The End of Quantum Field Theory.

Top page (correct and new Bohr model including helium)
"Spin g-factor". Dirac equation for hydrogen.
Quantum electrodynamics is "artificial" trick.
The current high energy physics have failed.
Superstring and Loop quantum gravity are real ?

Table of contents. (13/1/6)

Quantum field theory is NOT physics, but just "math".

(Fig.1) "Math" ?

The present quantum field theory is very complicated and takes much time to learn.
( I aimed to make this page simple and easy to understand. )

If you believe this quantum field thoery is true and can give fundamental answers about our nature, you will be surely disappointed that this theory has NO clear and physical images, and is only "mathematical" language after learning it for a long time.
On this page, we explain this fact as clearly as possible.

Special relativity → Klein-Gordon equations.

(Fig.2) Einstein energy-momentum relation.

Acoording to the special relativity, when the object is moving at the velocity of " u ", its mass becomes heavier.
Considering this relativistic mass, they defined four-momentum.

As shown on this page, relativistic four-momentum (= momentum p, and energy E ) transform like spacetime under Lorentz transformation.
This is natural, because four-momentum is made from four vector (= ct, x, y, z )
(Eq.0-1)

These four momentums satisfy energy-momentum relation of Fig.2, from which all quantum field theories start.
And the equation of Fig.2 holds in any veclocity "u", so Fig.2 is called "Lorentz-invariant scalar".
( "Scalar" means they do not change under Lorentz transformation. )

In the relativistic quantum field theory, we learn about Klein-Gordon ( K-G ) equation, first.
Because the meaning of Klein-Gordon equation is just equal to special relativity of Fig.2.
Klein-Gordon equation is gotten by replacing momentum and energy of Fig.2 by operators of quantum mechanics.

(Eq.0-2) Klein-Gordon equation.

As I said, Fig.2 is invariant under Lorentz transformation, so Eq.0-2 must be Lorentz-invariant, too.
As a result, the wavefunction φ of Klein-Gordon equation must contain infinite momentums in the interval of integration , which diverge to infinity.
If it contains finite momentums, the interval of integration changes under Lorentz transformation, which means Eq.0-2 is NOT Lorentz invariant.
So it must contain infinite momentums.

Klein-Gordon → Dirac equation.

As seen in Eq.0-2, Klein-Gordon equation is second-order time derivative and very inconvenient to use.
So they got first-order Dirac equation based on Klein-Gordon equation.

(Eq.0-3) Dirac equation.

In Eq.0-3, γ repsresent 4 × 4 γ matrices, and p0 represents E/c.
As you see in Eq.0-3, to get the first-order relativistic equation, we have to rely on "mathematical" matrices.

So the present relativistic quantum field theory is only a "mathematical" thing, which doesn't have physical images at all.
Next we prove Dirac equation of Eq.0-3 is just equal to Klein-Gordon equation.

(Eq.0-4) Dirac equation = Klein-Gordon.

As shonw in Eq.0-4, when we multiply Dirac equation by its pair operator from the left side, we know Dirac equation is just equal to Klein-Gordon equation (= special relativity ).
To get the relation of Eq.0-4, the 4 × 4 γ matrices must satisfy
(Eq.0-5)

where σ is 2 × 2 Pauli matrices.

Pauli exclusion principle and antiparticles.

So they insist relativsitic Dirac equation inludes quantum mechanical "spin".
In fact, Dirac equation is most influential in QFT, because all fermions such as electrons, positrons, neutrinos, and quarks are Dirac equation.

And all of QED, standard model and string theory rely on this Dirac equation.
Dirac wavefunction (= ψ ) of Eq.0-3 can be expressed as
(Eq.0-6)

Like Klein-Gordon, the interval of integration is infinite in Eq.0-6, which causes serious divergence.
And operators "c" and "d" denote "particle" and "antiparticle", respectively.
And as I explain later, to get the positive energy, these operators such as "c" and "d" must obey anticommutation relation.

So they insist Dirac equation shows the mysterious force of Pauli exclusion principle.
Due to the anticommutation, the same two operators become zero, as shown in Fig.3.
( But unfortunately, there are NO physical images at all here. )

Anticommutation = Pauli exclusion principle ?

(Fig.3) Pauli exclusion principle = anticommutation = Math ??

This anticommutation relation is also used in string theory.
But as you see in Eq.0-6, we can NOT know the real nature of Pauli exclusion principle only from this abstract "mathematical" expression.
So even if you learn about the string theory, you cannot know what the mysterious force of Pauli exclusion principle really is.

There are almost infinite academic papers about spin in the fields of spin electronics and condensed matter.
But there is NO single paper which shows what the spin really is in them, even though the world has advanced greatly in science and technology now. This is strange.
As the Pauli exclusion principle and Schrodinger's wavefunction remain a riddle, the present quantum chemistry is useless and only a "mathematical" thing.

Dirac equation with photon is QED.

(Eq.0-7) Dirac Lagrangian + photon → QED.

As I explain later, if we consider classical Lorentz force, we need to replace the momentum operators in Dirac's Lagrangian ( Eq.0-7 ).
The vector potential "A" of Eq.0-7 represents "photon".

From this, we obtain the interaction term between photons and fermions, which is used in the quantum electrodynamics (QED) and gauge theory.
As the relativistic restriction is too strict, this very abstract "mathematical" term is the only interaction.
( And can you actually imagine "what the photon really is" only from this mathematical expression ? )

(Eq.0-8) Weak force interaction.

