Top page ( Quantum mechanics and Einstein relativity are wrong )

Anomalous magnetic moment by QED is wrong.

Standard model is full of fictions.

- Propagator of Klein-Gordon virtual scalar field.
- Propagator of Dirac virtual fermion and electrons.
- Propagators of virtual photons.

*(Fig.1) unphysical Feynman diagram, internal lines = propagators of virtual electrons and photons.*

The present relativistic quantum field theory, QED and standard model **lack** reality, because they describe their **fictitious** particle interaction as unphysical propagators exchanging unreal virtual particles ( this p.5 ) in the abstract Feynman diagram ( this p.6-7 ).

These virtual particles of propagators **disagree** with Einstein relativistic mass-energy relation ( this p.8-10, this p.2-7 ). ← Relativistic QED is self-contradictory.

As shown in the upper Fig.1, these unphysical particle interaction = propagators consist of virtual electrons (= fermions ) and photons connecting unphysical interaction term or vertex between mathematical photon (= A ) and Dirac electron (= ψ ).

*(Eq.1) Relativistic Klein-Gordon unphysical particle's solution.*

Eq.1 is the plane wave solution of relativistic Klein-Gordon or scalar zero-spin particles ( this p.17, this p.3 ) such as unphysical Higgs.

Some textbooks use the coefficient of 1/2E or 1/2ω instead of the square root of 2ω in the above Eq.1 scalar field wavefunction.

But these coefficients can be freely adjusted by modifying the particle exchange relation's coefficient of Dirac delta function ( compare the difference between this Eq.4.19 and this p.2-(10) ). ← After all, total equation or Hamiltonian energy are artificially designed to become the same form in any cases, so no problem.

Here we use (-1, 1, 1, 1) version metric tensor.

If you want to use (1, -1, -1, -1) version, change notation like ( kx → -kx, k^{2} → -k^{2} ).

*(Eq.2) virtual Klein-Gordon or scalar-field propagator.*

First, we derive QED Feynman propagator of scalar-field (= Klein-Gordon particles such as Higgs ), which is the basis of all other propagators of fermions (= virtual electrons ) and virtual photons, so important.

This QED virtual propagator (= violating Einstein relativity ) basically consists of two virtual particles' fields (= φ(x')φ(x) ) and mathematical Heaviside step function = θ(t).

They put these two virtual particles' fields in order of time where right φ is earlier and the left φ is later using the artificial step function (= θ(t) = 1 when t >0, θ(t) = 0 when t < 0 ).

The most serious **wrong** math trick for deriving this artificial QED propagators is that the irrelevant variable ω ( this p.5 ) used in expressding the nonphysical step function is unreasonably switched into the **unreal** energy (= k^{0}, p^{0} or E ) of virtual particles ( this p.2, this p.2-left ), as I explain later.

This is clearly a **wrong** illegitimate math trick, so QED and its indispensable tool = propagators are completely wrong equations or theories.

*(Eq.3) Heaviside step function θ*

This mathematical step function is used for the time-ordered (virtual) propagator.

In the step function (= θ ) of Eq.3, when (t-t') is positive, θ becomes "1", and when t-t' is negative, θ becomes zero.

Using the complex integral, step function can be expressed as ( this p.5, this p.4 )

*(Eq.4) Step function θ represented by integral*

*(Eq.5) Step function θ, integral form-2*

where we make the exponential function parts the same.

↑ This variable ω ranging from negative infinity to positive infinity will be artificially switched into unreal infinite energies of virtual particles in QED propagators by the **wrong** math trick.

This mathematical step function is originally unnecessary, but this unnecessary step function of the excessive "**1/2π**" was indispensable as one of artificial illegitimate manipulations of QED anomalous magnetic moment result.

--------------------------------------------------

First, we prove the relation of Eq.4.

If you want to skip this proof, please proceed to the next section.

To prove Eq.4, we have to use Cauchy's residue theorem.

In this Cauchy residue theorem, only the coefficient of 1/(z-a) is left by the complex integration.

*(Eq.6)*

*(Eq.7) Only the coefficient of 1/(z-a) remains*

where integral is counterclockwise closed contour.

And when the closed contour doesn't contain "pole", Eq.7 becomes zero.

