Quantum field theory and QED are just an unphysical useless math irrelevant to our real world.

From Eq.4-12 and definition of Eq.1-21, the fictional canonical momentums (= Not real momentum ) is
(Eq.4-14) Relativistic Klein-Gordon (fake) generalized canonical momentum = π

↑ This generalized (canonical) momentum of relativistic Klein-Gordon equation is the artificially-created imaginary concept, hence, Not a real momentum, due to its wrong math.

The real momentum (= p ) must be expressed as a particle's mass multiplied by its velocity (= p = mv,  this-(2.3.7),  this p.4-(6.20),  this p.7 ).

This Klein-Gordon's ( fictional canonical ) momentum (= π ) is expressed just as the time derivative of the field (= dφ/dt or overdot-φ ), which does Not contain the mass (= m ) of the real momentum of mv, hence, completely a different and fake momentum !

[ Getting Klein-Gordon fictitious Hamitonian energy (= H ) ]

By substituting the artificially-created Klein-Gordon Lagrangian (= L ) into Eq.1-22, the fictitious Hamiltonian (density) or total energy of Klein-Gordon field ( this p.4 ) is
(Eq.4-15)

Substituting the Klein-Gordon wavefunction φ of Eq.4-4 into Eq.4-15, and integrating it in all space, we can get the following final form of the fictitious relativistic Klein-Gordon's Hamiltonian energy ( this p.2~4,  this p.13 ), using this calculation

(Eq.4-16) Fictitious Klein-Gordon's Hamiltonian (= final form of H ) or total energy

After all, the final form of Hamiltonian (= total ) energy (= H ) of the relativistic Klein-Gordon equation consists only of unphysical particle (= expressed just as abstract math symbols such as a or a^† ) with infinite kinds of energies and momentums (= integral over all momentums k from -∞ to +∞ ), which is just a nonphysical math theory clarifying No detailed physical mechanism ( this p.6-12,  this p.4-5 ).

The nonphysical symbol " a^† " means the particle's creation operator (= increasing one particle ), and " a " means the particle's annihilation operator (= decreasing one particle ), which abstract operation is the only thing the useless quantum field theory can express as particles' motion.

In the process of obtaining this fictitious relativistic Klein-Gordon Hamiltonian (= total ) energy, the positive (unrealistic) infinite energy term (= Dirac delta function δ(0) = infinity ) appears due to the illogical quantum mechanical relation or rule ( this p.4-5, ).

Some physicists claim this (unphysical) vacuum positive infinite energy (= expressed as infinite delta function = +δ(0)) appearing in relativistic Klein-Gordon scalar-spin-0-particle total energy can be canceled out by another unrealistic negative infinite vacuum energy (= -δ(0) = -∞, which I explain later ) appearing in Dirac spin-1/2-particle's Hamiltonian energy using the hypothetical nonphysical theory called "supersymmetry ( this p.23,  this p.7,  this p.13 )" or ad-hoc renormalization artificially removing infinity ( this p.23,  this p.26 ).

↑ But both relativistic Klein-Gordon's spinless particle (= bosons ) and Dirac's spin-1/2 particle (= fermions ) are free particles ( this p.2 ) with No potential energies, which cannot be negative energies (= free particles with positive kinetic energy and No negative potential energy must always have zero or positive total energies ), so the ficititious negative infinite energy canceling positive infinite energy between free spin-0 and spin-1/2 particles stated by the imaginary supersymmetry is unrealistic and wrong.

Here we explain about the detailed calculation of the upper Klein-Gordon's Hamiltonian.

We substitute the fictitious Klein-Gordon's wavefunction of Eq.4-4 into φ² of Eq.4-15 (= Klein-Gordon Hamiltonian energy equation ).
Each momentum k or k' ( from -∞ to +∞ ) is independently integrated from each other in the two φ².
So we use
(Eq.4-17)

Using Eq.4-17,  φ² of Eq.4-15 becomes
(Eq.4-18)

The terms of a^†a^† (= particle creation-creation ) and aa (= particle annihilation-annihilation ) in the upper equation are zero due to the zero coefficient of the final Hamiltonian energy, so they are eliminated.

Here we use
(Eq.4-19) Artificial rule of Klein-Gordon's particles.

The rule of the upper Eq.4-19 is the artificially-created commutation rule imposed on fictitious Klein-Gordon or spin-0 scalar bosons ( this-middle,  this p.3,  this-(7) ).

The artificial rule of Eq.4-19 means the interchange of the (nonphysical) particle creation operator (= a^† ) and annihilation operator (= a ) gives the meaningless infinity as delta function δ(k-k') where this delta δ function becomes infinity when these two momentums are equal (= k-k' = 0 ), or this delta function becomes zero when k is not equal to k'.

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

↑ Dirac's delta function (= δ(k-k') which could be infinity, when k = k' ) has to be used in the relativistic quantum field theory which must contain all kinds of momentums (= k ) continuously from minus infinity to plus infinity, which is why QED results always diverge to meaningless infinity ( this p.22 ).

In the last line in Eq.4-18, we exchange k and k' of one term, and unite it with another.

Integrating Eq.4-18 ( first term ) with respect to space variables, we obtain
(Eq.4-20)

where delta function's formula is used.

From the definition of Eq.4-5,
(Eq.4-21)

Using Eq.4-15, Eq.4-20 and Eq.4-21, the final form of (fictitious) Hamiltonian becomes
(Eq.4-22) Fictitous Klein Gordon Hamiltonian (= H ) final form

The final form of this upper Hamiltonian (= H should be total energy E or ℏω ) is Not a true Hamiltonian, because this fake Klein-Gordon's Hamiltonian originally gives twice the total energy (= 2E² = 2(ℏω)² instead of the true Hmiltonian energy = E, as shown in the upper Eq.4-22 last line ) and this the wrong total energy (= 2(ℏω)² ) value is artificially canceled out by the coefficient parameter (= 1/2ω = this parameter "2" can be artificially changed into other values ) inserted in the artificially-created Klein-Gordon's wavefunction (= φ,  this p.7-9,  this p.15 ) in advance.

In the second line from the last in Eq.4-22, we use the relation ( this-(A-45) ) of
(Eq.4-6)

As a result, adding delta function of Eq.4-18, we get the final version of fictitious relativistic Klein-Gordon Hamiltonian of
(Eq.4-23)

Fictitious relativistic Klein-Gordon Hamiltonian energy includes infinite vacuum energy (= delta function δ(0) = +∞ ).

When the relations of Eq.4-19 is satisfied, Eq.4-4 and their fictional canonical momentums (= π ) in Eq.4-14 satisfy the commutation relations of
(Eq.4-24)

Eq.4-24 can be confirmed by a little long calculations ( this p.13-17 ).

As you know, the famous commutation relation of Eq.4-24 is based on de Broglie's relation ( quantum mechanical momentum p is expressed as the space derivative ) of

(Eq.4-25) original quantum mechanical commutator relation deos Not contain infinite delta function.

Relativistic Klein-Gordon's fictional canonical momentum (= π ) is Not a real momentum, so blindly applying (and modifying ) the original quantum mechanical commutation rule (= Not including infinite delta function ) between the space coordinate (= q ) and the momentum (= de Broglie wave operator ) to this fictional Klein-Gordon momentum is baseless.

[ Relativistic particles violate causality ?? ]

In this section, we explain violation of relativistic causality briefly.
Basically the propagator contains the two processes of particle and antiparticle.
If we consider the propagator of only particle ,
(Eq.4-26)

Suppose t = t'.
If Eq.4-26 doesn't include ω_k as a coefficeint, Eq.4-26 becomes delta function, as follows,
(Eq.4-27)

Eq.4-27 means that when x is different from x', Eq.4-27 is zero.
So Eq.4-27 satisfies relativistic causality.

But Eq.4-26 includes ω_k as a coefficeint.
So the calculation result of Eq.4-26 is
(Eq.4-28)

Eq.4-28 is NOT zero in the space-like region of
(Eq.4-29)

So according to the relativistic quantum field theory (QFT), the particles can be faster than light ( this p.3 ), even though they use the special relativisty !
This is clearly self-contradiction.

But if we combine both directions (= particle and antiparticle ) at the same time, this satisfies causality.
So the quantum mechanical physicists are satisfied with these results.
( How do you think about it ? )
Klein-Grodron fields express π - meson and Higgs particle.
As you notice, there are no concrete images in these particles.

Relativistic Dirac equation describing all spin-1/2 particles is just a nonphysical theory irrelevant to real world.

[ Dirac equations allegedly describing all electrons, quarks, antiparticles, neutrinos.. as nonphysical math symbols must contain wrong energies and unphysical matrices, so the relativistic quantum field theory is unable to describe real phenomena. ]

(Fig.D) Relativistic quantum field theory represented by Dirac equation contains unphysical γ matrices and the paradoxical energies contradicting Einstein relativistic energy-momentum relation, so Dirac fermions don't exist.