In the quantum electrodynamics, they artificially remove and renormalize infinite values into the charge and mass of the particle. ( So bare charge and mass is infinite. )
Eq.0-8 represents the interaction between W boson and fermions. ( See this page. )

If this W boson has no mass, it is similar to the photon interaction of Eq.0-7.
As a result, Weak interaction becomes renormalizable.

But do you think these imaginary infinite bare charge and mass really exist ?
The most serious thing is that the present relativistic quantum field theories such as QED, standard model and string theory have NO clear physical images, which we have expected.

Virtual particles = self-contradiction of relativity.

Suppose one electron (= p ) at rest absorbs one photon (= q ) and becomes p' electron.
(Fig.4)

The energy of an electron at rest (= p ) is only rest mass energy ( E/c = mc ).
( The momentum of this electron at rest is zero. )
And an electron (= p' ) after absorbing photon has the momentums.

As shown in Eq.0-1, the electrons ( before and after the interaction with photon ) need to satisfy energy-momentum relation,
(Eq.0-9)

From Fig.4 and the conservation of energy and momentum, the photon's energy and momentum is
(Eq.0-10)

As you know, photon's mass is zero, but from Eq.0-10 and Eq.0-9,
(Eq.0-11)

Eq.0-11 means this photon's mass is imaginary number !
Because energy-momentum equation of Eq.0-11 must be zero or less than zero for real particles.
(Eq.0-12)

So Eq.0-11 shows this photon is tachyon, which disobey special relativity.
In spite of this fact, they often use this tachyon-like particles in the accelerator and QED.
As a result, relativistic quantum field theory include self-contradiction.

Though they insist uncertainty principle of energy and time allows this virtual particles to appear for very short time.
Do you think these contradictory particles really exist ??

The recent experiment using the charged muon has showed that the proton shrinks in size (radius) , and is about 4% smaller than previously thought. (Pohl, R. et. al. Nature 466, 213-216 (2010).)
If this is true, the quantum electrodynamics (QED) itself is wrong or has something missing, because this result is different from that of QED.

Notations and introduction.

[ Notations.]

(Fig.5) Lorentz-invariant scalar.

In Fig.5, the light is emitted from the origin ( x=x'=0, t=t'=0 ) and after a while, detected at some time of t or t' in each frame ( K and moving K' frames ).

According to special relativity, its speed muest be always "c" in any reference frames.
So the equation of the last line in Fig.5 does NOT change in any reference frames.
This equation is called "Lorentz-invariant scalar".

In the relativistic quantum field theories, we start from the invariance of the action S ( or Lagrangian ) to get the equation of motion.
So Lorentz-invariant action (or Lagrangian ) is most important concept in the relativity.

Relativistic Four-vector.

(Eq.1-1) Four-vector.

To express scalar such as the equation of Fig.5, it is convenient to define four-vector such as Eq.1-1.
In Eq.1-1, when the index μ is at the lower position (= xμ ), only the zero component ct becomes negative ( x0 = - ct ).
Other components ( μ = 1, 2, 3 ) do not change their sign depending on their index position.
(Eq.1-1')

This notation rule is called ( -1, 1, 1, 1 ) version metric tensor.
( If you want to change to (+1, -1, -1, -1) version, see appendix.)

Using Eq.1-1, the equation of Fig.5 can be expressed as
(Eq.1-2)

The notation of Eq.1-2 is often used in the relativistic theory.
The scalar can be expressed as a combination of variables of lower and upper indices.

And then the same variable (= for example, μ ) is used twice in one term, it means the sum of four terms with respect to μ = 0, 1, 2, 3.
( Even when Σ symbol is omitted, its meaning remains the same. )

Here we define Minkowski metric tensor, as follows,
(Eq.1-3)

Using this metric tensor in four vector of Eq.1-1, we can change the index position ( xμ → xμ ), as follows,
(Eq.1-4)

In Eq.1-4, "ν" is used twice in one term, so it means the sum of four-terms with respect to ν = 0, 1, 2, 3.
Using the relation of Eq.1-4, the scalar equation of Eq.1-2 can be expressed as
(Eq.1-5)

According to the special relativity, the relativistic momentum and energy change as four momentum.
(Eq.1-6)

Using Eq.1-6, the Lorentz-invariant energy-momentum relation of Fig.2 can be expressed as
(Eq.1-7)

Next we explain the derivative.
The partial derivative symbols of space coordinates (i = 1, 2, 3) are expressed as,
(Eq.1-8)

The sign of the space derivative doesn't change depending on the index position.
The partial derivative of the zero component (= time ) is
(Eq.1-9)

where x0 = "ct".

Different from Eq.1-1, when the index " 0 " is at the upper position, the time derivative becomes negative.
Because the normal derivative can be treated as covariant vector as shown on this page .
( In this page, you may consider these notations just as rules. )

As other four vectors, four current density is known.
(Eq.1-10)

where ρe is "charge density", and J means current density.
Eq.1-10 means ρe and J transfrom like the spacetime ( t, x, y, z ) under Lorentz transformation,
(Eq.1-10')

The definition of Eq.1-10 is indispensable for the relativistic Lorentz force law and the relation between current and charge density.

But Eq.1-10 causes observer's supernatural power, as shown on this page.
Four currents of Eq.1-10 change harmonizing with four vector potentials ( A ).
Of course, these definitions were "artificially" introduced to combine special relativity and Maxwell equations.

Lagrangian is most important concept in QFT.

