An “ab initio” Model for Quantum Theory and Relativity

The paper introduces a theoretical model aimed to show how the relativity can be made consistent with the non reality and non locality of the quantum physics. The concepts of quantization and superposition of states, usually regarded as distinctive properties of the quantum world, can be extended also to the relativity.


Introduction
The quantum theory and the relativity have stimulated influential ideas and experimental efforts to investigate and understand a huge number of natural phenomena from atomic to cosmic scale [1] [2]. However, with space ranges spreading from ~10 −18 m to ~10 26 m by about 44 orders of magnitude, is comprehensibly problematic the attempt to unify in the frame of a unique theory the whole variety of related natural phenomena. Yet, is symptomatic the fact that similar difficulties often arise even in formulating a more selective class of specific physical problems. In the case of the relativity, for example, something relevant should be still missing even at the mere cosmic scale; despite the great amount of its previsions and discoveries, remain problematic crucial topics like the progressive acceleration of universe expansion, the MOND (modified Newton), the dark matter and dark energy. A possible hint to overcome these difficulties is to identify an appropriate background of ideas that integrate or modify the preexisting ones; for example, at the quantum level, the major problem of the relativity is its link to the non-reality and non-locality of the quantum theory [3]. Also, the "handwritten" cosmological constant reluctantly introduced by Eins-tein after the Hubble experimental hint, is a further example of necessary revision of the general relativity even in its most representative cultural frame i.e. the cosmology. Is reasonable the suspect that focusing greater attention on the existing conceptual background is not an additional difficulty but a possible solution? To provide a contribution to this problem, the paper [4] has introduced an operative definition of space time i.e. the estimated age and radius of the Universe, from now on quoted with shortened notation u t and u r , the Einstein cosmological factor Λ and the Hubble constant u H . Two quantities of interest are the mass ob m detectable in the universe counting the stars only [5] and the critical density cr ρ of Friedman equations [6]  refers to stars only and because it concerns by definition stars whose light has in fact reached us during the life time of the Universe. The present theoretical model implements systematically uncertainty ranges to calculate quantum and relativistic quantities according to the logical step "local values → uncertainty ranges". From a formal point of view this statement can be acknowledged reminding the standard concept of measure errors: just as no one trusts the reliability of a single value measured in its experimental error bar, likewise (1.2) waive the signification of a local dynamical variable in its uncertainty range. Yet the true physical meaning of this replacement is one among the crucial points of the model, as it will be more thoroughly shown below; some examples of calculated results are also reported in the Section 6 to confirm the concepts exposed in the Sections 1 to 5. The uncertainty ranges are defined via the standard notation ( ) , being f ′′ and f ′ the range boundaries defined by two arbitrary values allowed to the concerned function; these values are arbitrary, unknown and unknowable by definition of quantum uncertainty. In general, both of them can be variables or constants. As introduces the ratio f x δ δ that in turn takes physical meaning under appropriate conditions. A further way to implement the ranges is that already highlighted about δε , i.e. n n n ω ω ω ′ ′′ ≤ ≤    , which means , n n n ′ ′′ ≤ ≤ (1.9) being of course n′ and n′′ arbitrary and unknown integers. In the following the shortened notations . Eventually note that in principle both signs are allowed for any range; for example nothing hinders that x x x x x δ =± = ± ± being both x and 0 x arbitrary. Sometimes in the following text a given result is obtained more than once in different contexts: this is not a redundant repetition, rather it must be intended as a check confirming that all conceptual steps progressively exposed are consistently linked each other. Despite the agnostic way to introduce (1.2), the remainder of this paper is able to formulate a self consistent theoretical physical model. The text is exposed in order to be as self contained as possible.

Preliminary Considerations
This section introduces some considerations having general character of straightforward corollaries of (1.1) to demonstrate that this definition of space time is physically sensible. All concepts introduced below are listed sequentially without calculations, while emphasizing their physical meaning; the validity of the various formulas inferred through the model will be concerned in the next section 6.

Energy and Energy Density in the Space Time
Implement first the dimensional analysis of (1.1), (1.3) and (1.2) defining , where  stands for energy and  for length to introduce the concept of energy density η . Then multiplying and dividing side by side these two equations one finds having identified  pertinent to η with that defined by the second equation. Equation (2.1) yields two equations. Owing to (1.4) the first one is according which the value of η depends on that of  via fundamental constants only: i.e.

( )
It is necessary to make (2.  , is such that dividing both sides by t δ one finds The Planck length appears to be the smallest space range physically inferable through the definition (1.1) of space time, whereas the Planck time and energy can be nothing else but P P t c =  and =   P P t according to (1.4), whence the Planck mass

Classical Newton law
Since  c in (2.2) has physical dimensions of acceleration, write thus  δ δ = ± = ± = ± ×    m m mass m G G G mass length (2.11) the double sign being in agreement with (2.5). The energy  is here identifiable with the analytical form of the classical Newton law, if δ  is regarded as the uncertainty range corresponding to the random distance between 1 m and 1 m .
First of all, with this available dimensional information only, it is impossible to identify the respective role of either mass; otherwise stated, as 1 m and 2 m were both formally inferred from a unique 2 m , the quantum uncertainty re-  (2.7). This property of c is a corollary of (1.2), not a postulate; as such it must hold in any reference system.

Quantum Uncertainty and Space Time
The energy density (2.2) is inherent the concept of space time according to its own origin (1.3) and has several implications, first of all the existence of a pressure st P internal to space time volume 3 x δ previously symbolized as 3 length ; the subscript means space time to emphasize that it is inherently based on the definition (1.1) only. As sketched in the Appendix A [4], one finds 1 2 , , ,  where m  is the reduced Compton length of m. It appears reasonable to assume that the range size δ  is an integer number n of reduced Compton wavelengths, here regarded as the shortest wavelength relatable to one particle. All of this agrees with the results preliminarily obtained in section 1. A full paper [7] has been devoted to highlight the implications of (2.20), which in fact transfers n from its basic definition in (1.2) into the specific physical problem of a bound particle moving circularly around a central force field.
On the one hand (2.20) is justified by its direct corollaries (2.22) and (2.23), which however represent a particular case of boundary condition allowing steady wavelengths. While ξ δ  is uniquely definable as the radius of a circumference, it must be replaced by a combination of minor and major semi-axes of an ellipse in order that (2.23) describes still the integer number of steady wavelengths along an elliptic perimeter. Are known in this respect various formulas, e.g. [8], that calculate this perimeter; the next subsection 5.7 will show how to infer through this reasoning the perihelion precession of orbiting planets. On the other hand the definition (2.23) of momentum is merely formal, being based on a dimensional assessment compliant with the condition v c < via the arbitrary factor ξ . Nevertheless also in this reasoning energy quantization and De Broglie momentum are contextual. It is shown soon below that n of (2.17) is the refractive index of a dispersive medium. These considerations, crucial for the birth of the old quantum mechanics and here inferred as corollaries, suggest the necessity of defining more in detail the actual physical meaning of v.

