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An Analysis of States in the Phase Space From Quantum Mechanics to General Relativity_图文

AN ANALYSIS OF STATES IN THE PHASE SPACE: FROM QUANTUM MECHANICS TO GENERAL RELATIVITY

Sebastiano Tosto ENEA Casaccia, via Anguillarese 301, 00123 Roma, Italy sebastiano.tosto@casaccia.enea.it

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ABSTRACT The paper has euristic character. It exploits basic concepts of quantum physics to infer on a selfconsistent basis the properties of the gravitational field. The only assumption of the theoretical model is the quantum uncertainty: the physical properties of quantum systems depends on the delocalization ranges of the constituent particles and not on their local dynamical variables. The conceptual approach follows the same formalism already described in early non-relativistic papers [S. Tosto, Il Nuovo Cimento B, vol. 111, n.2, (1996) and S. Tosto, Il Nuovo Cimento D, vol. 18, n.12, (1996)]. The paper shows that the extended concept of space time uncertainty is inherently consistent with the postulates of special relativity and that the most significant results of general relativity are achieved as straightforward consequence of the space time delocalization of quantum particles.

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1 Introduction. The papers [1,2] have shown that the quantized angular momentum and the non-relativistic energy levels of harmonic oscillator, many electron atoms and diatomic molecules can be inferred utilizing one basic assumption only: the quantum uncertainty, introduced explicitly and since the beginning as conceptual requirement to formulate the respective physical problems. The theoretical model underlaying these papers starts from a critical review of the concept of local dynamical variables. In general, positions and momenta of the particles in any quantum system change in a complex way because of mutual interactions; the physical information accessible about the system, e.g. its time evolution, is inferred solving the appropriate wave equation. Since the solution providing the eigenvalues defines the wave function as a function of the space coordinates, these latter are in general regarded as necessary input of any quantum problem. Yet it is sensible to expect that a further distinctive feature of the system are the ranges of values allowed for coordinates and momenta of each constituent particle; also these ranges depend on the kind of interaction and are distinctive of the global configuration fulfilling the condition of minimum energy. The importance of this reasoning is evident even from a classical point of view, considering that the dynamical variables are mere limit cases of vanishingly small intervals of variability around the corresponding local values. Hence appears rational in principle and even more general, the possibility of describing the physical systems through their phase space. This kind of approach, here introduced as mere alternative to the usual formalism of the classical physics, is in fact essential to formulate the quantum problems: moving the physical interest from the conjugate coordinates and momenta of the particles to their respective variability ranges appears particularly suitable to explain the behaviour of particles subjected to the Heisenberg principle. Consider for instance the radial distance ρ of an electron from the nucleus defined by 0 < ρ ≤ ρmax , being ρmax the maximum distance. If ρ changes randomly, then ρmax cannot be uniquely defined by a particular value specified “a priori”; yet is relevant in principle its conceptual significance: ρmax , whatever its explicit numerical value might be, introduces the range ?ρ = ρmax ? 0 allowed to the random variable ρ . Moreover also the change of local momentum 0 < pρ ≤ pρ max is likewise defined in the range ?pρ = pρ max ? 0 . Even in lack of detailed information about ρ and pρ , these ranges enable the number of allowed states in the phase space for the electron radial motion to be calculated; if so, ?ρ only is of interest, not any partial range ?ρ § = ρ ? 0 defined by random values ρ < ρ max that would exclude radial distances in principle also possible for the electron. Although in the present case ?ρ ≡ ρ max and ?pρ ≡ pρ max , i.e. the total ranges coincide in practice with the maximum values

ρmax and pρ max , the notations ?ρ and ?pρ better emphasize their consistency with any local
coordinates and momenta in principle possible for the electron. Consider now the energy E = E ( ρ , pρ ) of electron radial motion; it is possible to write E = E (0 < ρ ≤ ?ρ , 0 < pρ ≤ ?pρ ) , whatever the local ρ and pρ might be. Yet the previous considerations suggest regarding this energy as E = E ( ?ρ , ?pρ ) ; this last step is non-trivial because the unique information available is now the number of states in the phase space consistent with the ranges allowed to the variables of the system, whose calculation becomes then necessarily the central aim of the physical problem. These ideas clearly hold in general also for more complex systems, e.g. for the distances rij between the i-th and j-th electrons in a many electron atom. If so, instead of attempting to increase the accuracy of some existing computational model through a new kind of approximation or some new hypothesis to handle the local terms, appears more useful to exploit a leading concept very general and thus appropriate for any physical system: the quantum uncertainty. Then the basic reasoning to describe the electron moving radially with respect to the nucleus consists of the following points: (i) to introduce the ranges ?ρ and ?pρ ; (ii) to regard them as radial uncertainty ranges of the electron 3

randomly delocalized; (iii) to exploit the concept of uncertainty according to the ideas of quantum statistics. If so, the local values of ρ and pρ do not play any role in describing the electron radial motion: considering uniquely the phase space of the system nucleus/electron, rather than describing the actual dynamics of the electron through the wave equation, it is possible to disregard since the beginning the local values of the conjugate dynamical variables considered random, unpredictable and unknown in principle and then of no physical interest. This is a conceptual requirement, not a sort of numerical approximate method to simplify some calculation. The question rises however about the effective importance of these states in describing the physical properties, since E ( ?ρ , ?pρ ) cannot be longer calculated according to the formalism of wave mechanics. The paper [1] shows that only the concept of quantum delocalisation is essential to calculate “ab initio” and without any further hypothesis the energy levels of many electrons mutually interacting in the field of nuclear charge; this idea was proven more useful than a new numerical algorithm also to treat the diatomic molecules [2]. Despite the apparently agnostic character of such a theoretical basis, where any kind of local information is considered worthless, the results are in fact completely analogous to that of wave mechanics in all the cases examined, thus showing that the degree of knowledge possible is in fact that inferred considering the particles of the system randomly and unpredictably delocalized within the respective uncertainty ranges. According to the previous considerations, the basic assumption of the quoted papers is summarized as follows 1,1 E ( x , p x , M 2 ) → E ( ?x , ?p x , ? M 2 ) → E ( n , l )
The logical steps 1,1 do not require any hypothesis or constraint about the motion of the concerned particle and even about its wave/particle nature. The first step simply replaces the local dynamical variables x, px with the respective ranges ?x and ?px , arbitrary and linked by the relationship

?x?px = n? 1,2 with n in principle arbitrary for each freedom degree of the system defined by its pertinent couple of conjugate variables; here x denotes a set of generalized coordinates. The second step calculates the numbers of states through elementary algebraic manipulations, thus determining for instance energy and angular momentum of the electron in the field of nucleus as a function of the numbers of states n and l related to its radial and angular uncertainties; the latter number, in particular, determines the angular momentum M2 and one of its components M w along an arbitrary direction defined by the unit vector w . Three examples formerly considered in [1] are sketched in section 2 to clarify the assertions so far introduced and highlight how to exploit in general the positions 1,1 and eq 1,2, which are the unique postulate of the present theoretical model. This section aims mostly to show their validity and reliability by comparison with well known results of elementary wave mechanics; the approach is so simple that it is reported also here to make the present paper self-contained and clearer. The early non-relativistic papers [1] and [2] were progressively extended to more complex problems. The reference [3] concerns the relativistic free particle, the reference [4] the many electron atoms/ions, the reference [5] a thermodynamic system composed by an arbitrary number of metal ion cores organized to form a lattice surrounded by a sea of delocalized electrons. In these papers the compliance of the positions 1,1 and 1,2 with relativity was only shortly sketched; the connection between quantum mechanics and relativity requires a more specific examination, out of the mere quantum character and purpose of the quoted papers. The physics of the quantum world rests on the uncertainty, which replaces the concept of position with that of probability density; in general relativity, however, position and velocities of particles have definite values, the spacetime metric is defined by a set of numbers associated with a given point with respect to which is defined the distance of any other point. The quantum theory is non-local, the general relativity exploits the local realism. The basic assumptions of relativity concern light speed, reference systems and equivalence of inertial and gravitational mass through which are defined scalar curvature and tensor properties of spacetime in the presence of matter; in quantum theory the
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electric and magnetic fields are subjected to the uncertainty principle. The gravity does not obey therefore the rules of field theory and appears conceptually far from the weirdness of the quantum world; yet relevant coincidences with the electromagnetism have been remarked, e.g. the gravitational waves propagate at the light speed. The quantized results of wave mechanics and the outcomes of the continuous space-time of relativity seem apparently related to two different ways of thinking the reality. The modern theories of gravity regard the spacetime metric as a field and attempt to quantize it; certainly, this approach makes the relativistic formalism closer to that of quantum mechanics, yet the quantum gravity is still today a puzzling problem. The string theory is one of today’s leading theories for its internal coherence of approach to the quantum gravity, yet it is not completely clear how it relates to the known physical universe. On the one side the conceptual gap just sketched acknowledges implicitly two different ways of describing the reality, on the other side it suggests a possible alternative to the current efforts aimed to introduce ideas and formalism of one theoretical frame into the other one: to search a key principle underlaying both microscopic and macroscopic theories to reveal their intimate connection. The sought idea should exploit an essential principle of nature, so fundamental that even the principles of relativity are conceptually rooted and hidden in it together with the probabilistic character of quantum mechanics. This path is effectively viable if the quantum origin of the gravity force could be demonstrated: if so, the task of harmonizing the two theories would be replaced by that of organizing various hints sequentially inferred from a unique conceptual basis common to both theories. In the present paper the sought basic principle is tentatively identified with the quantum uncertainty, already proven effective in [3,4,5]. This choice is decisive to organize the paper: the main aim of the next sections is to demonstrate the quantum nature of the gravitational field. The first step to this purpose is to show the link between the present approach and the operator formalism of wave mechanics; the reasoning extracted from [3] and exemplified in appendix A for the simple case of a free particle, highlights that the operator formalism is inherent the concept of uncertainty and can be inferred from this latter together with the concept of indistinguishability of identical particles; it shows that the approach starting from eq 1,2 thanks to the positions 1,1 is more general than that based on the operator formalism of wave mechanics and then more suitable to be extended to the problems of quantum gravity. The second step is to show that the examples of section 2 are susceptible to more profound generalization. The reasoning introduced in section 3 extends the results of section 2 to the special relativity with the help of a further uncertainty equation involving the time. Once having proven that the conceptual frame underlying the examples of section 2 is compliant with the basic principles of special relativity, an analogous procedure is followed in section 4 to describe the behaviour of quantum particles subjected to the gravity force. The sections 3 and 4 are then the most important ones of the present paper, in that they provide the generalization of the positions 1,1 to the special and general relativity with the help of eq 1,2 and next eq 2,5 introducing the time uncertainty. 2 Simple non-relativistic quantum systems. 2.1 Angular momentum. Let p = p and ρ = ρ be the moduli of the random momentum and radial distance of one electron from the nucleus having charge Ze . The steps 1,1 require that only ?ρ and ?p must be considered in the phase space of the system nucleus+electron. No hypothesis is necessary about ?ρ and ?p to infer the non-relativistic quantum angular momentum and one of its components M w = (ρ × p) ? w ; any detail about the actual electron motion is unessential. As shown in section 1, the first step 1,1 calculates the number of states allowed for the electron angular motion through the positions ρ ≡ ?ρ and p ≡ ?p ; putting ? M w = ( ?ρ × ?p ) ? w = ( w × ?ρ ) ? ?p , then ? M w = ?χ ? ?p , where

?χ = w × ?ρ . If ?p and ?χ are orthogonal ? M w = 0 , i.e. M w = 0 ; else, writing ?χ ? ?p as
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( ?p ? ?χ / ?χ ) ?χ

with ?χ = ?χ , the component ±?pχ = ?p ? ?χ / ?χ of ?p along ?χ gives

? M w = ±?χ?pχ . In turn this latter equation gives M w = ±l ? where l = 1, 2 ? ? ? according to eq 1,2.

In conclusion M w = ±l ? , with l = 0, 1, 2, ? ? ? : as expected, M w is not defined by a single value function because of the angular uncertainty of the electron resulting in turn from the uncertainties initially postulated for ρ and p . Being ?χ and ?pχ arbitrary, the corresponding range of values of l is arbitrary as well; for this reason the notation ? M w is not longer necessary for the quantum result. It appears that l is the number of states related to the electron orbital motion rather than a quantum number, i.e. a mathematical property of the solution of the pertinent wave equation. The quantisation of classical values appears merely introducing the delocalisation ranges into the classical expression of M w and then exploiting eq 1,2; the physical definition of angular momentum is enough to find quantum results completely analogous to that of wave mechanics, even disregarding any local detail about the electron motion around the nucleus. The quantity of physical interest to infer M 2 is then l , since only one component of M can be actually known: indeed, repeating the procedure for other components of angular momentum would trivially mean changing w . Yet, just this consideration suggests that the average values of the components of 2 2 angular momentum should be equal, i.e. ? M 2 x ? = ? M y ? = ? M z ? . Each term is averaged on the number of states summing l 2 ?2 from ? L to L , being L an arbitrary maximum value of l ; then L 2 2 3 2 2 ? M i2 ? = ? 2 ∑ llii = =? L li (2 L + 1) gives M = ∑ i =1 ? M i ? = L ( L + 1) ? . Clearly these results do not need any assumption on the specific nature of the electron and have therefore general character and validity for any particle; in effect, after the first step 1,1, the unique information available comes from the very general eq 1,2 not longer involving local coordinates and momenta of a specific kind of particle. In this first example ?ρ was in fact coincident with the maximum value ρ max once having defined the random variable ρ in the range of values 0 < ρ ≤ ρmax . More in general, ′ ? ρo whitout changing the however, the radial uncertainty range could be rewritten as ?ρ ′ = ρ max result; ρo is the coordinate that defines the origin of ?ρ ′ . This is self-evident because neither ρo nor ρmax need to be specified in advance and do not appear in the final quantized result. In other words, the quantum expression of M w does not change whatever in general ρo ≠ 0 might be, since turning ?ρ into ?ρ ′ means trivially defining the radial coordinates in a different reference system: yet the considerations about M w hold identically even in this new reference system, since the key idea of quantum delocalisation and the physical meaning of the steps 1,1 remain conceptually ′ ′ ′ identical. In effect M ′ w = ( ?ρ × ?p ) ? w gives M w = ±?χ ?p χ ; yet this equation provides the same result previously found, because the postulated arbitrariness of the ranges in eq 1,2 entails again arbitrary values of l ′ . Of course the same holds for the radial momentum range and, with analogous reasoning, also for any other uncertainty range. On the one side ?ρ does not compel specifying where is actually located the origin ρ o of the radial distance range, e.g. somewhere within the nucleus or in the centre of mass of the system or elsewhere; being by definition ρ = ρo + ?ρ , any local coordinate is the limit case of ?ρ → 0 through the arbitrary value of ρo . On the other side this property of the ranges bypasses puzzling problems like how to define the actual distance ρ between electron and nucleus; in lack of any hypothesis about the local coordinates this distance could be even comparable with the finite sizes of these latter, which however are not explicitly concerned. Then, as reasonably expected because of eq 1,2, the conclusion is that the number of allowed states depends upon the range widths only, regardless of the reference systems where these latter are defined; hence the results inferred here hold for any reference system simply by virtue of 6

the first step 1,1. This statement, sensible in non relativistic physics, becomes crucial in relativity for reasons shown in the next sections 3 and 4. In this respect is interesting, in particular for the purposes of the next section 3, a further comment about the limit case where the angular momentum tends to the classical function; for l >> 1 the quantization is not longer apparent and both M 2 and M w are approximately regarded as functions of the continuous variable l . This limit case suggests considering the classical modulus of ?M = ?r × ?p , which reads ?M = ?r ?p sin ? , being ? the angle between ?r and ?p . This way of regarding ?M is consistent with the quantized result and emphasizes that ?M is still due to the range widths ?r and ?p determining l . It has been also shown that even considering different ?r′ and ?p′ the quantized result is conceptually analogous, while being now ?M ′ = ?r ′?p′ sin ? ′ . Both expressions must be therefore also equivalent in the classical limit case; hence ?r 2 ?p 2 sin ? 2 = ?r ′2 ?p′2 sin ? ′2 whatever ?r′ and ?p′ might be. Although in general ?r 2 ≠ ?r ′2 and ?p 2 ≠ ?p′2 , because the vectors defining ?M and ?M′ are arbitrary, the classical equivalence is certainly ensured by a proper choice of ? and ? ′ . The last equation, expressed as a function of the local dynamical variables included within the respective uncertainty ranges, reads (rp sin ? ) 2 = ( r ′p′ sin ? ′) 2 and gives the conservation law of angular momentum of an isolated system in agreement with the result already found M w = M′ w . This result holds regardless of the analytical form of p . In general, the reasoning above is summarized by ? M ′ 2 ?r ′ 2 = ?p 2 sin ? 2 = ?p′2 sin ? ′2 ?p 2 ≠ ?p′2 ? ≠ ?′ 2 2 ?M ?r The first equation is fulfilled by arbitrary ?p 2 and ?p′2 , as it must be, and could be rewritten with the ratio ?p′2 / ?p 2 at the right hand side; in this case ?r 2 and ?r ′2 would appear in the second

equation. If both ? M 2 and ? M′2 are calculated with equal ranges of values of l and l ′ , then ? M′2 / ? M 2 = 1 , which also entails ?r 2 = ?r ′2 . Since one component only of angular momentum can be defined in addition to the angular momentum itself, taking advantage of the fact that 2 2 2 M′ 2 ≥ M′ w and M ≥ M w the first equation is rewritten as follows M ′2 ? M 2 ?r ′2 w = M2 ? M2 ?r 2 w
2 2 ′2 One would have expected M′ w at numerator of this equation; yet the position M w = M w fulfils the

limit condition M′2 → M 2 for ?r ′2 → ?r 2 . Hence 2 ′2 M ′2 = M 2 ? (M 2 ? M 2 w )(1 ? ?r / ?r )
2 2