*(Eq.4) Heaviside step function = θ*

In Eq.4, when t-t' is positive ( t-t' > 0 ), the exponential parts of Eq.4 becomes zero in the region of ω = -i∞, as follows,

*(Eq.8) When t-t' > 0, the integral is equal to clockwise comlex integral.*

So we can treat the integration from -∞ to +∞ along the real line as the **complex integration** in the clockwise direction like in Fig.2.

*(Fig.2) Step function ( complex integration ) of Eq.4. ( t-t' > 0 )*

As shown in Fig.2, "pole" exists ( in Eq.4, ω + iε = 0 → pole = ω = -iε ) in the minus imaginary area, so it doesn't become zero.

Using Eq.7 and the fact that the coefficient c_{-1} of the step function θ becomes " 1 (= exponential function e^{-ω(t-t')} → 1 when ω = -iε → 0 )", when ε approaches zero,

*(Eq.9)*

Eq.8 is **clockwise** closed contour, so the result becomes opposite ( 1 → -1 ).

*(Eq.10)*

As a result, when t-t' > 0, the step function θ becomes "1",

*(Eq.11)*

When t-t' is negative ( t-t' < 0 ), the exponential part of Eq.4 becomes zero, as ω → i∞,

*(Eq.12)*

So the integration of Eq.4 is equal to the **counterclockwise** closed contour in the complex field,

*(Fig.3)*

As shown in Fig.3, this closed contour doesn't contain a pole.

So the step function θ becomes zero, when t-t' is negative, which satisfies the condition of the step function (= θ(t) = 0, when t < 0 ).

Next we think about the step function of Eq.5.

*(Eq.5)*

*(Fig.4) Step function of Eq.5.*

In Eq.5, "pole" exists in the **plus** imaginary area, as shown in Fig.4.

In the same way as Eq.4, we can get the results of

*(Eq.13)*

This result agrees with Heaviside step function ( this p.5 ).

---------------------------------------------

*(Eq.1) Unphysical relativistic Klein-Gordon scalar particles *

In order to obtain Klein-Gordon virtual propagator, we use the ( unphysical ) Klein-Gordon scalar particle's plane wave solution like the upper Eq.1 ( this p.3-6, this p.1-5 )

The creation and annihilation virtual particle's operators of Eq.1 satisfy the following commutation relation (= which is just one of artificially-created rules ).

*(Eq.14)*

where δ is Dirac delta function which is used just as one of artificial rules (= the coefficient of the upper delta function can be freely changed ) that have **nothing** to do with the real particles or principle at all.

The notation about the above commutation relation of (unphysical) particles is like in this p.1-lower, this-(8.2.10).

Using two different wavefunctions of Eq.1, we calculate the propagator,

*(Eq.15)*

Due to the relation between the vacuum (= |o> or <o| ) of both sides and the particle's annihilation operators,
only aa^{†} term (= adjoint-creation = a × creation operators = a^{†} ) is left ( this p.2-last ), which is also one of artificially-created rules ( this p.23-(3.28-3.29) ).

*(Eq.16)*

Using the commutation relation of Eq.14 and Dirac delta function (= δ ), the result of Eq.15 becomes ( this lower )

*(Eq.17)*

As shown in the upper Eq.17, if the unnecessary step function is not used, the energy ℏω_{k} satisfies the relativistic energy-momentum relation (= not virtual ), but the coefficient of this resultant equation simply connecting two scalar fields becomes 1/(2π)^{3} which can**not** obtain the convenient QED anomalous magnetic moment which needs the coefficient of 1/(2π)^{4} incorporating the artificial virtual step function energy.

So adding the unnecessary step function
is one of QED artificial math tricks for **manipulating** the anomalous magnetic moment even by relying on unreal virtual particles disobeying Einstein relativity.

In the same way, exchanging x and x' of Eq.17,

*(Eq.18)*

In Eq.18, ω_{k} (= ℏω = relativistic energy E ) is an *even function* of k_{i} (= ℏk_{i} = relativistic momentum p_{i}, i = 1,2,3 components ), as follows,

*(Eq.19)*

So if k_{i} (= momentum, k = 1,2,3 ) values are determined, ω_{k} is also determined, and it's always *positive*.

