revised by P.F.Parshin, Prof. Dr.

The purpose of this paper is to show that the main principles which physics has been based on can be regarded as possible consequences of a unique mentally consistent axiom. This leads to the unification of the GRT with the QM since they should express the same thing as they both could derive from the same axiom. This Paper leads to the same consequences as "The hypothesis of the Quantum Space Time-Aether" [1] which is based on the unification of the physical meaning of the notions which derive either from the GRT or the QM; many physical phenomena and laws which have not been explained according to what until now has been accepted, can be explained on the basis of this hypothesis.

The analysis following in a first stage is based on the aspect we have for the notions, space, time, exist, matter, energy. In a second stage these notions are extended - not arbitrarily - and regarded as stochastic. Time implies the existence of earlier and of posterior; space, too. If I say 10 cm I mean the existence of 1,2,...,9,10 i.e. the existence of earlier and of posterior. Therefore, the existence of earlier and posterior is the condition of space and time to exist and vice - versa. When we say that matter exists, we mean that every part of it exists in space and it is measurable. This means that matter has everywhere the property of earlier-posterior and therefore, according to what was mentioned, it can be described in spacetime terms. Thus, we can state the following statement I.

A matter system in general has differences within its various areas. This means, according to statement I, that a matter system in general is characterized by different rates of earlier - posterior within its various points. More practically, it means that, with respect to an observer, a clock has different rates of its successive hits at various points. Since space is also locally affected by the local rate of earlier-posterior, it is expected to be deformed due to different rates of earlier-posterior. For the purposes of this paper the following definitions are necessary [1].

i. As reference spacetime we define a euclidean spacetime to which, through transformations of deformity, any field can correspond. This reference spacetime is not only a geometrical notion because, according to the present hypothesis, it is also matter. Any magnitude of it will be denoted by the subscript ₀. A point A₀ of the reference spacetime occupies, by the action of the field, a position A¹A₀

ii. As Hypothetical Measuring Field (HMF) is defined a hypothetical field, which consists of the reference spacetime, in which at every point A₀ the real characteristics of the corresponding point A of the real field exist. [1]

iii. In a HMF we define as relative spacetime magnitude sr the ratio of a real infinitesimal spacetime magnitude ds to the corresponding magnitude ds₀ of the reference spacetime: i.e. sr=ds/ds₀.

The energy of a matter system measures its ability to exist. If a system starts to degenerate, its ability to exist is getting lower until it disappears. According to this paper, an infinitesimal spacetime element dU itself is matter and its energy dE, i.e. its ability to exist, is the ability of an interval between two successive hits of a clock connected to the element to exist. This ability can be measured by an observer who can measure the duration dt between two successive hits at various points. With respect to the observer, some intervals between two successive hits "live" more than other intervals. Thus, we can state that the ability of dU to exist is expressed equivalently by its energy dE and by its duration dt. Thus, taking into account the definition ii, we can state the following statement II.

Statement II: "In the HMF the energy of an infinitesimal spacetime element is equivalent to its internal time " .

where as internal time we define an infinitesimal time of a phenomenon of comparison.

An immediate consequence of statement II is that:

Even though this statement is mentally consistent, it is regarded as principle. The purpose of this paper is to show that the Plank law, the relativity theory and De Broglie's principles can be regarded as possible consequences of this principle. Thus, a matter spacetime system can be simulated by a particle wave which, as it is known, is interpreted statistically. It is noted that statement II does not imply any stochastic behaviour in spacetime. However, by the aid of Goedel's [2] and Rosser Barkley [3] work, it derives, in a mentally consistent way, that spacetime can be regarded as a stochastic magnitude(see discussion). This verifies the statistical interpretation of the QM and it leads to the modification of statement II and equation (1) to non relativistic forms [1]. The final consequences of this principle-statement II are the same as the ones of the hypothesis of the Quantum Spacetime - Aether [1] (see chapter IV). Many physical phenomena and laws which have not been explained according to what until now has been accepted, can be explained on the basis of this hypothesis (see chapter IV).

The purpose of this chapter is to show that the relativity theory can be regarded as a possible consequence of statement II on condition that any spacetime is considered as a continuum.