2,1

In the non-relativistic case ?r and ?r ′ are merely two different ranges by definition arbitrary. It will be shown in section 3 that eq 2,1 is also consistent with the Lorentz transformation of the angular momentum. The same reasoning and formal approach just described hold to calculate the non-relativistic electron energy levels of hydrogenlike atoms and harmonic oscillators. 2.2 Hydrogenlike atoms. The starting function is the classical Hamiltonian of electron energy in the field of the nucleus, which reads in the reference system fixed on the centre of mass 2 pρ M2 Ze2 Ze 2 2 E = Ecm + + ? U =? E = E ( ρ , pρ , M ) 2 ? 2 ?ρ 2 ρ ρ being ? the electron reduced mass and Ecm the centre of mass kinetic energy of the atom regarded as a whole. Also now E (n, l ) is obtained replacing the dynamical variables, unknown in principle, with the respective uncertainty ranges. In agreement with the previous discussion, also the uncertainty on U , due to the random radial distances allowed to the electron, concurs to define the 7

numbers n and l of quantum states unequivocally defined and necessarily consistent with the radial ranges ?ρ and ?pρ . This is in effect the physical meaning of the positions U ( ρ ) ? U (?ρ )
2 2 and pρ + M 2 / ρ 2 ? ?p ρ + M 2 / ?ρ 2 : putting pρ ≡ ?pρ and ρ ≡ ?ρ , the number of states allowed to the electron motion in the field of nucleus are calculated in agreement with the given form of the potential and kinetic energies. The energy equation turns then into the following form 2 ?pρ ? M 2 Ze2 * E = Ecm + + ? E * = E * ( ρ ≡ ?ρ , pρ ≡ ?pρ ) 2 ? 2 ??ρ 2 ?ρ The uncertainty on M 2 is taken into account by the range of arbitrary values allowed to l , whereas a further arbitrary value n is to be introduced through n = ( 2?ρ?pρ / ? ) 2 because of eq 1,2. The

factor 2 within parenthesis accounts for the possible states of spin of the electron, which necessarily appears as “ad hoc” hypothesis in the present non-relativistic example. The factor ? is due to the 2 fact that really p ρ is consistent with two possible values ± pρ of the radial component of the momentum corresponding to the inwards and outwards motion of the electron with respect to the nucleus; by consequence, being the uncertainty range ?pρ clearly the same in both cases, the calculation of n simply as 2 ?ρ?pρ / ? would mean counting separately two different situations both certainly possible for the electron but actually corresponding to the same quantum state. These situations are in fact physically undistinguishable because of the total uncertainty assumed “a priori” about the central motion of the electron; then the factor ? avoids counting twice a given quantum state. In conclusion, the only information available in the energy equation concerns n and l consistent with the radial and angular motion of the electron; they take in principle any integer values because the uncertainty ranges ?ρ and ?pρ include arbitrary values of ρ and p ρ and then are arbitrary themselves. Replacing ?pρ with n? / ?ρ and M 2 with (l + 1)l? 2 in E * , the result is

E * = Ecm +
Trivial manipulations of this equation give

n2?2 l (l + 1)? 2 Ze 2 + ? ?ρ 2??ρ 2 2??ρ 2
2

1 ? n? Ze 2 ? ? (l + 1)l ?2 Z 2e 4 ? ? ? 2 2 E = Ecm + ? ? + 2? ? ?ρ 2??ρ 2 2n ? n? ?
*

E * is minimized putting equal to zero the quadratic term within parenthesis, certainly positive; being E = min E * the result is

( )

n 2 ?2 n? Ze 2 ? (l + 1)l ? 2 Z 2e 4 ? ?ρ min = 2 ?pρ min = = E = Ecm + ? 2 2 2,2 2 Ze ? ?ρ min n? 2 2 ??ρ min 2n ? Then the total quantum energy E (n, l ) of the hydrogenlike atom results as a sum of three terms: (i) the kinetic energy Ecm of the centre of mass of the atom considered as a whole, (ii) the quantum
rotational energy of the system consisting of a reduced mass ? moving within a distance ?ρmin from the nucleus and (iii) a negative term necessarily identified as the non-relativistic binding energy ε el of the electron. The values allowed to l must fulfil the condition l ≤ n ? 1 . Rewriting indeed E in a reference system with the centre of mass at rest, Ecm = 0 , and utilizing ?ρmin the result is 2 4 ? (l + 1)l ? Z e ? ε el = ? 2 ? 1? 2 2 l ≤ n ?1 ? n ? 2n ?

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If l ≥ n then the total energy ε would result ≥ 0 , i.e. the hydrogenlike atom would not entail an electron bound state. Since the stability condition requires the upper value n ? 1 for l , it is possible to write n = no + l + 1, where no is still an integer. Hence 2,3 2 2? 2 ( no + l + 1) In conclusion, all the possible terms expected for the non-relativistic energy are found in a straightforward and elementary way, without hypotheses on the ranges and without solving any wave equation: trivial algebraic manipulations replace the solution of the appropriate wave equations. It is worth emphasizing that the correct result needs introducing the concept of electron spin to count appropriately the number of allowed states, whereas the non-relativistic wave equation solution skips such a requirement. The present approach requires therefore necessarily the concept of spin, although without justifying it. As concerns the positions 1,1, it is also worth noticing that only the first step E ( x, px , M 2 ) → E ( ?x, ?px , ? M 2 ) concerns the particles, whereas the second step E ( ?x, ?p x , ? M 2 ) → E (n, l ) concerns in fact their phase space; indeed E (n, l ) is a function of the number of quantum states, which are properties of the phase space like the pertinent ranges. This is especially important when considering many electron atoms: the fact that any specific reference to the electrons is lost entails as a corollary the concept of indistinguishability; ni and li of the i -th electron are actually numbers of states pertinent to delocalisation ranges where any electron could be found, instead of quantum numbers of a specified electron. The energy levels of many electron atoms and ions have been then inferred without possibility and necessity of specifying which electron in particular occupies a given state; in effect, the electrons cannot be identified if nothing is known about each one of them. The paper [2] shows that the same ideas hold also to calculate the binding energy of diatomic molecules. The lack of local information inherent the assumptions 1,1 and 1,2 entails then in general the indistinguishability of identical particles. 2.3 Harmonic oscillator. This case is particularly interesting for the purposes of the present paper and simple enough to be 2 also reported here. With the positions 1,1, the classical energy equation px / 2m + k ( x ? xo ) 2 / 2
2 2 2 becomes ?px with / 2m + k ?x 2 / 2 ; then, thanks to eq 1,2, one finds ?ε = ?px / 2m + ω 2 mn 2 ? 2 / 2?px

ε el = ?

Z 2e 4 ?

(min) ω 2 = k / m . This equation has a minimum as a function of ?px ; one finds ?px = mn?ω and (min) thus ?ε = n?ω , being n the number of vibrational states. For n = 0 there are no vibrational 2 (min) (min) states; however ?px ≠ 0 compels also ε 0 = ?p0 / 2m ≠ 0 . Therefore ?p0 = ?p x ( n = 1) defines
(min) ε 0(min) = ( ?p0 ) / 2m = ?ω / 2 , with ?p0(min) = m?ω . Being ?ε (min) = ε (min) ? ?ω / 2 , the result 2

2,4 is obtained considering uncertainty ranges only, once again without any hypothesis on these ranges. 2 2 (min) Note that ?px , in agreement with the virial / 2m = ω 2 mn 2 ? 2 / 2?px = n?ω / 2 with ?px = ?px theorem; ε (min) is given by the sum of kinetic and potential terms, whereas the zero point term has (min) (min) and ?p0 are merely particular ranges, kinetic character only. Also note in this respect that ?px among all the ones still possible in principle, fulfilling the condition of minimum ε and ε 0 ; analogous reasoning holds also for ?ρmin and ?pρ min of eqs 2,2. These results do not contradict however the complete arbitrariness of the uncertainty ranges, since in principle there is no compelling reason to regard these particular ranges in a different way with respect to all the other ones; yet, the comparison with the experimental data merely shows preferential propensity of nature for the states of minimum energy. In effect, it is not surprising that the energy calculated with extremal values of dynamical variables does not coincide, in general, with the most probable 9

ε (min) = n?ω + ?ω / 2

energy. In conclusion, these examples highlight that the physical properties of quantum systems are inferred simply replacing the random, unknown and unpredictable local dynamical variables with the respective quantum uncertainty ranges: the key problem becomes then that of counting correctly case by case the appropriate number of allowed states. Consider now that a further uncertainty equation conceptually equivalent to eq 1,2 can be inferred introducing the time range ?t necessary for any particle having velocity v to travel ?x [3]; since v ≠ 0 otherwise the particle would be exactly located somewhere, defining formally ?t = ?x / v x and then ?ε = ?px v x , eq 1,2 reads

?t = t ? to ?ε?t = n? 2,5 This result requires that v is finite. Eq 2,5 is not a trivial copy of eq 1,2, even if n is the same: it introduces new information through v and shows that during successive time steps ?t the energy ranges ?ε change randomly and unpredictably depending on n . To clarify this assertion, consider that 1/ ?t has physical dimensions of frequency; then the general eq 2,5 can be rewritten as ?ε n = n?ω § , being ω § a function somehow related to any frequency ω . If in particular ω § is specified to be just an arbitrary frequency ω , eq 2,5 reads in this case ?ε n = n?ω 2,6
Thus ω § ≡ ω enables an immediate conceptual link with eq 2,4; having found that n is according to eq 1,2 the number of vibrational states of harmonic oscillator and n?ω their energy levels, then without need of minimizing anything one infers that ?ε n = ε (min) ? ?ω / 2 is the energy gap between the n -th excited state of the harmonic oscillator and its ground state of zero point energy; the (min) condition of minimum and ?px are now replaced by the specific meaning of ?t . This conclusion shows that a particular property of the system is correlated to a particular property of the uncertainty ranges, thus confirming the actual physical meaning of these latter. In this case the random, unpredictable and unknown ε n falling within ?ε n are necessarily the classical energies of harmonic oscillator whose quantization leads to ε (min) . Note that ω was previously defined through the formal position ω 2 = k / m ; now eq 2,5 shows its explicit link with the time uncertainty ?t . Appendix B shows that the energy operator of wave mechanics is inferred from the time-energy uncertainty equation in complete analogy with the momentum operator of appendix A. It is well known that the uncertainty inherent the quantum world has been found by Heisenberg exploiting the operator formalism of wave mechanics, whereas the appendixes A and B show that the reverse logical path is also possible, i.e. that the uncertainty entails in turn the operator formalism. This bidirectional correspondence between total uncertainty and wave character of quantum mechanics, non-trivial although reasonably expected, on the one side explains why the results obtained through eq 1,2 and 2,5 are consistent with that inferred through the operator formalism, as in effect it has been shown in this section; on the other side, however, it confirms that the approach starting from the positions 1,1 is more general than that based on the formalism of wave mechanics: the indistinguishability of identical particles, pillar of quantum mechanics together with the exclusion principle, is a corollary in the former case and a postulate in the latter case. The quantum approach to relativity is then carried out in the sections 3 and 4 exploiting uniquely and directly the concept of uncertainty of eqs 1,2 and 2,5 rather than the usual operator formalism of field theory, with trust on the prospective effectiveness of the more general character of the former approach with respect to the latter. 3 The special relativity. The section 2 has shown that the concept of delocalization is the “added value” necessary and enough to plug the classical physics into the quantum world. A few comments highlight further this point and show that really the special relativity is straightforward generalization of the results therein shortly sketched. Although the approach based uniquely on eqs 1,2 and 2,5 is apparently more agnostic than that based on the formalism of wave mechanics, the physical information about 10

the quantum systems examined in [1,2] is in fact completely analogous. Actually this consistency could be expected because the local coordinates and conjugate momenta appear explicitly only in the wave equations providing all the possible information on any physical system, but not longer in their eigenvalues; on the one side the local dynamical variables are physically worthless in determining the allowed states the system, on the other side neglecting them in principle merely avoids handling variables conceptually inessential as concerns the eigenvalues and then simplifies considerably the way to infer the allowed physical information: the present approach, indeed, does not need solving any wave equation just because the local dynamical variables are disregarded since the beginning. The examples previously carried out highlight the reasons of it: the quantum properties are controlled by the uncertainty equations 1,2 and 2,5, whereas the quantum numbers are actually numbers of allowed states introduced since the beginning into the problem through these equations. Therefore the uncertainty is not mere restriction of knowledge but rather a sort of essential information, actually the only one available; it is also worth noticing that just the abstract concept of quantum delocalisation is significant in principle because, as it appears in the examples summarized above, size and origin of the uncertainty ranges are actually never specified. In the hydrogenlike atom ?ρ is mere notation to indicate a space range around the nucleus, whose actual size remains however undetermined; the only essential idea is that it conceptually exists and encompasses the possible radial positions expected for the electron in any given physical situation, these positions being however completely arbitrary in principle. For this reason even an infinite size is allowed to the ranges, in agreement with their complete arbitrariness, without divergence problems. Since M w , M 2 , ε (min) and ε el do not depend explicitly on these range sizes, the physical properties of matter do not diverge and even admit the infinite limit, in which case they tend to the respective classical quantities. Just because lacking any specific value, ?ρ plays a role similar to that of the dynamical coordinate ρ it replaces: e.g., an increasing radial range is consistent in principle with larger electron/nucleus distances, whereas the limit case of infinite distance from the nucleus is described by ?ρ → ∞ . Also, local derivatives like ? ρ are replaced systematically by

? ?ρ , as it is evident considering that ?ρ = ρ ? ρo is defined by a fixed value ρ o and a variable value ρ allowed to change when the radial coordinate of the particle is affected by any physical reason. These considerations, merely reasonable in a non-relativistic frame, have now central importance: they are in effect the starting point to introduce the basic principles of special relativity. The leading idea is still that the uncertainty ranges are not subjected to any hypothesis about their sizes and analytical form and about the coordinates xo and ρo defining their origin, in agreement
with the requirement that any coordinate ρ is effectively defined by ρo + ?ρ . Since however xo and ρo are defined by their own reference system only, it means that in fact the present physical description of quantum systems does not specify in particular any reference system of space coordinates. The present approach excludes therefore the existence of a preferential reference system to describe the physics of the quantum world. This conclusion holds identically also for the time; as shown in the case of the harmonic oscillator, the consistency of eqs 2,4 and 2,6 does not require any hypothesis about ?t and even to does not need to be specified whatever ω might be. Since to can be defined only introducing a time reference system, one concludes that there is in fact no preferential time reference system. This analogy between time and space coordinates, already emphasized also in appendix B the infer the momentum and energy operators, is too strong and significant to be merely accidental, rather it suggests their physical concurrence in describing the quantum properties. In effect this assertion is confirmed recalling that eqs 1,2 and 2,5 are direct consequences one of the other for a given number n of quantum states. Being physically meaningless to ask which equation is “more fundamental”, the angular momentum or the energy levels of hydrogenlike atoms formerly inferred from eq 1,2 could be identically regarded as a 11

consequence of eq 2,5 rewritten in the form 1,2, as explicitly done in the case of the harmonic oscillator. The physical equivalence of the time and space ranges entails the cooperative physical importance of the respective random coordinates as well. Hence the link between eqs 1,2 and 2,5 appears actually more profound than in non-relativistic physics; it shows that there is no reason to regard separately the local space and time coordinates, thus suggesting that the positions 1,1 concern actually a general principle where a unique space-time range ? xt involves and combines together both space and time uncertainties ?x and ?t , so far regarded separately for sake of simplicity and graduality of exposition only. This is possible because also the analytical form of the ranges is itself completely undetermined in eqs 1,2 and 2,5. Their link, expressed collecting together eqs 1,2 and 2,5 with the same n ? xt ?pxt = ?ε?t 3,1 shows that the time must be regarded together with the space coordinates of the system; the right and left sides of this equation differ only for the dynamical variable conjugated to either coordinate. Owing to the lack of specific constraints on the ranges, this conclusion has further consequences. Consider for instance ?x , being x regarded here as generalized coordinate, e.g. radial or Cartesian or curvilinear: in principle there is no reason nor necessity to conceive this range as a function of the space position only; rather the angular momentum or the hydrogenlike atoms could have been identically described considering instead of ?x another range ? xt defined as proper linear combination of ± v ?t and x ? x o
? xt = x ? x o ± v ?t

