This is called "form factor 2".

When q^{2} = 0 (= causing paradoxical situaion ), Eq.103 becomes,

*(Eq.104)*

Fortunately, Eq.104 converges, as follows ( this p.2-right, this p.5, this p.42 (or p.48) ),

*(Eq.105)*

As a result, Eq.89 becomes

*(Eq.106)*

And the correction of the g factor (= anomalous magnetic moment ) 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.

The most serious scam in this QED anomalous magnetic moment calculation is explained in Fig.15.

QED tries to **artificially remove** the unnecessary **infinite + finite** part attached to γ^{μ} matrix on the pretext of
renormalization.

Then, only the finite anomalous magnetic moment
part (= allegedly Not attached to γ^{μ} matrix, but this turned out to be **untrue** ) remains by avoiding being renormalized or removed.

The QED hidden **inconvenient** truth is that this alleged anomalous magnetic moment part **includes** matrices (= the alleged spin matrices = σ^{μν} = the product of **two γ matrices** = γ^{μ}γ^{ν} ) !

The reason why this QED anomalous magnetic moment calculation is **contradictory** and wrong is that after they try to free as many γ^{a}p_{a} and γ^{a}p'_{a} terms as possible from γ^{μ} matrix using this formula, they try to **return** all the p^{μ} and
p'^{μ} terms (= not attached to γ^{μ} ) to the original equation attached to γ^{μ} using Gordon identity.

And they artificially remove and renormalize all those terms attached to γ^{μ} matrix without removing only the anomalous magnetic moment part including σ^{μν} matrices which also contain γ^{μ} matrix ( this p.3, this p.4, this p.12-(51) ). ← Thse spin σ^{μ}^{ν} matrices consisting of two γ^{μ} matrices are Not removed or renormalized ! ← QED method is inconsistent and cotradictory !

If they artificially **remove all terms attached to γ ^{μ} matrix**, they have to

As shown in the last line of Eq.106, the alleged anomalous magnetic moment term iσ^{μν}q_{ν} can be replaced by γ^{μ}( γ^{a}p_{a} - γ^{a}p'_{a} ) using q = p' - p, which expression goes **back** to the original equation (= when still many terms including this anomalous magmetic moment σ^{μν} term were attached to γ^{μ} in the first stage of calculation ) before QED manipulation starts, which anomalous magnetic moment calculation process is contradictory and **meaningless**.

↑ This is the most illegitimate trick of QED anomalous magnetic moment calculation, and shows QED is completely wrong and inconsistent with **No** power to predict any physical values.

2022/8/6 updated. Feel free to link to this site.