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The quantum field theory based on the hypothesis of the absolute reference system is a classical non-relativistic theory, which is compatible with current quantum theory. This conclusion arises when one compares the theoretical results of quantum electrodynamics using the basic principles of this hypothesis. Wave equation, which replace s this of Schrodinger, is the classical wave equation of a peculiar electromagnetic wave, derived from the study of particle structure.

According to the hypothesis of the absolute reference system [

We will examine now the scattering of a photon by an electron, in the reference system of the laboratory (which is the earth’s frame of reference), that is Compton effect (ref. [

Based on what we have mentioned before about the absorption of a high energy photon from a free electron, the phenomenon studied will be accompanied by an increase in the mass of the electron equal to the equivalent mass of a bound photon

The total energy of the outgoing photon after the impact is equal to

Due to the conservation of momentum on the X axis, the following relation is taken:

The conservation of the momentum on the Y axis:

Of these two last relations:

According to the previous mentioned and the relation (1.1):

From these two latter relations, the change in the wavelength of the photon initially incident to the electron is calculated:

An experiment of controlling the correctness of a proposed dynamics, such as the dynamics of the absolute reference system, is that of W. Bertozzi1, which was carried out in the early 1960s, and refers to the measurement of the maximum speed of high energy electrons by a linear accelerator (ref. [

These electrical signals are transmitted on a down-turn oscilloscope via two wires of the same length (so that the signals need equal time to reach the oscilloscope). In this way the pulses displayed on the oscillator give the real time of transmission of the electron beam along the “drift space”.

While electron velocity measurements are determined directly using the oscilloscope, kinetic energy is determined from potential difference produced in the Van de Graaff generator and electric field in Linac. This is a strictly predetermined procedure, which has been tested in the laboratory by magnetic deflection methods.

To test any dependency of the electron velocity from the force exerted, due to the very strong electric field, an additional measurement acquired by the high energy electrons is made to a further embodiment comprising an aluminum disk on which impinge the electrons at the end of their path and a thermocouple to measure the temperature increase of the aluminum disc, so the added energy in the form of heat will be proportional to the increase in temperature. In addition, an additional device for measuring the charge collected in the disk is used, in this way to determine the energy transferred from each electron. Such energy measurements were made in the estimated accelerator energies at 1.5 MeV and 4.5 MeV (tested by the above-mentioned magnetic deflection methods), whereby the corresponding values, obtained with the heat increase measurement method in the aluminum disc, were 1.6 MeV and 4.8 MeV.

The results of the experiment, listed in

The comparison of experimental results with theoretical calculations of the special theory of relativity and of the hypothesis of the absolute reference system, is certainly the basic criterion of convergence of the experiment with these considerations. The theoretical values of kinetic energy of the special theory of relativity derive from the relation:

where m is the mass of the electron, u the electron velocity of the beam and

However, the transferred kinetic energy in the target molecules, i.e. the experimentally measured heat, is calculated based on the relative description of the electron collisions of the beam with the atoms of the material. In the absorbtion of a free high energy photon (i.e. of an, elementary, plane electromagnetic wave) from an electron, half of this energy is transferred as kinetic energy, while the other half is available for formation of additional elementary mass. Indeed, in the collision of the beam electrons with the target, the predominant image of the interactions is that of the polarized photons, as an image of elementary plane-waves, that act as interaction photons. Under these conditions the kinetic energy transferred to the target atom, in the form of heat, will be equal to:

kinetic | flight | electron | |
---|---|---|---|

energy | time | velocity | |

K, MeV | |||

0.5 | 3.23 | 2.60 | 6.8 |

1.0 | 3.08 | 2.73 | 7.5 |

1.5 | 2.92 | 2.88 | 8.3 |

4.5 | 2.84 | 2.96 | 8.8 |

15 | 2.80 | 3.00 | 9.0 |

In an intermediate state, where the electrons of the beam move at speeds that are not very close to the velocity of light in the vacuum, a part of the total number of force carriers will transfer the total kinetic energy to the target atoms, while the remaining force carriers the half of the kinetic energy. In this case, the energy transferred in the form of heat to the target atom will have a value between

The values of the first two lines in the table correspond to speeds equals to 87% and 91% of the light velocity in the vacuum, not so close to 100%. Since, according to the hypothesis of the absolute reference system, the heat transferred to the target has values between