3,2

As concerns ?pxt nothing changes aside from its conceptual meaning, now consistent with the space-time definition of uncertainty. At this point, however, the question rises about why the results of section 2 even ignoring the time are so reasonable, of course in the limits of non-relativistic approximation. None of the papers [1 to 5] has considered the time evolution of the concerned systems, rather the displacement of each constituent particle throughout different regions of space has been described via a unique uncertainty range large enough to encompass any position allowed by its random motion; being the space ranges arbitrary by definition, there is no reason to exclude in principle this way of thinking, which in effect gives sensible results. A trivial answer could be that, ′ as concerns the eigenvalues, one having shown the equivalence of ?x = x ? xo and ?x′ = x ? xo ′ = xo ? v ?t . If so, ?t remains hidden could also define ? xt = x ? xo ± v ?t simply admitting xo ′ and does not explicitly affect the overall uncertainty of the system. within the arbitrary choice of xo However this explanation is not completely exhaustive: it would be difficult to justify in a coherent way and within a unique conceptual frame why the time does not affect the steady cases of section 2, whereas for instance it should also control the evolution of unstable systems; clearly the position ′ = xo ? v ?t cannot discriminate itself whether or not the time is to be purposely introduced into xo the specific problem, e.g. by any kind of external transient perturbation. In this respect eqs 3,1 and 3,2 change drastically and constructively this non-relativistic point of view: since the time is actually a dynamical variable like the space coordinates, then likewise arbitrary in general, a more comprehensive answer to explain the results of section 2 necessarily concerns a new definition of “steady” system, i.e. the one whose total uncertainty is statistically time independent. To clarify this point, consider for instance an isolated many electron atom. Although the most probable electron configuration is unambiguously defined by the minimum energy consistent with the mutual coulombian repulsion and nuclear attraction, quantum fluctuations of the local dynamical variables from the most probable values are likely to be expected; hence the random interactions of each electron with the neighbours modify also its numbers n and l of allowed states. If the energy of the whole atom deviates from the minimum value within a range ?ε § , then the quantum fluctuations of position and momentum of electrons are defined within a time scale ?t § = n? / ?ε § ; an analogous 12

reasoning holds also for l , because eq 2,5 is physically equivalent to eq 1,2 through which have been calculated the eigenvalues of angular momentum. A steady time average of all the possible instantaneous electron configurations is then to be expected for any ?t > ?t § , since by definition ?t § includes any kind of local time instability consistent with ?ε § ; under this condition the time does not longer play an explicit role in determining the average n and l of each electron, which remain statistically unchanged and identify the energy eigenvalues of the global steady state. Hence the eigenvalues calculated in section 2 are indicative of stable states effectively observable during a time range long enough to define an average steady extent of overall uncertainty. The position ′ = xo ? v ?t is then justifiable under the implied condition ?t > ?t § , otherwise the transient xo character of the quantum fluctuations would be explicitly evident. If so, the role of time is in fact hidden by the randomness of local internal interactions weak enough and short enough to evidence their constant average only. Hence, just the arbitrariness of both time and space ranges enables the actual existence of steady states with the condition ?t > ?t § implicitly assumed and effectively allowed, to which correspond average numbers of states statistically stationary describing physical observables like the quantum energy. This reasoning is easily extended to the case where the quantum state of the many electron atom is perturbed by any external force filed, in which case ?t is related to the energy range ?ε due to the perturbation. In general the time inequality, rewritten in the form ?ε < ?ε § , is not fulfilled because ?ε or ?ε § or even both depend on the kind of interaction. Consider a real gas of many electron atoms interacting through anelastic collisions. The quantum fluctuation ?ε § of each atom is still controlled by the allowed changes of n and l , whereas its thermal fluctuation ?ε is due to collisional momentum exchanges. Since the extent of ?ε § depends on whether the corresponding ?n is related to small or large values of n , whose values are anyway arbitrary, does not exist a unique criterion to discriminate unambiguously whether or not a given thermal energy fluctuation violates the time inequality. The obvious conclusion is that an external interaction acting on the system produces in general a time dependent perturbation; the stationary state of the isolated atom is not longer allowed, the explicit time dependence of the uncertainty ranges is necessary. From a non-relativistic point of view, therefore, the choice of ? xt alternative to ?x appears relevant only to discriminate the transient or steady character of a given physical problem; since the time is a mere parameter purposely introduced, its role changes depending on whether ?ε § characteristic of the system, whatever its reason might be, fulfils or not the condition above. Yet according to the relativistic point of view, the ranges formerly introduced in section 2 are actually mere space-like or time-like particular cases of ? xt , in general and regardless of the particular aim of the specific problem. Moreover the conceptual equivalence of eq 2,5 with eq 1,2, the only one previously considered, entails that also ? xt , whatever its analytical form might be, must have the features evidenced by the examples of section 2. In particular ? xt must fulfil the indistinguishability of all the space-time reference systems, already emphasized and necessary to regard the arbitrary local values of the dynamical variables as particular limit cases of the respective ranges. All the ranges having the form ? xt = x ? x o ± v (t ? to ) are effectively indistinguishable because the quantities appearing in its definition, in particular x o , to and of course v , are arbitrary; no distinctive property discriminates one of them from the others. Clearly however also ?′xt =
?x 2 ± v 2 ?t 2 , for instance, could legitimately replace ?x without changing any step of the approach leading to eqs 2,3 or 2,4 or 2,6. The importance of these considerations rests on the quantities that define conceptually ? xt , rather than on its specific analytical form; it holds in particular for v that links space and time. It is also possible to consider a particular value c of v having some distinctive property; if c is uniquely

13

identified by a well defined numerical value, then, whatever this latter might be, it can be nothing else but the finite upper limit allowed to v . Indeed c cannot be infinite otherwise ? xt would express exclusively an infinite uncertainty, which however would entail an exact value of its own conjugate momentum within a vanishingly small uncertainty range; yet the basic hypothesis underlying the present approach simply excludes an exact local value of any dynamical variable. Although an infinite value of ? xt is certainly possible, owing to the complete arbitrariness of the range sizes, the concept of uncertainty requires however that any other finite value be also possible. On the other hand, v cannot be uniquely equal to zero, which would trivially turn ? xt into the non-relativistic case. Then one concludes that the only valid alternative of physical interest to the non-relativistic definition of uncertainty range is a combination of space and time coordinates through a finite coefficient v or finite c ≥ v ; this property of c ensures the validity of the reasoning for any value of v . If so, then it also follows immediately that c must be the same in any reference system; otherwise, any specific value of c replacing v could somehow identify the uncertainty range ? c with respect to any other ? c′ characterized by its own value of c′ . Of course this conclusion identically holds for ?′ c = ?x 2 ± c 2 ?t 2 . It appears therefore that the invariant character of c in all of the inertial reference systems is intimately linked to the concept of their indistinguishability, i.e. to the lack of a preferential reference system to describe the quantum properties through ranges replacing the local variables. The relativistic concept of their equivalence, intrinsically inherent the arbitrariness of the uncertainty ranges, entails a further consequence: the reference systems are in fact indistinguishable provided that some ranges exist that are invariant themselves, which just for this reason have distinctive importance to infer universal physical laws. Hence the existence of an invariant interval is not only a property but mostly a necessary conceptual requirement once merging space and time coordinates into a unique four dimensional space. Clearly these remarks summarize and correlate the basic hypotheses of special relativity: finite value and invariance of c , invariance of physical laws for different inertial reference systems, non-relativistic physics as a limit case. The non-trivial fact is however that the properties of the uncertainty ranges introduce these basic assumptions as straightforward corollary of the quantum delocalization inherent any physical particle. Appears particularly interesting therefore any expression of ? xt function of c , whereas there is no reason to expect that the same holds for v . It is known that the invariance of interval entails in turn the Lorentz transformation of time and space in different inertial reference systems in reciprocal constant motion. Considering “a priori” only uncertainty ranges as proposed here, i.e. conceptually and not as a sort of approximation to simplify some calculation, these transformations are inherently fulfilled and appear consequences themselves of the quantum uncertainty. The following relevant example highlights this concept and extends the invariance of c to the interval invariance rule too. Consider a photon travelling a range c?t = x ? xo ; the coordinates xo and to identify a well defined space-time reference system. A slower massive particle would travel a smaller range xs ? xo ; it is possible then to write the formal identity ′ , obtained c?t = xs ? xo + δ X with δ X = x ? xs . Consider now the further range c?t ′ = x′ ? xo

′ and x to x′ during the time range ?t . Also xo ′ and to ′ define another reference shifting xo to xo ′ ? xo ) / ?t ; this is clearly the system, displacing with respect to the former at average rate V = ( xo case of two identical ranges initially overlapped and then mutually displaced by sliding one along ′ ? xo ′ + δ X ′ with δ X ′ = x′ ? xs ′. the other at constant rate. The same reasoning gives then c?t ′ = xs Since the range sizes are arbitrary, whatever their boundary coordinates might be, it is possible to ′ ? xo ′ ; being in general ?t ′ ≠ ?t , this position is fulfilled simply with put by definition xs ? xo = xs
14

δ X ′ ≠ δ X . Yet it is not possible ?t ′ = ?t too, which would entail also δ X ′ = δ X and thus trivially
identical stationary ranges. Regarding again the same ranges, yet considering now c 2 ?t 2 = ( x ? xo ) 2
′ ) 2 , then c 2 ?t 2 = ( xs ? xo ) 2 + ?X 2 and c 2 ?t ′2 = ( xs ′ ? xo ′ ) 2 + ?X ′2 , with ?X 2 and and c 2 ?t ′2 = ( x′ ? xo ′ ? xo ′ ) 2 of the slow particle. Hence a range invariant ?X ′2 properly defined by ( xs ? xo ) 2 and ( xs under translation of two reference systems reciprocally moving at constant rate V exists indeed and has the form δ s 2 = c 2 ?t 2 ? ?X 2 despite ?t and then ?X are necessarily different in each one of them. This form of δ s 2 deserves attention for Lorentz’s transformations of the quantities appearing in eq 3,1. Rewriting c 2 ?t 2 ? ?X 2 = c 2 ?t ′2 ? ?X ′2 as c 2 ?t 2 = c 2 ?t ′2 ? ?X ′2 , which is certainly possible because the condition δ X ≠ δ X ′ does not exclude ?X = 0 , gives the known result ?t = ?t ′ 1 ? V′2 / c 2 with V′ = ?X ′ / ?t ′ ; moreover, the position V ≡ V′ consistent with a unique ′ ? xo ) / ?t = ?X ′ / ?t ′ . translation rate of the reference systems where are defined the ranges gives ( xo

′ ? xo ′ ? xo ) / ?X ′ = ?t / ?t ′ = 1 ? V 2 / c 2 is the sought transformation property between xo Then ( xo and ?X ′ corresponding to that between ?t and ?t ′ . For sake of clarity, this conclusion is further exploited here very shortly as concerns the angular momentum of a system of particles considered in eq 2,1; in general ?r and ?r ′ are two independent and arbitrary ranges. If however ?r ′ = r ? ro′ can be rewritten as ?r ′ = r ? ro ? v ?t , then ?r and ?r ′ are defined in reference systems in
reciprocal motion; in this case ?r ′2 / ?r 2 = 1 ? v2 / c 2 because of relativistic reasons, i.e. even though eq 2,1 is calculated with l = l ′ . In these reference systems, hold for the given physical system the steady uncertainty conditions previously discussed; then, replacing the ratio in eq 2,1, M′2 results to be Lorentz’s transformation of M 2 defined in a reference system where the center of mass is at rest. Also, the reasoning introduced in [3] shows the result obtained when the quantization of angular momentum is again inferred from the definition (ρ × p) ? w , yet replacing ρ and p with the respective uncertainty ranges fulfilling Lorentz’s transformations; it means that ?ρ and ?p are now invariants in two inertial reference systems R and R ′ in reciprocal motion. The local variables can be again disregarded because noting must be specified about the respective range sizes in either reference system, and then wathever the transformation law from ?ρ , ?p to ?ρ ′ , ?p′ might be. It is found in this way that M w is not longer given by ±l ? only, rather appears a further component ±l ′? / 2 , with l ′ = 0,1, 2, ?? . The spin of free particles is then inferred without any hypothesis “ad hoc”; the interval invariance rule, consequence itself of eqs 1,2 and 2,5, confirms the selfconsistency of the theoretical model and the conceptual link between quantum mechanics and relativity. Yet the spin can be more shortly and intuitively inferred through a simple reasoning about the invariant range δ s = c 2 ?t 2 ? ?X 2 , noting that in general ?t and ?X are arbitrary and that a 2 ? b 2 = a ? ( b / a + (b / a )3 / 4 + (b / a )5 / 8 + ??) b / 2 . Thus, regardless of whether c 2 ?t 2 > ?X 2 or

c 2 ?t 2 < ?X 2 , the series expansion of δ s reads δ s = ρc + ρo / 2 ; the former inequality, for instance,
3 would correspond to ρc = c?t and ρ o = ?X ? ?X / ρc + ( ?X / ρc ) / 4 + ??? expressed in principle ? ? with any number of higher order terms. Since δ s is arbitrary, it is possible to repeat identically the reasoning sketched in section 2, yet utilizing now the invariant vector δ s = ρc + ρo / 2 in the

definition M rel w = (δ s × p ) ? w . The invariant form of p , once replaced by its own uncertainty range, does not need to be specified; as previously shown, the local momentum is not really calculated at any position or time, rather it is simply required to change within a range of values, undetermined ′ itself. Thus it is immediate to infer that M rel w = l ? + l ? / 2 . The spin appears therefore because the simple space range of section 2 is replaced by a more complex range consisting of both space and 15

time parts, which explain why the series development defines δ s as sum of two addends. This reasoning is further extended considering again the eq 3,1 and requiring that the link between ?px and ?ε be invariant. This is possible if ?x / ?t = c , hence ?px c = ?ε is a sensible result: it means of course that any ε within ?ε must be equal to cpx of the corresponding px within ?px . If however ?x / ?t < c , the fact that the arbitrary v x is not longer an invariant compels putting, for
k +1 k k +1 , with k arbitrary exponent; therefore (?px v ? instance, v k = ?ε shows in x ?x / ?t = c x )c k general an invariant link between ?px v ? and ?ε through c k +1 . Since this equation must x correspond to a sensible non-relativistic limit, is mostly interesting the particular case k = 1 ; then ( ?px / v x )c 2 = ?ε , which means also px = ε v x / c 2 . This result contains as a particular case px c = ε

and entails also ε / c 2 = m to fulfil the non-relativistic limit px / v x → m . To find other well know results of special relativity is so trivially obvious that it does not deserve further attention here. The quoted paper [3] concerned the free particle, as implicitly assumed also here to carry out the present considerations. Successively, the relativistic energies of hydrogenlike and many electron atoms and ions have been also calculated “ab initio” uniquely on the basis of these concepts and without any specific hypothesis about sizes and properties of the space-time and momentum uncertainty ranges in very good agreement with the experimental data [4]. These results, which critically depend upon the mutual interactions between electrons and nucleus, suggest that the same concepts should be in principle appropriate to describe also the interactions between atoms/ions in a body of condensed matter; if so, even the macroscopic thermodynamic properties should be inferred by exploiting appropriately the basic idea of the papers [1,2,3,4]. The specific heat of metals has been in effect calculated in [5] in very good agreement with the experimental data, thus showing that the present approach is suitable to describe also the thermodynamic properties of condensed matter. Replacing the local coordinates with the respective uncertainty ranges deserves further attention. Despite the compliance with the special relativity, seems however problematic the further extension of the concepts introduced above to the general relativity, where the gravity force is explained as curvature of the space-time and the invariant interval is replaced by a more complex local metrix to describe the mass induced curvature. The tensor calculus expressing mathematically these concepts entails, for instance, that the local scalars ρ 2 and p 2 have the form of sums ρi ρ i and pi p i . On the one hand, disregarding the local terms in principle as shown in the previous section means actually excluding the tensor formalism of general relativity and then the wealth of information inferred by consequence. Also, the positions 1,1 bypass familiar concepts of the gravitational field theories; covariant or contravariant derivatives, necessary in curvilinear coordinate systems, are in fact useless if the local dynamical variables do not longer play “ab initio” any physical role. On the other hand, is however relevant the fact that the basic assumptions of special relativity have been inferred as corollaries of the uncertainty; moreover the transformation properties of length and time of section 3 have been inferred without utilizing the 4 vector algebra. Appears also encouraging the possibility to infer in a straightforward way the most relevant results of special relativity as sketched above; the relativistic Dirac equation has been inferred in [4] in the particular case of hydrogenlike atoms. Besides, there is no reason to exclude that the results of tensor calculus can be also obtained via the formalism followed in the present model. An example is given by the Lorentz transformation of angular momentum of an isolated system, defined in general relativity as antisymmetric 4 tensor M ik = ∑( xi p k ? x k p i ) whose spatial components coincide with the components of the vector M = r × p . It is known that M′2 in a reference system whose centre of
2 2 mass moves with constant velocity v is related by the equation M ′2 = (1 ? v 2 / c 2 ) M 2 + M 2 w v / c to