Quantum Velocity and Space Time
The steps introduced by (2.9) are significant: whereas c in (1.1) is self-evident, it is a constant of nature, now rises the problem of clarifying further the physical meaning of velocity modulus v as a property of massive particles moving through the space time. To introduce v from a first principle, note that (2.12) yield ( ), Recall now the eq 2 r n δ λ π = found in (2.23), whose physical meaning introduces a crucial condition on any mass 2 m orbiting according to (2.11) around 1 m at constant distance having modulus r δ : owing to the dual nature wave/corpuscle of matter, a steady 2 m wave is required to describe a stable or- At the right hand side appears the momentum p λ of a De Broglie wave delocalized within 2 δ π r . At the left hand side appears the radial momentum range δ r p of a corpuscle delocalized in r δ : for example in the case of (2.11) it means that the space gap r δ between the masses 1 m and 2 m implies a steady wavelength λ along the orbital path of the running mass around the rest mass, which confirms the indistinguishability of gravitational and inertial mass introduced in section 2.2. Rewrite now 0 0 λ δ = − = − r r r p p p p of (2.27) with vector notation, noting that δ r p is radial momentum range around the rest mass, whe- , which suggests that not necessarily the orbit must be circular; even admitting δ r p con-

Relativistic Outcomes
The relativistic worth of these results follows straightforwardly, first of all because even the time is inherently involved by (1.2); multiplying side by side the first and third (2.9) one finds energy velocity m c and thus, owing to the first (2.17), Note that 2 n is here mere dimensionless multiplicative factor of v  , which however cannot be specifically calculated because both δ  and t δ are uncertainty ranges unknown and conceptually unknowable; so 2 n contributes in determining the resulting v  , whatever its value might be. These dimensional equations introduce the velocity and relativistic momentum components v  and p  . Moreover write according to (2.12) , i.e. just (1.2) merely with a different notation of the conjugate dynamical variables. These equations imply the indistinguishability of identical particles, because actually they concern the phase space rather than the particles themselves; in other words it is impossible to distinguish electron 1 from electron 2 delocalized in a region of space time if nothing in known about them. Indeed it has been shown in (2.29) to (2.35) that, for example, momentum and energy are directly related to the range sizes (2.40) regardless of any hypothesis about the particles themselves. On the one hand is remarkable the fact that the Newtonian definitions (2.9) imply the concept of uncertainty, thus confirming that actually even the classical gravity is rooted in the quantum equations (1.2). On the other hand the agnostic meaning of uncertainty, which implies lack of information about the boundaries of the ranges and about the local values of the dynamical variables allowed in their ranges, is not a postulate but a corollary of the way to introduce (2.40  , then it is possible to introduce f itself without defining explicitly its R, whatever the specific physical meaning of v and f might be. Nevertheless the lack of a specific reference system does not imply in fact any ambiguity, as it will appear in the following section 6 where are calculated some numerical outcomes of the present model; rather the physical meaning of v results from that of x δ and t δ or δε and p δ themselves according to (2.12). Although this physical model seems too agnostic to infer valuable information, note that these conceptual premises have been enough to infer the fundamental (1.2) from (1.1) and even preliminary relativistic results. The remainder of the paper aims to show that just this conceptual agnosticism allows to overcome the determinism of Einstein general relativity and plugs it into the elusive quantum world; the calculations will be carried only after having completed adequately the theoretical frame so far introduced.

Uncertainty, Covariance, Simultaneity
In general the choice of the reference system R is crucial in any classical physical model that implements deterministic local coordinates; in the Einstein relativity, the equations are required to be invariant with respect to the reference systems, including the non-inertial ones. Consider however a quantum problem formulated only via uncertainty ranges; in fact (2.41) shows that if all * j n and n are arbitrary, then there is no direct correlation between range sizes of dynamical variables and reference systems just because the former do not contain any information someway related to the latter. As it has been remarked for n, the only available information is that the product of two range sizes of conjugate dynamical variables must be quantized; i.e. both products * * j j n n ′ of (2.41) must yield an arbitrary integer whatever the local values of the respective dynamical variables j and j' might be. So, if the local coordinates are replaced by ranges that fulfill (1.2) and (2.41), then is missing "a priori" the existence of privileged reference S. Tosto systems; moreover it is easy to show that, by consequence, the requirement of the different form of equations in R and R' becomes inessential. Is instructive in this respect the classical example reported in various textbooks, e.g. [10], of a point mass m tethered by a massless and inextensible wire, so that the mass moves circularly around a fixed coordinate. This example becomes significant noting that if the wire is broken, e.g. by the centrifugal force itself, thereafter the motion of the mass is rectilinear uniform along the tangent to the circumference in the breakdown point. This is true in R with origin fixed on the rotation center of the mass. In R' fixed on the moving mass, instead, the mass is at rest; when the wire is broken the mass deviates from its initial path, it follows a curved tra- where the unique variable is  . This latter however is a local energy that must be implemented solely via its uncertainty range problem, so it is irrelevant the local disagreement a a′ ≠ and the fact that a' includes various additional terms with respect to a . The agnosticism implied by (1.2) compels considering these ranges before and after the wire breakdown regardless of how the corresponding local accelerations are made of; in other words the chance δε δε ′ = of including the respective a and a', in principle possible because a and  differ by a proportionality constant factor, bypasses the necessity of discriminating R and R' to describe the tethered system with or without its breakdown. Since this holds for any uncertainty range by definition, S. Tosto in fact (2.41) waives the necessity of specifying either reference system to concern the dynamical variables. Accordingly it is possible to regard all ranges of (1.2) independently of their definition in a specific R: rather it is possible to introduce x δ and t δ independently of the local space and time coordinates since the unprimed range sizes at the left hand sizes are arbitrary, the same must hold for the primed range sizes at the right hand side. Then it must be possible to implement identically both of them, regarding thus the lower boundary value 0 as a particular but not deterministic case. To infer physical information from these statements and check their validity, find the classical energy ′  of hydrogenlike atoms. Implementing (1.2) for r p′ , which actually owing to (2. Now it should be clear why range sizes and boundary coordinates are irrelevant as concerns the quantum problems, as in effect it has been demonstrated for various systems [11] [12]. In particular it is not necessary to specify R centered on the nucleus, it is enough to state that nucleus and electron are r δ apart; the radial range size is then defined by the non deterministic Bohr radius via the integer 1 n ≤ ≤ ∞ . Is clear thus the meaning of the coefficient 2 in the second (2.44): an electron r′ apart from the nucleus has total radial delocalization range 2r′ . In this model the quantum numbers are in fact numbers of quantum states.
(ii) Relativistic implication: the invariant equations. Write (1.2) as follows 2 2 , so that, subtracting side by side,  c t x δ δ − is particularly important as it has been demonstrated in [13] to be conceptual foundation of the special relativity. Special attention deserves in this respect the operator formalism of quantum mechanics, which regards since the beginning the particles as waves; instead the last equations have concerned the corpuscular properties of matter. The next section shows how to introduce in this conceptual frame also the wave formalism, in agreement with the corpuscular/wave nature of the particles.  (1.2). Since any integer n can be expressed as a difference of two integers n′ and n′′ one finds