Changing the sign of the momentum's 1-3 component k_{i} ( k → -k ) in Eq.18, the exponential function of momentum ( e^{ikx} ) parts become the same as Eq.17, as follows,

*(Eq.20)*

↑ The point is that artificially changing the sign of the remaining relativistic energy (= ℏω ) is the most difficult, and needs some **illegitimate** trick relying on unreal **virtual** energy originating from the step function θ(ω).

Using Eq.4 and Eq.17, the first term of Eq.2 is

*(Eq.21)*

As shown in the upper Eq.21, the step function's variable = ω is **artificially changed** into the energy (= k^{0} or E ) of the virtual particle's propagator by the illegitimate change of variables ( step function ω = fake energy is changed into ω - ω_{k} where ω_{k} is the original relativistic energy, and **eliminates** only this original relativistic energy ω_{k} ) !

This is why this fake virtual particle's energy originating from the irrelevant step function is **independent** of other momentums (= k_{i}, i = 1,2,3 ) variables, and this virtual particles in the propagator never satisfy Einstein relativistic energy-momentum relation.

From Eq.5 and Eq.18 ( Eq.20 ), the second term of Eq.2 is

*(Eq.22)*

Change of variables (= ω ) in the upper Eq.22 is different from Eq.21, which is one of artificial tricks, and the added coefficient 1/2π by the step function is clearly prepared only for influencing the dubious anomalous magnetic moment's value. ← Was it necessary to add the redundant step function even by using its ω as fake virtual particle's energy ?

QED founders changed the step function variable = ω into the (fake) virtual particle's energy k^{0}, as follows,

*(Eq.23)*

As a result, the **Klein-Gordon propagator** of Eq.2 is the sum of Eq.21 and Eq.22, as follows,

*(Eq.24)*

This is what is called "**propagator**" (= Green's function ) of scalar field.

Here (-1,+1,+1,+1) version of metric tensor is used.

If you want to change into (+1,-1,-1,-1) version, see this.

Other sites also use this step function or its similar Cauchy complex residue integral to obtain the same Klein-Gordon propagator ( this p.9-14, this p.2-5, this p.5-6, this p.4-13 ).

And this Klein-Gordon scalar propagator is a *basis* for all other virtual propagators for fermions (= electrons ) and photons, so this is very important.

*(Eq.24') Photon's propagator includes unreal zero-component scalar polarized photon (= g _{00} = - 1 ) with negative probability.*

Propagators of (virtual) photon field is very similar to Klein-Gordon scalar field's propagator ( this p.7-8, this p.46 ).

There are two different points between Klein-Gordon and photon's propagators.

First, a photon's propagator's mass should "appear to" be zero (= m = 0 ), though the virtual photon's mass violating Einstein relativity is Not zero. ← contradiction.

Second, a relativistic photon must contain unreal four types of polarizations = two transverse polarizations + one **unreal** longitudinal poalrization + one **unreal** zero-polarized scalar photon ( this p.6-7, this p.9-10 ).

To satisfy relativistic Lorentz invariant (= transform obeying Lorentz transformation, which condition is called Lorentz gauge or Feynman gauge, this p.7, this p.13-upper, this p.7-8 ), only this unreal zero-polarized scalar photon must have unreal negative probability (= g_{00} = -1 ) in photons' commutation relation ( this p.2-middle, this p.6 ) like other relativistic four vectors (= signs are different between zero-component vector and other three component vectors ).

So the original Klein-Gordon's propagator's (virtual) particles' exchange relation should be changed into the relativistic unreal four-polarized photons' versions ( 1 → g_{μν} or η_{μν} which represents metric tensor, this p.6, this p.3 ).

*(Eq.25) ↓ Unphysical Dirac equation's electron and antiparticle*

In this section, we derive virtual propagators of unphysical Dirac fermions or electrons.

The present particles physics, QED and standard model use the relativistic Dirac equation which describes all fermions such as electrons, quarks and neutrinos as nonphysical math symbols with No realistic figures.

And this unphysical matrix-Dirac equation's conjugate transpose (× γ^{0}) is ( this p.24 (or p.21 ) )

*(Eq.26)*

And this Dirac unphysical particle ( c, c^{†} ) and antiparticle ( d, d^{†} ) are supposed to satisfy the artificial **anticommutation** relations (= unphysical mechanism of Pauli principle, this p.2-3, this p.4 ) of

*(Eq.27)*

where δ(k-k') is Dirac delta function.