In the relativistic quantum field theory (QFT), Lagrangian is most important concept.
From Lagrangian, we obtain important equations such as Klein-Gordon and Dirac equations.
( So it is safe to say that Lagrangian is everything to QFT. )

(Fig.6) Lagrangians of classical and QFT

Fig.6 left shows classical Lagrangian ( L = kinetic T - potential V energies ).
Fig.6 right shows Klein-Gordon Lagrangian, which has less physical meanings than classical.

You may remember this form of Lagrangian is decided only for getting Klein-Gordon equation.
Of course, as Klein-Gordon equation is Lorentz invariant, this Lagrandian needs to be Lorentz invariant scalar, too.

(Fig.7) Euler-Lagrange → equation of motion.

In the classical mechanics, when the action S is invariant under some coordinate transformation, Lagrangian gives the equation of motions through Euler-Lagrange equation.

Also in the quantum field theory, the same logic is used.
The difference is they differentiate Lagrangian with respect to some field operator ( φ ) in QFT, so its derivative includes from zero to three components ( μ = 0, 1, 2, 3 ).

(Fig.8) Canonical momentum → Hamiltonian.

Hamiltonian means the total energy in the classical mechanics, and can be gotten from the canonical momentum and Legendre transform, as shown in Fig.8
The same logic is used also in QFT (= lower line of Fig.8 ).

In QFT, the contents of the field operator are important for Hamiltonian.
Because they include creation and annihilation operators, which form positive number operator (= aa or bb ) .

As I explain later, Dirac Hamiltonian gives negative number operator of antiparticles.
To avoid this, Dirac Hamiltonian needs anticommutation relations, which express Pauli exclusion principle, they insist.

Classical Lagrangian (L)

In classical mechanics, the Lagrangian is defined using kinetic energy T and the potential energy V, as follows,
(Eq.1-11)

where "dot" means time derivative.

Lagrangian is a tool for getting the equation of motion and its Hamiltonian (= energy ).
And the relativistic quantum field theory completely depends on these Lagrangians, though each meaning such as canonical momentum is unknown well in QFT.
They just believe these abstract Lagrangian without connecting them with "real" objects.

When we substitute the Lagrangian (L) of Eq.1-11 into the Euler-Lagrange equation of
(Eq.1-12)

We can get the equation of motion.

By the way, how can we get the above Euler-Lagrange equation ?
The time integral of the Lagrangian is called the action denoted by S.
(Eq.1-13)

In the quantum field theory, we use Lagrangian density instead of Lagrangian, so this action needs to be the integral with respect to all spacetime (= d4x ).

When the action S is stationary, the first-order change of
(Eq.1-14)

must be zero.

In the relativistic theory, the stationary action S needs to include Lorentz invariance.
Considering the changes of q and the time derivative of q, Eq.1-14 is expressed as
(Eq.1-15)

where
(Eq.1-16)

where q is a generalized coordinate.

The endpoints ( t1 and t2 ) of the action are fixed, so Eq.1-16 becomes zero, as follows,
(Eq.1-17)

As a result, when the change of the action (= Eq.1-14 ) is zero, the following condition is satisfied.
(Eq.1-18)

Finally, we get Euler-Lagrange equation of
(Eq.1-19)

Euler-Lagrange equation of quantum field theory.

In case of quantum field theory, we do variation with respect to some field operator ( for example, φ ), as follows,
(Eq.1-20)

In Eq.1-20, we should use the derivatives of φ with respect to xμ = ( ct, x, y, z ), because φ is a function of all of them.
And like Eq.1-16 and Eq.1-17, the endpoints of the action are fixed ( δφ = 0 at a1 and a2 ), so total differential of the second line is zero.

Hamiltonian and canonical momentum.

Getting Hamiltonian through Legendre transform from Lagrangian is one of the most important ways in the relativistic quantum field theory (QFT).
The major difference between classical and QFT is QFT has less physical images than classical mechanics.
So we have to accept them only as rules in QFT.

In Lagrangian, the canonical ( or generalized ) momentum (= p ) is defined as
(Eq.1-21)

Hamiltonian (= total energy ) is expressed as the Legendre transform of the Lagrangian, as follows,
(Eq.1-22)

When Lagrangian is Eq.1-11 (= classical ), Hamiltonian becomes
(Eq.1-23)

So we get the classical Hamiltonian (= kinetic and potential energies ) of
(Eq.1-24)

Maxwell equations and interaction with charges.

[ Lorentz force and interaction with photons. ]

(Fig.9) Lorentz force → QED interaction term ?

In the quantum electrodynamics (QED) and the standard model, the interaction terms among particle + (anti) particle + photon is the most important concept in Feynman diagram.

This interaction term is based on the classical Lorentz force.
But as I said above, this interaction term causes "virtual" photon, which disobeys special relativity.

(Fig.10) Overview.

First, we find Lagrangian which gives Lorentz force relation through Euler-Lagrange equation.
Second, using usual definitions of canonical momentum, we obtain Hamiltonian from the Lagrangian.

Third, we consider momentum and energy operators change by external electromagnetic fields, and find their replacements in Hamiltonian.
These replacement are also used in Dirac equation, which causes the interaction with photon in QED.

When the charge ( e > 0 ) is moving in the external electric (E) and magnetic field (B), the equation of motion is
(Eq.2-1)

This force is called "Lorentz force".

