Top page ( Quantum field theory is wrong ← 6/30/2024 )

Calculation of QED g-factor.

*(Fig.1) Tiny g-factor is gotten by "artificially" removing "infinity" (= renormalization ) !?*

Though it is said that quantum electrodynamics (= QED ) is the most successful theory in predicting anomalous magnetic moment (= g-factor ) and Lamb shift, these explanations are **NOT** true.

In QED, they just **artificially** remove **infinity** to leave only **convenient** *finite* terms by renormalization.

Because they argue that **infinite** *bare* charge and mass of an electron conveniently **cancel** infinity to generate *finite* physical values.

In fact, Paul Dirac said, "I am **dissatisfied** with the situation, because QED **neglects** infinities in an *abitrary* way, it is **NOT** sensible mathematics". Feynman also called this QED "**hocus-pocus**".

The magnetic moment of an electron spin is **equal** to Bohr magneton (= μ_{B} ).

They defined spin angular momentum as **1/2 ħ** instead of classical "*1 ħ*", so g-factor becomes twice ( *1* → **2** ).

The magnetic moments (= g-factor × angular momentum ) are the **same** in Bohr's orbit and electron's spin.

Because, magnetic moment = *1 ħ* × *1* (= Bohr's orbit g-factor ) = **1/2 ħ** × **2** (= spin g-factor ).

It is **natural** that there is some small **difference** between **ideal** g-factor 1 ( or 2 ) and the actual value.

They argue this **tiny** difference (= 0.001596 .. ) can be explained by **unrealistic** QED, which always gives **infinity**.

*(Fig.2) Artificial trick of QED = leaving only "convenient" finite values. *

When QED calculation contains "**loop**" in Feynman digram, it always become divergent to **infinity**.

Of cource, we have to **remove** these divergence somehow to get physical **finite** values such as magnetic moment.

So physicists **justified** the unreasonable equation of " *infinity = infinity + finite* value ", and have applied this math **trick** to various important values since 1940s.

As you ( and Dirac ) noticed, these are only *artificial* and "**stopgap**" measures.

Because "**infinity**" can *absorb* ( or *eject* ) any "**finite**" values, so they **abuse** this logic to obtain **convenient** finite values.

*(Fig.3) Renomalization of one-loop Feynman diagram.*

In QED Feynman diagram, they connect electron-electron and photon-photon at each **vertex** consisting of 1 photon (= ends of dotted line ) and 2 fermions (= ends of solid line ). The internal lines show propagators and virtual particles **violating** special relativity.

In 1947, Schwinger obtained α/2π ( α = fine structure constant ), which is **close** to actual g - 2 factor. Because this Schwinger's g-factor is the **first** prediction by QED, they could *establish* **new** rules to get this value.

So this g-factor correction is **NOT** natural result of QED, but an **artificial** result *thanks to* **new** rules such as propagators, "gauge" and renormalization.

Of course, using only these new rules, they could **NOT** obtain good result, so they had to rely on
the wrong math **trick** (= artificial change of variables ) to be close to the experimental value.

*(Fig.4) 2-loop correction summing 7 Feynman diagrams was a mistake in calculation ? *

An year after Schwinger's one-loop g-factor, Karplus and Kroll calculated its two-loop corrections consisting of 7 Feynman diagrams in *1949*.

According to this site, their results were **wrong**, and Petermann corrected its error **eight** years later ( in *1957* ).

But as you think, it is very **unnatural** that the physicists *all over* the world **left** the error as it was, for the whole **8** years ! Because **just** 7 Feynman diagrams do **NOT** need as many as 8 years for its calculation.

Because, even if you calculate two-loop Feynman diagrams in the **same** way as the one-loop corretion gotten by Schwinger, you **cannot** obtain the corret results.

They needed to invent new **other** artificial rules to get satisfying results.

**8** years was necessary for developing the **new** methods, **NOT** only for its calculation.

*(Fig.5) "k" is from -∞ to +∞ → Change of variable → "l" is also from -∞ to +∞.*

One of the most artificial tricks in one loop correction is "change of variable" from "k" to "l".

Here the part of "yq - zp" is **finite** variable, and the original virtual *relativistic* **momentum** "k" is from **-infinity** to **+infinity**.

These propagators including "**infinity**" values in momentums give ultraviolet **divergence**. See this and this (p.2). They replaced the variable "k" with "l", their difference is **finite** (="yq-zp" ).

The virtual momentum "k" is from **-∞ to +∞**.