The notation of this unphysical anticommutation relation ( which nonphysical abstract relation represents Pauli principle ? ← No way ! ) is like in this p.3-(16), this-(8.2.11).

Other combinations become zero at the right hand sides. ← This anticommutation relation allegedly representing Pauli exclusion principle is just one of artificially-created rules which have **nothing** to do with real physical principles or particles.

The coefficient of the delta function of the right-side of the upper anticommutation equation can be freely changed ( this p.2-upper ) together with other relations.

By imitating Klein-Gordon propagator, **Dirac fermion propagator** is expressed as

*(Eq.28)*

Dirac field satisfies the artificial *anticommutation* relation, so the "**minus**" sign is added to the second term of Eq.28.

In the first term of Dirac propagator of Eq,28 connecting Dirac fermion field (= ψ ) and its adjoint multiplied by γ^{0} matrix (= bar-ψ, this p.61 ), by considering annihilation by the vacuum at both ends (= artificial rule ), only "cc^{†} (= electron creation-creation oprators, this p.2-last )" is left.

*(Eq.29)*

This is just one of many artificial rules in QED.

From Eq.29 and Eq.27, we get

*(Eq.30)*

where Dirac delta function is used.

The important point is that **unphysical** matrix parts (= u, this p.8 ) of Eq.30 remains a strange matrix = **Not** a number, as follows,

*(Eq.31)*

To put it simply, the two matrices u and bar-u **only stand in line** in Eq.31.

But the strange matrix of Eq.31 is Not a right math but an artificially-created weird matrix (= 4×1 column matrix u is multiplied by 1×4 row matrix bar-u from the right ! this p.8 ) only for obtaining the illegitimate QED anomalous magnetic moment using these artificially-created virtual propagators.

According to the artificial rule, the unphysical matrx-matrix state of Eq.31 can be expressed as ( this p.12, this p.26 ),

*(Eq.32) Relativistic Dirac equation formula (= on-mass-shell )*

We use (-1,+1,+1,+1) version of metric tensor, so in (+1,-1,-1,-1) version of metric tensor, the sign of γ^{μ}p_{μ} should be changed.

Other sites may use different coefficient in Eq.32 ( this p.1-(4), this p.5 ). But by artificially adjusting coefficients of other relations ( this p.9-lower ), we can obtain the same (artificial) Dirac relation.

Compare this site's (3.17) in p.23 (3.18),(3.22-25) in p.24 and this, this.

If two nonphysical matrices u of Eq.32 only stand in line (not united into number),
when we multiply Eq.32 by another u(k) from right side,

*(Eq.33)*

where we use the relation of

*(Eq.34)*

Eq.33 is equal to

*(Eq.35) Relativistic Dirac equation formula = on mass shell*

Eq.35 is equal to Eq.5-12 (= Dirac electron's relation ) of this page.

As a result, Eq.32 is proved.

In the same way, we can define the antiparticle's relation of

*(Eq.36) Dirac equation antiparticle formula*

Adding another v(k) from the right side of Eq.36 and using this relation, we get

*(Eq.37)*

Eq.37 is equal to Eq.5-13 (= Dirac antiparticle's relation ) of this page.

Using Eq.32, Eq.30 becomes

*(Eq.38)*

In calculating the second term of Dirac propagator (Eq.28), " c^{†} c " term (= electron's part ) is annihilated.

*(Eq.39)*

And only the unphysical antiparticle = " d d^{†} " term is left.

*(Eq.40)*

As you notice, the two-matrix part v(k) of Eq.40 is **a row matrix (= bar-v ) × a column matrix (=v)** = the total result is just a non-matrix or a number.

So Eq.40 is completely different from Eq.30 (= u × bar-u = the total result is matrix ).

In order to artificially make virtual Dirac's fermion propagator used in fraudulent QED, they had No choice but to rely on **wrong** math trick of forcibly changing both of them (= product of u and bar-u = matrix, and product of bar-v and v = non-matrix or number ) into the **same** kind **nonphysical matrix** form.

Instead of the original non-matrix number (= bar-v × v ), they **wrongly** use Eq.36 (= matrix, v × bar-v ), and Eq.40 is unreasonably changed into

*(Eq.41)*

↑ This artificial unreasonable switch is clearly one of **wrong** math tricks for manipulating QED calculation results.