The Wave Formalism
and thus also having multiplied both sides of these equations by * δψ ′  and * δψ ′′  with the purpose of obtaining again via (2.55) a real value of momentum consistent with the relativistic p  of (2.49). Subtracting side by side (2.55) and writing explicitly It is easy to acknowledge that for  . An identical reasoning holds of course here; trivial algebraic steps analogous to that from (2.55) to (2.57) yield compatibly with the existence of states of negative energy. Moreover  , whereas (2.57) would consist of two primed and double primed functions like this From these considerations inferred as corollaries of (1.2) and (2.32) without need of postulates, was born the early wave mechanics and the modern quantum mechanics.

Relativistic Corollaries
This subsection examines four main implications of (2.2) noting that if too. Owing to (1.4) and (2.9), the definition (1.1) of space time appears compliant with the idea of a dynamic system characterized by matter, energy and forces; also, the equivalence of mass and energy of the special relativity inferred in (2.33) agrees with the feature of space time characterized by the energy density η of (2.1) inherent to its definition (1.1). Without these results the space time would be an empty concept unavoidably abstract and unphysical. Instead, for reasons shown in the appendix B, does exist in principle an outwards pressure corresponding to the energy density η in (2.2), which in turn can be partially or totally counterbalanced by the attractive gravitational effect of matter/energy possibly present in a given volume of space time according to (2.32) and (2.33). The space time is therefore a dynamical system, in principle in equilibrium or non-equilibrium conditions, which evolves as a function of time. This point in particular, which anyway governs its dynamics, is now concerned to justify the possible presence of mass in a volume 3 x δ of space time. Implement (1.1) to find a further result based again on a dimensional reasoning. Note the possible correlation between quantities having the same physical dimensions; m is an arbitrary mass confined and delocalized within the arbitrary size δ  of an uncertainty space time range, thus without chance of information about its exact position. This section concerns just the physical conditions consistent with the delocalization of m in an uncertainty range, in agreement with (1.2).

Real and Virtual Mass
Are reasonably conceivable two conditions on the correlation (3.1), here expressed as follows being ξ an appropriate proportionality factor.
(i) One concerns the Lorentz invariance of both definitions (3.1): for the first one this condition is self-evident because it is a constant, for the second one the condition must be purposely required. Write owing to (2.35) since both δ ′  and m′ are Lorentz transformations of δ  and m, it must be true that With the given definition of β , for i.e. m is the v dependent dynamic mass corresponding to the rest mass m′ defined in (2.33), whereas δ ′  is the space contraction of the proper length δ  . It is significant that (3.3) confirms the result (2.36) obtained via (2.35).
(ii) Consider now the limit of (3.2) for 0 m → ; it is reasonable to expect that this limit is nothing else but the definition (1.1) of empty space time, i.e.
This limit ensures the consistency of the definitions (3.1) in agreement with the idea of m delocalized in δ  : if no particle is delocalized, the range size is null. This suggests putting by dimensional reasons Note that owing to ( the last step of the chain means that whatever λ′ might be, it is possible to define a corresponding r δ ′ that must identically fulfill the condition (2.26) being inessential the primed notation. Hence (3.7) yields On the one hand multiplying and dividing the right hand side by o m m (3.9) i.e. still holds a Newton-like law but with positive sign, yet in principle still consistent with (2.11). On the other hand, examine (3.9) that holds in general for any the last position holds because n′ is due to the integer number of wavelengths consistent with 2 r δ π ′ according to (2.26), whereas n is clearly due to (1.2). In turn, via the Planck force. So bh r δ takes the form of a zero point energy of a mass m oscillating with frequency bh ν corresponding to (3.15) in its confinement range δ  .

Invariant Equations of Special Relativity.
Rewrite Identically (1.3) as being v c ≤ the modulus of an arbitrary velocity allowed in the space time containing mass, concerned in the previous subsection. In principle v could be the group velocity (2.25) of a wave packet propagating through space time volume filled with dispersive medium, or it could be the expansion rate of the boundary of space time volume compatible with (2.14), or eventually it could even be simply the velocity of a body of matter moving through the space time; it depends on how is defined v. To examine this point regard v as a possible velocity allowed in the space time, whatever it might represent in any reference system, and consider that (3.19) identically rewritten as describes the swelling of the early space time volume introduced in (1.3), here indicated as 3 x δ along with the factor 2 2 c v . This equation is justified by (2.13), (2.14), (2.2) and (2.9) and will be further implemented also in the next subsection 3.5. Rewriting explicitly (1.1) as a function of v one finds therefore accordingly the identity (3.19) becomes compliant with the space swelling rate during the time lapse t δ , whereas (2.1) yields In principle this result is compatible with (2.4) and (2.14). A corollary of (3.22) follows starting again from (3.21) to write Journal of Applied Mathematics and Physics according which trivial manipulations yield, as shown in (3.3), The algebraic steps show that Lorentz transformations and invariant interval in inertial R and R', here introduced for simplicity via a one dimensional approach but immediately referable to a 4D formulation, are intrinsically inherent the space time definition (1.1) yield again space contraction and time dilation of special relativity. Equation (3.27) is particularly important because it is shown in [13] that the invariant interval is the conceptual basis of the special relativity, whence the chance of obtaining in particular (3.28).

Relativistic velocity
The results hitherto achieved compel explaining the concept of velocity. Multiplying both sides of (2.10) by 3 v c one finds by consequence of (2.9) and according to (2.29) whereas it is possible to define that allows the last equation. Once more it is worth emphasizing that the inva- and thus, taking the reciprocals of both sides, First of all eliminate n; it could be put equal to 1 by definition, yet it is easy to follow a general procedure valid for any n. With the positions , , , Let us elaborate further this result in order to obtain a significant equation; is useful in particular the position where ξ is arbitrary proportionality factor. With V δ in principle arbitrary as well because of the uncertainty, as previously stated, this position is allowed. This step appears important rewriting (3.36) via (3.37) as where the resulting * V is still an arbitrary velocity. This formula is actually well known, as it relates in special relativity the sum of c plus any velocity returns always c.
This reasoning is not at all redundant repetition of a result already known: (3.30) and (3.31) are quantum properties obtained contextually to (3.29) that is the invariant definition of relativistic momentum. Hence the reasoning implies merging of quantum and relativistic results concurring to the definition of c as an invariant limit velocity: this crucial statement of relativity is here required by (1.2). Note that (3.38) has been obtained via V r and V, which are arbitrary like the respective v r and v but leave out n: i.e. the quantization is not essential to infer (3.38), as it has been emphasized while obtaining (3.36). On the one hand it explains why the relativity was formulated without suspecting the underlying quantization, which indeed appears hidden in (3.35) in the present model. On the other hand it means that the positions (3.35) are not merely formal, as it is evident rewriting (3.33) as the problem of (3.33) is that the left hand side vanishes for n → ∞ incompatibly with the right hand side that never vanishes because v c δ ≤ . Owing to (3.35), instead, at the right hand side of (3.33) appear just the velocities leading to the result (3.38) of actual interest without contradicting the arbitrariness of n.
So (3.38) completes the conclusion (2.18), where a well known quantum inequality was inferred just from a physical property of c; here also this property of c appears as a further corollary of (1.1) and (1.2). Once more, as already shown in further papers [4] [11] [12], relativistic and quantum principles appear in the present approach as harmonically coexisting concepts without "ad hoc" hypotheses.