So the resultant "l" is **also** from **-∞ to +∞**, because the *finite* value is **absorbed** into infinity, and **gone**.

*(Fig.6) The integral with respect to "k" ( or "l" ) becomes "zero".*

In Fig.6, the denominator becomes "**even**" function, because ±∞ **remain** ±∞ even after "change of variable" ( k → l ).
And the numerator is "**odd**" function with respect to "k" ( and "l" )

As a result, this integral from *-∞ to +∞* becomes **ZERO**, irrespective of "change of variable".

This is a **right** math. You can confirm this, performing integration by parts.

*(Fig.7) Change of variable ( k → l ) "changes" the result ! ( zero → NOT zero ).*

But in QED calculation which Schwinger performed, **wrong** mathematics is used.

After the change of variable ( k → l ), **only** the denominator used right math.

In numerator. they consider only "l" included in k = "l" - yq + zq as odd function.

As a result, in this wrong math, the **finite** value (= -yq+zp ) is **left**, which gives a part of g-factor, they insist.

Originally, the numerator "k" is **genuine** *odd* function from -∞ to +∞

So the **remaining** *finite* value (= -yq + zp ) in the numerator is very **artificial** trick, and just **wrong**.

About the actual calculation, see this (= Eq.7, 14, 15 ) this (p.98) and this.

*(Fig.8) Mathematical trick = Change of variable affects the calculated results !?*

As shown in Fig.8 simple example, "change of variables" ( k = l - β ) **artificially** changes the *result* itself, though it's just calculation method.

This is completely **mistake** in calculation. Of course, "intentional" mistake.

Lamb shift also uses this g-factor value as its part, so QED Lamb shift is **wrong**, too.

And they try to add various **artificial** effects to fit them with experimental results ( ex. virtual excited energies in Lamb shift **non**relativistic part ), which means this result is **NOT** natural.

Of course, if special relativity and Dirac's hydrogen are wrong, QED Lamb shift must be **replaced** by other interpretations.

*(Fig.9) New rules = "mass" and "propagator" renormalization are artificial trick in 2-loop correction.*

In one-loop correction of anomalous magnetic moment, they could obtain value of **α/2π**, counting on wrong math.

And this one-loop correction is the **first** QED calculation, so we can say this *whole* method itself is **new** rule.

In two-loop correction of g-factor, they had to **rely** on **other** new concepts such as "**mass**" and "**propagator**" renormalization.

**Depending** on *new* rules means this 2-loop calculation is also **artificial**, and **NOT** first-principle at all.

You can download 2-loop correction paper from this site.

Click "L7-KarpusKrollPhysRev.77" button, then you can get this paper (= Karplus and Kroll ).

The mass and propagator-renormalized terms in this Karplus paper are the **same** as latest Petermann's paper ( in IId and IIe two-loop diagrams ).

*(Fig.10) Separating "finite" values into "unneeded" and "needed" parts ← artificial trick.*

The most important tricks in 2-loop correction is the **redefinition** of propagator ( of photon and electron ).

As shown in this paper (= Eq.11, 12, 13 ), they **replaced** the original photon propagator (= D_{F} ) with **new** artificial one by **removing** *unnecessary* terms.

The problem is that they **eliminated** not only infinity but also a part of **finite** values to obtain g-factor.

There is **NO** fixed rule about this elimination, so they could use almost "**free**" ways, as they liked.

And of course, this new method is **NOT** seen in the original 1-loop correction.

So it is clearly an **artificial** math trick to get correct g-factor value also in 2-loop correction.

*(Fig.11) Redefinition of electron propagator (= S _{F} ) is one of artificial tricks.*

As shown in this paper (= Eq.15-23), they **replaced** electron's propagator (= S_{F} ) with **new** one.

In this process, they **removed** not only infinity but also **unneeded** "*finite*" values.

Again this renormalization (← ? ) in propagators is an artificial trick.

So this anomalous magnetic moment is **NOT** first-principle, and does **NOT** prove the validity of QED at all.

*(Fig.12) Choosing and removing "unneeded" finite values from photon propagator.*

As shown in the above paper ( Eq.13, Fig.IIe diagram of FIG.1 ), they **eliminated** not only infinity (= P ) but also **finite** values in calculating photon self-energy.

Furthermore, they did **NOT** mention the **content** of the finite "P_{0}" term. There is **NO** fixed rule here.

About the detailed calculation, see this section.

*(Fig.13) Unnecessary "finite" values are removed, too !*

The way of **separating** infinity into "*infinite*" and "*finite*" terms is completely **arbitrary**.