Here we use "four-vector potential" like current density, as follows,
(Eq.2-2)

In Eq.2-2, "A" is vector (magnetic) potential and "φ" is scalar (electric) potential .
"Four-vector" means they transform like spacetime ( ct, x, y, z ) under Lorentz transformation.

(Eq.2-2') Lorentz transformation.

From the historical viewpoint, these artificial definitions were introduced for combining special relativity and Maxwell equation.
Because as you know, the electromagnetic waves, which speed "c" is gotten from Maxwell equation, always travels at the velocity of "c" in any reference frames.

The transformation of Eq.2-2 is indispensable for Maxwell equation to be invariant in any frames.
So the contradiction between Maxwell equation and relativity means the special relativity is wrong. ( See also this page. )

Using these vector and scalar potential, electric (E) and magnetic (B) fields can be expressed as
(Eq.2-3)

(Eq.2-4)

These "artificial" definitions were indispensable for relativity and Maxwell equation.
( Frankly speaking, vector potential "A" itself was introduced for special relativity. )

Lagrangian, which leads to Lorentz force relaton of Eq.2-1, is known to be
(Eq.2-5)

First we consider only the x (= 1 ) component of motion.
Substituting L of Eq.2-5 into Euler-Lagrange equation of Eq.1-19,
(Eq.1-19)

the first term is
(Eq.2-6)

where A and φ are functions of time and space coordinates.

And the second term of Eq.1-19 is
(Eq.2-7)

and
(Eq.2-8)

Eq.2-8 is NOT partial differentiation with respect to t.
And we need to consider that the space coordinates ( xj ) of A also include time t.
Because the vector potential A at the point of the moving particle is related.

From Eq.2-6 and Eq.2-8, Euler-Lagrange equation of Lorentz force becomes
(Eq.2-9)

Next we confirm that Eq.2-9 means Lorentz force relation (= Eq.2-1 ).
Substituting Eq.2-3 and Eq.2-4 into Eq.2-1,
(Eq.2-10)

Eq.2-10 is just the same as Eq.2-9.
So we can prove that Eq.2-5 means Lagrangian of Lorentz force.

Using Eq.1-21, the canonical momentum (= p ) is
(Eq.2-11)

This is equal to Eq.2-7.

From Eq.1-22, Hamiltonian of Lorentz force is
(Eq.2-12)

So the magnetic force (= vector potential A ) doesn't contribute to the total energy.

And the momentum of Eq.2-12 can be expressed using the canonical momentum of Eq.2-11, as follows,
(Eq.2-13)

So canonical momentum (= p ) means de Broglie's waves of the charged particle.
(This may need more discussion...)

So in the quantum field theory considering Lorentz force, we need to use the following important replacement,
(Eq.2-16)

The replacement of Eq.2-16 in the momentum and Hamiltonian operators is most important concept in the quantum electrodynamics (QED) and gauge theory.

When the vector potential A transform like four-vector under Lorentz transformation, this replacement can be used in Klein-Gordon equation.
( Eq.2-16 is considered as classical limit. )

This transformation of vector potential is linked to the transformation of four-current density.
So if the four-current is wrong, this replacement in QED is wrong.
As shown in Eq.5-46, Dirac Lagrangian uses this replacement, which leads to QED.
But Dirac Lagrangian is first-order, in which electric current is inconsistent with usual current, as I explain later.

Maxwell's equations and its Lagrangian

(Fig.11) Overview.

Also in Maxwell equation, first we find Lagrangian which can give Maxwell equations through Euler-Lagrange equation.

As shown in Eq.2-2, Eq.2-3, Eq.2-4, they "artificially" introduced new concepts such as four vector potentials to agree with special relativity.
So we need to express usual classical Maxwell equation using vector potential "A".

First we define the new concept of antisymmetric tensor ( Fμ ν ), as follows,
(Eq.3-1)

where A means the vector (and scalar) potential.

Lagrangian for Maxwell's equations can be defined as
(Eq.3-2)

where μ0 means vacuum permeability.
As I said above, we need to sum up 0, 1, 2, and 3 components in both μ and ν, because both those varaibles are used twice in one term.
And as I said in above section, Eq.3-2 is scalar form (= a pair of upper and lower indices in μ and ν ).

When we substitute Lagrangian of Eq.3-2 into the following Euler-Lagrange equation ( see also Eq.1-20 ), we get the usual Maxwell equations in the form of vector potentials.
(Eq.3-3)

We can get the usual Maxwell's equation.
( Replacing E and B fields by A and φ, this proves to be usual Maxwell equation. )
Here we explain about the detailed calculation method of Eq.3-3.

In case of μ=1, using Lagrangian of Eq.3-2, the first term of Euler-Lagrange equation of Eq.3-3 is
(Eq.3-4)

The differentiation with respect to A1 acts only on the second term.

And we think about the case of ν = 2 ( μ = 1 ) in Eq.3-3.
In this case, two terms including F12 or F21 are related.
So the second term of Euler-Lagrange equation of Eq.3-3 is
(Eq.3-5)

where we use
(Eq.3-6)

As shown in Eq.1-1 and Eq.1-8, the sign of 1,2,3 component does not change depending on the index position ( upper or lower ).
We should worry only about the zero componet about the index position.

For example, when μ = 1 and ν = 0, terms including F10 and F01 are involved, so the second term of Eq.3-3 becomes
(Eq.3-7)

where we use
(Eq.3-8)

The sign of Eq.3-8 is opposite to the result of Eq.3-5.
But considering Eq.3-8, the final derivative becomes the same form as Eq.3-3,
(Eq.3-9)

As a result, in all cases we obtain the same form of Maxwell equation.