Euler-Lagrange Equations and Gravitational Potential
Start from (1.2) that yields also, recalling the considerations of subsection 2.6, let us define

S. Tosto Journal of Applied Mathematics and Physics
Note now that (4.1) can be rewritten as The step from (4.1) to (4.3) is not trivial. As anticipated in the subsections 2.6 and 2.7, the chance of exchanging the place of x δ and p δ  fulfills the specific concept of derivative in the physical frame of the quantum uncertainty; in fact x δ  is mere ratio of arbitrary ranges finite by definition, to be regarded as independent differentials possibly but not necessarily tending both to zero. This subsection aims just to show that this way of intending the quantum derivative is physically sensible. Multiplying now both sides of the second equality (4.3) by x δ , one finds Define now a function f consistent with this result, i.e. such that f δ fulfills in turn (4.4) and (4.5) are consistent with the positions As concerns the first equation, the first (4.2) yields .
Hence, merging the last two results, one finds According to (4.4) the function f has physical dimensions length 2 /time and fulfills the same kind of equation of the Lagrangian  of a physical system; in fact f is proportional to  a multiplicative constant 3 c G apart. Since is an energy, this is in principle just the sought Lagrangian. Yet the way to obtain this equation via the proportionality constant does not require the condition The Euler-Lagrange equations are well known; yet the non-trivial fact is that they have been inferred here as corollaries of (1.2) and (1.1), which are the conceptual root of both relativistic and quantum physics. Moreover this result supports the present way to regard the concept of derivative as a ratio of uncertainty ranges. Follow now two checks of the present way of reasoning.
(i) The classical Newton law inferred in the section 2.2 seemingly does not account for the finite propagation rate of any perturbation or interaction. Actually such information is explicitly available writing The force defined in this way is related to an energy ′ ′′ ≤ ≤    , vanishes with of the space time range x δ , is positive or negative depending on whether x δ swells or shrinks as a function of time and vanishes for 0 s δ → ; i.e. the force is defined within 0 s δ ≠ . It reasonably means that a time range t δ is necessary in order to allow its propagation at distance s δ , outside which the force in null. The fact that s δ has been defined via c means the carrier of the force must be a virtual photon or a graviton or anyway a massless particle propagating at speed c. Note that instead the classical Gm m  has the form of a force propagating instantaneously because it is based only on (2.9) and (2.10) without implementing (1.2). Actually (2.11) avoids itself this error because it is expressed via the uncertainty range δ  , not via the deterministic  ; since (1.2) involve inherently t δ , (2.11) could have been written itself as thus involving anyway the time range t δ  governing its propagation. This holds of course for any force. As concerns the gravity note that also now it is possible to repeat for (4. might be, it is reasonable to regard its series expansion whose first order term is a constant; if so, then neglecting for the moment the higher order terms, it is possible to write This formula is formally similar to (2.11), yet it incorporates the idea of a non instantaneous long term force that worried Newton himself. The form of the higher order terms will be concerned later, see next (5.114).
(ii) Consider eventually that (2.12) yields with the help of (1.2) and (4.2) In summary it is possible to write this result as where ϕ is a new function having physical dimensions velocity 2 ; with vector notation the first equation reads This definition, inferred here as a corollary, was taken in [13] as a basis to infer special and general relativity; the sign of δϕ depends on whether 1 δε δε −   . Moreover (4.11) also reads according to (2.8) where x a has physical dimensions of acceleration. It is immediate to acknowledge that ϕ of (4.11) is the definition gravitational potential [13], which will be more specifically concerned in the next subsection 5.5. Also, v  is not simply being a acceleration by dimensional reasons. Hence (3.20) yields Summarizing therefore this result as by definition of uncertainty range ( ) 1 δ −  , there are in principle three chances. The first one is that with notation emphasizing that a is a constant. Moreover are also possible for (4.15) appear natural the positions (4.14). In the particular case (4.16) where a is constant, it is possible to write whereas (4.15) becomes being  the Laplace-like curvature radius of space time with principal curvature radii 1 δ ±  and 2 δ ±  . In general the signs of these radii depend on the specific problem [14], e.g.: for a liquid droplet in a gaseous environment are both positive, for a gas bubble in a liquid environment both negative, for a liquid meniscus between solid cylinders with saddle-like geometry one positive and one negative. So it is not surprising that in principle all chances have been found in the present general approach starting from first principles as concerns the space time swelling. It is significant anyway that the concept of space time curvature is definable in a natural way even in the present quantum/relativistic context through the concept of uncertainty range. It is instructive in this respect the crucial role of (1.2) in linking quantum and relativistic points of view. Consider two remarks. ( that in turn reads ( ) the last equality is legitimated in analogy with (4.19) once having defined by dimensional reasons

The General Relativity as a Corollary
Some relevant concepts of general relativity are quoted in this section to show how to generalize the approach hitherto followed for the special relativity. Are examined in particular further significant implications of the quantum uncertainty ranges, to show how both special and general relativity contextually merge in a unique non-local and non-real conceptual frame. Some hints in this respect have been early examined in [16]; further topics are here reminded along with new considerations just to point out what have to do these typical concepts of quantum theory with the gravitational field. Indeed the problem of quantum gravity involves non only the quantization of this field according to the distinctive concept of superposition of quantum states, but also the inherent concepts non-reality and non-locality.

