Top page ( Quantum mechanics and Einstein relativity are wrong. )
QED anomalous magnetic moment calculation uses wrong math trick.
(Fig.1) Quantum electrodynamics (= QED ) artificial manipulation of anomalous magnetic moment. ← QED is "hocus-pocus" !
It is often said that quantum electrodynamics (= QED ) could successfully obtain the tiny anomalous magnetic moment or g-2 factor of (unphysical) electron spin.
But in fact, QED is just a fraudulent theory artificially manipulating values and creating ad-hoc inconsistent rules to forcibly get the dubious anomalous magnetic moment even by using wrong math called renormalization (= hocus-pocus, dippy process ) removing infinity caused by unreal virtual particles for getting the tiny finite magnetic moment value.
One of major artificial inconsistent QED rules is that they deliberately apply different inconsistent rules to the original electron spin's magnetic moment (= 1 = spin-1/2 × g = 2 factor ) by relativistic Dirac equation and the dubious tiny anomalous magnetic moment (= 0.00116 or (g-2)/2 factor ) as the result of removing infinity, as shown in the above Fig.1.
Relativistic QED uses the unphysical abstract interaction term between unphysical Dirac electrons (= ψ = incoming electron and bar-ψ = outgoing electron ? ← just artificial nonphysical rules, this p.27, this p.8 ) and unreal virtual photon (= A, this p.8 ), when calculating (anomalous) magnetic moment ( this p.36 ).
In the original Dirac equation, when calculating the major part of the magnetic moment (= 1 = spin-1/2 × 2 ), they change the mathematical γμ matrix allegedly representing the interaction vertex ( this p.11 ) into the two kinds of terms using Gordon identity ( this p.18, this p.3, this p.12 ).
↑ One of these two kinds of terms is expressed as the sum of momentums (or energies ) of incoming (= p ) and outgoing (= p' ) electrons like (= p'μ + pμ ), which terms are Not attached to γμ matrix.
The other kind of term is the so-called (unphysical) spin magnetic moment term expressed as nonphysical 4×4 σμν matrix composed of two γ matrices ( σμν = γμγν, this p.3-lower, this p.7-upper, this p.4 ) multiplied by ( p' - p ) = q (= virtual photon's momentum and energy ).
↑ The coefficient of this unphysical spin term which allegedly represents Bohr magneton (= unit of magnetic moment ) is " 1 (= g/2 ) ", so they insist that the original Dirac equation's intearction term with (virtual) photon containing just one γμ matrix could give the major part of spin magnetic moment (= 1, or g-factor = 2 ).
The serious problem is that QED artificially changed this original rule for obtaining the magnetic moment when calculating the "hocus-pocus" anomalous magnetic moment !
In the calculation of the dubious anomalous magnetic moment, they first obtain two kinds of terms.
One kind of these terms is terms attached to γμ (= called form factor F1 ), and the other term is (= p'μ + pμ ) freed from γμ matrix which are seen also in the original Dirac equation's magnetic moment (= 1 ) calculation.
If QED applied the consistent common rule to both the major magnetic moment (= 1 or g=2, this p.40 (or p.46 ) ) and tiny anomalous magnetic moment (= α/2π or 0.00116, where α is fine structure constant = about 1/137 ), they had to change all the terms attached to γμ matrix into two kinds of terms of (= p'μ + pμ ) term and the spin magnetic moment term = σμν(p'-p) using Gordon identity ( this p.2 ).
↑ But only in the dubious anomalous magnetic moment case, the terms attached to γμ matrix is known to miserably diverge to infinity ( this p.18-upper ), so if QED uses the consistent same rule as the original magnetic moment (= 1 or g=2 ) also in the anomalous magnetic moment calculation, this tiny anomalous magnetic moment value diverges to the meaningless infinity !
So QED illegitimately changed the rule, and then, switched the (= p'μ + pμ ) term (= remain in the original Dirac equation g=2 magnetic moment ) instead of γμ terms into the two kinds of terms of γμ terms and the ad-hoc spin magnetic moment (= σμν ) terms which are also attached to γμ matrix ( this p.2 ).