M 2 in another reference system where the centre of mass is at rest. Yet this result, obtained via transformations of the components of momentum 4 tensor, has been already inferred through eq 2,1
16

simply regarding ?r ′2 and ?r 2 as ranges in reference systems mutually displacing at velocity v , as previously shown. In conclusion, extending further to the general relativity the procedure summarized previously by the positions 1,1, i.e. considering eqs 1,2 and 2,5 as unique postulate, appears in principle possible provided that the physical basis of the gravity force be somehow rooted itself in the concept of uncertainty; if so, despite the mathematical formalism necessarily different with respect to the current field theories, the reasoning followed in section 2 should include also the gravitational interactions in the present conceptual frame. For this reason the next section 4 exploits again and identically the approach outlined in section 2; the aims are: (i) to show that the gravity force is effectively rooted into the concept of quantum uncertainty; (ii) to consider various isolated systems of two particles whose interaction is uniquely due to the gravitational force; (iii) to formulate for each system the classical gravitational problem; (iv) to plug into the respective problems the positions 1,1 and then eqs 1,2 and 2,5; (v) to describe the behaviour of the interacting particles with the formalism of the quantum uncertainty. The belief underlying such an approach rests on considering also the properties of space-time, including its curvature, as mere consequence of the quantum uncertainty. If so, the formalism of tensor calculus is not required to exploit the points (i) to (v); attention is payed to the quantum states inherent the uncertainty, rather than to the way of describing the mass induced space-time local curvature after having hypothesized its connection with the gravity field. In other words, if this latter would really have quantum origin, there would be no reason to exclude that the space-time curvature is itself a quantum phenomenon. The simpler mathematical formalism of section 3 will be then exploited also in section 4 in agreement with the idea that the latter is actually consequence of the former through the space-time definition of uncertainty. The next section shows that the quantum delocalization is effectively the sought “added value” to the classical Newton physics enough to infer even the most relevant results of general relativity in a surprisingly simple and straightforward way. 4. The gravity field. The theoretical basis of the papers [3,4,5] rests entirely on the concepts sketched in section 1, subsequently elaborated as shown in the examples of section 2. The physical information inherent the agnostic logic of the quantum phase space has shown that a unique hypothesis, the random delocalization of particles in uncertainty ranges, describes the properties of matter from the microscopic scale (particle systems) up to the macroscopic scale (thermodynamic systems). Since in non-relativistic physics the time and space coordinates are regarded separately, the role of time appears explicitly in specific cases only where it is deliberately introduced, e.g. if a transient external interaction perturbs the system. Opportunely however the time is already inherent the definition of relativistic uncertainty range according to the considerations of section 3. Since ?t concurs to define the uncertainty of any system, the role of time in describing the quantum systems is self-legitimated regardless of any further explanation. For instance the short remarks of section 3 about the angular momentum have introduced the spin as mere consequence of considering together time and space uncertainties. Thus, after having formerly introduced ? ?x , it appears natural to consider also ? ?t in agreement with the idea of regarding the change of t ? to in conceptually analogous way as that of x ? xo ; now is t the variable coordinate in an arbitrary time reference system where is defined to . The physical meaning of space derivative has been highlighted in section 2 when calculating the particular space ranges that correspond to the minimum energy of the system; let us show now that relevant consequences are also inferred from the concept of time derivative. On the one side the physical equivalence of eqs 1,2 and 2,5 removes the necessity of introducing an external interaction to explain the possible time evolution of the uncertainty of a system; on the other side just this equivalence reverses the non-relativistic point of view, suggesting the possible existence of an internal interaction between particles due to the space-time definition of uncertainty. The elusive character of such an interaction, unexpected in the conceptual frame of 17

section 2 and then not considered in the cases therein examined, could be easily understood; the simple fact that it explicitly involves both space and time parts of the uncertainty ranges, explains why it was inevitably missed in the non-relativistic conceptual frame. If for instance this interaction depends on ?? xt / ??t , then it becomes immediately clear why it was skipped in the steady cases of section 2. Nevertheless the effects of such an interaction can be regarded in the phase space exactly as shown in section 2, i.e. simply replacing the coulombian force of the hydrogenlike atom or the harmonic spring of the oscillator with the new force, whose analytical form must be however known. In other words, it is necessary: (i) to show that effectively the space-time dependence of the uncertainty ranges defines this force, (ii) to infer the analytical form of this latter through the positions 1,1 and (iii) to calculate data to be compared with experimental observables. In principle, it is correct to say that only the experimental observation legitimates the existence of such a force. Yet, since in the quoted papers the gravity has been never considered, one suspects that just the gravity could be the sought kind of interaction concerned in particular by the point (i): for instance, it could be active even between electron and nucleus in the hydrogenlike atoms, although neglected in the Hamiltonian of section 2 with respect to the Coulomb interaction. The next sub-section 4.1 examines the points (i) and (ii) in a merely speculative way, i.e. regardless of any preliminary information about the forces of nature and without any conceptual hint provided by known experimental evidences. The idea of time deformation rate of the uncertainty ranges will be introduced in abstract way, i.e. simply because nothing hinders in principle its effective occurring once having introduced their time dependence; this idea will be legitimated “a posteriori” by the results of the following sub-sections 4.2 to 4.8 that concern the aforesaid point (iii). All of the considerations hereafter carried out are therefore developed on deductive and self-contained basis, once again starting from the classical physics implemented with the concept of quantum time-space uncertainty. It is interesting the fact that the experimental verification of the results not only validates the whole theoretical model, but also highlights the hierarchical significance of the concept of quantum delocalization among the known universal principles of nature; for instance, the question rises about whether the space-time uncertainty requires as additional hypothesis the concept of space-time curvature or infers it as a consequence, in the same way as the indistinguishability of inertial reference systems entails by necessity the invariance of c . The next subsections aim to answer this question by introducing first the gravity force in the same conceptual frame introduced in section 1, i.e. simply describing the behaviour of quantum particles delocalized in time-space dependent uncertainty ranges of the phase space. For simplicity the particles are assumed having zero spin and zero charge, in order to consider their gravitational interaction only; if so the quantum results describe also the behaviour of macroscopic bodies, e.g. planets, through a proper mass, time and length scale factor. 4.1. Quantum basis of the gravitational interaction. Les us consider first an isolated system of two non-interacting free particles constrained to move within their respective space-time dependent uncertainty ranges ?x1 and ?x2 . Being the problem one-dimensional by definition, let ?P 1 and ?P 2 be the momentum ranges conjugate to ?x1 and ?x2 including any local values of the components P 1 and P 2 of the respective momenta P 1 and P2 . Before interacting, the particles are delocalised in the respective ranges independently each other; two separate uncertainty equations hold therefore for each particle ?Pi ?xi = ni ? i = 1, 2 4,1a Moreover, let us write the corresponding time uncertainty equation as done for eq 2,5 in section 2 ?ηi ?τ i = ni ? 4,1b The notation of eqs 4,1b emphasizes that the time ranges ?τ i here introduced and ?t of eq 3,2 have different physical meaning: the former define energy ranges encompassing the possible ηi of free particles, the latter introduces the time into the definition of total uncertainty of a particle. The 18

fact that two variables, space and time, determine the size of the uncertainty ranges has relevant consequences. Consider for instance ? xt = x ? x o ± v ?t where is delocalized the first particle only. Regardless of the actual distance between the particles, an appropriate value of ?t certainly exists such that the overall width of ? xt becomes large enough to include the second particle too. Since
v is finite because of the reasons introduced in section 3, after an appropriate time range the

particles initially free are delocalized in a common range. The initial situation before interaction is therefore described by eqs 4,1 until when, for ?t large enough, both particles share the same ? xt , from now on shortly called ?x ; in other words, the separate ranges ?xi merge into the unique uncertainty range ?x including all of the possible random distances between them. The same reasoning holds from the point of view of the second particle; then changing the size of ?xi as a function of time means that the phase spaces of both particles merge together, i.e. their dynamical variables cannot be longer regarded separately because they share the same range of allowed values. In other words the particles interact. This is in fact the physical meaning of the range ?ρ including the possible random distances ρ between electron and nucleus in the hydrogenlike atom, thus describing in this case the Coulomb interaction through the corresponding radial and angular motion numbers of states. The relativistic point of view, therefore, accounts also for the transition from non-interacting state to interacting state thanks to the concept of space-time dependent uncertainty. Let us show that in general a force is originated when the ranges ?x1 and ?x2 of eqs 4,1a merge into a unique ?x , i.e. the dynamical variables of each particle can randomly access the phase space of the other one. If the interaction changes the initial values of both dynamical variables, then the sizes of the respective uncertainty ranges must change as well. Accordingly, when ?xi turn the respective sizes into ?x , new uncertainty ranges ?Pi′ are also required to encompass the new local values Pi ′ resulting after interaction. From a physical point of view it

?i , which entail by consequence of eqs 4,1a the means introducing the deformation rates ?x ? of momentum uncertainty ranges as well. This reasoning holds also for the respective changes ?P i
local energies ηi of the free particles; the initial ranges ?ηi must change to encompass the new local values ηi′ modified by the interaction, e.g. when one particle initially free starts orbiting in the

?i are to be also expected gravity field of another particle. Thus, energy range deformation rates ?η ? = ? ( n ? / ?x 2 ) ?x ? ; regarding in because the total energy of the system is changed. Eqs 4,1a give ?P
i i i i

?i as quantities pertinent to the final situation particular this expression for ?xi = ?x , i.e. ni and ?x
where the merging of ?xi into ?x is completed, one finds ? = ? ni ? ?x ?i Fi = ?P 4,2 i ?x 2 Eqs 4,2 still express the quantum uncertainty, yet in a form involving the time derivatives of ranges of dynamical variables and related numbers of states; also now the local values of coordinate and momentum are conceptually discarded since the beginning. While showing that Fi do not depend ? as well, eqs 4,2 introduce force fields on the random time changes of local values P and related P
i i

? are spreading within ?x rather than local forces. The forces introduced by the respective ?P i

regarded separately because in general both ?xi are initially arbitrary and independent each other, so that the deformation rates allowing their merging into a unique common range are in turn independent as well. Each field is described by its own source particle and deformation rate: eqs 4,1a define then two corresponding equations for the respective interaction driven momentum ? = ?P ? is ?i . The force fields are in general not necessarily additive; ∑ i ?P changes consequent to ?x i 19

?i / ?xi2 = n?x ? / ?x 2 , which comes from ∑i ni / ?xi = n / ?x . If so, eqs 4,1a show that the true if ∑ i ni ?x

additivity is in effect fulfilled: whatever ?xi might be, being ?x arbitrary, the condition holds and ? . In general however it is more correct to expect ∑ n / ?x = ∑ n a / ?x k , defines ?P and thus ?P
i i i k k k

being ak proper coefficients; this is because there is no possibility to specify how ?xi actually modify to merge into ?x , as in effect it is obvious since nothing is known about each one of them. ?i / ?xi2 = ∑ k nk kak ?x ? k / ?x 2 k . While including a1 Even so, deriving with respect to time gives ∑ i ni ?x in the first order term ?x ?2 , the fields are actually additive to the first order approximation only, although the forces Fi are still defined at left hand side without changing the previous reasoning; in ? related to ?x ? only, whatever these latter and fact, the key quantities to generate F are again ?P
i i i

?i introduces unambiguously the resulting function of ?x might be. In conclusion, the concept of ?x ? simply as a consequence of eqs 4,1a. Furthermore eqs 4,2 also show that F and ?x ? the related ?P
i

i

i

depend on mi defining the respective Pi , whatever the local velocities of the particles might be. The fact that the local dynamical variables are random and unknown, suggests that the source of gravitational field can be nothing else but the mass; this conclusion justifies therefore the conceptual link of the forces just to the gravity. Then an isolated particle generates a gravity field ?i ≠ 0 is induced just by its mass, even within the pertinent ?x simply admitting that its own ?x regardless of the presence of other massive particles. Although the existence of the field has been formerly introduced through the concept of interaction between particles, it is also true that this interaction is consequence itself of the gravity field induced by each mass. It is clear that the left ? and thus for hand side of eq 4,2 is linear function of mi ; the same holds indeed for the range ?P i

?i itself, which therefore can be defined as mi c 2 /( ni po ) . The proportionality factor po has ?x physical dimension of momentum. Then ? Fi ?P ?c 2 1 c2 i ? = =? ?xi = mi 4,3a mi mi po ?x 2 ni po ?i is in fact related just to mi ; the first equation, As expected, the last equation shows that ?x regarded classically, shows that any mass behaves in the same way as a function of ?x when ? i . Clearly po does not depend on mi in eqs 4,3a; it subjected to this kind of force, being Fi / mi = v is instead peculiar feature of the interaction and must have necessarily the form po = h / λo , which in fact associates a wavelength to the gravity field. This result highlights that in fact the only masses appearing in these equations are mi of the actual kinetic momenta Pi ; then the reasonable coincidence of gravitational and inertial mass follows by consequence, since just these masses are also the sources of the respective gravitational fields. To better understand the physical meaning of this position let us extend first eqs 4,3a to the case where one of the particles is massless. Both Fi are still defined even in this case, since none of the considerations so far carried out excludes this possibility: no hypothesis has been made about the actual nature of the particles to infer eqs 4,2, whose validity is straightforward consequence of eqs 4,1 and then absolutely general. The particle 2 with m2 = 0 is a photon having speed c and momentum P2 = h / λ2 = ?ω2 / c , i.e. eqs 4,2 regard an isolated system formed by a light beam in the gravitational field of the mass m1 . Replacing in eq
4,3a m2 c 2 with ?ω2 , the second eq 4,2 reads now

? 2ω2 1 ? F2 = ?P2 = ? po ?x 2
20

?2 = ?x

?ω 2 n2 po

m2 = 0

4,3b

while F1 is still given by the first eq 4,3a; since F1 does not depend on the mass of the second ? is still that already introduced although the particle on which it acts, the physical meaning of ?P
1

kind of interaction is clearly different. At this point, three preliminary considerations are useful before considering in detail eqs 4,3. The first is that F1 and F2 are coupled, being generated when both ?x1 and ?x2 merge into a unique ?x ; it means that even the light interacts with the gravity field generated by massive particles. The second is that, since we are considering an isolated system ? of the photon can be only explained admitting that its momentum P changes of two particles, ?P 2 2 because of the interaction with m1 ; it entails then two possibilities: the gravitational field affects (a) the wavelength or (b) the propagation direction of the photon. The third is that the gravity field must propagate with light speed for the interaction with the photon be allowed to occur. These effects are completely outside of the previsions of Newton’s law initially formulated merely to describe the dynamics of massive bodies attracted by Earth’s gravity; yet the quantum origin of the force fields introduced above entails these effects as a natural corollary. Returning now to the case of two massive particles, the reasoning to infer F1 and F2 has evidenced that the present explanation of the ?i < c according to the last eq 4,3a; indeed ?x ?i = c for gravitational interaction requires in general ?x

mi ≠ 0 would define po as a function of mi , whereas it must be instead mere proportionality factor ?i and mi . Hence: (i) the gravity field propagates like a wave having frequency between ?x

ν o = c / λo defined by the momentum po = ?ωo / c ; (ii) po defines also the field energy ε o = ?ωo ; ?i are described by the (iii) merging ?x1 and ?x2 into ?x entails time dependence of ηi , whose η 2 ?i = ?ni ? / ?τ i since η ?i = ?η ?i = ?η / ??τ i according to eqs 4,1b; (iv) the actual physical ranges ?η ? = ?P / ??t and ?x ?i = ??xi / ??t according to eq meaning of ?ηi is related to that of ?τ i ; (v) ?P i i
3,2. These assertions are correlated and now clarified. If the gravitational field generated by m1 ?1 , then necessarily some field energy is lost outside ?x1 during propagates in ?x1 faster than ?x interaction: the field spreads indeed beyond the range allowed to the source particle 1. When the ?2 is controlled by the residual field second particle starts interacting, its initial energy η2 changes; η energy ε o still in ?x1 during merging with ?x2 . To examine the consequences of the points (i) to (v) and highlight how ε o generated by one particle affects ηi of the other particle initially free, let us consider the case where the particles form an orbiting system and specify in particular the field angular frequency ωo = ni / ?τ i as a function of the revolution period ?τ i of the orbiting i -th ?i = (?ωo ) / ?τ i : it means that the energy ?η ?i ?τ i released from the system to the particle. Then ?η field is irradiated by this latter outside ?xi at rate c , and then lost by the system via pulses ?ωo of ?i / c)ηi be the new value of gravitational waves having characteristic frequency ωo . Let ηi′ = (1 ? ?x ?i → c and ηi due to this specific loss mechanism; this position fulfils the conditions ηi′ → 0 for ?x

?i → 0 : the former limits evidence that just ?x ?i < c allows the gravitational waves, ηi′ → ηi for ?x ?i ≠ 0 triggers the gravity field. the latter evidence that the i -th particle is actually still free unless ?x ?i = (ηi′ ? ηi ) / ?τ i reads also ?η ?i = (?x ?i / c)ηi / ?τ i , then ε o = ηi ?x ?i / c . Let us check this result Since η
noting that eq 4,3a gives ηi = ni ( ?ωo ) 2 /( mi c 2 ) . A simple dimensional analysis elucidates the further
?i = 2ni ? 2ωoω ? o /( mi c 2 ) as η ?i = 2ni2 (G ? 2ωo2 / c 5 )( ?ω ? o ?x ?i / Gmi2 ) , where G is the reasoning: rewriting η gravitation constant preliminarily introduced here and more thoroughly justified in the next subsection, the factors within parenthesis have physical dimensions ? and t ?2 . As expected, the ?i agrees with ?η ?i = ? ni ? / ?τ 2 of eq 4,1b. It suggests that ?ω ? o ?x ?i / Gmi2 ∝ ?(η §i / ?) 2 and form of η

21

2 §2 ?i ∝ ?Gωo then η ηi / c 5 , where ηi§ is reasonably the orbiting kinetic energy: it specifies that the field energy loss is just due to an orbiting system. To complete this reasoning consider the particular case where the i -th particle orbits circularly with angular frequency ωo at constant distance ?xi from its 6 ?i = ? wo Gmi2ωo gravity centre; so ηi§ = ?miωo2 ?xi2 / 2 gives η ?xi4 / c 5 , where the factor wo summarizes the constants of proportionality so far introduced. This result, pertinent to the specific case described by the given definition of ?τ i and ηi§ , highlights the link between these latter and ε o ,

thus explaining why during the change ηi → ηi§ time instability is also to be expected for the
6 orbiting system. To infer the value of wo , let us rewrite ??ηi / ? ( wo ?τ i ) = Gmi2ωo ?xi4 / c 5 : the right hand side describes the orbital motion of the particle and thus is related to the force acting on the ?i ; the left hand ?i ni ?ωo = mi c 3 into η particle, as shown by G introduced together with the identity ?x side concerns the energy per unit time released to the field. The former depends on ?t of eq 3,2 ?i , the latter instead depends on ?τ i according to eq 4,1b. Having because of the time derivatives ?x ?i = (?ηi / ??t )(??t / ??τ i ) = 2π (?ηi / ??t ) . defined ωo = ni / ?τ i one expects ?τ i = ?t / 2π and then η