An infinitesimal area of a spacetime continuum can be regarded as an area with constant rate of earlier-posterior and therefore it has not any spacetime deformity. Time is independent of space in this infinitesimal area and, since its rate is different in various points of the field, it can be regarded as a 4th dimension. Thus, in Riemans's 4-dimentional space with x₄=ikt, where k a constant with units of velocity so that x₄ will have units of length, we can write [4]:

For k=c, where c is the speed of light, eqn (2) implies the Lorentz transformations which are the basis of the GRT. An immediate consequence of Lorentz transformations is that :

where v the velocity of a spacetime element with time of a phenomenon of comparison dt with respect to a spacetime element with time of the same phenomenon of comparison dto. Because of statement II and eqn (3), we obtain:

We consider a flat matter spacetime whose all spacetime magnitudes equal the mean values of the same magnitudes of the field under study. For this flat matter space time it will be valid:

where i indicates a point of the HMF. Because of eqns (3,4,5) we will have:

Considering the following :

we obtain:

Eqn (9) is compatible with the SRT which therefore can be regarded as possible consequence of principle - statement II. Thus, a spacetime system can be simulated, in some of its properties by a relativistic point mass.

For the purposes of this paper the following elucidation is useful.

Elucidation: Eqns (8,9) are valid for any flat matter spacetime; therefore, they are valid also for a flat spacetime whose all spacetime magnitudes have values equal to the mean values of these magnitudes of a stochastic matter spacetime system.

A particle according to this paper can be regarded as a spacetime formation, which changes in time. This change must be an oscillation and not a non periodic process because of the property of particles to be in general stable. For the simple case of one dimension this oscillation can be analysed according to Fourier analysis in harmonic oscillations. The wave function Ψ=Ψ(x,t) can be written in the form Ψ=Ψ(x',t) where x'=x'(x,t). For a given t, according to Fourier analysis, Ψ(x',t) can take the form:

where L is a proper interval which will be defined in the later and n=1,2,....

The same form is valid for any t but with different A₁, A₂,…A_n, B₁, B₂,…B_n. Thus, in general we may assume that Ψ has the form of eqn (10) on condition that A₁, A₂,…A_n, B₁, B₂,…B_n are functions of t. It is noted that ψ of eqn(10) can be extended to complex forms when A₁, A₂,…A_n, B₁, B₂,…B_n are complex. By using the exponential form of cos and sin, eqn(10) can be written:

If x'=x-c_nxt=x-l_n(w_n/2p)t=(l_n/2p)(x(2p/l_n)-w_nt) where c_nx, l_n, w_n/2p are the velocity, the wave length and the frequency of the n^th harmonic wave, we will have that:

Thus, eqn(11) can take the form :

1) l_m=L/m, i.e. that L is the wave length of the first harmonic

2) both l_m=L/m and w_m=2pc_x/l_m can take values with both signs  so that all terms of eqn (11) can be included. (For the possibility of having negative values of l_m and w_m see in section 2 of this chapter and in the discussion). In the general case of waves which are propagated in various directions we can write:

where is a unit vector which has the direction of wave velocity. This wave function is valid on condition that space-time has not any deformation. When the vibrating medium is spacetime itself, we may assume that this wave function describes the HMF in which, by definition, there exist only local deformations. Thus, ψ describes the changes of relative spacetime magnitudes i.e. the changes of the rates of earlier-posterior at various points (r,t) of the HMF (see definitions ii, iii ). It is noted that Ψ wave function can take the form of eqn(13) only on condition that it is stochastic; this will be shown in paragraph 3 of this section.

Eqn (1) can be viewed in two ways:

a. when dto is a unit of time, eqn (1) describes the duration dt, with respect to an observer and, as was mentioned, it leads to the relativity theory.

b. When dt is a constant period of time in the HMF, then eqn (1) can be written in the form:

where n is the frequency of a periodical phenomenon of comparison and f an arbitrarily constant factor through which we can change the scale of n, n₀. If n=1, n₀ must be different in various points of the HMF. If this is the case, n₀ represents the number of hits of a clock connected with the spacetime element dU in the unit of time which is observed in the reference spacetime and eqn (14) can be written in the form:

where dE_j0=dE_i0 and n_i0¹n_j0 for i¹j and where i, j indicate points of the HMF. Since, according to this paper energy-matter is nothing else than a system with different and changing rate of earlier - posterior, eqn (15) shows the way through which a field exists and acts at various points.