In the original 2nd term of Dirac fermion propagator, "bar-ψ" exists on the **left**-side of the normal-ψ = the order is bar-ψ-ψ ( this p.168 (or p.169 ), this p.62-lower, this p.4-(4.25), this p.45 (or p.46) ).

So in this 2nd term, the inside bar-v-v is the original right order. ← But they artificially **flip** this order from bar-v-v to v-bar-v ( this p.29-(3.63), this p.2-middle, this p.8-(5.27) ) ! This is clearly **wrong** manipulation, and QED propagators are **illegitimate** and false.

Note that coefficient p_{μ} of Eq.41 originally means Dirac's *differential operators*.

So if the exponential function changes (at the final form ), they also change as

*(Eq.42)*

After the wrong illegitimate manipulation, basically, this artificial Dirac propagator resembles Klein-Gordon propagator of Eq.24 except for their coefficients, as follows,

*(Eq.43) Klein-Gordon propagator.*

*(Eq.44) Artificial wrong switch from relativistic energy into virtual energy.*

QED had to artificially make the exponential function parts (+ coefficients ) of the 1st (= particle or electron part u ) and 2nd (= antiparticle part v ) the same common form by flipping the sign of the energy and momentum of the antiparticle part.

The sign of the momentum term (= k_{j}, j = 1,2,3 ) can be easily changed (= k_{j} → -k_{j} ), but the sign of the relativistic energy part (= k^{0} or p^{0} or E = a function of momentums ) remains the same, because the relativistic energy is the square root of the sum of all the squared momentum components and rest mass energy.

And unphysical Dirac equation's (pseudo-)relativistic energy, momentum and rest mass energy relations of particles (= electrons ) and antiparticles are originally **different**, so QED had to change these two different-types of Dirac pseudo-relativistic energies into the **irrelevant common unreal virtual** energy obtained from somewhere else (= step function's ω ).

They tacitly and **wrongly** changed this original relativistic energy (= a **function** of relativistic momentums or on-mass-shell ) into the irrelevant unreal **virtual** energy (= originating from the artificial step function ) independent of (= disobeying ) relativistic momentums and Einstein relativity (= virtual or off-mass-shell ).

*(Eq.45) ↓ Illegitimate switch from original relativistic energy obeying relativity (= a function of relativistic momentums ) into unreal irrelevant virtual energy (= p ^{0} ) disobeying relativity*

Eq.45 is Dirac fermion propagator ( this p.15-(7-94) ) transmitting unreal virtual electrons and antiparticles **disobeying** Einstein relativity.

The original relativistic energy (= p^{0} ) was a function of other relativistic momentums and rest mass obeying the relativistic Dirac equation.

But in the final process of obtaining artificial Dirac propagator in Eq.45, they illegitimately changed this original relativistic energy obeying relativity into the **irrelevant virtual energy** disobeying relativity originating from the irrelevant step function.

*(Eq.45') Dirac propagator should contain antiparticle, but it disobeys antiparticle's relation ! ← self-cotradiction !*

Eq.45' is two ways of representing Dirac fermion propagators ( this p.28-(7.20), this-(6.69), this p.45-(3.62) ).

In Dirac fermion propagator using the one-degree (virtual) energy-momentum relation in its denominator, the Cauchy complex integral with respect to the virtual energy (= p^{0} or k^{0}, this p.15 ) gives one of relativistic Dirac equations which is satisfied only by particles (= electron ), Not by antiparticles.

↑ This clearly contradicts the fact that the antiparticle's relation was also used to obtain this artificial Dirac fermion propagator.

This paradox or discrepancy is caused by wrongly and artificially changing the original relativistic energy into the unreal virtual (common) energy originating from the irrelevant step function.

As a result, Dirac fermion propagator used also in calculating QED (infinite) anomalous magnetic moment is obtained by wrong math, so relativistic QED itself is a wrong theory.

As shown in this-lower-equation, the Cauchy complex integral was basically done (= pole where the denominator is zero gives the relativistic energy satisfying relativistic momentum and mass relation ) only in **one** virtual propagator (= integral with respect to only **one** variable k or its energy component k^{0} ) in all three virtual propagators (= Cauchy integral with respect to other two virtual energy variables k' or (kp-k) is not done ) in each QED vertex-virtual-loop calculation.