And then, QED artificially removed only the first terms attached to γμ matrix including infinity using the hocus-pocus renormalization trick ( this p.6-7, this p.3 ), and deliberately left only the ad-hoc spin magnetic moment (= σμν ) terms, which are also attached to γμ matrix, though.
QED deliberately ignores the fact that this nonphysical σμν matrix allegedly indicating "spin" also contains the γμ matrix ( this p.3 ) ! ← But only this nonphysical σμν terms attached to γμ matrix is Not removed or renormalized, while other infinite terms attached to γμ matrix are artificially removed !
This is clearly an inconsistent artificial rule, and shows QED adopting different inconsistent rules in different situations is illegitimate and wrong.
The point is this dubious QED anomalous magnetic momnet value can be freely changed if an arbitrary part of this spin magnetic moment term is changed into (= p'μ + pμ ) terms like the original Dirac magetic moment (= 1 ) calculation.
↑ Three kinds of terms = terms attached to γμ matrix, p'μ + pμ terms, and magnetic moment σμν terms can be freely changed into each other, which means the QED anomalous magnetic moment value can be freely changed, and QED prediction is meaningless.
QED cannot show the legitimate reason why the terms of (= p'μ + pμ ) necessary for obtaining the original spin magnetic moment (= 1 or g = 2 ) instead of unnecessary γμ terms have to be artificially switched into the unncesessary γμ terms and the spin magnetic moment terms only in the ad-hoc anomalous magnetic moment (one-loop) case.
On this page, we use (-1,+1,+1,+1) version of Minkowski metric tensor "g"
(Eq.1)
And unphysical γ matrices are
(Eq.2)
" I " means 2 × 2 identity matrix.
(Eq.3)
where σj mean 2 × 2 Pauli matrices of
(Eq.4)
As shown on this page (Eq.5-12), the relativistic (nonphysical) Dirac equation satisfies the following relation (= if you want to use (+1,-1,-1,-1) version of metric tensor, change the sign of γμpμ → -γμpμ, this p.16, this p.9 ),
(Eq.5)
The artificial notation of bar-u is the complex conjugate transpose u† multiplied by unphysical γ0 matrix.
This artificial notation is necessary for unphysical relativistic Dirac equation to fix the fact that the unphysical 4×4 γ matrices are Not Hermitian (= anti-Hermitian ) unlike 2×2 Pauli matrices.
Using the upper γ matrices, we have
(Eq.6)
The unphysical 4×4 σμν matrix allegedly indicating Dirac electron version of "spin" of Eq.6 is defined by, (This σ is 4 × 4 matrix, unlike Pauli matrices. )
(Eq.7)
These nonphysical 4×4 σμν matrices consisting of two γ matrices ( this p.7-upper, this p.3-lower ) are supposed to represent artificial "spin" in the relativistic Dirac equation ( this p.5, this p.16, this p.4, this p.2 ).
But this weird spin σμν matrix includes the "zero" component (= μ = 0 or ν = 0 ), which is irrelevant to the spin magnetic moment ( μ, ν = 1,2,3 components )
So QED treating these nonphysical σμν matrices including irrelevant states as fictitious spin is illegitimate and false.
Here we use
(Eq.8) incoming and outgoing electrons
p (= incoming electron ) and p' ( = outgoing electron ) are the energy (= zero component, ν=0 or μ=0 ) and momentums (= ν, μ = 1,2,3 ) of electrons.
u(p) and bar-u(p') are unphysical Dirac electron's spinors ( this p.16 ).
Multiplying Eq.6 by the first operator of Eq.8 from the right.
(Eq.9)
where the artificial Dirac energy equation of Eq.5 is used.
Multiplying Eq.6 by the second operator of Eq.8 from the left.
(Eq.10)
The last term uses antisymmetric property of σ ( see Eq.7 ).
And dividing the both sides of Eq.9 by -mc
(Eq.11)
Exchanging the variables μ and ν of Eq.10, and dividing it by -mc,
(Eq.12)
Multiplying Eq.11 by the first operator of Eq.13 from the left, and multiplying Eq.12 by the second operator of Eq.13 from the right.