The different physical meaning of ?t and ?τ i formerly emphasized entails the time scale factor

2π between ?ηi / ??τ i and ?ηi / ??t . This factor has been so far ignored and included into the ?i ; it is summarized by the final coefficient wo . Then it proportionality constants introduced after ?x ?i , between is reasonable to regard wo as the sought time scale coefficient 2π , linear like ?x
6 ?ηi / ??τ i and Gmi2ωo ?xi4 / c 5 . Assuming then wo = 2π gives ?ηi / ?( wo ?τ i ) = ?ηi / ??t , so that both sides are expressed as a function of ?t , i.e. 6 ?ηi Gmi2ωo ?xi4 ? = 2π ??τ i c5 ?i inferred through intuitive elementary considerations agrees substantially with The formula of η that found in general relativity to describe the emission of gravitational waves in the presence of weak fields; since 2π differs from 32 / 5 by less that 2%, the present result compares reasonably well with that of the approximate solution of Einstein’s field equation. Analogous result will be more rigorously obtained in subsection 4.7 as a function of the reduced mass of the orbiting particle. The reasoning has been shortly sketched here to emphasize that the field momentum po describes the kind of interaction: regarding ωo = po c / ? equal to the actual angular frequency of orbital motion means specifying the experimental situation that characterizes the consequent frequency of gravitational waves removing field energy out of the system. It also clarifies the importance of eqs ? and its link to the radiation of gravitational waves, whose 4,1b to justify the time instability η quantum origin appears then evident like that of F1 and F2 themselves. The simple Newton law formulated without reference to the underlying quantum meaning cannot explain neither effect, conceptually important although irrelevant from a practical point of view as concerns the expected consequences, e.g. the contraction rate of the reciprocal distance of the orbiting masses. The remainder of the present paper describes in detail the consequences of eqs 4,1 to 4,3, thus showing that the most remarkable results of general relativity are inferred through simple considerations having entirely quantum mechanical character: it confirms the legitimacy of regarding the gravitational forces as consequences of deformation rates of uncertainty ranges rather than through the local metrics of curvilinear coordinates of macroscopic bodies. 4.2 Newton’s law. Consider an isolated system formed by two particles having masses m1 and m2 interacting in ?x . In agreement with the idea that each force is defined by its gravitational mass only, eqs 4,3a read

22

? ωo m1 m2 ?c 2 po = po (λo , ?x) F = ? χ χ = po = 4,4 2 2 2 ?x ?x po c The minus signs mean that the fields are attractive. The last eq 4,4 is understood noting that χ does not depend on the mass of the particles, being property of their interaction only; yet there is no reason to exclude that it depends in general on ?x through po , as it will be more comprehensively F1 = ? χ
explained in subsection 4.6. Both Fi show that the gravity force decreases with x ?2 law; of course they are written as a function of the squared range replacing the random and unknown distances x 2 from the respective mi by consequence of eqs 4,1, i.e. as a function of the respective numbers ni of states of the system consistent with ?x allowed to both particles. The potential energy corresponding to Fi is U i = ∫ Fi dx′ = ∫ Fi d ( ?x ') ; if the integration limits are ?x and ∞ , where U i vanishes, one finds of course U i = ? χ mi / ?x = Fi ?x . The effects of each force on the other particle, so far regarded separately, can be also combined into a unique general law F describing their mutual attractive interaction. Clearly F must be function of the product m1m2 in order that it reduces as a particular case to F1 or F2 for unit m2 or m1 respectively. It suggests introducing a unit reference mass mu defining F

m1m2 χ χG = u 4,5 2 m ?x12 The notation ?x12 highlights that now the total range allowed to both particles expresses their maximum distance; any random mutual position, of no physical interest, is then encompassed by ?x12 . Introducing mu means that both m1 and m2 are expressed in the same mass unit system, F = ? χG
which was instead in principle unnecessary when considering separately F1 and F2 ; in this way the numerical values of χ G and χ coincide and depend on the measure units fixed by ?x12 and mu , ? . Note that it is not longer possible to distinguish with the time unit also fixed in agreement with ?P
i

in eq 4,5 which particle is the source of the field and which one is that interacting with the field itself; actually this equation combines together two forces F12 and F21 with the particles 1 and 2 formally interchangeable in the role of gravitational and inertial masses. Then the physical equivalence of these masses requires F12 + F21 = 0 to exclude a net resulting force simply regarding the unique interaction of eq 4,5 from two different points of view. Before concerning in more detail the physical meaning of po , see eq 4,21 of the next subsection, note that approximating this latter
0 with the constant value po gives χG = const ; in this case F reduces to Newton’s gravity law

2 2 written as a function of the range ?x12 replacing random distances x12

m1m2 ?c 2 FN = ?G G= u 0 4,6 2 ?x12 m po A final remark concerns G . The quantum nature of this constant, self-evident because of the presence of ? in its definition, appears more clearly noting that with trivial manipulations FN reads

FN (m / m )(m2 / mP ) ?c ?G c4 u 0 2 =? 1 P m = l = F = 4,7 m p = m c P P o P P FP (?x12 / l p ) 2 G c3 G It is crucial to observe that introducing here the Plank units is not mere dimensional exercise to combine G with the fundamental constants of nature, because both ? and c are already inherent the conceptual definition of gravity constant; the fact that the dimensionless force F * has the proportionality constant normalized to 1 supports its definition in eq 4,6 and thus the approach so F* =
23

far followed. The dependence of the value of G on the choice of the current measure units is obvious; yet, without evidencing the quantum origin of the force, the possible combinations of ? , c and G would have mere formal meaning. Eqs 4,6 and 4,7 show that G enters into the equation of FN only because the usual choice of mass and length units does not take properly into account ? and c , as instead it would be physically appropriate according to the fourth eq 4,4 and whatever mu might be. Simple manipulations of eqs 4,5 evidence an interesting analogy between the factors u ?2 defining F . Since m u / po c has physical dimensions of reciprocal squared velocity, (v o ) , and
u (m u v o ?x12 ) 2 that of squared angular momentum, it is formally possible to rewrite eq 4,5 as follows

?c 3 Mm F =? 2 ?o ?M12

u 2 (mu vo ?x12 ) 2 = ? o ?M12

M = m1 + m2

m = m1m2 / M

4,8

2 is the squared angular momentum defined with ?o dimensionless proportionality constant; ?M12

by a unit mass moving along a proper orbit at average distance ?x12 from the gravity centre with
u velocity v u o : being ?x12 arbitrary, the pertinent v o is arbitrary as well. Eqs 4,8 are interesting because F does not longer contain dynamical variables of the single particles, but only properties 2 of the system regarded as a whole: characteristic angular momentum replacing ?x12 , total mass and

reduced mass. Rewriting m1m2 as mM emphasizes that a reduced mass travels around a fixed
2 centre where is concentrated the total mass of the system; hence one would have expected ?M12

expressed as a function of m . The second eq 4,8, related to the value of v u o , requires then
u mu vo = mv o in order to introduce in F the expected reduced mass; this is certainly possible

because, by definition, the new value v o leaves F unchanged. Being m arbitrary as well, it means ? = 0 for F = 0 , mv = const . This result, momentum conservation, agrees in particular with v inherent eqs 4,2 when no external force acts on the system; since v o is arbitrary, it holds regardless
rel 2 of its actual analytical form. If for instance v o is rewritten as v o = v rel then, repeating o / 1 ? (v o / c ) rel 2 the reasoning above, the result reads m v rel o / 1 ? (v o / c ) = const , with mere numerical requirement
rel vo ≠ v o . The momentum conservation is then inherently valid also in special relativity. Moreover
2 eq 4,8 shows that once having fixed M and m , a given value of F determines uniquely ?M12 ; the

fact that ?x12 does not longer appear explicitly in the first eq 4,8 means that if F is constant, i.e.
u without external forces perturbing the system, (m u v o ?x12 ) 2 is also constant. Hence, owing to the previous result, u ′ 2 F = const (mu vo ?x12 ) 2 = ( mv o ?x12 ) 2 = ( mv′ o ?x12 ) i.e. the angular momentum is conserved whatever m vo , m v′ o and ?x12 might be, in agreement with the result already found in subsection 2.1. It is significant that eqs 4,8 enable these results to be inferred without integrating any equation of motion and without any hypothesis ad hoc, but simply owing to the form of G in eq 4,6. In principle the reasoning carried out for the dynamical variables of the system can be extended to the constant factor, thus obtaining ( ?c )3 Mm 2 ′Mo F =? 4,9 ?o = ? o / ?2 2 2 ′ Mo ?o ?M12 2 ′ is a further dimensionless proportionality factor. The notation emphasizes that M o where ?o is

inferred from po only. Eq 4,9, with both factors inversely proportional to the respective squared angular momenta, evidences the formal analogy between the variable and constant factors of F , respectively describing the dynamics of the masses in the gravitational field and the gravitational 24

′ justify the positions 4,8 and 4,9 on mere dimensional basis; yet the field itself. Both ?o and ?o former is a formal position, whereas the latter introduces in eq 4,5 new physical information through the field angular momentum. Eq 4,9 will be further concerned in the next subsection 4.8. Usually proportionality factors very different from unity show that some relevant physical effect is ′ ≈ 1 , whereas hidden in the dimensional analysis; it will be found in subsection 4.8 that in fact ?o
instead ? o ≈ (2π ) 2 . It supports the idea that eq 4,9 is not a trivial way to rewrite eq 4,5, since it introduces the field andular momentum M o . 4.3 Red shift. Consider an isolated system formed by a photon moving in the gravitational field of m1 . Let us rewrite the second eq 4,3b considering the energy F2 ?x within the uncertainty range ?x

? 2ω2 m2 = 0 = ?? ω ?ω < 0 4,10 po ?x With the negative sign due to eq 4,3b, eq 4,10 describes the loss of photon energy ??ω related to momentum change ?P2 = ??ω / c according to the aforesaid effect (a) already introduced in subsection 4.1. Let us define an analogous energy for the particle 1 ?c 2 m1 F1?x = ? 4,11 po ?x The fact that ? /( po ?x) appears in both eqs 4,10 and 4,11 confirms that the photon momentum
F2 ?x = ? change is correlated to the force field F1 . Eliminating ? /( po ?x) between these equations gives F1?x = m1c 2 ?ω

ω2

4,12

The arbitrary value of ?x determines ?ω as a function of the corresponding F1?x , whatever the values of the former and the latter regarded separately might in principle be. Since both eqs 4,10 and 4,11 concern ?x , the propagation direction of the photon is by definition within the same range where is also located particle 1; hence the frequency change is that of a light beam moving radially with respect to the gravitational source m1 . Eq 4,12 rewritten as ?? ? ω F ?x m2 = 0 = ?? = 1 4,13 2 c ω2 m1 shows that the change of gravitational potential ?? within ?x is a property of the particle 1 only. In effect ω2 does not depend on ?? , being by definition the photon proper frequency. Instead to

?? is related the frequency shift ?ω , which is to be expressed with respect to ω2 through the reasonable position ?ω = ω ? ω2 ; indeed for ?? → 0 eqs 4,13 and 4,12 give both ?ω → 0 and
?ω /( ?? / c 2 ) → ω2 in any point of ?x . Moreover according to eq 4,12 ?x → ∞ entails ?ω → 0

because F1?x → 0 , i.e. photon and m1 are infinitely apart; as expected ?? → 0 is also compatible with a vanishingly weak field within ?x , in which case the frequency ω → ω2 tends obviously to that of a free photon exempt of gravitational effects. Thus eq 4,13 can be also regarded in a formally different way: it calculates the frequency shift ?ω with respect to ω2 due to the gravitational potential ?? = ? ? 0 generated by m1 , which means ? / c 2 = (ω ? ω2 ) / ω2 with ? < 0 because

?? < 0 according to eqs 4,4. In any case, ?ω < 0 means ω < ω2 i.e. the proper frequency ω2 is shifted by ?ω down to the lower value ω when the photon moves through ?x towards a gravitational potential decreased by ?? . No hypothesis is necessary about ?x , which in effect does
25

not appear in the final result. Note that regarding eq 4,10 only, (??ω / c)?x = n2 ? , the uncertainty does not prevent in principle ?x → 0 and then ?ω → ∞ , which of course is nothing else but ?P2 → ∞ in eq 4,1a. Yet, the crucial concept that discriminates any option mathematically possible from the actual physical behaviour of the system is the coupling of F1 and F2 , i.e. the interaction between the photon and m1 . Actually eqs 4,10 and 4,11 cannot be regarded separately; they describe a physical event when merged together into eq 4,12, which effectively concerns what really happens. The interaction is the boundary condition of the physical system that eliminates in fact the infinities admitted in principle by the unavoidable arbitrariness of any uncertainty range. 4.4 Time dilation. Since the first eq 4,4 gives ?? / c 2 = ? ? /( po ?x ) , it follows that ?? ?τ ?x ? =? τ0 = ?τ = 2 c τ0 c cpo The notation emphasizes that the time range ?τ is related to the gravitational potential through po , whereas τ 0 does not by definition. Moreover ?τ → 0 and ?τ /(?? / c 2 ) → ?τ 0 for ?? → 0 ; hence

τ 0 is the reference time in the absence of gravitational potential. Since ?τ < 0 if ?? > 0 , considerations completely analogous to those previously carried out show that the time τ deviates from τ 0 by ?τ = τ ? τ 0 < 0 as a result of the rising of gravitational potential ?? = 0 ? ? > 0 , because ? < 0 . The equation ?x ? ? ? 4,14 τ = ?1 + 2 ?
c ? c ? shows therefore that time dilation occurs in the presence of gravitational potential. 4.5 Light beam bending. The section 2.2 has described the frequency shift of a photon moving radially with respect to the gravitational mass m1 ; the link between F1?x and F2 ?x , eq 4,14, was exploited to this purpose.

This result can be further extended; the system photon/ m1 can be also described by the force FN
2 writing eq 4,6 in the form ?Gm1 (?ω2 / c 2 ) / ?x12 , i.e. concerning the interaction of m1 with the

photon as if this latter would have virtual mass m2 = ?ω2 / c 2 . Eq 4,13 suggests considering the ratio F ?x m ?ζ = ? N 122 = G 1 4,15 ?ω 2 / c ?x12 conceptually analogous to ?? ; the sign has been chosen in order to define the range ?ζ positive for reasons shown below. Now the photon is not longer constrained to travel within the same range including m1 , as explicitly requested in the previous case by the simultaneous conditions 4,10 and 4,11. Eq 4,15 introduces the force between particles ?x12 apart, yet without necessity that this latter coincides with the range ?x where the photon is allowed to move; then, in lack of specific boundary conditions, one must assume that in general the photon moves outside ?x12 where is located m1 . Since ?ζ has physical dimensions of squared velocity, it is certainly possible to write

?ζ = ξ c 2 , being ξ a dimensionless proportionality factor: ?ζ =ξ 4,16 c2 It is reasonable to expect that by analogy with eq 4,13 also ?ζ controls the photon momentum change, yet in this case the aforementioned effect (b) is expected to occur: the fact that the uncertainty range ?x of photon position does not include the source of gravitational potential,
26

? of the photon concerns now the ? to which is related ?P suggests that the deformation rate ?x 2

stretching of ?x due to its bending by effect of FN ; otherwise stated, the photon deflection reproduces the curvature of its allowed uncertainty range ?x caused by the gravitational potential ?ζ / c 2 . Since no hypothesis is necessary about the uncertainty ranges, even the curvature of ?x does not contradict or rule out any result so far obtained: the crucial idea is still the conceptual impossibility to know where m1 and the photon are exactly, irrespective of whether their position uncertainty is described by linear or curved ranges. One sensibly expects that the local interaction increases as long as the photon approaches the gravitational mass m1 ; hence the next step should seemingly be the estimate of curvature radius at any point of ?x . Yet such a calculation is ruled out by the quantum uncertainty assumed since the beginning: this radius could be effectively calculated only introducing the local position of the photon, which instead is considered “a priori” unknown, random and physically meaningless. Even discarding the local coordinates, however, it is possible to estimate the average curvature in any point x0 of ?x through the angle δφ between the tangents in two arbitrary points x′ and x′′ around x0 ; without concerning the actual time dynamics of the progressive bending process, it is not necessary to specify these points and the actual local value of δφ for x′′ → x′ . Accordingly, also the concept of distance of the photon from m1 is disregarded and replaced by the uncertainty range ?x12 ; being indefinable the local coordinates of the particles, a range of possible distances is in fact consistent with the unknown and random positions of m1 and photon. Actually this is the only approach conceptually possible, regardless of any simplifying aim to bypass the mathematical difficulty of the dynamical problem. Examining the quantities that appear in eq 4,16, the only way to introduce δφ into eq 4,15 is through the parameter ξ , whose physical meaning has been not yet specified. In principle it is possible to put δφ = a + bξ + ??? , where a and b are appropriate power series coefficients. Replacing into eq 4,16, the result at the first order of approximation is Gm1 /(c 2 ?x12 ) = δφ / b ? a / b owing to eq 4,15. It gives a = 0 because if m1 = 0 there is no gravitational effect and then no photon deflection. Hence Gm b δφ = 2 1 4,17 c ?x12 Clearly δφ depends on the particular choice of the points x′ and x′′ if the size of ?x is finite, i.e. δφ = δφ ( ?x, ?x12 ) . Consider instead the total deflection angle δφtot defined for an infinite range ?x∞ with the points x′ → ?∞ and x′′ → ∞ where the gravitational field vanishes; δφtot is then by
min δφtot is also defined by the nearest approach distance ?x12 where the gravitational effect of m1 on

definition greater than any local value δφ calculated between two arbitrary points of ?x . If so,

the photon is reasonably strongest. The asymptotic tangents to the actual photon path define two ′ and ?x∞ ′′ crossing in a point xcr , whereas the total deflection angle δφtot infinite linear ranges ?x∞ ′ and ?x∞ ′′ . Further information comes of the photon path is given by the relative slope between ?x?∞ from the quantum uncertainty: wherever the points x′ and x′′ might be, the travel directions of the photon cannot be regarded separately depending on which point it moves from; whatever size and local curvature of ?x might be, the paths from x′ and x′′ or from x′′ to x′ are indistinguishable. This means that ?P2 within ?x is given by h / λ2 ? (?h / λ2 ) = 2h / λ2 ; therefore ?P2 = (2 / c)?ω2 gives (2 / c)ω2?x = n2 , being n2 the number of states allowed for any frequency ω2 of the quantum system photon/ m1 . Hence for any ω2 and n2 one expects that δφ is a function of c / 2 , being the ? within ?x ; it means that factor ? the fingerprint of the quantum uncertainties ?P2 and ?P 2 27