The So-Called "EPR Paradox"
It has been shown in the subsection 2.8 that the wave formalism is a corollary of (1.2) together with the relativistic properties inferred in subsections 2.5, 3.1 and 3.2; these results make in principle the present model compatible with the standard answer of wave quantum mechanics to the paradox. Yet, although it would be legitimate to skip additional comments to the ample literature already existing on this topic, it is instructive to emphasize the distinctive contribution provided to the paradox by the present model based on the quantum uncertainty. Deserve attention the following crucial points of this theoretical framework: (i) the concept of uncertainty ranges replacing the local dynamical variables is in principle compatible with the concept of entanglement; (ii) the difficulty of superluminal distance is bypassed, because the deterministic concept of distance between physical objects is unphysical; (iii) the concept of non-locality reduces to that of unpredictable randomness of particles confined and delocalized in quantum uncertainty ranges and excludes any kind of local information; (iv) by consequence of (iii), the concept of "non-locality" is strictly related to that of "non-reality".
Consider two particles, whose delocalization is in principle possible either in their own independent uncertainty ranges or in one shared uncertainty range. In the first case the particles in holds identically also for two particles in their own uncertainty ranges. Hence do not exist "spooking actions at a distance" but rather "actions at a spooky distance": once having renounced to the classical determinism and accepted (1.2) there is no way to distinguish the behavior of particles far apart or close each other confined in a given delocalization range, whatever their interaction mechanism might be. Particularly interesting is the former case of two entangled particles born within a unique uncertainty range where, for example when one γ photon decays by interacting with a nucleus or via vacuum fluctuation, e.g.
Accordingly, in the conceptual frame based on (1.1) and (1.2) the EPR paradox shouldn't even be formulated: the present model is inherently non-local by definition. Moreover the agnosticism of (1.2), not purposely invoked here but assumed since the beginning as the unique leading idea of the present physical model, implies a conceptual gap in (3.1) between elusiveness of (1.1) and reality of (3.9); the former is mere dimensional definition of the framework allowed for latent events, the latter made feasible by the measure process breaks the latency of possible events. Since nothing is "a priori" known about v m and * v m , e.g.
number of pairs or energy and lifetime of pairs and so on, the present model is inherently also non-real by definition. In other words the physical agnosticism implied by the concept of uncertainty as hitherto exposed, corresponds to the non real essence of the quantum world before the experiment; hence one must accept the idea that also the relativistic properties hitherto inferred are subjected to the same non-weird but logical consequences of (1.2) without need of postulating any "collapse" of wave function into a well defined quantum state. It means that n introduced in (1.2) and next appearing in (3.9) remains arbitrary and undefinable until when the measurement converts it into a specific obs n ; in turn, the wave formalism allows calculating the probabilities inherent the superposition of allowed states. Consider now the orientation of the possible spins of the particles with respect to an arbitrary direction. When measured, their spin orientation must yield a total angular momentum equal to zero like that of the empty space time (1.1) before the vacuum fluctuation (3.11). Physical information in this respect is provided only by the angular momentum conservation law, which however presupposes a measurement process. In general this is a perturbation action that affects the quantum state of any particle. In particular, being both particles in the same x δ , the measure process perturbs the system of entangled particles wherever they might be, not either particle only. If for any physical reason the shared x δ is modified, then the consequent x δ  implies p δ  and thus a force field in x δ that in turn perturbs the couple of particles. from a deterministic metrics, whatever it might be. To confirm that all of these considerations hold also for the relativity, the next subsections concern a few selected topics purposely chosen to emphasize the role of the quantum uncertainty in the general relativity: the latter is in fact a corollary of the former. The most important point in this respect is the equivalence principle, which is soon examined first in the section below.

The Equivalence Principle
Two relevant results previously obtained, Equations (2.9) to (4.25), address directly to Einstein's equivalence principle, as it has been explained through the simple reasoning early concerned in [9]: the reasoning is so crucial and short to deserve being sketched here for completeness. Think a space time uncertainty range in agreement with the second (2.19). Clearly  is the space time Laplace-like curvature radius corresponding to the attractive gravity force − of (2.11), as explained in (4.19).
Once having expressed the deformation of space time in terms of range size change rate x δ  , return now to the Einstein equivalence principle considering for simplicity the change of ( ) 1 x t only with constant 2 x ; this is enough to account for the rising of a force field inside x δ and highlight the reasonable conclusions of two independent Newtonian observers sitting on either boundary of ( ) The key points are: (i) the observer 1 sitting on 1 x experiences an acceleration since his variable coordinate is defined with respect to the origin of R, i.e. this observer moves far from or towards to the origin of R during the deformation of x δ ; (ii) the observer 2 feels anyway a force field inside x δ although he is at rest in R.
Therefore the observer 2 concludes that an external field is acting on x δ , whereas the observer 1 acknowledges an acceleration as if his position in R would be perturbed by the force field in x δ . Once more the consistent conclusion is that in fact the space time deformation rate x δ  causes itself the rising of a force field and that an accelerated reference frame is equivalent to such a force field. Only for 0 x δ → the force field appears as a local classical force. It is immediately evident the role of the quantum uncertainty in this explanation of the concept of force, required by the physical equivalence of the boundary coordinates in lack of any discriminating information about their behavior: indeed p δ  is nothing else but a corollary of

Quantum Angular Momentum
This topic has been concerned in [11] [16] [20]. Here are sketched for completeness some selected reminds only, useful later. By definition the component of angular momentum along an arbitrary direction defined by the unit vector z is M z = × ⋅ r p z , which reads in the present conceptual frame as Let us sketch some properties of quantum angular momentum, which will be useful in the next subsection, assuming that L l L − ≤ ≤ ; i.e. l ranges between two allows values −L and L, of course arbitrary, whereas (5.4) holds for any L, exactly as done in (1.9). The following considerations emphasize the reasoning carried out in [11], although here the steps to calculate M 2 differ slightly from that therein exposed: consider here that if M z is the only component knowable, then M 2 must be somehow related to M z only. Note that Since the angular quantum number l is actually a number of allowed quantum states likewise n of (1.2), the idea is now that M 2 should be defined by its own quantum angular number of l states and that in turn this latter is related to the sum of all l-th states allowed to its unique defina- In effect, once having written , the knowledge of the three components (5.6) reduces in fact to that of one component only; hence it is natural that this result coincides with that of (5.5) expressed in 2  units and confirms (5.8). Follow now three important corollaries.  Replace thus M z c with 2 q , as both have physical dimensions energy length × ; the same reasoning yields now (ii) A further corollary concerns the spin of particles and the Pauli principle. i.e.

( )
Note that the left hand side of (5.15) defines an angular momentum in fact allowed, so  in (5.16) is a half integer angular momentum due to 2  , which is clearly by analogy with l the component along an arbitrary direction of a new half integer angular momentum. In [9] is concerned the spin of particles more in detail starting from (5.15). Here this topic is not further concerned for brevity, e.g. to show why actually J L S = ± ; it is interesting instead to remark that the Pauli principle follows as a corollary of (5.16) [20]. This interpretation of the Pauli principle is a crucial consequence of the fact that l and L are not mere quantum numbers, but numbers of allowed quantum states likewise n of (1.2). (iii) Consider the following definition of M z , which reads  having merged ζ with the arbitrary range size of δ  . This simple reasoning has defined via M z the fine structure constant, the Coulomb law, the magnetic potential  and the definition of Planck charge.