(Eq. 13)
And then summing these two results, and dividing the sum of them by 2, we can get
(Eq.14)
(Eq.15)
This "q" is the energy and momentums of the ( virtual ) photon = external magnetic field.
According to this definition, when the electron's momentum (= velocity ) is constant ( p'-p = 0 ), external magnetic field (= qA or ∇×A ) is zero, even if it exists.
This is contradictory, and QED using this notation as (fictitious) electron's spin magnetic moment interacting with the external magnetic field (= virtual photon q ) is false.
Using the relation p' - p = q (= energy difference between outgoing p' and incoming electrons p equals virtual photons q or external magnetic field ? ) and defining c = ħ = 1, Eq.14 becomes
(Eq.16)
QED argues that nonphysical 4×4 σμν matrices mean "spin (with No grounds ) or its angular momentum ℏ", hence σμν/2m indicates Bohr magneton, which is a unit of (spin) magnetic moment.
The coefficient "1 (= spin 1/2 × g=2 factor )" of this nonphysical σμν/2m is supposed to indicate electron's magnetic moment ( this p.18, this p.3 ), whose derivation contradicts the artificial rule obtaining QED tiny anomalous magnetic moment, which means QED is illegitimate.
(Eq.17) QED unrealistic magnetic field B = ∇×A (= A is magnetic vector potential )
QED uses the paradoxical external magnetic field = B = ∇×A (= A is unphysical magnetic vector potential, this p.3, ∇ is the differential operator ).
As shown in above, QED's unrealistic magnetic field consists of the virtual photon's energy q (= correspond to the differential operator ∇ ) + external magnetic potential A ( this p.13-upper, this middle, this p.16-(79) ), which causes a serious paradox, hence, QED is false.
For example, when the electron with spin magnetic moment is moving around in a circle in the static uniform external magnetic field (= B ), this static (= constant ) uniform magnetic field qA (= including virtual photon q absorbed into the incoming electron ) must be the constant value which is uniform irrespective of the electron's position.
But an electron's momentum is constantly changing (= electron must alwys absorb the changing virtual photon's momentum q ) by its circular motion even under the static uniform magnetic field, which is clearly contradiction, and QED magnetic momnet calculation is based on the false assumption.
The unphysical QED spin matrix σ part is said to indicate the following Bohr magneton, though σμν includes the irrelevant zero compoent.
(Eq.18)
"z" component of QED unphysical 4×4 spin is
(Eq.19)
Even in nonrelativistic Schrodinger's form, the quantum mechanics uses the mathematical illegitimate tricks like
(Eq.20)
where p (= electron's momentum ) includes derivative (= electron's de Broglie wavelength related to momentum ), which acts on before and after the magnetic potential Aj.
So de Broglie wavelength (= derivative ∇ ) of electron is unreasonably changed into derivative of the external magnetic vector potential A to get the external magnetic field (= B = ∇×A, this p.11-12 ), though the electron's momentum or de Broglie wave is a completely different thing from the independent external magnetic field B = ∇×A.
This quantum mechanical magnetic field representation is clearly wrong.
In this section, we explain about the time evolution and interaction used in quantum electrodynamics.
As you know, the time dependent Schrodinger equation is expressed as
(Eq.21)
So the eigenstates ( Φ(t) ) of Schrodinger pictures contain the time variable (= t).
The initial state ( t=0 ) is defined as
(Eq.22)
So when the Hamiltonian (H) is time independent, the solution of Eq.21 is
(Eq.23)
The quantum field theory (QFT) often uses Heisenberg pictures instead of Schrodinger pictures.
Because the main places of renormalization are operators rather than eigenstates.
(The eigenstates themselves are not very important in QFT and QED.)
So the operators ( A(t) ) are time dependent in Heisenberg pictures, as follows,
(Eq.24)
( Instead, the eigenstates (Φ) are time independent in Heisenberg pictures. )
From Eq.24, we can know the A(t) of Eq.24 satisfies the following Heisenberg equation of motion.
(Eq.25)
where each operator ( OH(t) ) in Hamiltonian is expressed like Eq.24.