According to Eq.2-4, the magnetic field B can be expressed, as follows, (using F)
(Eq.3-10)

And according to Eq.2-3, the electric field E is
(Eq.3-11)

where A0 = - A0 = φ/c. ( See Eq.2-2. )
( As I said above, these artificial definitions are to combine relativity and Maxwell equations. )

Replacing Fμ ν by elecrtomagnetic fields like Eq.3-10 and Eq.3-11, we can prove that Eq.3-3 means the usual Maxwell's equations.
For example, μ = 1 in Eq.3-3,
(Eq.3-12)

F11 is zero in Eq.3-1.
Eq.3-12 is just equal to Maxwell equation of
(Eq.3-13)

In the same way, when μ = 0,
(Eq.3-14)

where J0 = -J0 = cρe. (See Eq.1-10).

As shown on this page, this definition of four current causes our supernatural powers to "teleport" charges without touching them.

Next we explain a little about the calculation process of Eq.3-3.
As I said, two Fs of Eq.3-2 means
(Eq.3-15)

In the case of
(Eq.3-16)

Differentiating Eq.3-2 (Eq.3-15) with respect to Eq.3-16,
(Eq.3-17)

Considering Eq.3-15, we also need to calculate
(Eq.3-18)

Summing Eq.3-17 and Eq.3-18, 1/4 value of Eq.3-2 vanishes in Eq.3-3.
When you deal with zero compoment, you need to care about the relation between the sign and index position.

Klein-Gordon ( scalar ) field

[ Theory of Relativity → Klein-Gordon equation ]

As shown on this page, the theory of relativity satisfies the relation among four-momentum,
(Eq.4-1)

(-1,1,1,1) version notation of Eq.1-6 is used.

In the quantum mechanics, the relativistic energy (E) and momentum (p) are replaced by the operators like
(Eq.4-2)

The momentum of Eq.4-2 is de Broglie relation.
Using Eq.4-2, Eq.4-1 can be expressed as
(Eq.4-3)

where ∂0 = - ∂0 and ∂i = ∂i ( i = 1, 2, 3 ).

Eq.4-3 is called Klein-Gordon equation.
The wavefunction φ(x) is called the Lorentz scalar field (= Klein- Gordon field ).
This scalar field expresses Higgs particles. ( Other particles are Dirac fields, except for photon. )

(Fig.12) Overview of Klein-Gordon field.

Fig.12 shows the overall picture of Klein-Gordon field.
In relativistic quantum field theory, basic tools are Lagrangian, Euler-Lagrange equation, canonical momentum and Hamiltonian.

The field φ(x) satisfies Eq.4-3, so it can be expressed as
(Eq.4-4)

where
(Eq.4-5)

and
(Eq.4-6)

which is equal to Eq.4-1.

And as shown in Eq.4-4, in quantum field theory, one particle is created (or annihilated ) in all space at the same time to keep Lorentz invariance (= symmetry).
And Eq.4-4 contains infinite particles of every momentum and energy (from -∞ to + ∞) to Lorentz symmetry.

So the quantum field theory doesn't mean the real world.
And coefficient angular frequency ω of Eq.4-4 is important for Lorentz invariance and getting for the energy values.

Relativistic wavefunction = faster-than-light tachyon !

As you know, the velocity of the wave is gotten from frequency f × wavelength λ ( v = fλ ).
So the velocity of this wavefunction becomes
( Fig.T-1 )

This speed is faster-than-light.
This unreasonable wavefunction is caused by the relativistic invariance of the equations.
Ironically, the relativistic restriction causes imaginary "tachyon".

They insist if we combine negative and positive energy solutions in Dirac equation, this tachyon vanishes.
But each electron or positron is faster-than-light. This is strange.

[ Lorentz invariance in propagator ]

The equations from Eq.4-7 to Eq.4-11 ( complex scalar field ) are moved into this page.
(Eq.4-11)

where θ means "step function".

As shown in Eq.4-11, the propagator contanins the integration of d4p, and the interval is from minus infinity to plus infinity to be Lorentz-invariant, which causes serious divergence.

Getting Hamiltonian

Lagrangian for getting Hamiltonian is defined as
(Eq.4-12)

where "dot" means time derivative.

Substituting Eq.4-12 into Euler-Lagrange equation of Eq.1-20, we get Klein-Gordon equation.
(Eq.4-13)

From Eq.4-12 and definition of Eq.1-21, canonical momentums is
(Eq.4-14)

From Eq.1-22, Hamiltonian (density) of Klein-Gordon field is
(Eq.4-15)

Substituting Eq.4-4 into Eq.4-15, and integrating it in all space, we can get the Hamiltonian of
(Eq.4-16)

You need to remember that quantum field theory is just very abstract "math" like Eq.4-16.
In Eq.4-16, aa represents the number operators (= N ).

The order of creation and annihilation operators in the number operator is very important, and it causes Pauli exclusion principle in Dirac equation.
Next we actually calculate Hamiltonian and get Eq.4-16.

First, we substitute the scalar wavefunction of Eq.4-4 into φ2 of Eq.4-15.
Each momentum k or k' ( from -∞ to +∞ ) is independent from each other in the two φ2.
So we use
(Eq.4-17)

Using Eq.4-17, φ2 becomes
(Eq.4-18)

where we use
(Eq.4-19)

Eq.4-19 is commutation relation of scalar field.
As I explain later, this commutation relation is necessary for de Broglie relation between canonical momentum and field operator.
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, Hamiltonian becomes
(Eq.4-22)

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

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

And the last term of Eq.4-23 is divergent term.
But QFT neglects this divergent term ( or it can be eliminated by supersymmetry ).
These divergent terms are caused by the fact that relativistic QFT must contain every momentum (=k ) to keep Lorentz invariant.