The theoretical values derived from the special theory of relativity are confirmed only in the first line of the table, while the rest are much smaller than the corresponding experimental ones. These values might be somewhat acceptable if they were larger than the corresponding experimental ones. Therefore the experimental results are in accordance with the special theory of relativity only in that the higher speed in nature is that of the velocity of light in the vacuum. However, it is not in agreement with the corresponding experimental values of energy.

u/c | Experimental | values of energy | values of energy |
---|---|---|---|

ratio | values of energy | of the absolute system | of special relativity |

(MeV) | (MeV) | (MeV) | |

0.87 | 0.5 | 0.39 - 0.78 | 0.52 |

0.91 | 1.0 | 0.61 - 1.22 | 0.72 |

0.96 | 1.5 | 1.48 | 1.30 |

0.99 | 4.5 | 4.67 | 2.62 |

If photons are the structural component of matter, then a self-evident conclusion is that particle movements should obey a wave equation similar to the differential equation of the electromagnetic wave. It seems, according to the relative theoretical analysis of this section, in the light of the hypothesis of the absolute reference system, that the theoretical results of this assumption are acceptable, since they are in accordance with those of modern physics.

A particle, as previously described, is composed of a number of bound photons and its total energy will be determined as the sum of total energies of these photons. The total energy derived from the mass frequencies of all the bound photons of the particle shall be equal to:

where

where

The kinetic energy of the particle will result from the difference:

We define the quantity

Since the mass frequencies of the bound photons are different from each other, the transfer frequencies of all the bound photons, based on the latter relation, will be different from each other. Therefore, the sum of the transfer frequencies of all bound photons will be equal to the particle frequency. This frequency (which is actually a sum of frequencies) accompanies the movement of each particle and is a characteristic of its reference system.

Therefore the kinetic energy in relation to the particle frequency is:

Also the corresponding particle wavelength

where k (the wavenumber) is equal to

The wave function

which propagates at a velocity

This wave function as a solution of the wave equation will be of the form3:

where the constant A is the amplitude of the particle wave,

the equation of momentum in relation to energy for a free particle

which is the aforementioned differential wave equation.

The operator of momentum is the same as that of the Schrodinger quantum mechanics, but the operator of energy is differentiated by a factor equal to 1/2, since the operator of energy of the Schrodinger quantum mechanics is equal to

If the constant A of the solution of the differential equation of electromagnetic wave is the electric field amplitude derived from the sum of all the electric fields of all the elemental photonic electromagnetic waves accompanying the movement of the particle, then the quantity A^{2} is proportional to the energy density of these electric fields, and also it is proportional to the density of the mass inside the particle-space. Since the amount of this energy is constant for each particle, in a parallel beam of same particles with the same number of bound photons per particle, the magnitude

The particle current corresponding to the wave function

and therefore:

so, based on the continuity equation

According to relations

When the particle moves within a potential, then as we shall see later, the solution of the differential equation will surely vary as compared to that of the free particle. In this case, for a plurality of same particles, under the influence of this potential, the quantity

As we will see in Section 4, on quantum field theory based on the hypothesis of the absolute reference system, the Klein-Gordon equation and the Dirac equation, which are used in the case where the velocity of the particle is comparable to the velocity of light in the vacuum, are acceptable by the hypothesis of the absolute reference system. All the theoretical results of quantum field theory (for example, the theoretical results of quantum electrodynamics) are accepted by the hypothesis of the absolute system, if interpreted in the basis of this theory.

Starting from the assumption that an initially free charged particle, for example a free electron or a parallel electron beam, enters a space with stable electrical potential, with some initial conditions of the problem, we reach a differential equation. The solution of this equation gives the ability to determine the number of particles, as a function of the location. If the dynamic energy V of an electron is positive but less than the initial kinetic energy T, then the kinetic energy in the space of existing potential is

The equations to which it generally obeys a particle motion to an external potential V is the energy conservation equation

By including the time evolution of this particle’s state, we arrive at a more general form of the wave function,

It also, of course, satisfies the wave-equation:

For an electron moving in an atomic scale space, under the influence of the Coulomb field, the corresponding force exerted on it will have values that correspond to the same scale (that is, this force can not be enormous), and therefore, its velocity will not be comparable to the speed of light in the vacuum. So, the calculated contraction factor value is very close to 1 (

According to mentioned in the previous subsection 2.1 about the kinetic energy in relation to particle frequency, according to the relation (2.5) presented in subsection 2.1 the frequency of the particle is proportional to kinetic energy. Therefore, in a closed periodic motion of an electron, the average value of the kinetic energy, over a time period equal to that required for a complete closed orbit of the particle, will be proportional to the average value of this frequency. This results in the following relation:

where

and respectively is defined as “equivalent time period of oscillation” of the particle-wave the amount

Since the electron must behave, based on its structure, like a wave, in order for the movement of its closed orbit to be stable, this wave should be a stationary wave. Therefore, an additional binding condition is introduced which governs the periodic motion under consideration and is that the length of the closed orbit should be an integer multiple of the equivalent particle-wavelength.