Relativistic Dirac equation is the most important equation constituting the quantum field theory and QED (= quantum electrodynamics ) allegedly describing all spin-1/2 particles such as electrons, antiparticles, unseen quarks, neutrinos.. ( this 2nd-paragraph,  this p.2 ) as nonphysical math symbols with No concrete figure.  ← So relativistic quantum field theory and QED lack reality.

Relativistic Dirac equation is said to be a square root of Einstein's relativistic energy-momentum relation equal to the failed Klein-Gordon equation (= Dirac equation = square root of relativistic E² = (pc)² + m²c⁴, where E is relativistic energy,   p is relativistic momentum,  mc² is rest mass energy,  E, p are denoted by time and spatial differential operators based on de Brolgie wave theory in quantum field theory,  this p.1-2 ).

But the square root of the original relativistic mass relation consisting of the second-order energy and momentum is mathematically impossible as seen in the failed Klein-Gordon equation.

So relativistic Dirac equation had to rely on some "cheating" which artificially divided the second-order relativistic energy-momentum relation into two different energy equations (= ① and ② in the upper Fig.D ) using nonphysical γ matrix ( this p.22-23,  this p.29-30,  this p.12-13 ).

The fact that relativistic Dirac equation cannot describe any particles without the nonphysical γ matrices (= Dirac equation cannot describe each particle using the ordinary numbers without matrices ) means all the current relativistic quantum field theory, QED and particle physics heavily relying on this nonphysical Dirac equation lack reality.

Relativistic Dirac equation's nonphysical γ matrices consist of Pauli σ spin matrices, hence, physicists claim this relativistic Dirac equation could successfully express electron's "spin ( this p.8 )" without showing any detailed mechanism or figure of this (illlusory) spin.

But actually, these Pauli spin σ matrices included in relativistic Dirac γ matrices have nothing to do with "spin", shwoing quantum spin is illusion.

The problem is these two relativistic (pseudo-)equations using nonphysical γ matrices or the alleged (pseudo-)square root of the original relativistic energy relation give two types of the completely different energy-momentum relations ( this p.3 ), one of which contradicts the original Einstein relativistic energy-momentum relation.

① of the upper Fig.D gives the normal relativistic energy E, which is approximately equal to E = pc + mc² (← energy E, momentum p and the rest mass energy mc² have the same sign ).

But ② of the upper Fig.D gives the wrong energy E, which is equal to E = pc - mc² (= total relativistic energy E and momentum p have the opposite sign from the rest mass energy mc² ), which contradicts the original Einstein relativistic energy relation where the relativistic total energy E must be the sum of momentum p and rest mass energy mc².  ← Dirac's (pseudo-)square root of original Einstein relativistic relation contradicts the relativity !

Relativistic Dirac wavefunction consists of infinite numbers of (virtual) particles (= c^† ) such as electrons, quarks, and antiparticles like positrons ( = d^† ) expressed as the nonphysical abstract math symbols with No concrete particle's description ( this p.7,  this p.6-second-paragraph ).

These Dirac's nonphysical particles such as electrons (= c or u ) have the upper ① energy relation, and antiparticles such as positrons ( d or v,  this p.12-13 ) have the upper contradictory ② energy relation ( this p.16,  this p.19,  this last ), which fact proves relativistic Dirac equation is a failed wrong theory (← only the antiparticles must have the positive energy E and momentum p with paradoxical negative rest mass energy, like the upper ② ).

And like all other relativistic quantum field theory, Dirac equation must consist of infinite numbers of (virtual) particles and antiparticles with infinite amount of energies, which cause QED calculation results to diverge to meaningless infinity ( this p.4-6 ), which must be artificially canceled by another artificial infinity (= free parameters,  this p.4-5 ) by the illegitimate method called "renormalization ( this p.13-14,  this p.15-16 )" which wrong-math renormalization (= ∞ - ∞ = finite value ?  ← infinity is Not a definite value, so this QED calculation is Not the prediction of the definite physical values at all ) was criticized even by founders such as Dirac himself ( this 1st-paragraph ) and Feynman.

So all the relativistic quantum field theory, QED and standard model based on the artificial manipulation of the meaningless infinite parameters ( this p.2-3,  this p.3,  this p.11-13 ) have No power to predict any physical values ( this p.36,  this p.8 ), contrary to the media false narrative.  ← Actually, Nobody uses this useless QED in our daily lives.

The point is the media and academia just repeatedly and baselessly say the tired cliche "successful QED predicting anomalous magnetic moment !", but they Never explain those "dubiously successful" calculation process of how to obtain such tiny finite physical values (= anomalous magnetic moment,  this p.10-13 ) from the meaningless infinity which is the original QED calculation result ( this p.7 ).

Various artificial notations are used in quantum field theory.
On this page, (-1,+1,+1,+1) version of metric tensor is used.

Calculation of nonphysical relativistic Dirac equation

[ From the arrtificially-created fictitious Dirac Lagrangian (= L ), fictional Dirac Hamiltonian (= total ) energy (= H ), Dirac equation are obtained. ]

(Fig.13) Aritificially create fictitious Dirac Lagrangian (= L ) → Euler-Lagrange equation → Dirac equation.  → fictional canonical momentum → fictional Dirac Hamiltonian energy (= H ) consisting of infinite (virtual) fermions and antiparticles.  ← This meaningless concept is what the relativistic quantum field theory is.

Like other nonphysical relativistic quantum field theory and QED, first, physicists artificially created fake Dirac's Lagrangian (= L ) including unphysical Dirac wavefunction and its adjoint (× γ⁰ matrix ) form filled with infinite free relativistic (unphysical) particles (= c or b ) and antiparticles (= d,  this p1~2 ) with No potential energy which contradicts the original authentic Lagrangian consisting of kinetic and potential (= V ) energies.

Next, by imitating classical Lagrangian, inserting this artificially-created Dirac Lagrangian into Euler-Lagrange equation gives the Dirac equation (= which means this fictitious Lagrangian was created and chosen to give Dirac equation ).

Then, again, by imitating and modifying the classical momentum (= p = mv ) and Hamiltonian energy (= H ), they obtain the fictional (= canonical ) momentum (= π ) and fictional relativistic Dirac Hamiltonian (= total ) energy (= H,  this p.3-5 ).

↑ This final form of relativistic Dirac Hamiltonian (= total ) energy consists of infinite numbers of nonphysical math particles ( this p.2-5 ) with unrealistic infinite energies ( this p.6,  this p.23-second-paragraph,  this p.98-102 ) with no physical shapes, which relativistic quantum field theory lacking realistic particle pictures cannot be used for practical science at all.

In the upper figure, the nonphysical math symbols such as c^† and c mean the electron or particle's creation and annihilation operators, and d^† and d mean the antiparticle's creation and annihilation oeprators, respectively ( using other letters such as a and b instead of c and d as particles or antiparticles is OK ).

u(k) and v(k) are the unphysical 4×1 matrices.
u^†(k) (= or bar-u(k) ) and v^†(k) (= or bar-v(k) ) are 1×4 matrices.

All calculations of relativistic quantum field theory, QED and particle physics with these unrealistic infinite (virtual) particles' energies are known to give only unrealistic divergence or meaningless infinities, which must be artificially removed by the ad-hoc renormalization ( this 3rd-paragraph ) using freely-adjustable-infinite-parameters ( this p.62-72 ), which is useless, cannot predict any physical values (= physicists cannot define or predict the value of indefinite "infinity" to cancel the original QED infinity ), contrary to the media-hype.

↑ So all calculations (= giving only meaningless infinities ) in the current relativistic quantum field theory based on Dirac and Klein-Gordon equations are physically meaningless and useless.

The artificial notation of bar-ψ in the above Lagrangian means the complex conjugate transpose or adjoint Dirac wavefunction (= ψ^†,  this p.8 ) times unphysical γ⁰ matrix.

↑ This artificial ad-hoc notation is necessary in (pseudo-)relativistic Dirac equation, because the nonphysical 4×4 γ^j matrices (= j = 1,2,3 components ) are anti-Hermitian or Not Hermitian (= complex conjugate transpose of γ^j matrices γ^{j †} are Not equal to the original γ^j matrices,  this p.2-lower-p.3-upper,  this p.3-(75) ).  ← This meaningless math is a serious problem for quantum field theory, and add the ad-hoc γ⁰ matrix to Dirac equation is necessary for producing the adjoint form.

So the γ matrices allegedly representing 4×4 matrix-spin are completely different from 2×2 Pauli matrices (= the complex conjugate transpose of Pauli matrices are equal to the original Pauli matrices like σ^† = σ = Hermitian,  this-(2.4.12) ), but they try to associate this γ matrix to irrelevant "spin" allegedly expressed by Pauli spin σ matrix.

Anyway, there is No physical reality in particles and antiparticles allegedly expressed by the current relativistic quantum field theory.

[ Relativistic Dirac equation is Not physics but just an abstract meaningless math. ]

(Eq.5-1) Nonphysical relativistic Dirac equation cannot describe real particles !