? = ?(2 / c ) ?ω ?x ? / ?x = ??P2 ?x ? / ?x , which δφ = δφ ( ?x12 , ?ζ /(c / 2) 2 ) . This result agrees with ?P 2 2 ? = ? n ?? x ? / ?x 2 ; as expected, the bending effect of the photon path is due to the force reads also ?P 2 2

? and vanishes for ?x → ±∞ , where the gravitational field is null, with ?x ?2 that corresponds to ?x law. It confirms the sign in eq 4,15. For x′ and x′′ ranging from ?∞ to ∞ , i.e. for ?x → ∞ , the min min limit δφ → δφtot also requires ?x12 → ?x12 ; then δφtot = Gm1b /(c 2 ?x12 ) . In this way δφtot is defined uniquely by the asymptotic paths of the photon, regardless of any detail about the local curvature of ?x and actual photon position under gravitational potential progressively increasing; δφtot depends only on the boundary conditions of the system at infinity and is clearly a constant parameter of the present dynamical problem. Let us plug these considerations into eq 4,17: since the dimensionless constant b must be equal to 4 in order that δφtot depends on c / 2 , the result is
Gm1 ? 2 ? δφtot = min 4,18 ? ? ?x12 ? c ? Let us emphasize once again that this formula is expressed as a function of the reciprocal range min min ; of course it means that δφtot is inversely proportional to the minimum approach distance x12 ?x12 conceptually unknown and thus, whatever it might be, expressed here through its uncertainty range. min 2 In other words, eq 4,18 is the quantum expression of the well known result ?4Gm1 /( x12 c ) found in general relativity. Analogous considerations hold for the results of the next subsection. 4.6 The Kepler problem. The previous cases have been discussed without specifying in detail the analytical form of the field momentum po introduced in eqs 4,3; it has been merely emphasized that if po = h / λo is regarded
0 approximately as a constant po , then F of eq 4,5 takes the form of Newton’s law, eq 4,6. Yet there

2

is neither reason nor necessity to assume po constant; rather, any consideration about how it could possibly change as a function of a suitable parameter has been so far ignored merely because eqs 4,5 to 4,18 did not require closer insight about its actual form. To formulate correctly the present 0 ′ , where po ′ = po ′ ( ?x ) denotes a problem, however, it is necessary to write explicitly po = po + po
0 proper correction to po function of ?x ; for simplicity of notation, ?x without subscript indicates

the distance uncertainty range between the particles. The power series development of po must
0 + γ ′ / ?x + ??? , with γ ′ constant, having neglected for simplicity the higher have then the form po

order terms ?x? k : indeed the correction is expected to vanish for weak fields, i.e. for ?x → ∞ , where the classical Newtonian mechanics is quite accurate. The dependence of po on ?x is
? / ?x ) 2 γ ′ / po at the first order. sensible; recalling the second eq 4,3a, it gives for instance ??? x = ( ?x ? should be either zero or a constant value. Since nothing in nature Hence ??? x = 0 for γ ′ = 0 , i.e. ?x changes instantaneously, seems more reasonable the idea of ??? x ≠ 0 enabling a gradual increase of ? up to the value consistent with eq 4,2, which therefore reads also Fi = ?(n?po / γ ′)(??? ?i ) : at xi / ?x ?x

infinite distance, ?x → ∞ , the force vanishes because ??? x → 0 ; moreover the m?x ?2 dependence of ? and ??? F is replaced by the deformation rates ?x x of the uncertainty range ?x of the phase space, and by a weak dependence upon ?x through po ; the mass does not longer appears explicitly. This conclusion confirms the idea that the gravity force is nothing else but the experimental appearance of the deformation rate of the time/space dependent phase space uncertainty. There is no reason to 0 ′ << po exclude that even the first order correction po is conceptually important to account for small gravitational effects unexplained by the simple Newton law. This is the classical case of the perihelion precession of orbiting bodies. Let E < 0 be the total energy of a system of two bodies of 28

masses m1 and m2 subjected to gravitational interaction. The elementary classical mechanics shows that perihelion precession is allowed to occur if the potential energy U of a reduced mass m orbiting in the gravitational central field of M in agreement with eq 4,8 has the form U = U eff + U ′ : here U eff = ?GMm / x + M 2 / 2mx 2 is the effective potential and M2 the squared angular momentum of the particle at distance x = x(t ) around the gravitational centre; U ′ = U ′( x) is a proper correction to the effective potential. The particular case U ′ = β / x 2 , with β arbitrary coefficient, is explicitly reported in several textbooks; the solution of the “Kepler problem” with ζ M 2 + 2β m ζ = m1m2G = MmG U =? + 4,19 x 2mx 2 is summarized then by the following relevant equations:
U
cl min

=?

ζ 2m
2M 2

M2 p= = (1 ? e2 )a = b 1 ? e2 mζ
T = 2π a am

b=

M2 2m E

ζ = 2a E

4,20

2πβ ζ ζp where δ? is the precession angle, b and a are the minor and major semi-axes of the ellipse having eccentricity e and T the revolution time. Here U ′ is arbitrarily introduced as mere additive term to U eff , so that β appears in the solution only as separate additive term as well. Yet the pure

δ? = ?

Newtonian case where G is merely the constant inferred from the experiment does not justify terms additional to U eff ; relativistic concepts must be introduced since the beginning into the problem to explain why δ? is actually observed. To regard this point, let us run again the quantum approach of section 2 and rewrite first to this purpose eq 4,5 considering the most general form expected for po

F =?

m1m2 2 0 mu po ∑ ak (γ ′ / ?x) k ?x
∞ k =0

?c 2

4,21

in agreement with eq 4,4; ak are proper coefficients of series development with a0 = 1 in order that with γ = a1γ ' , for uncertainty ranges much larger than γ . Hence, owing to eqs 4,8,
0 0 0 0 ′ ( ?x) << po ′ ? po ′ (?x ) at the first order. Then the condition po gives po + po (1 + γ / ?x) , po ? po + po

?c 2 γ γ ? ζ ? G = << 1 F = ? ?1 ? ? 2 u 0 m po ?x ? ?x ? ?x 0 The third equation recalls that the constant value po fits the experimental gravitation constant, i.e.

G is by definition the zero order constant term such that F tends to FN of eq 4,6 when γ = 0 . Significant consequences are inferred introducing γ / ?x to approximate the most general form possible for po , since the present expression of F justifies in principle the additional potential term U ′ that describes the precession. The potential energy of the reduced mass m in the gravitational field of M reads ?ζ / ?x + ζγ /(2?x 2 ) , to which must be summed the angular orbital term M 2 /(2m?x 2 ) to obtain U . Let us follow now the same quantum formalism already shown for the hydrogenlike atom, since the distance x = x(t ) between the orbiting body and the gravitational centre has been already replaced by the corresponding ?x = ?x(t ) of eq 3,2; we are describing therefore a quantum system of two particles orbiting at random distance included within the uncertainty range ?x by effect of their gravitational interaction only. Also in this case it is essential that ?x be a function of space-time: if x(t ) changes with t , then ?x must depend on t as well in
29

order to include any possible change of x at various times. This agrees with the fact that just the ? of the uncertainty ranges defining the time dependence of ?x introduces the deformation rate ?x ?(t ) , are gravitational force. It is easy to show that the classical eqs 4,20, obtained integrating x inferred as limit case solving the present quantum problem exactly as shown in the examples of section 2. The kinetic and total potential energies as a function of m read ?Px2 ζ M 2 + γ mζ πγ Ekin = U =? 4,22 + δ? = ? 2 2m ?x 2m?x p The last equation is found comparing U of eqs 4,22 and 4,19; γζ / 2 corresponds to the coefficient β defining δ? in the last eq 4,20. The perihelion precession is then explained simply through the most general form possible for po , without any specific hypothesis and in agreement with the Newtonian limit 4,6; for this reason eq 4,21 has relevant theoretical interest, even considering the first order correction to Newton’s law only. Let E = Ekin + U < 0 be the total energy; then Ekin > 0 and U < 0 define the positive term 1 ? E / U > 0 . Replacing thus ?x with n? / ?Px in eqs 4,22 and minimizing E = ?Px2 / 2m ? ζ?Px /(n?) + (M 2 + γ mζ )?Px2 /(2mn 2 ? 2 ) with respect to ?Px , one finds

?Px min =
and then also

mζ n? 2 (n?) +M 2 + γ mζ

?xmin =

(n?) 2 +M 2 + γ mζ mζ

4,23a

Emin = ?
It appears that

1 ζ 2m 2 (n?) 2 +M 2 + γ mζ

U min = ?

mζ 2 2(n?) 2 +M 2 + γ mζ 2 ( (n?) 2 +M 2 + γ mζ )2

4,23b

Emin = ζ / 2?xmin

4,24 p / 1 ? e2 = M 2 /(2m Emin ) 4,25

Moreover, introducing also now a suitable parameter p , eqs 4,23 define the following quantities

p = M 2 / mζ

1 ? p / ?xmin = 1 + 2 Emin M 2 /( mζ 2 )

The revolution period is also calculated easily from eqs 4,23 to 4,25. In lack of local information about the coordinates of the orbit, let us introduce the average momentum Pav of the orbiting reduced mass and define the pertinent uncertainty range ?Pav including it as ω = ?Pav /(m?xmin ) ; of course this equation is obtained from ?Pav = P§§ ? P§ , with P§§ = mω x§§ and P§ = mω x§ putting the range x§§ ? x§ equal to ?xmin . Since ?Pav must be consistent with the energy Emin , it must be true
2 ?1 that ?Pav / 2 m = Emin ; hence ω = ?xmin 2 Emin / m . The angular frequency of orbital motion is

?1 ω = 2π / δ t , being δ t the time range of one revolution. Eqs 4,24 give ω = ?xmin ζ /(m?xmin ) , i.e.

δ t = 2π?xmin

?xmin m

ζ

4,26

As expected, the quantities just found are expressed through uncertainty ranges within which are delocalized the quantum particles, rather than through orbit coordinates applicable to macroscopic bodies. Yet, it is interesting to compare eqs 4,24 to 4,26 with eqs 4,20; we note that

?xmin ? a

M2 ?b 2m Emin

1+

2 Emin M 2 ? e2 2 mζ

δt ? T

Emin ? E

p ? a(1 ? e2 )

The correspondence between ?xmin and a is evidenced comparing eqs 4,24 and 4,26 with the fourth and fifth eqs 4,20 and confirmed by the correct calculation of revolution time δ t , here appearing as characteristic time range as well; also b is related to the range of arbitrary values 30

allowed to M 2 = nor (nor + 1)? 2 , where nor is the quantum number of orbital angular momentum. Analogous considerations hold for the other quantities; indeed also now (a 2 ? b 2 ) / a 2 = e2 . The quantum approach evidences that the uncertainty ranges have the features of the classical orbital parameters exactly defined for macroscopic massive bodies, yet without contradicting the postulated uncertainty of the quantum approach; being n arbitrary, it actually means that the plane of the orbit trajectory and the local orbit distances between m and M remain in fact unknown. However these result are not peculiar of the quantum world only, since they do not require ? → 0 and γ → 0 to infer the aforementioned classical results; it follows that the gravitational behaviour of a particle is in principle analogous to that of a planet, mass, time and length scale factors apart. The reduced mass m moving around m1 follows an elliptic orbit. Note that the orbit parameters a ,

b , T and related Emin , given in eqs 4,20 and summarized by the positions above, entail an
cl expression of minimum potential energy U min slightly different from the classical U min

2 2 2 U min ( 2(n?) +M + γ mζ ) M = 2 cl U min ( (n?)2 +M 2 + γ mζ )

cl cl is not surprising: U min is calculated simply minimizing U of eq 4,19, thus it The result U min > U min

is mere consequence of its own analytical form; here instead U min is by definition calculated in connection with Emin , i.e. minimizing the global energy of the system that includes also the orbital kinetic energy. The explicit expression e 2 = (( n?) 2 + γ mζ ) /(( n?) 2 +M 2 + γ mζ ) inferred from the second eq 4,25 shows that e2 → 0 for M 2 >> (n?)2 , whereas e2 → 1 for M 2 << (n?) 2 ; hence the orbit eccentricity is a quantum effect defined by the angular momentum M2 and by (n?) 2 , thus controlled by the relative values of nor and n in principle arbitrary. For classical bodies like planets

M2 is overwhelmingly much larger than ? 2 , so that M 2 << (n?) 2 requires n → ∞ i.e. a number of 2 states n >> nor >> 1 allowed to the orbiting system; it in turn also means that nor ( nor + 1) → nor , i.e.
2 M2 → M2 z ; the classical Newtonian orbit lies then on a plane corresponding to M z . The classical

cl of eq 4,20 coincides with U min putting γ equal to zero, as it is obvious, and when U min

M 2 >> (n?)2 , i.e. for circular orbits that entail in effect the minimum value of total energy; in this particular case, i.e. for ?xmin → M 2 /( mζ ) , eqs 4,24 to 4,26 read
?r 3ωr2 =
r Emin =?

ζ
m =?

= (m1 + m2 )G

?r =

M2 mζ

e2 = 0

4,27

1 ζ ?r m?r 2M 2?r Also the form of these equations, well known, is a further check of the present approach and will be profitably used in the next subsection. In principle eqs 4,27 require simply proper values of n and nor , which are however so large for macroscopic bodies that in practice γ mζ is expected to be completely negligible. Then, eqs 4,20, 4,22 and 4,25 give πγζ m πγ mMG δ? = ? = 4,28 2 M 2?xmin Emin p For γ ≠ 0 the major axis of the ellipse rotates by an angle δ? after one revolution period; then the mass m not only moves along its orbit but also rotates because of the angular precession within concentric circles 2e?xmin apart. From dimensional point of view it is possible to write

ζ 2m
2

ζ

r = U min

ωr =

31

γ = 2q′?xmin

m = q′′Emin / c 2

4,29

being q′ and q′′ proper dimensionless coefficients: the former measures the parameter γ in 2?xmin units, the latter relates m to the constant energy Emin of orbital motion in the field of M and is therefore negative by definition. Being both masses arbitrary in principle, eqs 4,29 are only a formal way to rewrite δ? more compactly as MG q = q′q′′ 4,30 δ? = qπ 2 pc Although q′′ is defined by the second eq 4,29, q remains unknown; owing to eq 4,24 one finds indeed q′′ = ?2mc 2 ?xmin / ζ , i.e. q′ = ? qζ /(2mc 2 ?xmin ) and γ = ? qζ /( mc 2 ) = ? qMG / c 2 . The further reasoning necessary to define q exploits the fact that γ is related to the precession angular momentum of m . From a quantum point of view, this situation is described introducing an angular 2 momentum M 2 and specifically due to the precession effect, i.e. such that prec additional to M
2 2 2 2 ′ δ? ∝ M 2 prec / M , through q = M prec / ? = l prec (l prec + 1) ; if so, q = ? l prec (l prec + 1)ζ /(2 mc ? xmin ) and

q > 0 . Then γ and δ? read M2 MG γ = ? prec MG δ? = l prec (l prec + 1)π 2 l prec = 0,1, 2, ??? 2 (?c) pc Since γ < 0 , eq 4,23a can be rewritten as ?xmin = ((n?) 2 +M 2 ) / mζ ? γ , i.e. the right hand side has
the form x§§ ? x§ expected for any distance uncertainty range; in other words, γ is actually the lower boundary of ?xmin . As concerns the value of l prec , the trivial case l prec = 0 has been already considered: this value is acceptable as Newtonian approximation only, being however irrelevant and unphysical in the present context. Considering therefore l prec > 0 only, one finds 2 MG 4,31 2 c2 M M One expects in general the condition ?xmin > rSchw , because rSchw defines the classical boundary where the escaping velocity from M is c : necessary condition to enable the orbiting system of massive particles is therefore that ?xmin including all the possible distances between M and m be
M γ = ? rSchw M rSchw =
M m1 m2 larger than rSchw . On the other hand, since γ is the lower coordinate of the range ?xmin = rSchw + rSchw allowed to both particles, then by definition accessible to these latter, it follows that necessarily M M γ > rSchw . Consider now eq 4,31: if l prec = 1 then γ would be equal to rSchw , which is not

l prec (l prec + 1)

acceptable; hence it must be true that l prec > 1 . On the other hand, increasing l prec means that both

