Reply To: Einstein

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Gyula Szász

Unfortunately, the editors and reviewers of the physical journals, Phys. Rev., EJPC, ZNA and Foundation of Physics could not begin somewhat physically with my new quantum field theory and rejected all my emitted articles. Therefore, the new theory came not in the circulation of physical science.

A comparison of quantum field theories

The currently used quantum field theories quantize the energy and the interaction with the Planck constant h and the special relativity theory states the energy-mass equivalence relation, E = m∙c^2. I have defined another kind of quantum field theory in which only the sources of the interactions are quantized. The energy and the interactions are not quantized and the energy is not equivalent the mass. We have obviously to do with two physically quite different kinds of quantum field theories.

I collect the key features of the new quantum field theory:
– The sources of the interactions are the conserved elementary charges. We know two kinds of elementary charges, they are the elementary electric charges qi = {± e} and the elementary gravitational charges gi = {± g∙mi}. Assumption: Physically no more elementary charges exist.
– The elementary charges generate two kinds of continuous fundamental interaction fields which propagate with c. These fields are at the presence of charges non-conservative and the two fields are independent from each other, they do not influence each other. The universal gravitational constant is G = g^2/4π.
– The elementary charges are distributed on four stable elementary particles, e, p, P and E. Each particle has two kinds of elementary charges. The elementary pariticles are the electron (e), the positron (p), the proton (P) and the elton (E). The elton is usually called as “antiproton”. Assumption: Physically no more stable elementary particles exist.
– The masses me and mP are the elementary masses of electron (e) and proton (P). The inertial and the gravitational masses of the elementary particles, e,p,P,E, are in each case equal and only for the stable elementary particles are the inertial and the masses equal. Assumption: These stable particles are not composed of any other particles.
– There exists a general uncertainty: neither the positions, nor the velocities of the elementary particles can be ever exactly observed. Infinitely large and infinitely small relative distances between particles do not belong to a physical description.
– Time and space are homogeneous and the space is isotropic. Because the interaction propagation with c the time and space are connected; the Minkowski space describes the connectivity of space and time.

Results of my new quantum field theory:

An action integral

I = ∫Ω L(x) (dx)^4 (1)

can be constructed in finite ranges {x}εΩ of Minkowski space with the key features. The elementary charges, qi and gi, generate the vector fields A(em)ν(x) and A(g)ν(x). The Lagrange density L(x) is the sum of the free particles parts, of the free fields parts constructed with the Faraday tensors, F(em)νμ, F(g)νμ, and of the interaction parts

L(int)(x) = + j(em)ν A(em)ν(x) – j(g)νA(g)ν(x), (2)

whereby j(em)ν and j(g)ν are the probability density currents of the charges. All parts of L(x) are constructed Lorentz-invariantly. Since all parts of L(x) are caused by the conserved elementary charges the action integral does not depend on the boundary conditions of the surface of Ω containing some numbers of particles, Ni. The action integral is not an expression of energy. Within Ω there exist different kinds of subsidiary conditions, one kind for the fields and another kind for j(em)ν and j(g)ν. The subsidiary conditions of the fields are known as Lorenz conditions. The Lorentz conditions express that the fields propagate with the constant velocity c within Ω. With these subsidiary conditions for the fields and applying the Hamilton principle on the action integral, I, we get the Maxwell equations for the fields. The Maxwell equations for the electromagnetic field and the gravitation field differ only on the sign of the probability density currents of the charges.