Nonphysical relativistic Dirac equation is expressed as Eq.5-1 where ψ(x) is the 4-component Dirac particle's wavefunction (4 × 1 matrix), γ^μ is 4 × 4 matrix, whose particles and antiparticles are just a nonphysical math with No concrete physical picture ( this p.2,  this p.7 ).

Here we use (-1,+1,+1,+1) version of metric tensor (= only zero component or time variable ct = x⁰ = -x₀ changes its sign between subscript and superscript, while space variables = x,y,z or 1,2,3 components do not change their signs in any index positions x¹ = x₁ ), so if you want to convert it into the other version, see this page.

In quantum field theory, the energy (= E ) and momentum (= p ) are expressed using the time and space derivatives (= de Broglie relation ), so Eq.5-1 is equal to

(Eq.5-2) Relativistic Dirac equation expressing energy (= E ) and momentum (= p ) by the time and space derivatives (= ∂ )

Eq.1-6 and Eq.4-2 are used.

The upper Eq.5-2 uses (-1,+1,+1,+1) version of metric tensor, so this expression has the different sign from this (+1,-1,-1,-1) version of notation.
Various artificial notations are often used in quantum field theory.

Dirac equation of Eq.5-2 must satisfy the 1st-failed relativistic Klein-Gordon equation (= second-order energy and momentum which are closer to the original Einstein relativistic relation ) to be equal to special relativity.

In order to obtain the Einstein relativistic equation and the 1st-failed relativistic Klein-Gordon equation, we need to add the following equation containing the energy (= E = cp⁰ ), momentum (= pⁱ, i = 1,2,3 ), nonphysical γ matrices, rest mass (= m ) and light speed (= c ) from left side of Dirac equation of Eq.5-2 ( this p.23 ),

(Eq.5-3) Relativistic 1st-order Dirac equation → 2nd-order Klein-Gordon equation

To obtain the original Einstein relativistic equation and relativistic Klein-Gordon equation with 2nd-order energy (= E² or (cp⁰)² ) from Dirac equation with the 1st-order energy, the relativistic Dirac equation must rely on the nonphysical 4 × 4 γ matrices

↑ Since about this time in 1920s~, quantum field theory has been "mathematical" theory rather than "physical" ( this p.7-8 )

Relativistic Dirac equation's nonphysical 4×4 γ matrices (= containing four matrices, γ⁰, γ¹, γ², γ³ ) need to satisfy the following condition ( this p.13 ) of
(Eq.5-4)

Eq.5-4 means the interchanging different γ matrices becomes the opposite sign (= ex. γ²γ³ = - γ³γ² ) called "anticommute" , and the product of the same two γ matrices become γ⁰γ⁰ = 1 (= or I, which is indentity matrix = " 1 " in the matrix version ) or γ^jγ^j -1 (= or -I,  j = 1,2,3 ).

We often use metric tensor "g" to express the relations of Eq.5-4.
(Eq.5-4')

(metric tensor)  -1,+1,+1,+1 version

4×4 γ marices satisfying Eq.5-4 are
(Eq.5-5)

where I means 2 × 2 identity matrix.

(Eq.5-6)

where σ_j mean 2 × 2 Pauli matrices of
(Eq.5-6')

↑ This is why physicists started to baselessly claim relativistic Dirac equation's nonphysical 4×4 γ matrices including nonphysical 2×2 Pauli matrices (= σ ) might indicate unrealistic "spin" or spinor, though these nonphysical matrices or Dirac equation have nothing to do with spin or actual physical phenomena or objects.

For example, γ⁰γ⁰ = I ( = 4× 4, identity matrix ),

And γ matrices satisfy anticommutation relation (= interchanging two different γ matrices gives the opposite sign ) like,

Using above relations of γ matrices, Eq.5-3 is just equal to the following Klein-Gordon equation (= original Einstein relativistic second-order energy-momentum-mass relation ),

(Eq.4-1)  original Einstein relativistic energy (= E ), momentum (= p), mass (= m ) relation

Other terms such as p¹p², p²p³, p¹p³, vanish due to anticommutation relation of γ

Multiplying Dirac equation of Eq.5-1 by cγ⁰ from left side and using γ⁰ γ⁰ = 1 and x⁰ = ct ,
(Eq.5-7)

Eq.5-7 will be used, when we try to get fictitious Dirac's Hamiltonian (= total ) energy later.

Relativistic Dirac fermion's wavefunction or plain wave solution can Not represent real particles or antiparticles.

[ Relativistic Dirac wavefunction just imposes the artificial, unfounded rule of nonphysical particles and antiparticles having the opposite energies. ]

Nonphysical relativistic Dirac fermion's wavefunction ( of nonphysical mathematical particles and antiparticles ) and plane wave solution ( this p.16 ) can be expressed as
(Eq.5-10)

In the upper Eq.5-10, ψ⁺ is what they call a "positive energy solution" of the alleged Dirac nonphysical particles, and ψ^- is a negative energy solution of the nonphysical anti-particles ( this p.2-3,  this p.4-5 ).

The signs of the energy (= E ) and momentum (= p ) included in the exponential part are the opposite between the particle (= ψ⁺ with e^ikx ) and antiparticle (= ψ^- with e^-ikx ) parts, which is just the artificially-created rule irrelevant to the real world's phenomena ( this p.6,  this p.1,6 ).

And u(p) and v(p) mean the nonphysical 4 × 1 matrices attached to particle's (= c ) and antiparticle's (= d^† ) mathematical operators, respectively ( this p.3 ).

The time (= energy E ) and spatial (= momentum p ) differential acting on exponential functions ( kx ) of Eq.5-10 is equal to relativistic Klein-Gordon of
(Eq.4-5)

and
(Eq.4-6)  E = energy,  p = momentum,  c is light speed

Substituting the solutions of Eq.5-10 into Eq.5-1 of Dirac equation, the positive energy of a particle is
(Eq.5-11)

we use this.

And the negative energy part of an antiparticle (= ψ^- ) is
(Eq.5-11')

These results show that the fictitious relativistic Dirac particle of Eq.5.11 and antiparticle of Eq.5.11' have the paradoxical energy-momentum-mass relation (= Not simply positive or negative energies ).

The original Einstein relativistic particle's (= electron ) energy E must be approximately the sum of particle's momentum (= p ) and the particle's rest mass energy (= E = cp + mc², where m and c are the rest mass and the light speed c ).

But the alleged antiparticle's (= positron ) energy E of Eq.5-11' is unrealistic, paradoxical, inconsistent with the original relativistic energy-mass-momentum relation, because this paradoxical antiparticle's energy E is equal to the particle's momentum minus the particle's rest mass energy ( E = pc - mc²,  this p.7,  this p.16,  this last,  this p.19 ).

↑ The rest mass energy must be always positive with respect to the total energy E (= relativistic Dirac equation represents "free particles", so negative energy or negative rest mass energy are unrealistic ), whether it's a particle or antipaticle, so relativistic Dirac equation contradicts the original Einstein relativity, so wrong.

Dirac equation deals with free relativistic particle with No potential energy ( this p.3 ), so its relativistic total energy E must be always the sum of the particle's kinetic or momentum energy and (= plus ) the particle's rest mass energy (= the absolute value of the total energy E must be always greater than the rest mass energy by the positive kinetic or momentum energy ).

But the paradoxical antiparticle's energy E is the kinetic or momentum energy minus the rest mass energy.  ← The absolute value of the total energy E of the free antiparticle (or positron ) becomes paradoxically less than the rest mass energy, which contradicts the original relativity.

↑ This unrealistic energy of the antiparticle has nothing to do with the issue of negative energy. The real problem is the signs of the energy-momentum (= E,p ) and the rest mass energy (= mc² ) are always opposite only in the fictitious antiparticles.

From Eq.5-11 and Eq.5-11', matrices of u(p) and v(p) need to satisfy the following relation in (-1,+1,+1,+1) version of Dirac equation,
(Eq.5-12)

(Eq.5-13) antiparticle part satisfies

Here we used Eq.5-14 and Eq.5-15
(Eq.5-14)

In Eq.5-14, u^†(p) (1 × 4 matrix) means the " complex conjugate transpose" (= change the sign of the imaginary number i → -i, and change it into the transposed matrix, this middle ) of u(p).   u^†(p) = (u(p)^*)^T  this p.8

The bar-u or bar-v of the upper Eq.5-12-Eq.5-14 means the complex conjugate transpose u^† multiplied by γ⁰ matrix from the right side ( this p.12 ).

The artificial notation of bar-ψ or bar-u in the above Dirac equation means the complex conjugate transpose (= ψ^† or u^†,  this p.8 ) times unphysical γ⁰ matrix, which are just the artificially-created concepts irrelevant to the real world's phenomena.