Emin and U min become less negative; then the condition of minimum energy suggests MG M 4,32 δ? = 6π 2 l prec = 2 γ = ?3rSchw pc as it is well known. Here the coefficient 6 is the fingerprint of the quantum angular momentum related to the orbital precession effect. Note that from the point of view of the orbiting particle the central mass M appears rotating at rate ? = δ? / δ t , which therefore also defines an orbital momentum M M related to the precession angular velocity ? = ?u around the direction of an arbitrary unit vector u . The Poisson relationship that links M M in the perihelion precession reference system (where the central particle does not rotate) and in the reference system where the central particle rotates with angular rate ? equal to the precession rate gives then the known result
32

dM M = ?× MM 4,33 dt Hence the perihelion precession of an orbiting particle entails also the existence of a drift force in the gravitational field of a rotating body. 4.7 Gravitational waves. Let us return now to the field energy loss related to the emission of gravitational waves from an orbiting system. The equations found in subsection 4.1 were inferred considering explicitly that the ?i , i.e. faster than gravitational waves remove energy through pulses ?ωo propagating at rate c > ?x
the deformation rate itself originating the force, with ωo related to the orbital period. The result was ? η mc ?x h ?ωo 6 ?i = ?2π Gmi2ωo ε o = ηi i = i i po = = 4,34 η ?x 4 / c 5 c ni po λo c A better and simpler calculation is now carried out starting directly from the results of the “Kepler problem”. The average loss of energy radiated after one revolution of the mass m , during which Emin changes by δ Emin and ?xmin by δ?xmin , reads at the first order δ Emin = (?Emin / ?xmin )δ?xmin ; eq 4,24 gives δ Emin / δ?xmin = ? Emin / ?xmin with good approximation for a small change of δ?xmin
2 during δ t = 2π / ω . It is easy to verify that this equation is also fulfilled by δ Emin = ? wEmin ω and

δ?xmin = ? wζω / 2 , where w is a proportionality constant: through the factor ω the former equation calculates the energy radiated showing reasonably that ?δ Emin is proportional at any time to the
2 current value of Emin , i.e. the loss is expressed by a negative quantity; the same holds also for

δ?xmin , since the loss requires also contraction of the minimum approach distance ?xmin between 2 the particles. Hence follow the positions ?δ Emin / δ t = wEmin ω 2 / 2π and δ?xmin / δ t = ? wζω 2 / 4π ,
where w must be proportional to G : in absence of gravitational field, i.e. for G = 0 , one expects that δ Emin and δ?xmin vanish. By dimensional reasons w is appropriately expressed as w = w′G / c5 being w′ a further dimensionless constant. It is easy at this point to show again the results in the r particular case of circular orbit; replacing Emin , ω , ?xmin with Emin , ωr , ?r of eqs 4,27 gives
r δ Emin Gm 2 ?r 4ωr6 ′ ? =w δt 8π c 5

The first eq 4,35 compares well with eq 4,34 previously inferred in subsection 4.1; also now appears a coefficient w′ . The connection between these equations is clear: the former is obtained starting directly from the energy Emin of the orbiting system, the latter was obtained instead from ?i of eqs 4,1b the specific energy the definition of field energy loss and then introducing into η
miωo2 ?xi2 / 2 of the mass mi of the i -th particle in circular orbit. It explains why eq 4,35 more

Gζωr2 G 3 m1m2 (m1 + m2 ) δ?r ′ ′ ? =w =w 4π c 5 4π?r 3c 5 δt

4,35

correctly replaces mi with the reduced mass m and confirms that the field frequency ωo really corresponds to the characteristic orbiting frequency ωr of the specific quantum system concerned r ?i inferred from eq 4,1b is just ?δ Emin in particular. It also confirms that, as expected, η / δ t of the orbiting system, in agreement with the idea that the energy lost by the orbiting system is effectively released to the pulse during each revolution period of m . The comparison between eqs 4,34 and 4,35 is legitimate to infer the numerical value of w′ ; therefore w′ = (4π ) 2 gives

?

The numerical agreement between 2π and the coefficient 32/5 of general relativity has been already emphasized in subsection 4.1; it is now significant the same agreement between 4π and the known factor 64/5 of the relativistic formula of orbit radius contraction related to the energy loss. 33

r δ Emin Gm 2 ?r 4ωr6 = 2π c5 δt

?

Gζωr2 G 3 m1m2 (m1 + m2 ) δ?r = 4π = 4 π c5 ?r 3c 5 δt

4,36

Eqs 4,36 show clearly that the present point of view does not concern the actual dynamics of radius contraction described point by point along the orbit path of m ; instead of describing an intuitive spiral motion progressively approaching the gravity centre, eqs 4,35 only show that after a time range δ t the current orbit radius ?r is contracted to ?r ? δ?r while a pulse of gravitational wave propagates at speed c with frequency corresponding to 1/ δ t . This is why in effect the orbiting frequency ωr appears in eqs 4,36. The emission of a gravitational wave pulse is conceptually analogous to that of a photon resulting by electron transition between two energy levels. Consider now that c5 / G has physical dimensions of power; then the first eq 4,36 reads also c δ Emin ?r m c5 π 2 ?5 = ?WPl f ρ ρ= M f = WPl = 4,37 δt 16 rSchw M G The average power radiated depends on the mass ratio and not on the masses themselves; hence it is M equal in principle for planets and quantum particles at proper orbital distances expressed in rSchw units. The value of WPl is extremely large, about 3.6 ?1052 watts; yet if ?r is of the order of
r planetary distances it is easy to realize that the factor ρ ?5 makes irrelevant δ Emin / δ t , and then δ?r / δ t as well, even for f ≈ 1 . Considering that δ t is the time range corresponding to one orbital revolution, the planet orbits are practically stable. However increasingly large powers are to be M . This is typically the case of quantum particles orbiting in their own expected as long as ?r → rSchw gravitational field; since the values of each mass do not appear explicitly in the equation, but only their ratio, strong radiation of gravitational energy is to be expected in an orbital system of quantum M particles at distances not very far from their own rSchw during a reasonably short time range δ t . 4.8 Newtonian gravitation constant. The results of section 4.1 show that the gravity force in a range of distances larger than or equal to rSchw is described by the following properties of the particle generating the field: (i) mass M , eq

M 2 4,31, (ii) angular momentum M 2 Schw (i.e. M12 calculated with ?x12 → rSchw ) and then possible spin, (iii) possible charge. Clearly there is no reason to exclude spin and charge of the particles concerned in eqs 4,3 and 4,5, although both have been so far neglected for practical purposes only, i.e. simply to focus the discussion on the gravity force. The following discussion aims to estimate the constant G , eq 4,6, through eq 4,9. Actully the constant factor of this equation is not G . Let us introduce then, for dimensional reasons only, the unit momentum Π u 2 into this equation in order that each mass be expressed as mass Πu / ? , i.e. mass per unit length, in analogy with eq 4,5 where F is in

effect function of mi / ?x12 ; once having replaced in eq 4,9 these latter with the angular momentum of the system, Π u 2 allows to express M and m in units consistent with G rewriting eq 4,9 as M * m* 1 ( ?c ) 3 Πu Πu * * F = ? χG 2 2 χG = M = M m = m 4,38 2 u2 ′ Mo M12 / ? Π ? ? ?o Being m1 and m2 arbitrary, m* and M * can reproduce desired values of experimental significance. Yet the interest of these equations rests also on the physical information inherent the field angular 2 momentum M o , which is further examined as follows. If it is possible to define M o then, in principle, it should be also possible to introduce its orbital and spin components L o and S o ; so
2 2 M o = Lo + S o . The fact that M o is a property of the field only, suggests M o = ( aλ o × p o ) 2 , where a is a proper multiplicative constant that identifies the field vector aλ o ; if this latter is normal to p o 2 then M o = ( aλo po ) 2 = (2π ) 2 a 2 ? 2 . On the other hand, it is known that for any quantum particle 2 Mo / ? 2 = lo (lo + 1) + so ( so + 1) + 2lo so . This expression is assumed true also for the field angular

34

′: a momentum. Let us recall now the final remark already introduced in subsection 4.2 about ?o proportionality constant significantly different from 1 means that some relevant physical concept is still hidden by its value. It means that the condition a ≈ 1 should be fulfilled, so it should be true that lo (lo + 1) + so ( so + 1) + 2lo so ≈ (2π ) 2 . To verify this condition, let us calculate the expression at
left hand side with various trial values of lo and so , including for the latter both half-integer and integer values; some outputs of interest in the present context among those calculated for arbitrary input values of 0 ≤ so ≤ 5 and 0 ≤ lo ≤ 7 are: … 20, 24.75, 30, 35.75, 42, 48.75, 56,…. The value closest to (2π ) 2 is 42, which is obtained with several couples lo and so , namely: [lo = 6 ? j , so = j ] for 0 ≤ j ≤ 6 . It is significant that the sought condition of making a as close as possible to 1 is fulfilled by integer values of so only. In any case, whatever the true combination of lo and so might be, the value 42 gives a = 1.03 . To check the physical validity of this result, let us substitute the 2 value M o = 42? 2 into the second eq 4,38, thus obtaining

χG =

1 ?c 3 ′ 42Π u 2 ?o

4,39

′ ?1 6.76 ?10 ?11 m3 Kg ?1s ?2 ; if the constant of Being Π u = 1 by definition, this equation gives χ G = ?o ′ is very close to unity, as it must be true, this value compares quite well with the proportionality ?o

experimental value Gexp er ≈ 6.67 ?10 ?11 m3 Kg ?1s ?2 . It confirms the validity of eqs 4,39 and 4,39. Moreover eq 4,6 gives po = 1.39 ?10?7 Kg m / s . Note that this estimate of G does not exclude its weak dependence on time, mentioned in some theories. 5. Discussion. The strategy followed in the present paper was to show first that the quantum point of view introduced in section 1 and highlighted in section 2 is conceptually consistent with the principles of special relativity; before carrying out any calculation, it has been shown in section 3 that effectively the invariance of interval and Lorentz’s transformation are inherent the concept of space-time dependent uncertainty. Only thereafter have been approached the problems of general relativity, formulated in the phase space with trust in the quantum nature of general concepts like the space? and time curvature. A revealing evidence was in effect the immediate connection between force ?P ? of the space-time uncertainty ranges ?x , inferred in agreement with the deformation rate ?x Newton law simply deriving the uncertainty equations 4,1a with respect to time under the condition of their conceptual equivalence with eqs 4,1b; in effect the Newton law is an approximation for 0 legitimated by the weak dependence of po upon ?x . In turn, the quantum origin of the po ≈ po gravity force supported the idea that also relevant relativistic effects should be explained in the theoretical frame previously outlined; in fact red shift and light bending were immediately acknowledged in subsection 4.1, effects (a) and (b), even without the specific discussion carried out in the following subsections. This surprisingly simple evidence suggests that the deformation rate of phase space ranges effectively includes also the space-time local deformation itself. For instance, the subsection 4.5 has shown that the curvature of the space-time uncertainty ranges ?x is simply a ? . There is a conceptual analogy particular case of the more general concept of deformation rate ?x between the ways to find quantum and relativistic results. In wave mechanics the eigenvalues are inferred through the wave function; yet, even without solving any wave equation, the real observables verifiable by the experiment are found in section 2 simply introducing the concept of total uncertainty into the pertinent classical problem and counting the number of allowed states. The same reasoning holds also for the relativistic results of section 4: instead of solving Einstein’s field equations through functions of generalized coordinates in curvilinear reference systems, which 35

would require the formalism of tensor calculus, the present approach introduces since the beginning the quantum uncertainty into the classical formulation of the various Newtonian problems of section 4, thus finding in effect the same results of general realtivity verifiable by the experiment. Consider again the beam light deflection in the gravitational field: the general relativity describes in detail the bending effect through the local space-time curvature, whereas the experimental observation concerns merely the overall deflection of star light in the field of sun. This final outcome, which is in fact the result experimentally available, is however just that calculated in eq 4,18 even without concerning the local bending dynamics of the light path. The only necessary assumption is that the uncertainty be a basic principle of our universe, even more fundamental than the geometry of the curved space time itself. For this reason the formulae obtained in the various cases of section 4 are significant although corresponding to the respective approximate solutions of Einstein’s field equations. Even concerning weak fields, the physical meaning of the present results is essentially euristic; once having shown that the gravity is rooted into the concept of uncertainty, is legitimate the idea that higher order terms could be also found to further improve the present results within the same conceptual frame. The main task in this respect is to demonstrate that the present theoretical model includes correctly the relativity without hypotheses ad hoc, but simply introducing ideas and formalism of quantum mechanics into gravitational problems. Are indeed remarkable to this purpose the initial considerations of section 3 about the preminent importance of the numbers of allowed quantum states: the essential physical idea is to calculate appropriately these numbers through simple algebraic manipulations of classical equations thanks to eqs 1,2 and 2,5. Just these results, obtained introducing first the classical formalism and then conveying this latter into the quantum world through the concept of particle delocalization inherent the positions 1,1, confirm once again that only the ranges of dynamical variables have essential physical meaning, not the random values of position and momentum, time and energy. This primary strategy showed that effectively the quantum mechanics is the “added value” to the mere Newtonian gravity enabling the most relevant outcomes of general relativity to be found. The subsection 4.6 has shown that the behaviour of an orbiting quantum particle in the gravity field of another particle is similar to that of an orbiting planet. The Kepler problem and the other cases quoted in section 4 (light deflection, red shift and time dilation as well) suggest that simply a scale factor discriminates quantum problems and cosmological problems, but the basic behaviour of light and matter subjected to a gravity field is substantially the same since the fields are with good approximation additive. This conclusion is actually possible because the gravity is deeply rooted into the quantum concept of uncertainty, so that in effect there is no reason to expect a different behaviour for single particles or aggregates of an arbitrary number of particles. Moreover, there is no evidence in the present model that the energy of quantum particles affects the gravitational behaviour; in other words, no approximation has been introduced that could suggest a different behaviour of high energy or low energy particles in the gravity field, thus limiting accordingly the conclusions of section 4. For these reasons the subsection 4.1 is the most important one of the present paper. The reasoning holds also for the gravitational waves, which could be evidenced even in an elementary particle experiment, provided that in a beam are allowed to form local orbiting systems emitting gravitational waves that wiggle the path of the neighbouring particles: this is a prediction of the present model according to eqs 4,37. The general relativity is therefore nothing else but the quantum aspect of the Newton mechanics: for this reason the positions 1,1 and eqs 1,2 and 2,5 are enough to infer the former from the latter. The arbitrariness of the ranges is not mere restriction of information but source itself of information; it is indeed remarkable the fact that none of the concepts typical of general relativity has been postulated in the present approach, e.g. invariance of light speed and the equivalence of inertial and gravitational mass. Another relevant example in this respect is that the inertia principle is a corollary of eq 3,2. This principle, inferred for the gravity force in section 4 through the ? i , holds in general for any kind of force. Any interaction requires by definition a equation Fi / mi = v finite distance between particles, otherwise it should be either non-vanishing even at infinity or its 36

propagation rate should be infinite. In fact an ideal free particle is a lonely body in an infinite uncertainty range. Yet the condition ? xt → ∞ , which excludes in principle any interaction with another body of matter in the same range, entails ?pxt → 0 at any time. Thus, regardless of the current value of local pxt , the vanishingly small momentum range allowed for an ideal free particle compels the inertia principle v x = const . Hence, owing to the essential concept of delocalization, basic hypotheses of relativity like the inertia principle or equivalence of inertial and gravitational mass are actually corollaries in the present theoretical frame. In fact, the curvature of the space-time uncertainty ranges has been introduced as a natural consequence of eq 3,2, even discarding the local coordinates as a function of which the tensor formalism defines the space-time curvature. Also the known properties of the gravity force, e.g. conservation of momentum and angular momentum shown first in subsection 2.1 and inferred again in subsection 4.2, are straightforward consequences of quantum uncertainty, being obtained without integrating any equation of motion based on a postulated interaction law. A heuristic aspect of the present conceptual frame concerns v of eq 3,2. Suppose that v has a finite number of components v j along specified directions of the space with respect to an arbitrary reference system; if so, being v related to a corresponding momentum Pv , are by consequence defined for each v j local momentum components Pvj . These latter are in turn defined withing ranges ?Pj , clearly conjugate to the scalars x j ? xoj of eq 3,2. From a physical point of view, therefore, if a finite number jmax would exist such that really 1 ≤ j ≤ jmax , this reasoning would introduce in fact a new jmax -dimensional space; each dimension identified by its own j would be in fact legitimated in the present theoretical frame by an uncertainty equation conceptually identical to eqs 1,2 and related 2,5. Certainly this conclusion is an open point of the present model as concerns the physical consequences of the further dimensions hidden by the overall uncertainty of eq 3,2, which seems however enough to explain itself a wide amount of experimental evidences as a function of a unique fundamental assumption. Clearly any reasoning about v would require a valid physical reason, for instance the space-time anisotropy, to justify the peculiar components v j . In principle nothing hinders thinking so as concerns the consistency of the results of section 4: according to the previous reasoning such an anisotropy concerns ? xt only, and not space and time ranges separately; moreover, even the uncertainty anisotropy should not affect c for the reasons sketched in section 3 and could be evidenced in a relativistic experiment distinctive of the space-time properties of uncertainty. Yet does not seem really legitimate to introduce anything without experimental evidences and without compelling necessity to clarify unexplained effects; the fact that nothing in principle hinders the existence of these extra dimensions, which would support the string theory, is not enough however to justify speculations on their effective physical reality. Finally, note that the present physical model does not exclude the infinities; time, energy and dynamical variables could take in principle infinite values because the respective uncertainty ranges defining them are completely arbitrary. Yet these infinities coexist with well defined results, the eigenvalues, which are in effect calculated just postulating the random and unpredictable nature of the local dynamical variables on the one side and the arbitrary character of the respective uncertainty ranges on the other side. Is challenging the idea that the finite size and, presumably, time length of our universe are explained from the microscopic quantum scale to the macroscopic relativistic scale by physical variables conceptually described by indefinable ranges. May be, the key of this intriguing paradox stems on the fact that the nature admits in principle infinite ranges allowed to its physical parameters, but in practice does not need them. As a first example, it has been emphasized in section 4.3 that the gravitational interaction between light and matter removes the infinite values of frequency shift ?ω and ?P2 of the photon by merging eqs 4,10 and 4,11 into eq 4,12; in this case is the interaction the way to eliminate the infinities. The 37