So they **abused** this "renormalization" of infinity (+ finite value ) to obtain **convenient** "finite" values.

*(Fig.13') Replaced by "artificial" photon propagator (= bar-D _{F}^{(2)} ).*

As shown in Eq.36e in the above paper, they **replaced** the 1-loop part of photon self energy (= Fig.12 ) by **artificial** photon propagator (= bar-D_{F}^{(2)}, Fig.13 ).

So it is clear that they **intentionally** tried to *leave* only **convenient** "finite" values in calculating anomalous magnetic moment.

*(Fig.14) The photon propagator in this paper is "different" from standard form.*

The photon self-energy consists of two photon's propagators (= dotted line, p ) on both sides and two loop electron's propagators (= solid line, k and p-k ).

The **standard** photon's self-energy forms are seen in this (p.27) and this (p.199).

The form used in this 2-loop g-factor calculation is completely **different** from the *standard* form.

So, there are **NO** fixed rules stipulating what parts are removed by renormalization.

These *free* rules **allow** arbitrary choice and artificial **adjustment** in calculating anomalous magnetic moment

*(Fig.15) Feynman diagram of electron self energy*

The 2-loop correction of g-factor introduced other **new** concpets such as electron self energy in the above paper ( Eq.15, 16 and IId diagram ).

Feynman diagram of electron self energy consists of two electron propagtors (= p ) on both sides and one loop of a pair of electron (= p-k ) and photon (= k ) propagators.

*(Fig.16) ↓ Artificial transformation of equations*

In transforming the equation from Eq.21 to Eq.22 in the above paper, they used **non** standard calculation method.

As shown in Fig.16, this transformation is **unreasonable** and **NOT** correct.

Probably they **misused** "infinity" for this "unnatural" transformation.

*(Fig.17)*

Here Λ_{0} is "**finite**" value.

"m" and "λ" are electron's and photon's masses, "u" is from 0 to 1.

When this photon mass λ becomes zero, it causes infrared divergence.

Depending on the **way** of removing this infrared divergence, the calculation result becomes **different**.

*(Fig.18) Not only infinity but also some "finite" term are removed !*

In Eq.22, 23 in the above paper, they **artificially** removed not only infinity but also *unneeded* **finite** values.

The first term (= "**finite**" ) of Fig.18 is **removed** by "mass" renormalization and the second term (= "**finite**" ) is removed by "propagator" renormalization.

*(Fig.19) ↓ Unnecessary "finite" terms are removed !?*

As shown in Fig.19, they **eliminated** (= renormalized ) "finite" term, too.

*(Fig.20) Finite value*

As shown in Eq.19 in the paper, the integral of Fig.20 gives "finite" value.

*(Fig.20') Replaced by "artificial" electron propagator (= bar-S _{F}^{(2)} ).*

As shown in Eq.36d in the above paper, they replaced the 1-loop part of electron self energy (= Fig.15 ) by **artificial** electron propagator (= bar-S_{F}^{(2)}, Fig.19 ).

So again, it is clear that they **intentionally** tried to *leave* only **convenient** "finite" values in using electron self-energy.

*(Fig.21) "Mass" renormalization is a part of vertex renormalization, so meaningless.*

In 2-loop g-factor correction, they first do **mass** renormalization to leave only "**convenient**" finite values.

But as you see, this *distinction* of "mass" renormalization is **meaningless**.

Because this mass renormalization (= Fig.21 left ) is only a **part** of the *larger* **vertex** renormalization (= Fig.21 right ).

So **distinguishing** *small* "mass" renormalizatin from *larger* "vertex" renormalization is just an **artificial** choice.

It is clear that our Nature does **NOT** depend on these **arbitrary** and **convenient** math trick and choice.

So the familiar phrase such as "QED is the most successful theory" is completely **wrong**, and one of **mind-controls**.

*(Fig.22) Photon's self energy is affected by "arbitrary" transformation.*

In the above paper (Eq.12), they used Schwinger's photon self-energy after a slight **modification**.

"*Modification*" means they can **change** the calculated results as they **liked**. So it is **NOT** first-principle.

This Schwinger's paper can be gotten from this site or this ( see appendix, p.25- ).

In this paper (= eq. A.13 in p.25 ), they defined photon-self-energy term (= G_{μν} ) using two electron propagators (= S_{F}, see Eq.1.66 in p.7 ).

*(Fig.23) Two ( virtual ) electron propagators.*

Fig.23 is ( virtual ) electron's propagators.