Thus, for the same equation we have the following correspondences:

From eqn(15) we obtain:

where ánñ _E the mean value of frequency in the HMF for an energy state E.

E₀ is the energy of a field with ánñ_E=1. Thus, E₀ can be regarded as a constant since eqn(16) is valid for any level of energy.

In order that formula (13) is valid, harmonic oscillations are needed which will not take into account the spacetime deformation i.e. oscillations which describe the HMF. In such an oscillation frequency ánñ_E which is valid in general will be the frequency n of this oscillation. Thus, we will have:

Eqn (17) is compatible with Plank's empirical law and its extension i.e. De Broglie's principle for energy when E₀=h. Thus, De Broglie's principle for energy can be regarded as a possible consequence of principle-statement II.

Taking into account the way through which eqns(15,16,17) are obtained, we conclude that they are valid in general i.e. for any matter spacetime system and therefore for a photon or for a particle in general.

In the case of a photon which is described by eqn(9) for m₀=0 we have:

As was mentioned, in order that formula (13) is valid, harmonic oscillations are needed which will not take into account the spacetime deformation i.e. oscillations which describe the HMF. In such an oscillation and more specifically in the case of a photon for the wave length we will have:

According to the present paper, the wave length has sense since it refers to something that can vibrate and this is the matter spacetime system. Thus, for a photon because of eqns(18,19) we have, relations which are compatible with De Broglie's principles. Therefore, those principles can be regarded as possible consequences of principle-statement II.

In the case of a particle in general, from eqn(9) we obtain:

where E_eq is the energy of an equivalent photon. Since the rest energy m₀c² does not change, we may assume that only the equivalent energy E_eq participates in the creation of a spacetime wave. Thus, we can write:

For energy the general formula of eqn (17) is valid i.e.

where c_w is the speed of a hypothetical wave in the HMF. Both n and l_eq relate to reality but a part of n , i.e. n-n_eq, does not "create" wave length because of the rest energy. n-n_eq behaves as a frequency of a wave with zero speed and length (0=(n-n_eq )0). However , for reasons imposed by the Fourier analysis, eqns(22) must be applied. For a particle in general, because of eqns(21,22), we have relations which are compatible with the De Broglie principles and, therefore, those principles can be regarded as possible consequences of principle-statement II. For the same reasons as was mentioned for a photon the wave length which corresponds to a particle in general has sense since it refers to something that can vibrate and this is the matter spacetime system.

Taking into account eqns (13,19,21,22) and considering that , we have:

As it is known according to the QM, this equation describes a De Broglie particle wave. Because of eqns (9,23) we obtain the Schroedinger relativistic eqn. It is noted that according to this paper there does not exist any potential which acts from a far distance because spacetime itself is matter and therefore it can be regarded as a self- acting medium.

Thus, according to eqns (9,19,21,22,23) we may draw the following conclusion I:

Conclusion I: "A particle - /spacetime system can be simulated by an equivalent particle (De Broglie) wave" or equivalently "Any infinitesimal spacetime can be regarded as a particle (De Broglie) wave".`

It is noted that according to the QM a particle wave can be only statistically interpreted[5]. This also can be derived from the present analysis. In fact Ψ can be written in the form of eqn(13) on condition that, in the one dimension problem, we can write: x'=x-c_nxt. From eqns(21,22) we have that c_w=E/ïPï and because of eqn(20) c_w¹c . Therefore in general, in the one dimension problem, we have: c_xi¹c_xj for i¹j.

Thus, in order to have the same variable x' in eqns(10,11,12) at time t we need different values of x i.e. we must have:

Because of eqn(24) there is an uncertainty of x or in general of position r, when we write Ψ in the form of eqns (10,11,12,13). Inversely for the same (r,t) there exist an uncertainty of Ψ when it is written in the form of eqns (10,11,12,13). Therefore Ψ(r,t) can only statistically interpreted when it is written in the form of eqns(10,11,12,13) which implies that spacetime is not continuum. Taking into account the way through which eqns (9,19,21,22,23) have been derived and the elucidation of chapter II, we notice that conclusion I is valid regardless of the matter spacetime being stochastic or not. In fact, for a stochastic spacetime from eqn(14) we obtain:

where is a frequency which corresponds to . For we obtain and so on.