If the

and finally a general equation is:

This result, taking into account the equation of motion of the electron, according to the relative examples exposed in the next section, leads to the conclusion that the energy of the electron and also all physical quantities involved in this problem are quantized.

In the subsection 2.2 it is stated that the quantity

By normalizing the function

the amount

The mathematical development of the subject here was done by Andre Kessler4. It turns out that the Fourier conjugate of a very localized waveform will be spread out. Thus, if position and momentum (or energy and time, etc.) are Fourier conjugates, and if you know the position to a high degree of accuracy, then you don’t know the momentum very well and vice versa.

If

The inverse Fourier transform is:

The momentum will be given by the relation

where

Like earlier, we should expect the classical position to be the average value, and other values to be less probable, and therefore the probability of position is expressed by a normal distribution. If we let

where

Since the function

The coefficients

this gets us:

Since the

After the integration of the second member of the last equation, the following equation is taken:

In the last equation, the coefficients and exhibitors must be equal. The two equations:

end up in exactly the same relation:

Due to the relation

This, of course, the latter is true only if the probability distribution is normal. If it isn’t

or else:

This is the Heisenberg uncertainty principle for position and momentum.

The time-dependent part of wave function

The inverse Fourier transform can be written as follows:

here the quantity

where

As above, in the general case:

In the case of a harmonic oscillator the total energy is equal to

This last relation is the Heisenberg uncertainty principle for energy and time.

We will then look at some of the best-known examples of quantum mechanics in the light of the hypothesis of the absolute reference system. The results obtained by solving these examples seem to be fully in agreement with the corresponding results of Schrodinger’s quantum mechanics.

If we consider as initial condition the relation

A simplified example of an electron’s motion in a Coulomb field is that of circular motion. In this case the field comes from a unique proton. In the general case, such as the Hydrogen atom, the orbit of the electron is elliptical, but this subject will be examined in a next example.

The total energy of the electron in our example is:

where m is the mass, u is the electron velocity, e is the elementary charge, and r is the position of the electron in a cartesian coordinate system, with origin the center of mass of the proton-electron system. Based on the centripetal force exerted on the electron

Since the speed remains constant, based on the relation (2.23) presented in subsection 2.4, the average value of velocity-squared is

The frequency

and therefore on the basis of the foregoing, the total energy, speed, the radius of the circular track and the angular momentum, are given by the following relations:

In the same relations one ends up, applying Bohr’s theory of circular motion.

We assume that the potential at the X axis is zero in the range

The kinetic energy of the electron is:

where m is the mass of the electron. Since the speed remains constant, based on the relation (2.23) presented in subsection 2.4, the average of the velocity-squared is

The speed is equal to

The kinetic energy is:

The momentum is:

The potential of the one-dimensional harmonic oscillator, in the X direction, is given by the parabolic form:

assuming x equal to

where

Therefore the total energy of the oscillator is given by:

The values

Our basic hypothesis here is that the electron trajectory is elliptical, and one focal point of the ellipse is the center of mass of the hydrogen atom (that is, lies in the nucleus). The orbital position of the electron in polar coordinates (reference [

where a and b are the lengths of the semi-major and semi-minor axis of the ellipse respectively. The angular momentum is conserved, and is equal to:

The force exerted to the electron is:

where e is the charge of the electron. According to the last relation the angular momentum squared is:

The kinetic energy is:

According to the relations:

and relation (3.20), the relation for kinetic energy becomes:

The dynamic energy, according to relation (3.22), is:

The total energy of the electron as a sum of kinetic and dynamic energy (and according to the relation (3.22)) is:

From the last relation, it appears that the total energy of the electron is inversely proportional to the length of the large axis of the elliptical trajectory.