Basically, the form of this unphysical relativistic Dirac equation does not change in its conjugate transpose (= bar-u and bar-v ) or adjoint version ( this p.8,  this p.6,  this p.89 (or p.60 ),  this p.19-(10.148) and p.23-(10.166) ).  ← If you want to change the notation from the this website's (-1,1,1,1) to (1,-1,-1,-1) version, change the sign like γ^μp_μ → -γ^μp_μ as shown in this.

↑ This artificial ad-hoc notation of adjoint × γ⁰ matrix is necessary in (pseudo-)relativistic Dirac equation, because the nonphysical 4×4 γ^j matrices (= j = 1,2,3 components ) are anti-Hermitian or Not Hermitian (= complex conjugate transpose of γ^j matrices γ^{j †} are Not equal to the original γ^j matrices,  this p.2-lower-p.3-upper,  this p.3-(75) ), which is undesirable in the mathematical world of quantum field theory.

So the γ matrices allegedly representing 4×4 matrix-spin are completely different from 2×2 Pauli matrices (= the complex conjugate transpose of Pauli matrices are equal to the original Pauli matrices like σ^† = σ = Hermitian,  this-(2.4.12) ) in their mathematical properties.

In spite of this discrepancy, the relativistic quantum field theory baselessly claims that Dirac equation have the property of (unphysical) spin, just because its nonphysical γ matrices contain Pauli σ spin matrices without showing any more detailed physical mechanism of "spin".

We multiplied the first equation of Eq.5-14 by γ⁰ from right side, and use the relations of
(Eq.5-15)  γ^† = complex transpose of γ

The relation of Eq.5-15 can be proved using Eq.5-5, Eq.5-6 and Pauli matrices σ^† = σ of Eq.5-6'.
( anticommutation of γ⁰γⁱ = - γⁱγ⁰, and γ⁰γ⁰ = 1. )
In all cases of μ= 0, 1, 2, 3, Eq.5-15 holds true.

(Eq.5-16) artificial definition of bar-u

As shown in the upper Eq.5.16, the bar-u means the complex conjugate transpose or adjoint (= u^† ) of u matrix × γ⁰ matrix.

Using Eq.5.14 and Eq.5.16, we can get the right equation of Eq.5-12.

4 × 1 Dirac matrices of u(p) (= particle ? ) and v(p) (= antiparticle ? ) are unphysical,

To satisfy Eq.5-12 equation, we can artificially adjust the coefficient parameter of the particle solution's unphysical Dirac matrix = u(p) = 4×1 matrix as ( this p.3-4 )

(Eq.5-17) Unphysical Dirac matrix inside Dirac wavefunction

( This Dirac particle's solution u(p) must include nonphysical matrices γ^μ. )

↑ As shown in this, we can freely change and redefine each coefficient parameter inside the Dirac wavefunction, so any calculations based on these freely-adjustable parameters of Dirac wavefunction, quantum field theory and QED are meaningless with No power to predict any physical values.

Substituting Eq.5-17 into the left equation of Eq.5-12,

(Eq.5-18) Dirac equation (parameter) is secretly adjusted to satisfy the failed Klein-Gordon (= Einstein relativistic ) equation.

Eq.5-18 means that Dirac spinor of Eq.5-17 satisfies the failed Klein-Gordon equation ( this p.20-21,  this p.33-34 ).

↑ So "Dirac equation obtained "spin = nonphysical γ matrices" due to its square root of relativistic second-order Klein-Gordon equation" is a lie, because physicists secretly inserted one of "square-roots" into the freely-manipulable coefficient parameters of the particle's (or antiparticle's ) Dirac solutions !

So the failure of the 1st relativistic Klein-Gordon equation means the failure of the similar Dirac equation.

In the same way (= for nonphysical Dirac antiparticle to secretly satisfy Klein-Gordon equation, while the fatal paradoxical energy-mass relation still remains ), the solution v(p) can be defined as
(Eq.5-19)

Substituting the upper Eq.5-19 into Eq.5-13, the solution Eq.5-19 proves to satisfy Klein-Gordon equation.

The problem is Dirac equation for getting relativistic spin-orbit fine structure of hydrogen atom does Not use these Dirac free particle's solution (= u, v = artificially satisfying relativistic Klein-Gordon equation ), instead, Dirac hydrogen uses irrelevant nonrelativistic Schrodinger's hydrogen's spherical Harmonics, hence, Dirac hydrogen is inconsistent with relativistic Einstein energy relation (= or Klein-Gordon equation ), and wrong.

Replacing γ^μ matrices of Eq.5-17 by Eq.5-5 and Eq.5-6, the numerator of Eq.5-17 can be expressed as
(Eq.5-20)

where we use Eq.1-6. And p_jσ_j means the sum of 1-3 components (= x,y,z components ) of p_jσ_j where p is the particle's momentum and σ is Pauli 2×2 matrix.

They artificially defined the matrices of u(0) and v(0) in Eq.5-17 and Eq.5-19 as ( this p.3,  this p.10-12 )

(Eq.5-21)  Unphysical Dirac 4×1 spinors

So u(0) is the upper two componetns, and v(0) is the lower two components.

↑ These artificially-defined nonphysical Dirac matrices have nothing to do with real physical phenomena or imaginary "spins ( this p.16-17,  this p.4 )".

As shown in this p.34, using Eq.5-20, Eq.5-17 is equal to the following (= its coefficient can be freely adjusted to different values,  this p.3 )
(Eq.5-22)

(Eq.5-23)

So the complex conjugate transpose of u(p) × γ⁰ (= Eq.5-16) = bar-u 1×4 matrix can be expressed as
(Eq.5-24)

where Pauli matrices σ^† = σ.

Using Eq.5-22 and Eq.5-24,
(Eq.5-25)

(Eq.5-26)

here we use Klein-Gordon equation and σ_j σ_j = 1 and anticommutation in Pauli matrices. (See Eq.5-6' ).

Substituting Eq.5-26 into Eq.5-25, we can get ( this p.4,   this p.3 )
(Eq.5-27)

(Eq.5-28)

where the subindex σ of φ_σ means nonphysical "spin ( σ = ↑ or ↓ )" of Eq.5-23.

And when we calculate in the same way as that of u(p), we can get
(Eq.5-29)

↑ All these calculation values are physically meaningless and useless, because all these coefficient parameters of the alleged nonphysical Dirac's particles (= u ) and antiparticles (= v ) can be freely changeable to obtain the desirable results.

And if we delete γ⁰ in Eq.5-24, we have ( this p.2-lower )
(Eq.5-30)

In the same way,
(Eq.5-31)

So Eq.5-30 is equal to Eq.5-31 in the case that γ⁰ is deleted.

Dirac equation's artificial Lagrangian

Artificially-created relativistic Dirac Lagrangian leading to Dirac's equation can be defined as
(Eq.5-32)  artificial Dirac Lagrangian (= L )

This Lagrangian is said to be Lorentz invariant scalar.

As I said, Lagrangian (= L ) itself has No physical meaning except for a "tool" to get some particle's equation of motion and Hamiltonian (= total ) energy through some artificial procedure called Euler-Lagrange equation, and these relativistic Dirac and Klein-Gordon equations' Lagrangians were artificially-created, and inconsistent with the original Lagrangian form (= L = kinetic minus potential energy ).

First, with respect to conjugate transpose of ψ (× γ⁰), Euler-Lagrange equation (= to obtain Dirac equation ) is
(Eq.5-33)

where the second term of Eq.5-33 is zero. (= because Dirac Lagrangian of Eq.5-32 doesn't include the derivative of the bar-ψ )

So the first term of Eq.5-33 is ( using Lagrangian L of Eq.5-32 )
(Eq.5-34)

Eq.5-34 is equal to Dirac equation of Eq.5-1.

So by applying Euler-Lagrange equation to the artificially-created Dirac Lagrangian (= L ) with respect to the adjoint-ψ, Dirac equation can be obtained.

And the nonphysical conjugate transpose (= adjoint ) of Dirac equation (Eq.5-1) is

(Eq.5-35)  Dirac equation's adjoint 2nd-version.

where we use Eq.5-15 and multiply it by γ⁰ from the right side.
The direction of the differentiation is opposite from Eq.5-1, but the meaning of Eq.5-35 is supposed to be equal to Eq.5-1 (= these mathematically-defined equations do Not represent real phenomena, anyway ).

When using the usual ψ(x) in Dirac Lagrangian and Euler-Lagrange equation, we can obtain the adjoint form of Dirac equation ( this p.8,  this p.7,  this p.7 )
(Eq.5-36)

Eq.5-36 is equal to Eq.5-35.

As a result, we can get Dirac equation from the artificially-created Dirac Lagrangian of Eq.5-32.

↑ In other words. in order to get Dirac equation (= there are two versions of original Dirac equation and its adjoint ), its unphysical Lagrangian (= L ) was artificially created and chosen.

So these procedures to get Dirac equation from the artificial Lagrangian have No physical meaning.