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

Eq.4-24 can be confirmed by a little long calculations.
As you know, the famous commutation relation of Eq.4-24 is based on de Broglie's relation of
(Eq.4-25)

Of course, various particles including electrons are known to obey de Broglie's relation.
But "anticommutation relation" of Dirac equation violates this basic rule (= de Broglie's relation ) even in the electrons.
(This is explained in the latter part.)

[ 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 ωk 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 ωk 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, 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.

What is Dirac equation ? (real meaning and its defects)

[ Derivation of Dirac equation ]

Dirac fields are the most important, because it expresses leptons such as electrons and muons.
And the quarks are also expressed as Dirac fields.
( So if special relativity is wrong, all these theories based on Dirac equation are wrong. )

But unfortunately, these particles are only "mathematical" products, which reasons are explained in this section.
(Of course, electrons really exist. But QM can describe them only as "mathematical" products such as spinors.)

[ Overview of Dirac field. ]

(Fig.13) Overview of Dirac field.

To change the Klein-Gordon (K-G) equation to the first-order in the time derivative "compulsorily", we must rely on the 4 × 4 matrices".
To rely on the matrices means that the Dirac equation has strong "mathematical" property rather than "physical" property.

Even in the final theory of superstring, the fermions need to be expressed using spinors + γ matrices.
So as far as we stick to quantum mechanics (+ spinor ), we can NOT express electrons as reality.

Dirac equation is,

(Eq.5-1) Dirac equation.

where ψ(x) is the 4-component wavefunction (4 × 1 matrix), γμ is 4 × 4 matirix.

Eq.5-1 is equal to
(Eq.5-2)

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

Dirac equation of Eq.5-2 must satisfy Klein-Gordon equation to be equal to special relativity.
Adding the following equation from left side of Eq.5-2,
(Eq.5-3)

To agree with Klein-Gordon equation, γ matrices must be 4 × 4 matrices.
( Since about this time, quantum field theory has been "mathematical" theory rather than "physical", I think. )
γ matrices need to satisfy the following condition of
(Eq.5-4)

Eq.5-4 means different γ matrices anticommute with each other, and the same two γ become 4×4 I or -I.
We often use metric tensor "g" to express the relations of Eq.5-4.
(Eq.5-4')

(metric tensor)

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

where I means 2 × 2 identity matrix.
And
(Eq.5-6)

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

The relations of Pauli matrices in Eq.5-6' are often used in QFT, so you need to remember them.
The problem is we do NOT know what these Pauli matrices really are in a "realistic" sense.

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

And γ matrices satisfy anticommutation relation like,

Using above relations of γ matrices, Eq.5-3 is just equal to the following Klein- Gordon equation of
(Eq.4-1)

Other terms 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 appears again, when we try to get Hamiltonian of Dirac field.

Plane wave solution of Dirac equation

Plane wave solution of Dirac equation can be expressed as
(Eq.5-10)

where ψ+ is a plus energy solution, and ψ- is a minus energy solution.
And u(p) and v(p) mean 4 × 1 matrices.

The meanings of the exponential functions ( kx ) of Eq.5-10 are equal to Klein-Gordon of
(Eq.4-5)

and
(Eq.4-6)

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

where p0 = - p0.
And the minus energy is
(Eq.5-11')

From Eq.5-11 and Eq.5-11', matrices of u(p) and v(p) need to satisfy
(Eq.5-12)

and
(Eq.5-13)

Using Eq.5-15, the conjugate transpose of the left equation in Eq.5-12 becomes
(Eq.5-14)

where u(p) (1 × 4 matrix) means the conjugate transpose of u(p).
u(p) = (u(p)*)T

Here we multiply Eq.5-14 by γ0 from right side, and use the relations of
(Eq.5-15)

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.
And
(Eq.5-16)

We can get the right equation of Eq.5-12.

Properties of matrices of u(p) and v(p)

The important point is that we can NOT get the concrete solution of Dirac equation even in the simple plane wave.
Because its solution must include matrices.
So it can be said that Dirac equation is a "mathematical" thing.

We define the solution of u(p) as
(Eq.5-17)

( This solution u(p) must include matirices γμ. )

Substituting Eq.5-17 into the left equation of Eq.5-12,
(Eq.5-18)

Eq.5-18 means that Eq.5-17 satisfies Klein-Gordon equation.
So Eq.5-17 is appropriate for the solution of Dirac equation.

In the same way, the solution v(p) can be defined as
(Eq.5-19)

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

Replacing γμ 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 pjσj means the sum of 1-3 components.

We define the matrices of u(0) and v(0) in Eq. 5-17 and Eq.5-19 as
(Eq.5-21)

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

Using 5-20, Eq.5-17 is equal to
(Eq.5-22)

where
(Eq.5-23)

So conjugate transpose of u(p) × γ0 (= Eq.5-16) can be expressed as
(Eq.5-24)

where Pauli matrices σ = σ.

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

where
(Eq.5-26)

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

Substituting Eq.5-26 into Eq.5-25, we can get
(Eq.5-27)

here we use
(Eq.5-28)

where σ means "spin" of Eq.5-23.