Then, in order to use the Equation (2.23) presented in subsection 2.4, we will calculate the time

The area of the ellipse, taking into account the area speed, is:

Since the area of ellipse is

Since the frequency of periodic electron motion in the closed elliptical trajectory is

The time average of kinetic and dynamic energy (that is, the values of

so, based on the virial theorem

which is the expected value, since, due to the virial theorem, the total energy is

so, the quantized semi-major axis of the elliptical trajectory is:

The quantized total energy of the electron is:

We will now examine the quantized term of kinetic energy

where

Due to relation (3.22) and the relation (3.32), the resulting angular momentum is:

where

From the last two equations we get the equation:

If

that is, it takes values:

We assume now that the elliptical orbit of the electron is on the level XY of a Cartesian coordinate system XYZ. If the plane of the trajectory has been rotated at an angle

where we have considered as an integer quantum number of the projection of angular momentum on the Z axis the number

We assume the existence of an electric potential

We also assume that the kinetic energy of an electron of the incident-beam is E. Due to the energy conservation, the total energy of an electron of the refracted beam is

while the wave number

We initially consider

In the case where

Another way to deal with the same problem is to calculate the reflection and refraction from the expressions for the particle currents6. Since the above wave functions refer to the incident, reflected, and refracted beam, the quantity

where, the

In the area of the negative semi-axis the total wave function (incident and reflected beam) is given by the relation:

while in the area of the positive semi-axis (refracted beam only):

The continuity boundary conditions at

from which relations emerge:

From the last two relations and from relations (3.44) and (3.45), we end up with the previous relations for reflection and transmission.

we consider in this example a fixed potential in the X direction in the region

In the case where the kinetic energy E of an electron of the incident beam is greater than the dynamic energy

where

Also, following the analogous procedure for the transmission, we get the relation:

We will now follow the methodology on particle currents and boundary conditions of continuity, as in the previous example. In this case the reflection and transmission are derived from the corresponding current ratios, according to the following relations:

The total wave function in region a is defined as

From these equations, four relations between the complex amplitudes are taken, which are the following:

From the last four equations we get the equality:

From this last equality and from the relations (3.55) and (3.56), we reach the same relation for the reflection, that is the relation (3.53). In the same way, the transmission, given by the relation (3.54), is also calculated.

In the case where the kinetic energy E of an electron of the incident beam is less than the dynamic energy

Replacing the k and

while for

We will examine in this section quantum electrodynamics, according to classical theory based on the absolute reference system. Can a theory of classical physics to include quantum field theory and give corresponding theoretical results?

The answer is affirmative, and this will be seen in this study, in this section. The waveforms used in this section, as in the previous ones, are solutions of this wave equation and do not express probability amplitudes, as they derive from a probabilistic view. In contrast, that waveforms express the propagation of a peculiar particle electromagnetic wave, according to the assumption of absolute reference system, which is a classic non-relativistic view of nature in the broader field of statistical physics.

The free-particle wave-equation for high velocities, according to the hypothesis of the absolute reference system, is the wave-equation, in which the force currier is massive and the factor

This equation looks like the Klein-Gordon equation7. The wave function that is a solution of this equation will be that of the plane wave8, which is in the form of:

where N is the normalization factor, which will be discussed below. By substituting this wave function in the previous differential equation we get the following relation:

so, due to the energy relation

According to this last relation the wave function (4.2) takes the form:

The operator which acts on the wave function

while the differential operator, which yields as an eigenvalue the energy E is

Multiplying the two members of the Equation (4.1) by the quantity

Adding the last two equations and multiplying with

The particle density and the particle current are given by the relations:

According to relations

because the speed

Now we will extend to the study of the interaction between heavier particles and the corresponding dynamics. This study has to be done from the point of view of the hypothesis of the absolute system of reference. Such a potential we will first examine is that which comes from relatively heavier particles, such as protons and neutrons. In particular, we will calculate the potential comes from the exchange of intermediate particles (force carriers) that give rise to forces between such heavy particles (see [

The set of carrier particles in the field around a proton, in addition to the photons that are carriers of electrostatic interactions, consists of larger photon packets, which are the intermediate particles of interactions between nucleons. These particles, which are the carriers of strong interactions, are the explanation of the small radius of force action between the nucleons inside the atomic nucleus. The radius of action of static interactions depends, on the mass of the carrier of the quantum field, and an explanation for this was given by Yukawa in 1935, in his effort to describe the above-mentioned forces.