Black Hole
Consider (3.2) rewritten according to (2.20) as where v is velocity by dimensional reasons. Let us define now a dimensionless parameter ζ such that with the same physical meaning of (5.28). So the second (5.28) is the well known  Now note eventually an interesting corollary of (5.33) that reads * * c δ =    and is identically rewritten as where z is a new arbitrary parameter not yet introduced in the present model to be appropriately defined in agreement with (5.18). Simply renaming z as z e = ± , where both signs are compatible with 2 2 z e = , it follows that e ± is the electric charge, whereas the proportionality factor α linking the Coulomb law hidden in *  via c  is actually the fine structure constant. This last result is closely related to the results from (5.17) to (5.20) previously found. Emphasize now that the particular condition (5.24) is sensible, although it has been introduced preliminarily just in order for (5.23) to match (3.17) and not as a consequence of a fundamental requirement; yet (5.24) can be generalized while regarding (5.28) as mere particular case. The key point is to replace (5.23) via a function  of ζ and bh r δ defined as follows The condition on , .
bh bh On the one hand it is reasonable to assume that these equations concern two different properties of bh r δ ; on the other hand it is also reasonable to guess that two properties of the mass * m can be charge and angular momentum due to its possible angular spinning or to its possible spin or both. Anyway, since the uncertainty ranges at left hand side represent square lengths, it is immediate to conclude that the same holds for the right hand size terms; in other words, to include the charge terms it is enough to express the space range sizes that appear in the Coulomb law of (5.36). So, in Planck units, with appropriate signs at the right hand side. This yields the well known result consistent with (5.14) ( )

From Special to General Relativity
Rewrite the first (3.32) as In this case write (5.40) as thus, apart from the factor in parenthesis appearing at both sizes and thus irrelevant, (5.45) implies again and merge themselves into The left hand side of (5.54) diverges for n → ∞ ; however this is not a problem, being allowed by 0 t δ → .
where it is possible to identify at the right hand side The right hand side is the Hamiltonian of the orbiting system, i.e. Newtonian binding energy − G , which agrees with the idea of harmonic oscillator as a bound system itself. The minus sign of  G means that the force constant f k defining the quantum oscillator frequency implies an attractive energy between two orbiting masses; a repulsive energy would be instead inconsistent with steady quantum oscillations. As expected, whatever 1 m and 2 m might be, an appropriate n shows the actual quantization of orbital motion: for large masses, n is so large that the quantization is hidden by the values i.e. classically 2 ω is proportional to 1 m − via the factor f k . Moreover, the link between a one dimensional oscillation and a two dimensional system orbiting on an arbitrary plane implies 2 degenerate states, as the clockwise and counter clockwise rotation are both allowed and in principle indistinguishable; this also holds in the quantum world, indeed l takes all values n l n − ≤ ≤ identically to l − . So the macroscopic measurable orbiting energy (2.11) is twice that 1 2 2 m m G δ  of (5.60). This degeneracy can be also regarded as a statement of equivalence between inertial and gravitational mass: the degenerate energies concern now the systems where m 1 moves around m 2 or m 2 around m 1 , depending on either reference system R 1 or R 2 where the respective mass is at rest. Without the equivalence principle, R 1 and R 2 would not interchangeable, as instead it is true according to the quantum (2.41). The macroscopic Newton law inferred from an oscillating quantum system reveals and requires the sought equivalence. 5) Implications of harmonic oscillations. As a closing remark consider now the following dimensional definitions The physical meaning of these results will be highlighted by calculating their numerical values in section 6. 6) Gravitational waves. This subsection aims to sketch that the gravitational waves are actually quantized and fit the result inferred in (5.60); details on the physical model and results have been already concerned in a paper [16] on this topic. To add further considerations in this respect and highlight this point, let us start from the Einstein formula ( )  Just this conclusion is the key to guess the dimensionless G t ω that appears to be just a correction factor: being n δ integer, E δ − can be nothing else but something like n hν ′ with n′ integer in order to fit (5.60) [21]. Also, as G t is introduced via δε and thus arbitrary, put then ( ) Otherwise stated, G t ω has been defined in order that (5.74) is consistent with (5.62). In synthesis, the initial Einstein formula, deterministic, becomes here a very simple quantum result, showing at the right hand side the number n′ of gravitational energy quanta lost. Also here E δ − expresses the fact that n′ must be intended as n n n ′′ ′ ′′′ ≤ ≤ , with n′′ and n′′′ of course arbitrary, once more according to (1.9).
Although for brevity this result has been introduced here as mere elaboration of Einstein's early achievement, reversing the steps from (5.74) to (5.70) one could find the initial E δ whose quantization is however hidden. The paper [9] concerns instead an "ab initio" model, where are also described further implications of this result. The Einstein formula is actually a quantum of gravitational energy dissipated by an orbiting system. In this quoted paper, published before the experimental evidence of the gravitational waves, it is remarked that not necessarily the gravitational system must collapse; rather both signs possible for n δ describe the exchange of gravitational quanta between orbiting systems, possibly the so called gravitons, could be regarded in principle in analogy with electromagnetic excitation and decay of atoms by exchange of photons. This supports the idea of gravitons inherent the gravity propagation rate (4.8). The black hole radii are thus the limit of (5.40) for 1 v c → suggest an interesting feature of a bound gravitational system where either mass is a black hole; merging of their masses occurs when the event horizon of the latter approaches the Compton length of the former.
(ii) It is usual to say that at the center of a black hole there is a space time singularity. Emphasizing that no singularity is explicitly required by or directly implied in the present conceptual frame, such a singularity is actually unknowable and thus unphysical: according to (1.2) and (2.41), by definition non-deterministic, no information is accessible about what happens inside an uncertainty range. Thus the concept of local singularity is merely an arbitrary extrapolation allowed in the classical world only; here instead the relativity is conceived in the quantum frame of (1.2).
(iii) Via (5.59) yields an expression for the force constant f k of harmonic oscillations (5.62) Where a is a further definition of acceleration being . This result, which clearly plugs the force constant of the quantum oscillator into the frame of the general relativity, will be further considered in the next subsection.
(iv) The reduced Compton length Clearly the right hand side is a constant, thus invariant by definition; hence the ratio at the left hand side is an invariant as well. It is known indeed that this ratio is defined by two invariant quantities. Since the first (5.46) demonstrates that x t δ δ is a relativistic invariant, it follows that the numerator is also inva- x c t δ δ − , in particular, has been stated in [13] as the conceptual foundation of the special relativity; just for this reason it is remarkable the fact that in the present model i(5.46) and (5.90) are actually straightforward corollaries of the quantum uncertainty. The crucial difference between (5.90) and (5.86) is that now ε and pc appear through their uncertainty ranges and not as deterministic values. This result not only demonstrates the link between special relativity and quantum physics, but also allows further important steps concerning directly the general relativity. Although this point has been examined in several previous papers, see e.g. [4] [11] [12], the next section reports some relevant considerations just on this topic. Consider once more (5.83) for 1 2 v v c = = , as already done to infer (5.86); the reasoning is still that already highlighted, but now extended to find a further interesting result. Write explicitly (5.83) with the help of (5.34) as follows Einstein's special relativity. In fact the additional term in (5.92), more general than (5.86), is a known result of quantum gravity that helps solve three cosmological paradoxes [22]. More details about (5.92) are reported in [12].