Fictional relativistic Dirac's Hamiltonian energy (= H )

Artificially-created Dirac Lagrangian (= L ) is also used to obtain Dirac version of Hamiltonian energy (= relativistic Dirac free particle's total energy ) and the fictitious momentum (= p or π ) by copying the classical mechanical Hamiltonian procedure called Legendre transformation ( this p.9-10,  this p.4-5 ).

The problem is this momentum obtained by this Hamiltonian procedure is Not a real momentum in Dirac and Klein-Gordon equation, unlike the ordinary classical mechanics.

Relativistic quantum field theory and QED like to use the fictional concepts called the generalized (= canonical ) momentum (= π or p, ) based on wrong math, which is Not a real momentum expressed as mv (= mass × velocity,  this middle ).

This fictional canonical momentum (= p or π ) obtained from the artificially-created Dirac's Lagrangian and this (classical) transformation becomes ( this p.16 )

(Eq.5-37) Dirac's fictional (canonical) momentum = π

↑ This artificially-created canonical momentum (= π = iℏψ^† = complex conjugate ) including No mass or velocity is completely different from the true momentum expressed as the particle's mass × velocity.  ← Relativistic quantum field theory lacks reality in their unphysical momentum from the beginning.

From this fictional momentum (= π ) and Lagrangian (= L ), using this procedure, the relativistic Dirac's fictitious Hamiltonian total energy (= H ) is supposed to be ( this p.3-5,  this p.2-4 )

(Eq.5-38)  Fictitious Dirac Hamiltonian (= total ) energy (= H )

where the last equation of γ^j∂_j means the sum of 1-3 components (= x,y,z components ) with respect to "j" like γ¹∂₁ + γ²∂₂ + γ³∂₃.

From Eq.5-7,  Eq.5-38 is equal to ( this p.2,  this p.17 )

(Eq.5-39)  Dirac unphysical Hamiltonian energy

And the Dirac wavefunction solution containing nonphysical math particles ( this p.5-6,  this p.98-100 ) used in Hamiltonian of Eq.5-39 is expressed as

(Eq.5-40) ↓ Nonphysical Dirac wavefunction, c or d is a particle or antiparticle.

As shown in the above Eq.5-40, the most widely-used important relativistic Dirac equation's solution just expresses each fermion such as an electron, a quark or an antiparticle just as nonphysical math symbols with No realistic figures ( this p.19,   ).

The coefficient of the upper Eq.5-40 Dirac solution can be freely changed together with other artificially-manipulable coefficients of unphysical spinor relations of u and v, and the coefficient of the particle's anticommutation relation (= as I explain later, this anticommutation represents unphysical Pauli principle ) to obtain the desirable Hamiltonian energy's form ( this-(5.4),(5-5),p.3-last,  this p.151-(X.4)-(X.5)-p.153 ).

↑ These artificially-changeable coefficients of Dirac equation is Not a prediction of the theory, but just a artificial manipulation or choice of coefficient parameters ( this p.22 ), so relativistic Dirac equation has No power to predict anything.

The exponential function in the upper Dirac wavefunction contains particle's energy and momentum

And Dirac wavefunction's conjugate transpose or adjoint form (× γ⁰) is

(Eq.5-41) Relativistic Dirac plane wave solution, bar-ψ = adjoint × γ⁰

The bar-ψ of the upper Eq.5-41 is the conjugate transpose or adjoint form × γ⁰ matrix of the relativistic Dirac wavefunction or spinors ( this p.8 ).

These u(p) and v(p) are 4 × 1 spinors in Eq.5-17 and Eq.5-19, respectively.
And c(k) (+ u ) and d(k) (+ v ) mean annihilation operators of particle and antiparticle, respectively.

So c^†(k) and d^† (k) are nonphysical creation operators with No particle's figures, which abstract math symbols tell us nothing about the detailed physical mechanism of relativistic quantum field theory.

Substituting Eq.5-40 into Eq.5-39, and using Eq.4-5, Eq.5-30 and Eq.5-31, we can get the nonphysical Dirac Hamiltonian (= H,  this p.5,  this p.3-5,  this p.26-28 )

(Eq.5-42) Fictional Dirac Hamiltonian energy, c^† = particle,  d^† = antiparticle

c^† (= particle creation operator ),  d^† (= antiparticle creation operator )
c (= particle's annihilation operator ),  d (= antiparticle's annihilation operator ).

n = c^†c (= or a^†a ) in Hamiltonian energy (= H ) is the particle's number operator (= which is just an artificially-introduced rule, hence, has No physical meaning,  this p.4 ).

Using the artificially-prepared Dirac wavefunction, only the antiparticle's part becomes the opposite of number operator (= dd^† ) attached to the negative energy (= -E = -ℏω )

For example, in the actual calculation to get this Dirac Hamiltonian energy, using Eq.5-40, the c^†c part (= particle energy part ) of Eq.5-39 is
(Eq.5-42')

↑ All these calculations of Dirac particle's energy are physically meaningless, because they can freely manipulate the coefficient parameters (= so the resultant Hamiltonian energy can be also freely manipulated ) of Dirac wavefunction

Due to the minus sign of antiparticle part = the first equation in Eq.5-42' (= red line ), the antiparticle's Hamiltonian energy = d d^† part of Eq.5-42 becomes minus (= negative energy problem ? ), when the time (= energy ) derivative (= iℏ∂_t ) acts on e^-ikx = e^iωt

↑ This antiparticle's negative energy was artificially-prepared (= Dirac wavefunction ψ itself was artificially introduced ), so this mathematical antiparticle has No physical meaning.

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

"Relativistic quantum field theory (= Dirac field )" does Not clarify Pauli principle mechanism at all.

(Eq.5-42'') ↓ Only antiparticle (= d^† ) energy has the minus sign (= negative energy ? ), which must be fixed using unphysical Pauli exclusion principle ?

To get the positive number operator like n = d^†(k) d(k) from the antiparticle Hamiltonian energy part of Eq.5-42, physicists insist the following artificial anticommutation relation (= Pauli exclusion principle ?  = exchanging two math particles gives the opposite sign ) must be satisfied.

(Eq.5-43) ↓ This relativistic quantum field theory just showing the meaningless abstract math relation of "anticommutation" tells us nothing about the detailed mechanism of Pauli principle !

Relativistic quantum field theory claims this abstract meaningless math relation called "anticommutation ( this p.3 )" is the only explanation for mysterious Pauli exclusion principle, without showing any more detailed mechanism ( this p.5,  this p.5,  this p.4,  this p.2 ).  ← Our science stops progressing by stopping delving into deeper detailed mechanism of how exactly this Pauli principle repulsive force is generated.

The notation about the above anti-commutation relation of (unphysical) particles is like in this p.3-(16),  this-(8.2.11-14).

δ(k-k') is unphysical Dirac delta function, which is used just as one of artificial rules (= the upper coefficient of δ function can be freely changed ) that has Nothing to do with any real physical principle.

Using this artificially-introduced unphysical anticommutation rule of Eq.5-43, the Dirac Hamiltonian's antiparticle negative energy part of Eq.5-42 becomes the following Eq.5-44

Quantum field theory has No ability to explain Pauli exclusion principle in the realistic way !

(Eq.5-44) ↓ Only this meaningless math proves Pauli principle mechanism ?  ← impossible !  No real physical picture or mechanism is shown !

In the last line of the upper Eq.5-44, we can get positive number operator of d^†(k) d(k) (= positive energy in both particles and antiparticles ) using the artificial rule of anticommutation allegedly indicating Pauli exclusion principle.

So they insist Eq.5-43 represents the mysterious force of Pauli exclusion principle ( this p.4.5,7,  this p.5-6,  this p.16-17,  this p.4-lower ).

↑ But they just show very abstract meaningless math relation called anticommutation (= exchanging two nonphysical math operators gives the opposite sign = proof of Pauli principle ?  ← No way ! ).

Furthermore, in this process, physicists are saying the contradictory thing about this nonphysical anticommutation Pauli principle.

Because to eliminate the negative energy of antiparticle, they artificially created new negative vacuum energy (= minus δ(0) delta function in the last line of the upper Eq.5-44,  this p.5-(5.17),  this p.4-lower,  this p.23-(4.35) ), which means the unrealistic negative infinite energy remains, whether the meaningless Pauli principle's anticommutation rule is introduced or not.

This annoying negative infinite vacuum energy (= -δ ) may be canced out by the positive infinite vacuum energy of the failed Klein-Gordon's equation using the imaginary nonphysical theory called "supersymmetry ( this p.23-second-paragraph,  this p.13 )".

↑ But we cannot use the failed Klein-Gordon equation to artificially cancel this Dirac's infinite negative energy (= first, Dirac's equation represents "free relativistic particle with postive rest mass + kinetic energies and No potential energies" which cannot generate the negative infinite energy !  So this unphysical negative vaccum energy is unjustifiable ).

And even if they manage to artificially cancel this infinite vacuum energy (= called infrared divergence ), they have to deal with other more serious infinity caused by infinite virtual particles (= called ultraviolet divergence,  this p.5 ).