**k'** and **k''** are each electron's four - **momentum**. We use "m" instead of "κ" as electron's mass.

*(Fig.24) "p" = photon's momentum on both sides.*

"**p**" is photon's momentum on both sides (= "in" and "out" ).

So the sum of k' and k'' should be equal to "p".

*(Fig.25) They utilized "convenient" change of variables.*

In this paper (= eq. A.28 in p.26 ), they performed **artificial** "change of variable".

As you know, "change of variable" is just "*means*", so it **cannot** change the result itself according to "**right**" math.

Here "p" means photon's momentum, and "**k**" is **vitual** momentum caused by the **difference** between two electron's momentums. ( we exchange "p" and "k" in the above paper. )

*(Fig.26) Convenient "infinity" can absorb "finite" values.*

Using Jacobian determinant of muti-integral, you find this "change of variable" doesn't change the **coefficient** of integral variables.

But of course, the *upper* (= +∞ ) and *lower* (= -∞ ) limits of this integral must **change**, when they are **NOT** infinity.

This means they **abuse** "infinity" to get convenient "finite" value like this section.

Because "**infinity**" can *absorb* (or eject ) "**finite**" values **freely**.

*(Fig.27) *

In this paper (from A.27 to A.29), they transformed the equation using this change of variables.

Furthermore, they performed partial integral with respect to "**v**" in A.32, and get

*(Fig.28) Photon's self-energy ?*

See Eq. A.32, A.33, A.35 in the above paper.

This result is almost same as two-loop corretion of Fig.12.

*(Fig.29) Artificial "change of variables", part II.*

Instead of using the original change of variables, we use Fig.29, in which **v = 0**.

Of course, "v" is variable ( -1 < v < 1 ), so "**v = 0**" does **NOT** change the **form** of the integral.

*(Fig.30) "Change of variables" affect the calculation results !*

But if we use the condition of "**v=0**" in "change of variables", this photon self-energy becomes **zero** instead of Fig.28 ! So the *means* such as "change of variables" **affect** the calculation results !

This means the results of anomalous magnetic moment (= g-factor ) **depend** on **artificial** "change of variables", and are **NOT** natural things at all.

*(Fig.31) Artificially dividing zero into two ± values (= two red lines ).*

Also in some (unphysical) Feynman diagrams of virtual particle's two loop correction ( this p.2 Fig.1 IIa, IIc, and p.7 Fig.5 ~ p.8, and p.9,10 M^{IIa} and M^{IIc} ), QED uses an illegitimate trick artificially **manipulating** a finite value (= used as a part of anomalous magnetic moment or g-factor of an electron ), which means QED has No ability to predict physical values.

In this p.8 (32) equation, they artificially divide zero into two ±parts (= same value with opposite signs ) and insert only the latter part into the finite part (= become a part of anomalous magnetic moment ).

The former part is artificially removed by being shoved into infinite part B (= removed by renormalization later ), as shown in this p.8 (32), p.9 (33),(36b),(36c).

↑ artificial trick: zero → +a -a (= only -a part is used as a part of anomalous magnetic moment, and +a part is removed by artificial renormalization ).

This is clearly one of artificial unphysical **trick** just manipulating electron's anomalous magnetic moment value as they like (= so QED is useless with No ability to predict any physical values ) by diving infinite value into another infinity and arbitrary finite value.

( ↑ In this paper p.8 (32) 2nd equation, the author seemed to forget the coefficient of -γ to the last term. By calculation we can confirm equations(28)(30)(32) are the same. This Eq.97 formula is used in (32)-last )

And in the conversion from Eq.(28) into Eq.(30) in this physically-meaningless anomalous magnetic moment calculation ( this p.8 ), they **artificially changed** a part ot variables p' and p'' (= both of these are unreal virtual electrons whose mass is different from the real electron's mass ) into the contant κ^{2} (= the square of electron's mass ). ← The resultant artificially-obtained parameter = 4κ^{2}(1-u-1/2u^{2}) of Eq.30 (= which will be artificially removed by ad-hoc renormalization in Eq.(32) and Eq.(33)) can Not be directly obtained from the original equation of Eq.(28) without introducing the artificial rule.

Different physicists can **freely change** the rules and parameters, which illegitimate QED methods can Not predict any physical values ( see the foornotes of this p.8 and p.4, and see this p.10-(1.94) and p.26-(A.28-A.32) )

2022/1/28 updated. Feel free to link to this site.