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

This minus sign is caused by the minus energy of exponential function in v(p).

And if we delete γ0 in Eq.5-24,
(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's Lagrangian

Lagrangian which leads to Dirac's equation can be defined as
(Eq.5-32)

Of course, this Lagrangian must be Lorentz invariant scalar.

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

where the second term of Eq.5-33 is zero. (See Eq.5-32.)

So the first term of Eq.5-33 is
(Eq.5-34)

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

And the conjugate transpose of Dirac equation (Eq.5-1) is
(Eq.5-35)

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 equal to Eq.5-1.

So if we use the usual ψ(x) in Euler-Lagrange equation,
(Eq.5-36)

Eq.5-36 is equal to Eq.5-35.
As a result, we can get Dirac equation from Lagrangian of Eq.5-32.

Dirac's Hamiltonian

The canonical momentum of Dirac equation is (like Eq.1-21)
(Eq.5-37)

And Hamiltonian is
(Eq.5-38)

where the first line means the sum of 1-3 components with respect to "j".

From Eq.5-7, Eq.5-38 is equal to
(Eq.5-39)

So this means Hamiltonian of Dirac equation.
(Of course, to get Eq.5-39, Lagrangian of Eq.5-32 is prepared.)

And the solution which leads to Hamiltonian of Eq.5-39 is expressed as
(Eq.5-40)

And its conjugate transpose (× γ0) is
(Eq.5-41)

These u(p) and v(p) are Eq.5-17 and Eq.5-19, respectively.
And c(k) and d(k) mean annihilation operators of particle and antiparticle, respectively.
So c(k) and d (k) are creation operators.

Substituting Eq.5-40 into Eq.5-39, and using Eq.4-5, Eq.5-30 and Eq.5-31, we can get
(Eq.5-42)

For example, c c part of Eq.5-39 is
(Eq.5-42')

Due to the minus sign of the first line in Eq.5-42' (= red line ), d d part of Eq.5-42 becomes minus.
( Because Dirac equation is the first order in the time derivative, which is different from Klein-Gordon. )
(Eq.5-42'')

To get the positive number operator like d(k) d(k) in the antiparticle of Eq.5-42, the following anticommutation relations must be satisfied.
(Eq.5-43)

Using Eq.5-43, Eq.5-42 becomes
(Eq.5-44)

We can get positive number operator of d(k) d(k) (= positive energy in antiparticle ).
So they insist Eq.5-43 represents the mysterious force of Pauli exclusion principle.

But unfortunately, we cannot know what this Pauli exclusion principle really is, only from this very abstract "math".
And the superstring theory uses the same math of anticommutation relation. ( It makes NO progress. )
So even if we learn about superstring theory, we cannot know its real meaning. This is strange.

And the sign of the vacuum energy in Eq.5-44 is opposite from Eq.4-16, which is used in the supersymmetric theory such as superstring.

Dirac equation proves anticommutation relation of fermions ?

So the quantum mechanical physicists claim that Dirac equation proves the antisymmetric property of fermions.
(= Pauli exclusion principle )

Dirac equation is first-order in the derivatives, which is different from Klein-Gordon equation.
So Dirac equation can not hide the minus energy solution.
At the starting point of Eq.5-10, we define the plus and minus energy solutions, which causes the minus energy of Eq.5-42.

And if Eq.5-43 needs to be satisfied, the relation of ψ and its canonical momentum π (= Eq.5-37) becomes
(Eq.5-45)

This anticommutation relation of Dirac's canonical momentum is different from Eq.4-24 of Klein-Gordon equation.
The important point is that Eq.5-45 is inconsistent with de Broglie's relation of
(Eq.4-25)

If we define the new property of canonical momentum like Eq.5-45, we have to find out the new theory which replaces de Broglie's theory.
But 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 strange.

Dirac's current density contradicts original electric current.

Using the Lorentz force of Eq.2-16, and Lagrangian of Maxwell equation of Eq.3-2,
(Eq.5-46)

Eq.5-46 means the combination of Dirac equation and Maxwell equation.
(And here, particle's charge is supposed to be +e )

The following part of Eq.5-46 is
(Eq.5-46')

So the replacement of Eq.2-16 is used in Eq.5-46 with respect to μ= 0, 1, 2, 3.
The scalar potential φ = cA0 = - cA0.

Interaction term ( of particle and vector potential A ) of Eq.5-46 is

(Eq.5-47) Interaction term.

The interaction term of Eq.5-47 is very important in QED.
Because Eq.5-47 is the ONLY interaction among fermions (= electron, quarks, positron.. ) and photon.
In the abstract relativistic QFT, we cannot think about various patterns of their interactions.
Of cource, Eq.5-47 causes virtual photons which disobey special relativity.

We compare Eq.5-46 with Eq.3-2 and Eq.3-3 of
(Eq.3-2)

and
(Eq.3-3)

As a result, Dirac's current (J) density becomes
(Eq.5-48)

So the physicists claim that Dirac equation has succeeded in expressing the current density as shown in Eq.5-48.
But as shown in Eq.5-6, γ matrices consist of Pauli matrices.

It means γ matirices means spin components, which has NO relation with the direction of current.
But in Eq.5-48, spin component decides the direction of the current Jμ.
So Eq.5-48 and Eq.5-47 are based on very strange interpretation.