So at least two virtual propagators give unreal virtual photons and electrons whose energies disobey Einstein relativity, because electrons can interact only with unreal virtual photons if the total energy and momentum are conserved ( this p.5, this p.4-lower ).

This virtual fermion or electron disobeying relativity in the Dirac propagator clearly contradicts the process of obtaining this ad-hoc propagator using the relativistic Dirac equation (= obeying relativity ).

This unphysical Dirac fermion proparagor relying on the obvious **contradiction** caused by unreasonably switching the original on-mass-shell relativistic energy satisfying the relativistic Dirac equation into the **unreal** virtual energy disobeying the relativistic relation shows that QED desperately needed "**fudge** factor" of the excessive 1/2π from the coefficient of the artificial step function in order to artificially **manipulate** the dubious anomalous magnetic moment.

As shown in this, Lagrangian of the electromagnetic field in the vacuum can be expressed as

*(Eq.46)*

In Eq.46, if μ and ν are the same, F_{μν} becomes zero.

So the zero component of the **canonical momentum** is zero, as follows,

*(Eq.47)*

But as is explained later, if the unreal zero component of photon can not be defined, the (virtual) photon propagator calculation becomes **impossible**.

So we have to produce the unreal zero component canonical momentum "compulsorily".

They artificially define the unphysical Lagrangian which includes the zero component ( this p.8, this p.13 ), as follows,

*(Eq.48)*

This is called " **Lagrangian of Feynman gauge** ".

( Here we use usual SI unit, so magnetic permeability μ_{0} is added. )

The second term of Eq.48 vanishes if we consider Lorentz gauge condition of

*(Eq.49)*

So the **fictional** cannonical momentum of the zero component is, ( x^{0} = ct )

*(Eq.50)*

where Lagrangian involved is

*(Eq.51)*

And the **fictional** canonical momentums of other components ( j = 1, 2, 3 ) are

*(Eq.52)*

where Lagrangian involved is

*(Eq.53)*

μ and ν of Eq.48 contain all 0 - 3 components.

So the coefficient 1/4 (and 1/2) is cancelled out.

Like de Broglie's relation, we can define the **commutation** relations of

*(Eq.54)*

If so,

*(Eq.55)*

where **g ^{μν}** appears.

Other combinations are all zero.

Using Eq.55, we can prove Eq.54, as follows,

*(Eq.56)*

*(Eq.57)*

Here (-1,1,1,1) version metric tensor g is used.

See this page.

As shown on this, the solution of fictitious photon's A_{μ} (x), which satisfies Maxwell equation is

*(Eq.58)*

The time derivative of Eq.58 is

*(Eq.59)*

And to satisfy Eq.55, the following relations must be valid.

*(Eq.60)*

Other combinations are all zero.

As shown in Eq.60, only the zero component of photon particles has "negative" energy ( = "**negative**" number ). ( g_{00} = -1 )

So this zero component photon is **NOT** a real thing.

Why we define the minus sign in Eq.48 (and Eq.60) ?

In fact, if we don't define the minus sign of Eq.60, we can NOT construct virtual photon's propagator to calculate the ad-hoc g-2 factor.

In other words, to calculate g-factor, the unrealistic relation of Eq.60 must be introduced.

Next we prove Eq.55 using the relations of Eq.60.

Substituting Eq.58 and Eq.59 into Eq.55,

*(Eq.61)*

*(Eq.62)*

where time t is the same.

From Eq.61 and Eq.62 we have

*(Eq.63)*

Eq.55 is obtained.

*(Eq.64)*

Using the unreal zero component scalar photon (= g_{μν} ), they obtained unphysical virtual photon propagator used in QED.

Here we explain Hamiltonian of Coulomb gauge violating the relativistic Lorentz invariance, briefly.

When the scalar potential (= A^{0} ) is zero,

*(Eq.65)*

Hamiltonian is

*(Eq.66)*

Of course, in this website and ad-hoc relativistic QED, we use Lorentz gauge, so we have to consider the unreal "minus" energy of the zero component photons in addition to this.

2022/8/23 updated. Feel free to link to this site.