According to what was mentioned Principle-Statement II implies the GRT if spacetime is continuum and the QM (HMF) if it is not. Thus, the question of whether or not spacetime can be regarded as a stochastic is raised. A philosophical-mathematical consideration is given in the discussion and shows that spacetime is stochastic.

2. The Hypothesis of the Quantum Space Time -Aether

The "Hypothesis of the Quantum Space Time (QST) - Aether" is based on the unification of the physical meaning of the notions which derive either from the GRT or the QM. The main principles which derive from this hypothesis are:

Principle II. "The energy of any oscillating infinitesimal spacetime is equivalent to its internal time'' where as internal time is defined an infinitesimal time of a phenomenon of comparison.

We notice that principle II is an equivalent expression of statement II of the present paper since any oscillating infinitesimal spacetime refers to the same point of the HMF. We also notice that principle I is the same as the conclusion I which, according to what has been mentioned, can be regarded as a possible consequence of principle - statement II.

It is noted that many physical phenomena and laws which have not been explained according to what until now has been accepted, can be explained on the basis of this hypothesis. Thus, we can explain the second thermodynamic law as a result of Universe's expansion, the gravitation and the manner of forces' unification, the Casimir effect, the attraction and repulsion of ions, the property of self similarity in matter systems, the black holes' radiation, the reason why there exist limits of an approximative validation of the QM and the GRT and the reason why the Lorentz transformations are non valid in the whole extent of the QST [1,6,7,8]. It is noted that a first approximative interpretation of the cold fusion, according to this hypothesis, has been given [9]. It is also noted that, according to this hypothesis the QST is matter and therefore it can be regarded as aether.

The mathematical formula of the probability density P(r,t) implies the existence of a logical structure; however, P(r,t) itself means that something exists and does not exist at the same time. The existence of negative l_m and w_m in eqns(12,13) means the existence of negative space and time which is incomprehensible, while the corresponding formulae show the existence of a logical structure. It is noted that negative spacetime magnitudes, according to previous works [1,8], can be regarded as antimatter. Eqn (9) is valid either for real or imaginary values of E, P, m. The notion of an imaginary magnitude is incomprehensible. However, the imaginary values of E, P, m imply the existence of imaginary spacetime which, according to previous works [1,8] can be regarded as electromagnetic (em) space. All these are logical and illogical at the same time. Thus, a deeper interpretation is needed.

Our basic communication system comprises of aristotle logic and of a hidden axiom which postulates the existence of earlier and posterior. In fact, every word or phrase is constructed in such a way that the letters or the words are put the one after the other.

According to the process through which Goedel's theorem is proved [2,3], two contradictory statements are valid at the same time. This leads to the following statement "A": "There exists no system including logic and Peano's arithmetic axioms which will not lead into contradiction." This statement does not constitute expression of any of Goedel's theorems which are based on the hypothesis that all axioms are consistent; this Goedel's hypothesis of course is arbitrary. The basic axiom of Peano's arithmetic declares the existence of earlier and posterior. Because of this axiom we can have the numbers 1,2,3,4...and , according to what was mentioned, this axiom implies the existence of space and time. Thus, the basic communication system obeys the statement "A". We notice that statement "A" cannot be stated because it is based on the basic communication system which, according to statement "A", is contradictory. Thus, statement "A" imposes the silence. When we communicate, we use another hidden axiom according to which "what is accepted as truth is what includes the minimum possible contradictions" since the contradictions cannot be vanished . According to this hidden axiom, which we could name as "axiom of the minimum contradiction"[10], we obtain the logical and the illogical dimension that is needed, according to what was mentioned, in physics. In fact, through this axiom we try to approach logic(minimum possible contradictions) but at the same time we expect something illogical since the contradictions cannot be vanished. The systems of axioms we use include the communication system and therefore their contradictions are minimized when they are reduced to the communication system itself. It is noted that this paper is based only on the communication system itself since the existence of earlier-posterior implies the existence of spacetime and vice-versa. Thus, the contradictions of the basic communication system can explain the illogical behaviour of spacetime i.e. its stochastic nature as well as the existence of negative and imaginary spacetime magnitudes.

The authors wish to thank Prof. J.V. Drigojias for his useful discussions.

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