The determination of the equations for the particle motions

In order to get a Lorentz-invariant action integral, we put for the particles parts in the Lagrange density, L(x), the continuity equations of particle numbers, ji(n)ν(x), for i=e,p,P,E, multiplied by the constants mi∙c

L(p)(x) = Σi=e,p,P,E mi∙c∙∂ν ji(n)ν(x). (3)

Additionally, the subsidiary conditions of the particles, Gi, are to be considered at the variation

Gi = ∫Ω ∂ν ji(n)ν(x) (dx) 4 = 0, i=e,p,P,E. (4)

Before we perform the variation we must express the j(n)ν (x) in a quadratic form, with the Dirac spinors and with the γν matrices

j(n)ν (x) = c ⋅ Ψi(x) γν Ψi(x). (5)

Ψi(x) = Ψi(x) γ0 are the adjoin spinors to the spinors Ψi(x) and the expression Eq. (5) has the correct transformation behavior under Lorentz transformation. γ0 γ0 = 1 is the unit four matrix. We have to use the spinors because neither the positions, nor the velocities of the particles are ever exactly known. Furthermore, the Noether charges

∫V Ψi(r,t) γ0 Ψi(r,t) d^3x = ∫V Σj=1,4 Ψ*i,j(r,t) Ψi,j(r,t) d^3x = Ni, i=e,p,P,E (6)

are used for the normalization of the spinors for each volume V and at each time t. The sum j is taken about the four components of the four dimensional spinor. Applying the Hamilton principle we get the equations of motions for the particles expressed with the spinors. The stationary of the variation of I, considering the subsidiary condition, Gi,

δI + Σj Σi=e,p,P,E λj∙δGi = 0, (7)

cause the appearance of Lagrange multipliers, λj, in the Euler-Lagrange equation of the spinors in using independent variations of the adjoint spinors Ψi(x) and Ψi(x) for each particle i=e,p,P,E.

Bound states of elementary particles

In order to get temporary stationary bound state of elementary particles, we have to consider conditional probabilities for particle density currents relative to center of mass system (COM) in the mutual fields of the particles: we are looking for the temporary stationary solutions for the particles and the mutual fields. The conditional probabilities depend on the relative distances of particles. Since the particle number conservations are further valid, the appearance of Lagrange multipliers, λi, is expected. Thus, the temporal stationary conditions are connected with the Lagrange multipliers in such a way that the probabilities to find particles in distances relative to COM are temporally stationary and the relative currents vanish. Simultaneously, the mutual interacting field is also temporary stationary. On this point enter the relativity in the physical description. We have to express the action integral, I, and the subsidiary conditions, Gi, relatively to COM. Thus, we have to use the spinors as conditional probabilities of particle density currents with relative coordinates. The finite range Ω must contains the unique point of COM. Generally, temporary stationary solutions of the variation for a λi are given with different values of an additional positive real parameter E. The largest possible discrete value of E belonging to a λi are labeled as the ground state with the bound energy E0(λi). The bound energies are always negative, therefore, we label with the positive E0(λi) the negative of the bound state energies. Generally, a set of discrete values of the parameter E exists with

E0(λi) > E1(λi) > E2(λi) > E3(λi) …. > 0 (8)

for each λi. A bound state is generally a superposition of temporary stationary solutions. Nevertheless, we don’t speak about energy quantization at the relation (8) because this superposition is connected not only to temporary stationary mutual fields, but also to the simultaneous presence of radiation component of the field. Bound states emit always radiation with continuous submission of radiation energy, until the energy E0(λi) is reached.

The physical interpretation of the bound state problem

We have to describe the problem of the capture of an electric charged particle, i, with the elementary mass, mi, in the electric field of other moving particles with elementary charges qj = { ± e} with the elementary masses {me, mP}. Thereby we know neither the initial positions, not the initial velocities of the particles (for instance for the captured electron by atoms). Generally, electric charged particles moving in an electromagnetic field radiate always electromagnetic waves and the waves (the fields) propagate with c. According to the subsidiary conditions of the particles, Eq. (4), Lagrange multipliers, λi, are appearing in the equations of particles motions and these constants ensure temporally stationary solutions at some real values Ei(λi) of the whole problem. Mathematically, we have to solve four coupled differential equations of the first order in space with the normalization conditions, Eq. (6), for the particles. Since the spinors describe the probability density currents of particles, in the four components of the spinors, Ψi,j(r), j=1,4, the three components of the velocities are coded. The result is finally the determination of the bound energies Ei(λi) of the particle system for the temporary stationary case. In the following we shall write for simplicity E0(λi) = E(bound) for the ground state belonging for one λi. and rename λi. with h.