We assume initially that the mass of an intermediate particle that is exchanged is m. From a physical point of view, this exchange gives momentum that justifies the existing force of interaction. Along with the capturing of this mass, the heavy body takes extra energy equal to the total kinetic energy of this intermediate particle. This energy is:

where

The differential equation of wave motion results from the replacement, in the last relation, of p and E with the corresponding operators

This last equation has two partial solutions. One partial solution is given by the Equation (4.4), but, at the present, we are not interested as much in the propagation of the particle-wave, as much we are interested in the examined here static potential. The other solution is of the form:

and is independent of time. The

The physical analog of Yukawa potential in electromagnetism is that resulting from the substitution of the constant g with the charge q. However, because of the very small mass of the interaction photon, the exponential part of the potential is very close to 1 and therefore the electrical potential is of the known form

According to the history of nuclear forces the Yukawa hypothesis predicted as carrier of strong interactions a spinless quantum of mass

Let us assume that a time depended potential

where

and

Also the “probable wave-function”

so

Assuming that before action of the potential

so we get the following differential equation:

By substituting the expression 15 in this last differential equation, since

By multiplication of the last equation from the left with

Using the orthonormality relation for

where we have defined

and what is sometimes called the transition matrix element:

According to the first order approximation, the Equation (4.22) gives us

Using that

for

For a potential that is time-independent the expression, for the transition amplitude, becomes

where

We define the mean transition rate in the limit for large T as

When the wave functions are those of the plane wave, that is

For the plane waves this gives

where, for plane waves

The phase space factor for a process with n final state particles is

The cross-section is

We will examine the case where the outer field comes from a point charge at the beginning of the axes, while the velocity of the incoming charged particle is low, that is

The wave function of the initial state and the conjugate wave function of the final state are:

The transition amplitude is:

according to the relations:

By setting

The time-averaged transition rate is:

Due to normalization of the plane wave function over a box with volume V, we have

while the phase space factor is

According to the relation (4.33) the cross-section is

Since

The kinetic energy of the incoming charged particle is

This is the well-known Rutherford scattering formula.

The Dirac equation with small variations, as will be shown below and with the help of what has already been mentioned in this section, is perfectly compatible with the hypothesis of the absolute reference system. We want to find a squared equation, which gives the wave Equation (4.1). This equation has the following form:

This is certainly a well-known problem whose solutions for

We also define the four components of the matrix

so the Equation (4.42) becomes,

where we have used the symbol

We can use Dirac spinors to write plane wave solutions9 of the Equation (4.43). Consider

where

By substituting this last wave function in the (4.43) equation we get the following solutions:

The second of the Equations (4.44) gives:

where

Since the space of the absolute reference system is the three-dimensional Euclidean space, the four-dimensional space-time will be an extension of this Euclidean space in the four dimensions. In particular, the space-time metric tensor will be the Kronecker

A very important remark is that the Dirac equation does not need to be relativistically covariant under a Lorentz transformation in the hypothesis of the absolute reference system, since this physics is not relativistic.

Considering all of this, the components of

The Dirac equation for anti-particles is

The plane wave solutions for the particles take the form

where the spinors

and

According to Dirac algebra we have

The Hermitian conjugate of

so the Hermitian conjugate of Dirac equation is

Now we multiply the Dirac equation from the left by

Consequently, we realize that if we define a current as

then this current satisfies a continuity equation,

which is always positive.

Substituting the plane wave solution

Consequently, in order to have one particle per volume V we choose

Quantum theory, based on the assumption of the absolute reference system, in the context of quantum mechanics and quantum field theory, is simpler than current quantum theory and gives, with minor differences, the same theoretical results. To sum up, the study so far, based on the hypothesis of the absolute reference system, concludes with the following conclusions:

1) The wave behavior of particles based on the classical wave equation, instead of the Schrodinger equation, gives theoretical results in agreement with those of quantum electrodynamics. It is also noteworthy that based on the hypothesis of the absolute reference system, there are no negative energy values such as those derived from the Klein-Gordon equation, and the Dirac theory.

2) The theoretical energy values in the W. Bertozzi experiment (section 1.2) agree with the corresponding experimental ones, in contrast to the theoretical energy values derived from the special theory of relativity for this experiment, which are not in agreement with the experimental results.

3) The dynamics of the hypothesis of the absolute reference system is confirmed experimentally.

The author declares no conflicts of interest regarding the publication of this paper.

Patrinos, K. (2019) Classical Quantum Field Theory Based on the Hypothesis of the Absolute Reference System. Journal of Applied Mathematics and Physics, 7, 747-780. https://doi.org/10.4236/jamp.2019.74052