This result shows that the commonly-used phrase "Dirac antiparticle could solve negative energy !" is a total lie (= They just artificially introduced these meaningless math equations with No relation to the real world ), and the antiparticle's paradoxical energy-mass relation remains unsolved.

[ Dirac's unphysical anticommutation (= origin of Pauli principle ? ) contradicts de Broglie wave theory, so false. ]

And if Eq.5-43 needs to be satisfied, the relation of ψ and its fictional momentum π or iℏψ^† (= Eq.5-37) becomes
(Eq.5-45) π = ψ^† = Dirac fictional momentum

This anticommutation relation of Dirac's fictional canonical momentum ( this p.1,6,  this p.2,5 ) is different from Eq.4-24 of Klein-Gordon equation, which means Dirac's fictional momentum of Eq.5-45 is inconsistent with the original quantum mechanical commutation between position (= q or x ) and momentum (= p or spacial derivative ) operators based on de Broglie's relation of
(Eq.4-25)

If we define the new property of canonical momentum like Eq.5-45 (= anticommutation ), we have to find out the new theory which replaces de Broglie's theory.
But the experimentally-verified de Broglie's theory is a basic one in the quantum mechanics, which can not be eliminated.

Can we use π of Eq.5-37 as a canonical momentum for getting Hamiltonian, though it does NOT obey de Broglie relation ?
It is unacceptable.

QED based on fictional interaction between nonphysical Dirac electron and unreal virtual photon is illegitimate.

(Eq.5-46) ↓ This nonphysical abstract equation represents (fictitious) photon (= A ) and electron (= ψ) interaction ?  ← No more detailed mechanism, so useless.

Physicists try to combine the fictional Lagrangians of electromagnetic theories and artificial Dirac electron's Lagrangian to produce the virtual interaction between electrons and unreal virtual photons for QED infinity calculation ( this p.9,  this p.10-13,  this p.1-5 ).

↑ Both these Dirac electron (= ψ ) and (virtual) photons (= A ) are expressed just as nonphysical abstract math symbols which tell us nothing about the detailed interaction between electrons and (virtual) photons.

(Eq.5-46') ↓ Wrong replacement for fictional interaction between Dirac electron and unreal virtual photons causing meaningless QED infinity.

Interaction term ( of particle and vector potential A allegedly containing unreal virtual photons ) of Eq.5-46 is
(Eq.5-47) Fictional interaction term between electrons (= ψ ) and virtual photon (=A ).

Relativistic quantum electrodynamics (= QED ) calculation using this fictitious interaction between photon (= electric and magnetic potential A ) and electron (= Dirac wavefunction ψ ) is mathematically wrong, because QED tries to obtain very tiny finite physical values (= anomalous magnetic moment or g-factor ) by the illegitimate renormalization method of artificially subtracting infinity from another freely-adjustable infinite bare parameters (= Indefinite "infinity" itself cannot be defined or used as the "predicted value", at all,  this p.8-9,  this p.11-13 ).

↑ Infinity minus infinity = uncertain and indefinite finite value, because "infinity" itself is an indefinite uncertain value.  So QED renormalization method based on this uncertain and unpredictable value is an illegitimate and wrong math.

The problem is the original Lagrangian between charged particles and the electromagnetic photon (= A ) was created based on nonrelativistic energies (= E = p²/2m, Not based on relativistic E² = p² + mc² ), so this interaction term cannot be used in relativistic Dirac electron or QED due to the lack of theoretical basis.  ← QED is false.

Furthermore, the theory for getting this fictitious photon-electron interaction inside electromagnetic Lagrangian or Hamiltonian energy (= H ) uses wrong math, so all the relativistic quantum field theory such as QED and standard model based on this wrong interaction term between Dirac electron and virtual photon is illegitimate and invalid.

Dirac electron causes unrealistic electric current, so false.

Comparing Eq.5-46 (= mixed Lagrangian consisting of photon and Dirac equation ) with Eq.3-2 (= electromagnetic Lagrangian ) and Eq.3-3 of

(Eq.3-2) Fictitious electromagnetic Lagrangian where A is electronc and magnetic potential containing fictitious photon, J is electric current.

and
(Eq.3-3) Euler-Lagrange equation for obtaining Maxwell equation

They rashly started to claim that (fictitious) Dirac's electric current (J) density (= correspond to the classical electric current, they thought ) becomes

(Eq.5-48) ↓ Electric current (= J ) represented by unreal Dirac electron wavefunction (= ψ ) and unphysical γ^μ matrices

Using the fictional current j in the upper Eq.5-48, physicists baselessly claim that Dirac equation has succeeded in expressing the (fictional) electric current density (= j ) ( this p.2-lower,  this 10th-paragraph ).

But as shown in Eq.5-6, γ matrices consisting of mathematical Pauli spin matrices have nothing to do with the direction of the electric current, contrary to Dirac fictional current J using this γ matrices as the current's direction (= J_μ = γ_μ, where the component μ means the fictional direction of the current J ) as shown in the upper Eq.5-48.

4×4 γ matrices are supposed to represent (nonphysical) spin components of Dirac equation, which have NO relation with the direction of current.
But in Eq.5-48, this spin component (= μ of γ_μ ) decides the direction of the current J_μ.
So Eq.5-48 and Eq.5-47 are based on wrong interpretation.

By the way, why have we gotten the wrong result of Eq.5-48 in Dirac equation ?
In fact, γ matrices (= σ matrices) are not peculiar to Dirac equation.
Even in Schrodinger equation (or classical mechanics with No spin ), σ matrices are included, as follows,
(Eq.5-49)

Here we use the following relations of Pauli (= σ ) matrices.
(Eq.5-50)

When a particle is under Lorentz force of Eq.2-13 (= causing fictitious photon-electron interaction based on wrong math ), the quantum mechanics uses the mathematical tricks like
(Eq.5-51)

where p includes derivative (= based on de Broglie relation ), which acts on before and after A_j.
So de Broglie wavelength of the particle is also used in derivative (= ∂ = ∇ ) of the vector potential (= A ) to get the magnetic field (= B = ∇×A = ∂_iA_j - ∂_jA_i,  ← this derivative ∂ originated from the irrelevant electron's de Broglie wave ).
This is strange, becuase they insist electron's de Broglie wave = external magnetic field.

and we also use the trick of
(Eq.5-52)

Considering Eq.5-50, Eq.5-51 and Eq.5-52, Eq.5-49 under the replacement of Eq.2-13 changes to
(Eq.5-53)

Eq.5-53 and Eq.5-52 mean that spin-magnetic field (= vector potential) interaction (= SB = 1/2σB where σ is Pauli spin matric ) exists in Schrodinger equation, too.
And to agree with Bohr magneton ( of Bohr model ), spin g factor g_e must be "2".

↑ The fact that the nonrelativistic Schrodinger equation without spin also tacitly includes the Pauli σ spin matrices means relativistic Dirac equation's unphysical γ matrices consisting of Pauli σ matrices have nothing to do with (fictitious) spin.

Partial differentiation of Lagrangian ( L_I = - H_I ) of Eq.5-53 by the vector potential (A) is
(Eq.5-54)

The electric current (J) in Schrodinger equation consists of charge and its velocity, as shown in Eq.5-54.
( Here Eq.2-13 is used. )
And σ matrices is hidden as σσ = 1 as shown in Eq.5-53, which is NOT used in Schrodinger equation current.

On the other hand, the (fake) relativistic Dirac equation is first order in the momentum as shown in Eq.5-46, unlike the original nonrelativistic Schrödinger equation's second-order momentum.
(So also in the vector potential A, it is first order in Dirac equation, which is different from classical electromagnetism. )

As a result, differentiation with respect to vector potential (A) erases the particle's momentum expressed as the derivative (= ∂ ) in Dirac equation !
(So there is NO charge's current at this point in Dirac equation. )
This means that the first order Dirac equation, which is unable to give real electric current, can NOT use the replacement of Lorentz force (= Eq.2-14 ) correctly !
(Interaction of Lorentz force is "originally" considered in the second order equations of classical mechanics. )

But this (fictitious) replacement of Lorentz force is used as the (fictitious) interaction term between virtual photon and electron of Eq.5-47 in the quantum electrodynamics and standard model, which means All these QED and standard model using wrong photon-electron interaction term are wrong, too.

Photon doesn't exist. It's only a "mathematical" product.

[ Relativistic quantum field theory and QED consist of the unrealistic contradictory photons. ]

(Fig.P) ↓ A relativistic photon is just a unphysical math symbol where "zero component photon" generates the contradictory negative probability.  ← QED is false.

In fact, all the relativistic quantum field theory, quantum electrodynamics (= QED ) and standard model are unable to describe the light wave's imaginary particle called "photon" in the realistic way.