Red Shift and Time Dilation
Starting from (2.10) and (2.9) consider ϕ [13], being  Owing to proper time 0 t t < , this result yields time dilation t δ due to gravity field with respect to field null.

Black Hole Entropy
Define the ratio of Planck length and mass,

Perihelion Precession
Consider the square ranges ( )  is now an "effective" radius, taking into account that the perimeter C of ellipse is actually a function of its semi axes a and b; an approximate formula is for example , which is reliable for the present purposes because for a b = it reduces to 2 C a = π . If b a > then ( ) ( )( )

S. Tosto
(ii) The second one lucidly shows step by step how to infer classically this famous Einstein formula of Mercury perihelion simply comparing two forms of potential energy of orbiting system; the mathematical formulation introduces first the mere Newtonian potential 1 N U r β = and then also assumes an extended potential form 1 : (5.114) the notation emphasizes that the time derivative x δ  is actually regarded as ratio of ranges ( )  in agreement with the definition (6.11). It appears that the quantum energy η ε directly implied by the Hubble constant is just that given by (2.2). This calculation is interesting as it emphasizes the direct link between Hubble constant and definition of space time (1.1) through the vacuum energy density, while also confirming (2.2) and thus the Newtonian (2.11) via the concept of acceleration (2.8). 6) Implement (3.23) that yields ( ) where  and ′  are in general time dependent energies related to the initial energy density η . The physical meaning of these energies results particularly significant rewriting the first (6.26) as follows The first equation is tested with the help of the universe timeline temperature vs time published by the Fermilab and reported in [26]. This point has been already concerned in [20], where it is shown through the plot of temperature vs time implementing the timeline data; it appears that actually  is a constant, it is the best fit coefficient of T vs 1  i.e. also now appears the dependence of T upon via the ratio m length ; so, equating the dimensional definition (6.34) and (6.27) via an appropriate proportionality constant ζ , it is possible to write contextually φ yields energy per unit mass i.e. the gravitational potential, which shows that the definition (6.52) of ζ is sensible. As stated in (5.70) and (5.71) the factor 4π fits well the numerical coefficient 64/5, the deviation being a few % only; so with the definitions of ζ and m′ (6.51) reads which is is nothing else by the Einstein collapse rate of two orbiting masses: r δ  has the correct relativistic form expected for the orbit radius contraction of a gravitational system due to its energy dissipation rate via gravitational waves, but now it is quantized in agreement with (5.74). Hence (6.49) is sensible starting point to calculate the dynamics of a gravitational system compliant at least in principle with the general relativity. Of course here the reasoning has been simplified and shortened for sake of brevity only; the aim of this last point is to justify the validity of (6.49) and its ability of defining the black hole energy density ( ) ciple possible, regardless of the specific explanation about their actual formation mechanism/process. In other words, merging quantum theory and relativity into a unique conceptual frame is problematic because the two-way correspondence "deterministic metrics  wave quantum theory" doesn't work.
This conceptual gap is in turn due to the initial purposes of either theory: the wave quantum theory was in fact born to explain why the electron does not fall into the nucleus, the relativity to formulate a covariant approach to the nature laws. The right direction to follow is thus to merge not the whole theories themselves, but rather their fundamental roots from which everything follows. It is intuitive that the physical frame able to account for the conceptual pillars of both theories consequently will also be able to account for their specific topics; in effect it has been easy to show throughout the exposition of the present model that relativistic and quantum outcomes are contextually inferred in a straightforward and simple way. Thus the strategy of the present model follows the idea of waiving the standard premises of both theories, not because they are wrong per se but because they are incompatible, at least in the usual form currently implemented: instead of thinking an advanced relativistic formulation of problems into which to include successively also the quantum requirements, or vice versa, seems more practicable the idea of identifying a common conceptual root to start with, in order to infer as a natural corollary the fundamental axioms of both theories. In principle it seems hard the idea of abandoning the deterministic metric able to formulate covariant laws of physics, although it conflicts with the Heisenberg principle and the non locality/non reality; likewise it seems equally hard to give up the corpuscle wave dualism capable of explaining the tunnel effect, although it has seemingly nothing to do with the perihelion precession.
In addition this preliminary intent is still not enough to outline adequately the physical problem, there is a further conceptual difficulty.
Usually, the idea of quantum relativity reminds concepts like quantization or gravitational interaction between particles moving at speed near c or even superposition of gravitational states. In this respect nothing hinders in principle to conceive the actual corpuscles as waves: then, since F p =  , it is anyway possible to introduce p h λ = and next to define Eventually, introducing the uncertainty (2.41), it is possible to proceed towards a gravity field valid in all reference systems.
This outline of alternative approach shortly sketched as a corollary of De Broglie momentum would certainly allow an innovative relativity without insurmountable efforts. But unfortunately this is not the true crucial point: the classical mechanics or standard relativity could not fit the conceptual character of the quantum world without accounting for two points that no mathematical code could ever introduce: the non locality and non reality, without which phenomena like the entanglement could never be explained or even conceived. Without these distinctive quantum features, would be out of our mind the Bell inequality, which instead is a fingerprint of the gap between relativistic and quantum theories as shortly sketched in Appendix C.
Actually the most important problem is to obtain a non-local and non-real general relativity.
These features seem oxymora when concerning real corpuscles that someway must be referred sooner or later to the Newton law, may be as a particular case or limit condition. In other words the crucial point is either to make relativity non local and non real or to demonstrate that quantum physics is local and real.
Yet the experimental data show that the second alternative is unphysical; so the only attempt to formulate a successful connection between the theories is the first chance, which however requires a new conceptual reformulation well beyond the mere mathematical strategies.
In this respect the results of the present model indicate that (1.2) are a simple and reliable candidate to account for both theories. The standard quantum mechanics implements the operator formalism that by definition is related to the wave behavior of particles; yet to match relativity it is more sensible to implement the corpuscular behavior of particles according to the uncertainty, which inherently imply both delocalized mass and wave behavior. This intuitive statement summarizes the basic idea on which has been conceived the strategy of this paper. Clearly the mathematical formulation of the theoretical model must be consistent with these premises.
The universe implies the uncertainty. Indeed the mere definition (1.1) of space time takes implementable physical meaning when written first as in (1.3), which in turn provides physical information when rewritten further as in (2.1) and then as in (2.2) and next as in (2.29).
Often the algebraic steps have been inspired by and based on initial dimensional relationships, rather than on mathematical equations: the former are actually conceptual similarities, the latter prospect specific local values. This is for example the case of Equations (2.9) or (3.1) or (2.29); yet (1.3) and (3.20) are examples of how the dimensional premises turn into a physical formulation to be compared with the experience. But just this comparison rises a further crucial point: the concept of measurement.
As in fact the strategy of the present paper has followed these ideas, anyway the resulting (2.40) of the section 2.6 can be nothing else but agnostic relationships between ranges of dynamical variables preliminarily introduced in (1.2); strictly speaking their agnostic essence is a corollary of the initial abstract considerations, in turn based on the physical dimensions of the fundamental constants of nature. The physical kernel of these constants contains however all ingredients necessary to "materialize" their dimensional implications: as a matter of fact the uncertainty ranges of dynamical variables inferred in this way, see e.g.
(2.29) and (2.35) or (5.45) and (5.46), fulfill not only the same relativistic transformation properties of the local dynamical variables but also the Heisenberg requirement. It is then evident the more general character of the present ap-ther implemented according to its own properties in the various sections.
On the other hand the uncertainty is regarded without having in mind only its original quantum implications, for this reason has been emphasized its immediate derivation from the operative definition of space time proposed in (1.1) as early exposed in [4]. In other words have physical meaning the uncertainty ranges, and not the random local dynamical variables.
To link quantum and relativistic physics implies a conceptual cost; for example the Lorentz transformation of x and x' does not read ( ) ( ) x Vt x Vt β β ′ ′ − → + because the local time and space coordinates are unknown, it is only possible to consider x x δ δ ′ → but only the origins of the inertial reference systems. As a first remark about the quantum theory in this respect, note that the previous considerations are enough to bypass a wave based quantum approach only, as it is currently done, to start instead from the (1.2) totally agnostic but just for this reason more general; nevertheless the wave formalism and all its well known implications are in effect a straightforward corollary of the quantum uncertainty.
As a further remark, is worth emphasizing that in this model are missing equations of motion to be solved; yet it is natural because (1.2) skip even the probability of local position. So also concepts like "comoving coordinates" are useless because is missing the concept itself of local coordinate, systematically replaced throughout this model by the physical concept of coordinate ranges; nothing is assumed known about these latter, while the same holds for any other dynamical variable. Nevertheless, just this agnostic approach allowed to obtain in a straightforward way relevant outcomes of general relativity and numerical results of the section 6 skipping crucial concepts like distances between objects, classically defined. Although the present model waives concepts definable in the frame of a deterministic metric, the conceptual limit put by the uncertainty selects the allowed knowledge actually accessible to the observer; e.g. by this reason one component of angular momentum is physically definable. Without being aware of this conceptual limit, relativistic and quantum theories would remain incompatible with each other.
Also note that aim and formulation of the present model are in principle different from that of Dirac in describing the relativistic hydrogenlike atoms: in His model, Dirac implements known relativistic concepts to infer a wave equation consistent with the ideas already formulated by Einstein. Here instead the fundamental principles of both theories are consistently cogenerated "ab initio" in a self contained way. The only common premise is that both (1.1) and in turn (1.2) merge together space and time, which therefore are meaningless separately: the former implicitly by dimensional reasons, the latter explicitly. Thus it is evident that (1.2) cannot imply any metrics, i.e the chance of defining lengths, angles and so on, just because size and orientation in the space time of all uncertainty ranges are completely unknown by definition; nevertheless the conceptual physical formulation of vectors follows by extrapolating physical concepts, i.e. simply guessing the meaning of dynamical variables corresponding to the range S. Tosto position of the object; it is nevertheless reasonable the fact that according to (5.30) and (5.94) in proximity of the massive object the local time run t δ due to the gravitational field is slowed down with respect the proper time range