Fortunately, in the case of two-particle systems we have the possibility to say something about the relative velocity and about the relative distance of the ground state of particle systems in the mutual temporary stationary interaction as a function of E(bound) and of the reduced mass m’ = mi∙mj/(mi+mj), without solving explicitly the variation problem.
For the Planck constant h, which describe the atomic shells, there exists an old expression set up by Sommerfeld

h = e^2/2c∙(m’∙c^2/2∙E(bound))^1/2 = e^2/2c∙1/α, (9)

for the hydrogen atom, where m‘ = me∙mP/(me+mP) ≈ me is the reduced mass of electron and proton and the bound energy of the ground state energy is E(bound) = 13.8 eV. However, in the current quantum mechanics is not understood why α = v/c has a value of α = 1/137.036. v is the relative velocity of the electron. With the same relation for h, Eq. (9), the positronium problem can also be determined with the energy of the ground state, E(bound; positronium) = 13.8/2 eV = 6.9 eV, since the reduced mass is m’ = me/2. We shall use the relation, Eq. (9), for each two-particle-systems with the opposite signs of electric charges in order to get other values for h, if we have different reduced masses m’ and different ground state energies, E(bound). With other words, we have with Eq. (9) a simple way to get expressions of the values of the constant h in cases of two-particle-systems. For two-particle-system we have the expression for the inertial mass

mi = mi + mj – E(bound)/c^2.

In case of an electron-positron system, (e,p), if we use the condition for the bound state that the inertial mass is zero, mi = 0, we have the condition

2∙me = E(bound)/c^2.

Since the inertial mass, mi, cannot be negative, this states the lowermost bound energy of the (e,p)-system. Setting this equation and the reduced mass m‘(e,p) = me/2 in the relation, Eq. (9) for h, we get another value

h0 = e^2/2c∙(1/8)^1/2 = h/387. (10)

Thus, we get a much smaller value for h0 as the Planck constant h. The relative velocities of the bound particle (e,p) can be calculated for instance according
(v/c)/(1-(v/c)2) = (2∙E(bound)/(me∙c2).
Since in the case of (e,p), we have two particles with the same mass, we must set in the half of the ground state energy, E(bound), in order to get (v/c) = 0.894. The particles move with the velocity 89.4% of c. We label this state of the electron-positron system as the electron-neutrino, νe = (e,p); the neutrino has as well the inertial as the gravitational mass zero.
We have also a simple relation for the calculation of the relative distance of particles, of the size of the ground states radii,

r = h^2/(4∙π2∙m’∙e^2), (11)

For the electron-neutrino, νe, we get r(νe) = 0.703∙10^-13cm. Besides the (e,p)-system, we can also calculate with h0, Eq. (10), for the (P,e), (p,E) and (P,E)-systems the ground state energies, the relative velocities and the sizes. The ground state energies of the (P,e), (p,E) are 2.04 MeV. The ground state energy of the (P,E)-system is E(bound) = 2∙mP∙c^2. The size are d(P,e) = 2∙r(P,e) = 0.702∙10^-13cm and the same size appear also for the (p,E)-system. With the h0 calculated ground state of (P,E) we get a size of r(P,E) = 0.383∙10^-16cm: we label this state as the proton-neutrino, νP = (P,E). According to the finite sizes of two-particle ground states, we can state: in the interaction of elementary particles no singularities can occur and particles with the same mass don’t annihilate.
In opposition to Einstein’s energy-mass-equivalence, E = m∙c^2, the electron-positron and the proton-elton pairs do not annihilate at their merging in our theory. The energies of particle systems and the electromagnetic interaction are not quantized with E = h ν in our new quantum field theory. The new quantum field theory is an atomistic theory of matter based on the four kind of stable particles e, p, P and E with the conserved charges qi and gi.

Unfortunately, the transmission of upper and lower indexes does not work. Double appearing indexes are summed.

Gyula I. Szász