Hamiltonian can be divided as
(Eq.26)
which mean free energy term and interaction term, respectively.
Here we define the operators of interaction pictures as follows,
(Eq.27)
The important part in QED is the interaction term, so we suppose the eigenstates evolve by the interaction Hamiltonian, as follows,
(Eq.28)
where U(t) is
(Eq.29)
U(t) is called the interaction picture propagator or time-evolution operator.
The conjugate operator of Eq.29 is
(Eq.30)
So U† U = 1. This means that U(t) is the unitary operator.
The interaction picture operator AI(t) of Eq.27 is expressed using Heisenberg picture operator AH(t) of Eq.24 and U(t) of Eq.29 and 30, as follows,
(Eq.31)
We have to know more concrete expressions of U(t) than Eq.29 for perturbation calculation of electron g-factor.
From Eq.29, the simple differential equation of U(t) is
(Eq.32)
where the initial state is U(0) = 1.
The meaning of Eq.32 is equal to
(Eq.33)
So,
(Eq.34)
Multiplying Eq.34 by U† of Eq.30 from right side,
(Eq.35)
Here we can define as follows,
(Eq.36)
( Because U† means "retrogression" to time t from zero. )
Using Eq.36, Eq.35 can be expressed as
(Eq.37)
Eq.37 is the infinitesimal time. When we connect U of Eq.37, as follows,
(Eq.38)
Generally, the interaction Hamiltonian ( HI(t) ) doesn't commute when the times t are different.
Note that various U(t, t') of Eq.38 stand in time order, later on the left.
So we can express Eq.38 simply using the following time-ordering symbol T,
(Eq.39)
Eq.39 means that when we remove this symbol T, each U(t, t') stand in time order like Eq.38 automatically.
( So we can exchange U(t, t') freely in symbol T, and connect them as one exponential function like Eq.39. )
A propagator is the simplist form of Green functions, because it contains only two field operators ( 2 × φ ).
General form of Green function is
(Eq.40)
where H means Heisenberg picture, which is explained in the upper section.
From Eq.31, the operators of Heisenberg and interaction pictures are
(Eq.41)
where the unitary operator U(t) satisfies
(Eq.42)
Substituting Eq.41 and Eq.42 into Eq.40,
(Eq.43)
In the time-ordering symple T, we can exchange the order of the operators freely, as I said in the upper section.
We define the vacuum energy is zero at time t = 0, as follows,
(Eq.44)
(Caution: The vacuum of Heisenberg picture is different from the vacuum, because it contains infinite virtual particles including infinite momentum. )
From Eq.44,
(Eq.45)
The complete set of eigenfunctions in Heisenberg picture is
(Eq.46)
Inserting Eq.46 into Eq.45,
(Eq.47)
where En > 0.
If we take the imaginary time of
(Eq.48)
when time → - ∞
(Eq.49)
So only zero energy of Heisenberg picture is left in Eq.47.
(This part is very difficult to imagine. But this "mathematical" world is QFT, as I said many times.)
From Eq.47 and Eq.49,
(Eq.50)
In the same way (like from Eq.45 to Eq.50 ), we can get
(Eq.51)
From Eq.50 and Eq.51,
(Eq.52)
so,
(Eq.53)
Using Eq.50, Eq.51, and Eq.53, Green function of Eq.43 becomes
(Eq.54)
As I said, if we insert U(t) operators into the time-ordering symple T, we can exchange the order of them freely, as follows,
(Eq.55)
Using Eq.36 - Eq.39, Eq.55 becomes
(Eq.56)
where Hden means interaction Hamiltonian density. So Eq. 56 integrate it by space coordinates.
Z0 of Eq.53 means infinite disconnected diagrams ( vacuum bubbles ) contained in U(t).
So in Eq.56, disconnected vacuum bubbles of the numerator are eliminated by this denominator Z0.
As a result, it becomes sum of all diagrams connected to field operators (φ).
Eq.56 is the basic equations wich express Feynman diagrams. So this is very important.
2022/8/25 updated. Feel free to link to this site.