But why we have gotten the result of Eq.5-48 in Dirac equation ?
In fact, γ matrices (= σ matirices) is not peculiar to Dirac equation.
Even in Schrodinger equation (or classical mechanics ), σ matrices is 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, the quantum mechanics uses the mathematical tricks like
(Eq.5-51)

where p includes derivative, which acts on before and after Aj.
So de Broglie wavelength of the particle is also used in derivative of the vector potential to get the magnetic field.
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 means that spin-magnetic field (= vector potential) interaction exists in Schrodinger equation, too.
And to agree with Bohr magneton ( of Bohr model ), spin g factor ge must be "2".
This is too good to be true ?

Partial differentiation of Lagrangian ( LI = - HI ) 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, Dirac equation is first order in the momentum as shown in Eq.5-46.
(So also in the vector potential, 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 in Dirac equation !
(So there is NO charge's current at this point in Dirac equation. )
This means that the first order Dirac equation can NOT use the replacement of Lorentz force (= Eq.2-16 ) correctly !
(Interaction of Lorentz force is "originally" considered in the second order equations of classical mechanics. )

Interaction term of Eq.5-47 is used also in the quantum electrodynamics and standard model.
So these theories include serious defect.

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

A photon particle is said to be exist, but in fact we can explain all about the photoelectric and Compton effect using only a concept of wave.
Actually, if we consider the photon as a electromagnetic wave ( not as a particle ), the strange faster-than-light transmission (entanglement) can be forgotten. (See this page!)
The quantum electrodynamics (QED) was made to express the "photon particle".

The electromagnetic fields (B and E) can be expressed by the vector potential (Ai) and the scalar potential (φ = cA0 = -cA0) as shown in Eq.3-4 and Eq.3-5.
When this Aμ is transformed as follows, (gauge transformation)
(Eq.6-1)

The B and E fields are unchanged if we take any function f(x).
( Try substituting Eq.6-1 into Eq.3-10 and Eq.3-11. )

So we can choose the f(x) which satisfies the following relation ( Lorentz gauge ),
(Eq.6-2)

The reason why we choose this Lorentz gauge condition is to keep the Maxwell's equations Lorentz-invariant.
Eq.6-2 is scalar form (= a pair of covariant and contravariant vectors ).

Using Lorentz transformation, Eq.6-2 changes to
(Eq.6-3)

So we can prove Eq.6-2 is Lorentz invariant.

Using Eq.3-1, Maxwell equations of Eq.3-3 can be expressed as
(Eq.6-4)

Using Lorentz condition of Eq.6-2, Maxwell equations of Eq.6-4 become
(Eq.6-5)

Eq.6-5 changes as covariant vector under Lorentz transformation.
As a result, the form of the equation of Eq.6-5 doesn't change under Lorentz transformation.

Here we consider the photon particle in the vacuum. ( J = 0 ).
This case is
(Eq.6-6)

Actually this equation turns out to be the same as the Maxwell's equation when B and E fields are replaced by Aμ (+ using Lorentz gauge condition).

The solution of this equation of Eq.6-6 is,
(Eq.6-7)

If we suppose the commutaion relation between field and canonical momentum operators, the annihilation and creation operators of this equation need to satisfy the following commutation relation,
(Eq.6-8)

here, only in the case of g00, the right side of Eq.6-8 becomes negative (= negative number ).
This serious problem originates in the property of the four-vector in the special relativity.
When we use the Lorentz gauge condition of Eq.6-2 again, this problem can be solved.
( But in QED, this unreal zero component particle is used to get the important g-factor. )

This is what the "photon" really means.
In other words, the photon particle is only a "mathematical" product created by the creation operators.

The quantum electrodynamics (QED) doesn't have the power to describe the photon particle clearly. Unfortunately, it only shows the "mathematics".
It is said that Coulomb force is caused by the "virtual" photons.
But QED can NOT describe this state more concretely than "mathematical" creation operators.
So we can NOT advance any more within the very strict restriction of relativistic quantum field theory.

[ Divergence and standard model ? ]

In the quantum mechanics and the quantum field theory, the states change as eiHt with time.
And to keep the equation "Lorentz invariant", this (interaction) Hamiltonian (H) must contain infinite particles including infinite momentum and energy.

If the equation contains only some particles which have "definite" momentum, Lagrangian changes under Lorentz transformation.
As a result, the form of the Dirac equation itself is broken under the Lorentz transformation

To avoid this, Hamiltonian must contain infinite particles with every momentum.
But this leads to the divergence problem when we consider the interaction between the particles using the propagation function.

To rule out the divergence, we must use the renormalization theory in which the infinity in the propagator is revomed by the new "artificial" infinity.
For example, the "bare" charge and mass of a electron must be infinite to cancel the "infinite stimulus" around the electron out.
So the relativistic QFT is very strange!
( See also this page and manipulated g-factor. )

Standard models of the elementary particles includes about 20 important parameters, which can be "intentionally" manipulated.
See also this page.
(So, these important parameters can NOT be gotten from the standard model. they are gotten from only the experimental results.)
So the standard model would be replaced by some other theories in the future.
I'm very grad if some specialists in the elementary particles find this new theory based on the real viewpoint like Bohr model !

As you know, "string theory" is very far from this real world, which can not be confirmed forever. ( See also this page. )
It is very unrealistic that string theory needs many dimensions (more than 10).
The "mathematical" string theory is just "Not even wrong."
Because we can not confirm whether the "unreal" string theory is wrong or not forever.
And the superstring theory includes infinite unreal particle models.

to

2012/9/22 updated. Feel free to link to this site.