examples of section 2 have evidenced another aspect of this problem: among the range sizes in principle possible, particular values exist that fulfil some appropriate selection condition not excluding or contradicting the total randomness of eqs 1,2 and 2,5. The harmonic oscillator and hydrogenlike atoms reveal propensity of nature to fulfil the condition of minimum energy; with this preferential condition, which is proven effective in general even though not explicitly required by any fundamental physical law, the eigenvalues of eqs 2,2 and 2,4 are definite even being in principle consequence of total arbitrariness of the uncertainty ranges. Is the condition of minimum energy the other way to pass over the infinities in principle possible? In any case, despite the unambiguous agreement with the experimental results, it is also true that the outcome of present theoretical model is less compelling than that of the wave formalism. Once writing the quantum Hamiltonian of the harmonic oscillator, the solution gives uniquely eq 2,4 without any other chance. The same equation obtained in section 2 is instead compatible with other results, easily calculated, simply admitting that the minimum condition is not fulfilled. The most intuitive physical concept evoked by this conclusion is that of non-equilibrium state of matter; for instance each oscillator of a 4 2 ?1 solid body could be described by ε ne = (α ne + 1)(2α ne ) n?ω + α 2 ?ω / 2 through the non-equilibrium parameter α ne ≠ 1 . This idea is actually more general and concerns even more fundamental laws of nature. Recall the way to infer the invariancy rule of interval and the Lorentz transformations in ′ ? xo ′ + δ X ′ reciprocally sliding at section 3 through the ranges c?t = xs ? xo + δ X and c?t ′ = xs

′ ? xo ) / ?t ; being possible to put ( xs ? xo ) 2 = ( xs ′ ? xo ′ ) 2 by definition, one finds constant rate V = ( xo
then c 2 ?t 2 ? ?X 2 = c 2 ?t ′2 ? ?X ′2 . This result was considered conclusive, being in agreement with the special relativity and thus with the experience. However there is no reason to exclude in this ′ ? xo ′ ) 4 + ?X ′4 ; if so, respect even further positions like c 4 ?t 4 = ( xs ? xo ) 4 + ?X 4 and c 4 ?t ′4 = ( xs
′ ? xo ′ ) 4 would be now consistent with the invariant δ s (4) = 4 c 4 ?t 4 ? ?X 4 , again with ( xs ? xo ) 4 = ( xs ?t ′ ≠ ?t and δ X ′ ≠ δ X . Of course such an invariant interval has no physical interest, at least as far as we know, although it introduces the factor (1 ? v4 / c 4 )?1/ 2 still in agreement with with the Galilean limit for v << c and with the consequences expected for v → c ; yet the reasoning shows that the invariant squared interval describing correctly the reality is not the only one conceptually definible, rather it is the simplest one and the right one among the many in principle possible. As in the case of the minimum energy, also now the actual physical laws appear to be the result of a preferential choice of nature, which however does not exclude other possible laws. The conclusion is that a subtle wire links apparently different concepts like steady eigenvalues, non equilibrium states of matter, interactions, weirdness of quantum world; actually, the overlaying concepts are the uncertainty and its necessary infinities that ensure maximum randomness and minimum local information. A possible hint to solve the paradox of infinities conceptually allowed in a finite universe comes also from the indication that the physical laws are the result of two opposite instances: eq 1,1 introduces first the total arbitrariness into the description of a physical system through the total delocalization of its constituent particles, as it reasonable for a quantum approach; yet fine tuning of this arbitrariness on particular values, leading to the eigenvalues, is provided by eqs 1,2 and 2,5, for which clearly holds the correspondence principle. This circumstance, which connects the quantum results with the macroscopic world (see e.g. momentum and angular momentum conservation, inertia principle, equivalence of gravitational and inertial mass, the examples of section 4 and the value of G ) has been highlighted in the case of hydrogenlike atoms: the available physical information, e.g. the energy and angular momentum eigenvalues, is the same regardless of considering the electron probability density or the total ignorance about its position and momentum. If so, even an infinite uncertainty does not prevent the existence of observables, since any particle free or bound is actually a wave moving from minus infinity to infinity; it appears in appendix A, showing the link between the probability Π x defined in any range x2 ? x1 with x2

38

and x1 completely arbitrary and not subjected to any hypothesis, and the complex wave function

ψ ≡ Π xt exp(i? ) from it inferred. Yet these indefinable boundaries are actually non elusive because the reference systems are arbitrary themselves, so that an infinite coordinate could be regarded likewise any other finite coordinate with respect to a proper reference system at infinity itself. For this reason the infinity is not a failure of the model but a possible chance for a quantum particle likewise any other allowed coordinate. Is then the total uncertainty a concept really more agnostic than the probabilistic knowledge provided by the wave mechanics? A possible answer could be that the question is badly posed and physically meaningless, because the uncertainty is defined itself with respect to an arbitrary space-time reference system. Yet, a better answer is probably that the history of our universe does not depend on its own physical limits. Would then an endless universe be the same as the actual one we are trying to describe? Would then an ever lasting universe evolve as the actual one we are trying to foresee?

REFERENCES 1 S. Tosto, Il Nuovo Cimento B, vol. 111, n.2, (1996), pp. 193-215 2 S. Tosto, Il Nuovo Cimento D, vol. 18, n.12, (1996), pp. 1363-1394 3 S. Tosto, “Recent Research Developments in Physics”, S. Pandalay Ed., 3, (2002), pp. 487-520 4 S. Tosto, “Recent Research Developments in Physics”, S. Pandalay Ed., 5, (2004), pp. 1363-1422 5 S. Tosto, “An analysis of states in the phase space: the specific heat of metals”, in press on “Recent Research Developments in Physics”, S. Pandalay Ed.

39

APPENDIX A Consider the uncertainty equation 1,2 for a free particle; let x1 , p1 and x2 , p2 be two arbitrary couples of dynamical variables ?p?x = n? ?x = x2 ? x1 ?p = p2 ? p1 A1 No information is allowed about current position and momentum of the particle in the respective ranges of the phase space. Yet the total uncertainty does not prevent in principle to define the probability Π x that the particle be in an arbitrary subrange δx inside the total allowed range ?x δx δx = x ? xo = Πx δx ≤ ?x A2 ?x xo and x are arbitrary coordinates within ?x unknown “in principle”, in the same way as x1 and

x2 themselves, i.e. there is no physical criterion to define the width of δx or its location inside ?x ; also, there is no possibility to distinguish δx with respect to any other possible subrange. Also, no hypothesis is necessary about the ranges so far introduced. If the width of δx is variable, regarding x as current coordinate while considering constant xo eqs A2 give ?Π x 1 Π x = Π x ( x) = A3 ?x ?x Let us introduce the probability Π x into eq A1 considering both possibilities that the particle be or not within δ x . To this purpose, the couples of coordinates x1 , x2 and momenta p1 , p2 are considered fixed. Also, let n+ and n? be two arbitrary numbers consistent with the respective probabilities Π x
and 1 ? Π x . Putting then

(?x ? δ x)?p = n? ? n+ + n? = n it appears that effectively n+ / n + n? / n = 1 ; moreover eq A4 gives the identity
2 2 2

δ x?p = n+ ?

A4

? ?Π x ? (1 ? Π x )Π x ?p = n? n+ ? ? A5 ? ? ?x ? With the position n′ + n′′ = n+ n? , where n ′ and n ′′ are further arbitrary numbers, eq A5 splits as ? ?Π x ? Π x ?p = n′? 2 ? ? ? ?x ?
2 2 x 2 2 2

A6a

? ?Π x ? Π ?p = ?n′′? ? A6b ? ? ?x ? Being n+ and n? by definition positive, at least one among n ′ and n ′′ or both must be necessarily positive. Eqs A6 are now discussed considering the following cases consistent with the possible signs of n ′ and n ′′ . (i) n′ > 0 and n′′ < 0 . Eq A6a and b read also δ x?p = (n′ / n)? and δ x 2 ?p 2 = n ′′ ? 2 respectively
thanks to eqs A2 and A3: hence (n′ / n) 2 = n′′ and Π x = n′′ / n′ . Both results are possible for any n because n ′ and n ′′ are arbitrary. Eq A6a is conceptually analogous to the initial eq A1, from which it differs only formally because of the widths of the uncertainty ranges: multiplying both sides by n§ n / n ′ , with n§ arbitrary integer, one finds ?x§ ?p§ = n§ ? , where ?x§ and ?p§ are any ranges related to the initial ones δ x and ?p through the condition ?x§ ?p§ = δ x?p(n§ n / n′) . It is clear that the physical interest of eq A6a rests on the integer value of n§ , regardless of the particular sizes of the uncertainty ranges appearing in its mathematical form; if so, the physical meaning of ?x§?p§ is the same as that of eq A1. Of course the same holds for eq A6b, identical to eq A6a with the given 40

2

choice of n′′ . In conclusion, nothing conceptually new with respect to eq A1 is inferred from this combination of signs of n ′ and n ′′ . (ii) n′ < 0 and n′′ > 0 . The right hand sides of both eqs A6 have now the minus sign, so that neither of them can have the same physical meaning of the initial eq A1. Since Π x must be real and positive by definition, it should be true that ?Π x / ?x = if , where f = f ( x) is a proper function; this
2 position gives Π x = af 2 and Π 2 x = bf , with a and b positive constants resulting from eqs A6, and

then Π x = b / a = n′′ / n′ . Yet this result is inconsistent with the definition of ?Π x / ?x just introduced. Also this combination of signs is of no physical interest. (iii) n′ > 0 and n′′ > 0 . Now eqs A6a and b are physically different because of their signs: their ratio would give Π x negative. Then it is reasonable to regard these equations as two independent ways to describe the free particle within the uncertainty range ?x . Let us consider therefore these equations separately. Eq A6a is conceptually analogous to eq A1, as already shown; the solution of eq A6b is Π x = const exp(±i? ) being ? = x?p ? n′′ = xn /(?x n′′ ) . Yet Π x is not real; to

(

)

overcome this difficulty let us introduce the complex function
2

Π x in place of Π x and rewrite eq

A6b as a function of the former instead of the latter: dividing both sides by Π x , eq A6b reads

? ? Πx ? 2 ?p A7 p§ = ± ? ±? ? = ? p§ Π x ? ? ? x ′′ 2 n ? ? § The notation p is due to the fact that n ′′ and ?p are in principle arbitrary, so that p§ is an arbitrary parameter not longer coinciding with the momentum range ?p and having the same

(

)

physical dimension of this latter. Clearly p§ is defined by the differential equation A7 that reads

? ? Πx = p§ Π x A8 i ?x whereas the probability to find the particle somewhere in its uncertainty range is defined by
Π x = Π x Π* x ; if so, putting
Π x = (δ x / ?x ) exp(i? ) one finds again

Π x Π* x = δ x / ?x and A9

p§ =

* ? Π x ? Π x / ?x i Π* Πx x

The agreement of eq A8 with the initial eq A1 is trivially evident despite their apparently different mathematical formulation; repeating backwards the same steps just shown, eq A8 leads to eq A6b, originated together with eq A6a from the unique uncertainty eq A1. In conclusion, as a consequence of having introduced the probability Π x into eq A1 one finds two distinct equations, inferred from the respective eqs A6 ?x§ ?p§ = n§ ? A10a

? ? Πx = p§ Π x A10b i ?x The signs of p§ correspond to the components of momentum along the direction where are defined δ x and ?x . It follows that the free quantum particle is described by: (i) eq A10a, which differs trivially from the initial eq A1 merely because of the width of the uncertainty ranges and related number of states; (ii) a differential equation defining the momentum through the probability that the particle be in a given point of its allowed range ?x§ .
41

The point of view of eq A10a does not consider explicitly the particle, but only its random location somewhere inside ?x§ ; the same holds also for the momentum, which does not appear explicitly because it is replaced by its uncertainty range too. The only information available through this equation concerns therefore the number of states n§ consistent with the delocalisation ranges ?x§ and ?p§ ; nothing can be inferred about the dynamical variables themselves. However, the papers [1,2] show that even renouncing “ab initio” to any information about these latter, the quantum properties of the particle are correctly described, as it is shortly reported also in section 2. The point of view of eq A10b is different. This equation considers explicitly the subrange δx through Π x and, even without hypothesizing anything about its size and position within ?x§ , concerns directly the particle itself through its properties Π x and p§ , both explicitly calculated solving the differential equation. Yet the common derivation of both eqs A10 from the initial eq A1 shows that actually the respective ways to describe the particle must be consistent and conceptually equivalent: this fact justifies why the same results are expected through both points of view. Since no hypothesis has been made on the physical nature of the particle, this conclusion has general validity. It is known that eq A10b introduces the operator formalism of wave mechanics, whose major result is that both energy and angular momentum depend on quantum numbers. In effect also the approach based on eq A10a gives identical results, although any reference to the dynamical variables is missing in principle; moreover the physical meaning of quantum number turns into that of number of allowed states. This coincidence evidences the conceptual link between properties of the particles and phase space; it explains why the quantum energy levels and angular momentum of the particles do not depend on the current values of the dynamical variables even when calculated through the operator equation A10b defining p§ and Π x . This latter has been initially introduced in eq A2 as mere function of uncertainty ranges of the phase space; thereafter, however, it has also taken through the steps from eqs A3 to A10 the physical meaning of wave function of the particle defining the momentum. Eq A10a considers uniquely the phase space, whereas eq A10b concerns explicitly also the particle; if for instance in ?x there are two identical particles, then their own subranges δ x1 and δ x2 define the respective Π x1 and Π x 2 . Hence eq A10b rises the problem of knowing which particle is in either subrange, whereas eq A10a skips such an information since it concerns only the total range ?x regardless of the local coordinates and momenta of both particles, unknown and then ignored. So the concept of indistinguishability is basically inherent eq A10a, being conceptually impossible to identify particles whose properties are in fact unspecified; eq A10b requires instead a specific rule, to be introduced “ad hoc”, to arrive to the same conclusion. In other words, eq A10a entails the corollary of indistinguishability of identical particles; the operator formalism of eq A10b needs introducing this requirement as additional postulate to ensure the same physical information of eq A10a. Without paying attention to the steps from eq A1 to eqs A9 the indistinguishability must be necessarily postulated; this fact has an important consequence: the approach starting directly from eq 1,2 has more general character than that utilizing the operator formalism of wave mechanics. This is because eq A10a contains less information than eq A10b.

42

APPENDIX B The steps to find the energy operator are conceptually identical to those reported in appendix A, yet now the probability that the particle be in the range δ x is regarded as a function of time; Π x can be also defined as ratio between the time range δ t = t ? to spent by the particle within δ x and the total time range ?t = t2 ? t1 spent within the total range ?x . Then, the probability Π x introduced in appendix A as a function of x is actually a function Π xt of space and time coordinates. With a more correct notation that does not contradict anyone of the steps of appendix A, Π xt = δ t / ?t ; eqs A3 and A5 read ?t ?1 = ?Π xt / ?t and (1 ? Π xt )Π xt ?ε 2 = n? n+ ? 2 ( ?Π xt / ?t ) . Replacing position and momentum with time and energy in eq A1 and following an identical reasoning, eqs A7 and A9 give now
2

? ? Π xt ? ? ±? ? = ? ε § Π xt ? ? ?t ? ?

2

(

)

2

ε§ = ?

?ε 2 n′′

B1

whereas eq A8 reads now

? ? Π xt = ε § Π xt B2 i ?t The notation of signs of ε § has been chosen in order to agree with the sign of eq B2, which in turn makes the solution of the quantum Hamiltonian written with the help of eq A8 consistent with the classical result ε § = p§2 / 2m in the particular case of a free particle having mass m and momentum ?

p§ . The couple of equations A10 becomes therefore ?t §?ε § = n§? B3a ? ? Π xt ? = ε § Π xt B3b i ?t The comparison between eqs A10 and B3 is interesting because it shows the strict analogy between time and space coordinates in defining the wave function ψ ≡ Π xt exp(i? ) . Note however that eq
B1 does not exclude in principle the possibility of handling the signs of ε § likewise as that of eqs A7 and A8, in which case also states of negative energy would appear. Yet such a solution of the Hamiltonian has been discarded, because not consistent with the classical physics unless hypothesizing a negative mass. These conclusions have here mere formal valence; yet their actual physical meaning will appear in the sections 3 and 4 dedicated to the special and general relativity.

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