All the current relativistic quantum field theoy can do is show very abstract math symbol or creation ( annihilation ) operator as an invisible imaginary photon particle (= each fictitious photon is expressed just as the abstract symbol a^† ? ), which can tell us nothing about the detailed behavior and shape of a photon ( this p.338 ).

Relativistic quantum field theory and QED claim that an infinite number of fictitious (mathematical) photon particles constitute the magnetic vector potential (= A_j, j = 1,2,3,  this p.2-left,  this p.3 ) and electric scalar potential (= φ ) with No more detailed description about the mysterious photon's figure. = Science stops delving into deeper truth.

In the current electromagnetic theory, the fictitious unobservable concept called "gauge" is introduced as a freely-changeable parameter, which allegedly doesn't affect the observable electric and magnetic field.  ← But this is untrue, the relativistic quantum field theory and QED destroy and drastically change the original Maxwell equation due to this meaningless restriction on the unphysical gauge ( compare this p.6 and p.14 ).

Physicists try to restrict this free fictitious parameter "gauge" to impose various additional rule on the original Maxwell equation.

One simplest artificial condition is called "Coulomb gauge" where the realistic electric potential (= φ caused by the charged particle ) is forced to be zero and unavailable (= electric scalar potential φ = 0 ).

This artificial restraint removing the zero component (= A⁰ = φ/c = 0, where φ is electric scalar potential,  this p.7,  this p.2 ) of electromagnetic field makes this Coulomb gauge violate Einstein relativity ( this p.4-lower,  this p.5-3rd-paragraph,  = artificial Coulomb gauge condition breaks relativistic Lorentz invariance ), hence, this contradictory Coulomb gauge is Not used in the authentic relativistic quantum field theory or QED.

In this simplest Coulomb gauge, each unphysical photon expressed as an abstract math operator (= a_j ) is supposed to have two types of transverse polarizations ( this p.6-8,  this p.6-7 ) with No more detailed description than this abstract meaningless math symbols.  ← How and what shape of polarization each (fictitious) photon has is unknown in this very abstract unphysical math symbol photon of quantum field theory.

Another widely-used gauge restraint condition is called "Lorentz gauge ( this p.4-6 )", which is equal to "Feynman gauge ( this p.8-(6.37),  this covariant approach )" which is said to obey Einstein relativity, due to its Lorentz invariant property (= any observers can see the same physical principle, this,  this p.8-upper,  this p.5,  this p.8 ).

Electromagnetic wave or photon is known to have two types of transverse polarizations (= 1,2 components ) where the longitudinally-polarized light wave or photon should Not exist.

But the relativistic quantum field theory or QED allegedly compatible with Einstein relativity must include these two more excessive unrealistic photon's polarizations, the longitudinally-polarized photon (= 3 component ) and the scalar (= time-like) zero-polarized photon (= 0 component,  this p.3 ) expressed as nonphysical math symbols with No concrete figure ( this p.5-left-lower,  this p.8-(51),  this p.2,7 ).

This so-called relativistic electromagnetic theory using Lorentz (= or Feynman ) gauge must include the unrealistic scalar (= time-like ) zero-polarization photon (= which means the unreal zero component photon with No polarization,  this p.47-50 ) constituting the electric potential (= φ ) = they baselessly argue Coulomb electric force is caused by this unreal scalar virtual photon.

To satisfy Einstein relativistic Lorentz invariance (or symmetry ), this unreal scalar photon must have the unphysical negative probability (= this p.2-right-upper,  this p.5-6,  this p.10-(6.50) ) !  This unreal scalar-polarized (= time-like ) photon with negative probability or negative norm is physically impossible ( this p.3-4,  this-6.542,  this 2nd-paragraph,  this p.15,  this p.3 ).

Relativistic quantum field theory and QED needs (unreal) virtual photons to explain the electric and magnetic forces, and they baselessly insist this unobservable unreal virtual photons can have (unreal) longitudinal polarization ( this p.28-last,  this p.25(or p.23 )-last-paragraph,  this p.8 ) and the scalar-time-like-polarization ( this p.9-upper,  this p.16-2nd-paragraph,  this p.8.  this p.7-1st-paragraph ).

So all the relativistic electromagnetic quantum field theory and QED are wrong due to this unrealistic longitudinally-polarized photon and the zero-polarization scalar (= time-like ) photon with negative probability that is necessary to explain some phenomena such as Coulomb elecric interaction based on (unreal) quantum virtual photons ( this p.4 3rd-paragraph,  this p.4-right-5th-paragraph,  this p.8-last-sentence,   this p.3-first-paragraph ).

↑ Physicists try to artificially redefine the "vacuum" to eliminate this unreal scalar photon from the relativistic QED ( this p.18-19 (or p.22-23) ), but this unreal scalar photon or Feynman (= Lorentz ) gauge is used and necessary for any relativistic QED calculations (= relativistic theory always needs four-components of all concepts also in imaginary polarizations ) as a propagator of (unreal virtual) photons ( this p.12,  this p.9,  this p.9,  this p.14(or p.26),  this p.9,  this p.12-right ).

If (unreal) longitudinal-polarized and negative-probability-scalar (= time-like ) photons are unnecessary, physicists should have created a new theory without them, but they couldn't.
Quantum mechanics and relativistic QED definitely need unphysical (virtual) photons with unreal longitudinal and zero-polarized photons to satisfy Einstein relativistic theory or Lorentz (= Feynman ) gauge that always needs four-kinds of photon polarizations like four-spacetime-vectors ( this p.97-98,  this p.8,  this p.4 ).

Contrary to the media and academic hypes, the relativistic quantum field theory and QED have No power to describe and clarify the exact detailed physical mechanism of how each electron interacts with a (imaginary ) photon, and QED has No power to predict physical values (= this is why the media just repeats the empty cliche "successful QED !", and never presents us with their dubious detailed calculation methods,  this p.3-left ).

QED just shows the abstract simple Feynman diagram ( this p.58,  this p.9 ) where each electron expressed as the abstract math symbol ( this p.11-13 ) is said to interact with infinite unreal virtual photon violating Einstein relativity ( this p.13 ) with infinite amount of energies ( this p.29-last-paragraph ).

All calculation values of this unrealistic QED miserably diverge to infinity ( this 21 5th-paragraph ) due to the infinite unreal virtual photons, which mathematically-meaningless infinity must be artificially removed by another unobservable infinite free parameters or counterterms called "bare charge and mass" which ad-hoc method is called renormalization ( this p.28,   this p.40-41,  this p.67-68(or p.60-61) ). This illegitimate ad-hoc QED is useless with No power to predict any values (= anomalous magnetic momenet or g-factor was obtained just by freely adjusting these uncertain infinities, which is Not math or predicting values,  this 1st-paragraph ) .

[ Invisible "Gauge (= f(x))" is a freely-changeable parameter, which is physically meaningless. ]

(Eq.6-1) Electric (= E ) and magnetic (= B ) fields are unchanged under the artificial gauge transformaiton (= f ) in the magnetic vector (= Ai ) and electric scalar (= φ ) potential ?

The present quantum field theory focuses only on the unobservable = physically meaningless concepts such as fantasy quantum parallel-worlds and extradimensions, so useless.

Relativistic quantum field theory and QED also consider the physically-meaningless fictitious concept called "gauge (= f )" as a important tool to manipulate QED parameters.

Under this artificial gauge transformation in the upper Eq.6-1, the observable electric (= E ) and magnetic (= B ) fields are said to be unchanged.  ← But this is a total lie, as I explain later.

Relativistic quantum field theory uses the unobservable fictitious concpets such as the magnetic vector (= three components of A1, A2, A3 = Ax, Ay, Az ) and electric scalar (= φ = zero componenet of A0 ) potentials to describe the observable electric and magnetic fields.

↑ So physicists claim this degree of freedom in choosing the fictitious gauge allows them to artificially choose various gauge conditions such as Coulomb gauge and Lorentz gauge in the quantum field theory, which is a meaningless theory.

But in fact, all these gauge conditions such as Coulomb gauge and Lorentz gauge used in relativistic QED break and drastically change the meaning of the original Maxwell equation, so all the relativistic quantum field theoies, QED and standard model relying on this fictitous gauge theory are inconsistent with the experimentally-verified Maxwell theory.  ← QED is false.

[ Relativistic Lorentz gauge condition generating a unreal scalar photon. ]

In the relativistic QED, physicists often choose the fictitious gauge f(x) which satisfies the relativistic Lorentz invariance (= form of equation is unchanged under Lorentz transformation ). which is called Lorentz gauge or Feynman gauge ( this p.4 ) like,

(Eq.6-2) Lorentz (or Feynman ) gauge condition

We use (-1,+1,+1+1 ) version of metric tensor.
So only the sign of the zero component vector is changed depending on the index position ( A⁰ = -A₀ = φ/c,  A^j = A_j when j = 1.2,3 )

The reason why physicists tend to choose this Lorentz gauge condition is to keep the Maxwell's equations Lorentz-invariant.
Eq.6-2 is scalar form (= unchanged under Lorentz transformation due to the form of a pair of covariant and contravariant vectors,  this p.8-upper ).