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
and for 3 k = in particular by definition F is force while k x ∆ and x ∆ are arbitrary displacements of the boundaries defining the initial volume defining η , so that k P and P are pressures because S and k S are surfaces. In other words the energy density inside V is assumed to change at constant  merely because of the change of space time volume. Hence, taking the ratios side by side to eliminate δ =  F x, the first and last terms of the two chains yield  This identity is inferred from and thus agrees with the initial (C.2). It is clear that reverting the order of these steps starting from this identity, one finds again (C.2). Let now C P take different values C′ and n C′ , in which case the equality does no longer hold with the same values pertinent to A P and B P . In general, with the same reasoning repeated for initial ( ) : an initial discrepancy cannot result in a final identity through the same steps that have converted an initial identity into a final identity. Repeat now the reversed steps (C.5) to (C.2) starting from this inequality; it is clear that now one finds the corresponding inequality (C.3), rewritten in general via two inequalities. To understand whether (C.7) is consistent with (C.5) it is necessary to specify the properties , , A B C P P P : the problem is comparing "determinism vs non-determinism". In this paper determinism has been referred to either existence or not of local space time coordinates; now this concept is extended to the properties of the values (C.1) and concern non locality and non reality of the quantum world. The problem of interest is thus to establish whether or not the physical properties of a system of particles do exist "a priori" or are created by the interaction with the experimental apparatus that perturbs the initial unknown system. (C.1) have been written in order that the property C P appears with not value n C only and the property A P with occurrence value A only, whereas B appears in both forms; to infer information about non local and non real systems in comparison with real and local systems, consider a possible scheme where B is a property like particle spin or photon polarization. In the case of spin let B and n B represent the respective chances of spins paired or not. The scheme implements the following attributions of values , , At the right hand side of the inequality is concerned the quantum theory, which is both non real and non local; indeed the sections 2.6 and 2.7 have shown that the uncertainty requires contextually these features. At the left hand side is concerned any non-quantum theory, which is either non local or non real only.
The symbol ≥ is understandable regarding ( ) ...,... N as probabilities, which is possible simply introducing a normalization factor to unity; obviously the sum of probabilities of either property verified is higher or at least equal to that describing both probabilities contextually verified. The first column represents therefore the values A and B of the properties A P and B P in a non-quantum local theory, because it is non real only, and thus without spin correlation; this correlation requires indeed a non local spooky action to occur. In the second column the values B and C of the properties of B P and C P represents again a non-quantum theory, because of its non locality only, but now with spin correlation just due to its non locality. The third column represents the quantum theory, which is both non local and non real. The symbol ≥ identifies the Bell inequality. Predetermined physical properties, typical of non-quantum physical theories, classical and even relativistic as well, fulfill the inequality: the determinism of relativistic metrics belongs to a classical vision of universe, although enriched by covariance of physical laws, four dimensional premise, invariant light speed introduced by Einstein. The violation of the inequality does not require the existence of "hidden variables" to bypass the difficulty of a superluminal action between particles. In fact, hidden variables are excluded in the present conceptual frame based on (1.2) and bypassing the wave functions where these hypothetical variables could be somehow encoded. A further remark on the Bell inequality is that it reads