Under Lorentz transformation, the form of Lorentz gauge condition of Eq.6-2 is unchanged like
(Eq.6-3)

So we can prove Eq.6-2 (= Lorentz gauge condition ) is Lorentz invariant ( this p.7-8 ).

Both Coulomb and Lorentz (= Feynman ) gauge conditions destroy the original Maxwell equation, so QED is false.

Using Eq.3-1 ( this p.2 ), Maxwell equations of Eq.3-3 can be expressed using magnetic (= Ai ) and electric potentials (= A0 ) as

(Eq.6-4) Original (unbroken) Maxwell equation ↓

↑ This form of Eq.6-4 is the original authentic Maxwell equation using magnetic and electric vector potentials (= A ) and the electric current (= J ).

But using the Coulomb gauge or the relativistic QED Lorentz gauge condition of Eq.6-2 ( this p.6,14 ), Maxwell equations of Eq.6-4 is destroyed and completely changed into

(Eq.6-5) Lorentz gauge and Coulomb gauge destroy the original Maxwell equation, so wrong.

where the magnetic potential A₁ is supposed to be constant (= time derivative of A₁ or ∂₀A₁ = ∂A₁/c∂t is zero, which means electric current J₁ is constant ), while the electric current of J₂ or J₃ is not constant (= space derivative of A₂ or A₃ is not zero ).

↑ This destroyed Maxwell equations by imposing fictitious Coulomb gauge ( this p.16(or p.12)-(1.6),  this p.5-(6.19) ) or relativistic Lorentz gauge rules ( this p.3-lower,  this p.11-(86),   this p.4-(16.16),   this p.4-lower ) are clearly different from the original Maxwell equations.

This means the restriction by Lorentz gauge or Coulomb gauge, which completely destroys and alters the original authentic Maxwell equation, is invalid, and unreal longitudinal and scalar-polarized photon with negative-propability cannot be eliminated by gauge restriction in the relativistic quantum field theory and QED.

↑ Only one-component magnetic vector potential A_μ or A_j ( in Coulomb or Lorentz gauge electromagnetic equation ) is Not enough to make the magnetic field B, which requires at least two different components like B_1 = ∂₂A₃ - ∂₃A₂ = two components A₂ (= Ay ) and A₃ (= Az ) are necessary for describing the magnetic field B.

Here we consider the photon particle in the vacuum. ( J = 0 ).
In this case, the relativistic Lorentz gauge's fictitious electromagnetic equation becomes

(Eq.6-6) Broken Maxwell equation by Lorentz gauge restraint in the electric current J = 0

The broken Maxwell equation of Eq.6-6 clearly disagrees with the original Maxwell equation of the first line of Eq.6-5, but this broken Maxwell equation is the only choice for getting the photon's or vector potential A's wavefunction in the relativistic quantum field theory and QED.

The second-derivative ∂^ν∂_ν in the upper Eq.6-6 acting on the exponential function of photon A's wavefunction gives the relativistic energy equation with massless photon = Klein-Gordon equation with zero mass version ( this p.9,  this p.3-4,  this p.2,  this p.9,  this last,  this p.6-7 ).

The most serious problem is this relativistic quantum field theory and QED using the quantized (unphysical) photons contained in the magnetic (or electric ) potential A can Not describe the ordinary constant magnetic field caused by the constant electric current J.

In Lorentz gauge or Coulomb gauge, the broken Maxwell equation consisting of the second-derivative with respect to the time and space allegedly gives the (virtual) photon's energy (= E ) and momentum (= p ).

In the constant magnetic field produced by the constant electric current, the time derivative giving the photon's energy E is zero (= due to the constant magnetic vertor potential A with respect to the time ), hence, the space derivative giving the photon's momentum p must be zero, too ( E = p = 0) in the vacuum, as shown in the above figure where the broken Maxwell equation's the second space derivative ∂²A = p²A = 0 becomes zero.  ← the photon's momentum p (= expressed as the space derivative ) must be zero in the vacuum despite the existence of magnetic field.

But there is (constant) magnetic field in vacuum, which magnetic field has a power to push or pull the electric charge, which means the magnetic field must contain photons with some amount of energy and momentum (= photon's energy in the magnetic field vacuum should Not be zero ).

↑ As a result, the relativistic quantum field theory or QED giving the false zero photon's energy in the constant magnetic field is wrong.

The original complicated Maxwell equation like in the upper Eq.6-5 can Not be solved (= hence, photon A's wavefunction cannot be obtained in the original Maxwell equation ) using the following ordinary relativistic photon's solution or wavefunction,

Only broken relativistic Maxwell equation can have (fictitious) photon solution.  ← Relativity is wrong.

(Eq.6-7) Photon wavefuntion satisfying the broken Maxwell equation

The fictitious photon's solution or wavefunction of Eq.6-7 satifies only the broken simple Maxwell equation of Eq.6-6, and does Not satisfy the original complicated Maxwell equation of Eq.6-5-upper.

↑ This means the fictitious photons used in the relativistic quantum field theory and QED contradict the experimentally-verified Maxwell equation of Eq.6-5-upper, so wrong.

These fictitious "photon particles" satisfying the broken Maxwell equation in relativistic quantum field theory or QED are just nonphysical math symbols (= ex. a photon is expressed just as a^† ) with No real figures ( this p.2-right,  this p.3-7,  this p.13-21 ).

Relativistic QED must contain "unreal scalar zero-polarized photons with negative probability", so false.

(Eq.6-8) Lorentz (= Feynman ) gauge compatible with relativity contains unreal time-like scalar photons with negative probability.  ← Einstein relativity is false.

To obtain the (broken) Maxwell equation of Eq.6-6 in the relativistic Lorentz or Feynman gauge conditions, it must always contain four types of (unrealistic) polarizations such as zero-polarized scalar ( time-like ) photon, x,y-two poralized photons, z-longitudinal photon ( this p.49 (or p.50),   ).

To satisfy the superstitious Einstein relativity or Lorentz invariance condition, the sign of the zero-component (unreal) photon and the other three component photons must be the opposite.

↑ This relativistic unreasonable constraint generates the unreal zero component scalar (= time-like ) photon, which must have the unrealistic negative probability ( this p.10-(6.50),  this 2nd-paragraph ) or norm ( this p.3-4,  this p.4-left,  this p.2-right-upper,  this p.49-lower,  this p.6-(6.34) )

As shown in the upper Eq.6-8, the number (= expressed as number operator N, this p.14 ) of this unrealistic zero-component scalar (= time-like ) photon in the relativistic QED magically decreases when this scalar photon is created by creation operator due to its extra minus sign (= a^†,  this p.3-right,   this p.100-last-paragraph ) which is contradictory and wrong.

But this contradictory zero-polarized scalar photon is indispensable for the relativistic QED calculation (ex. for anomalous magnetic moment ) to keep relativistically Lorentz invariant form ( this p.3 = Lorentz or Feynman gauge photon propagator including all four-component photons is used in QED,  this p.8 ).  → meaningless infinity in QED calculation.

Relativistic QED calculation uses unreal virtual photon propagator ( this p.3 ), which must include the unreal four types of polarized photons (= including unreal scalar zero-polarized photon ) as the form of the four-vector metric tensor = g_μν ( μ, ν, = 0,1,2,3,  this p.6,   this p.4~p.5-left-lower ) in calculating the anomalous magnetic moment.

So all the relativistic quantum field theory such as QED and standard model containing contradictory virtual and scalar photons with negative probabilities are illegitimate and false.

Fictitious electromagnetic Hamiltonian energy of Coulomb gauge violating relativity.

(Eq.6-9) Relativity contradicts Maxwell equation, unless it imposes many artificial rules to destroy the original Maxwell equation.

The important point is that Einstein relativity contradicts the experimentally-verified Maxwell equation, which proves Einstein relativity is false.

To artificially match (broken) Maxwell equation, physicists have to impose various aritificial rules such as Coulomb or Lorentz gauge conditions which break the original Maxwell equation even by creating the fictitious zero-component photon with unreal negative probability.

In order to obtain the Maxwell electromagnetic energy density in the vacuum, physicists have to give up Einstein relativity (= Lorentz invariance is violated,  this p.5-upper ) and impose the contradictory Coulomb gauge condition.

Coulomb gauge has to often make the electric potential zero (= A⁰ = φ/c = 0,  this p.4-(6.15),  this p.3-lower ), which means the electric field (= E ) of the fictitious Coulomb gauge can Not be generated from the ordinary electric charge (= electric charge or electrons should generate electric potential φ, but these electric charges must be unrealistically zero in the ad-hoc Coulomb gauge ).

As a result, all the present relativistic and nonrelativistic quantum field theory, QED and standard model are based on contradictory concepts and artificial rules which cannot predict anything, so they are a useless and wrong theory.

2022/8/1 updated. Feel free to link to this site.