1 Introduction

It is hardly possible to imagine more fundamental concepts than time, space and numbers. The description of space and time is formalized by introducing a reference frame, which proved to be a basic issue in the development of physics. A rather foundational question may be posed regarding the most primitive object—a free nonzero-rest-mass (let us suppose fundamental) particle—can its time and space be considered as existing per se or only in relation to some outer, externally defined reference frame? In other words, is it possible to introduce for a free, completely isolated entity—a particle—its proper spacetime by a system of spatial and temporal coordinates, which are defined by nothing else than the particle’s intrinsic property—its rest mass \(m_0\), as well as by the constants of nature? This would add to the understanding of such a basic constituent of quantum mechanics (QM) as a free particle’s wave function. The obvious challenging issue is that not only a particle’s temporal, but also its spatial coordinate, necessarily included in wave-like behaviour, has to be defined as proper, that is, without relation to any external reference frame, as it is considered in the special theory of relativity (STR).

Let us compare the starting points of description of a particle (an ‘object’) in STR and QM. The presentation of the STR in [1] begins by stating that, in order to describe an object, one needs a system of reference, by which ‘... we understand a system of coordinates serving to indicate the position of a particle in space, as well as clocks fixed in this system serving to indicate the time’. Further, “... at each moment of time we can introduce a coordinate system rigidly linked to the moving clocks” and the time “... read by a clock moving with a given object is called the proper time for this object”. It is, however, obvious that any ‘rigid linking’ necessarily involves interaction, which evidently contradicts the condition of remaining a free particle; clearly, the same contradiction relates to the spatial coordinate as well. One of the starting statements of QM is that an object is described by its wave function. Comparing the wording of STR and QM may incline one to suggest that it is just the pronoun “its” making the difference. Indeed, while the STR introduces an object (or its free motion) as essentially related to an external property, such as the transformation at changing systems of coordinates, QM should introduce the wave function as a particle’s proper feature. This however is not exactly the case, since the arguments of the wave function are likely supposed to be related to an external reference frame. What is more, external observation, including an observer, is necessary to ensure with certainty a particular result of measurements. From another point of view, it is worth recalling a statement by Dirac [2], that the particle interferes only with itself, and this should mean, within its proper space (or spacetime). An attempt to address this controversy makes the core of the presented analysis.

As follows from the above argumentation, the usage of formalism developed in the STR and QM should be invariably avoided in a consistent definition of a free, non-zero-rest-mass \(m_0\) particle’s proper spacetime. The approach suggested here is to base the definition on a more primitive level, namely, on the de Broglie frequency [3], which is uniquely determined by a particle’s rest mass \(m_0\) and constants of nature combined as \(c^2/h\). As mentioned in [4], this defines, for a free, nonzero-rest-mass particle an “ideal” proper time scale as a sequence of “marks” along the line of proper time, provided by a natural standard time unit—the de Broglie time period. The idea to create a clock that relies on a fundamental link between time and a particle’s mass was recently realized in [5]. Association of proper time defined by the de Broglie frequency with a sequence of natural numbers was suggested in [6]; we will refer to the proper time so defined as de Broglie time. Indeed, since evolving a particle’s proper time as an uninterrupted ordered sequence of “ideal” (with the precision of the constants of nature) de Broglie time units involves their uninterrupted counting, or numbering, such an “ideal sequence” may be considered isomorphic to a growing open sequence of natural numbers 1, 2, ... , (some remarks on the long-lasting discussion of the connection between time and numbers may be found in [6] and references therein).

The question remains: how to introduce a free particle’s space as proper, that is, without any relation to an external reference frame? The bottom line of the present approach lies in assuming that it is possible to define positive and negative space directions of proper space following the construction of all possible positive and negative integers as all possible differences of natural numbers belonging to the sequence 1, 2, ..., n. Clearly, this requires passing from the de Broglie time unit \(h/m_0c^2\) to a Compton wavelength \(h/m_0c\) as a distance unit. Intervals of proper space scaled by their common interval of de Broglie proper time are referred to as proper velocities, which are expressed by a rational fraction of c without any relation to an external reference frame. It is worth mentioning that such a definition of velocity may remind one of a consequence of Newton’s first law, according to which velocity is something that persists without cause, which means without any externally defined factor.

The paper is structured as follows. In Sect. 2 the flow of de Broglie time and the interval of de Broglie time are introduced as connected to the sequence of natural numbers. Section 3 includes the definition of the set of intervals of a free particle’s proper space along with the respective set of proper velocities; the geometrical properties of intervals of proper space are discussed. In Sect. 4 the temporal and spatial coordinates of particle’s spacetime are introduced as the respective orthogonal projections of de Broglie time; a common dimensionless unit of proper space and proper time is introduced to make the correspondence to numbers complete. The related expression in the form of an energy-momentum relation is analysed, revealing the appearance, aside from the rest energy term \(m_0c^2\), of an additional term of the same order of magnitude, which is related to large intervals of the \(m_0\)-particle’s proper space. In Sect. 5 the wave- and particle-related properties are treated within the numbers-based approach. The possibility to relate the present approach to two indistinguishable particles is suggested. Section 6 contains the remarks regarding the meaning of the obtained results; in particular, it is discussed to what extent the numbers-based approach to a free particle’s spacetime may provide a common initial origination point for foundations of the STR and QM.

2 De Broglie Proper Time

2.1 Basic Assumptions

Let us formulate the requirements, which would allow one to define the time course and an interval of proper time of a free, completely isolated, nonzero-rest-mass particle. Inevitably, this situation would require that the particle is able to function as a “proper clock” by itself, without any externally imposed influence. This would require [6]:

  • Existence of a ’natural’ unit of proper time \(T_0\) that is defined exclusively by an intrinsic characteristic of a particle—its rest mass \(m_0\), as well as by the constants of nature;

  • An uninterrupted numbering (counting) routine of \(T_0\)-units that is neither imposed nor influenced by any external factor, which is defined by (isomorphic to) a sequence of natural numbers 1,2, ... .

An interval of proper time has to be defined by two moments of time. The question appears: is a non-contradictory definition of an interval of proper time for a particle, which is considered free of any external influence, at all possible? The suggested approach is based on assuming that such uniquely defined moments of time are:

  • The particle’s present moment \({\text{P}}_{1}\);

  • The moment of the particle’s temporal beginning or start \(\text {S}\), which precedes \({\text{P}}_1\), thus introducing the notion of the past, and therefore, of duration.

  • The two moments of time define the interval of de Broglie proper time \({\text{P}}_1{\text{S}}\). Identifying any time’s moment other than \({\text{P}}_1\) and \(\text {S}\) would contradict the notion of the in principle undisturbed (uninfluenced) existence of the particle.

Let us analyse how the suggested requirements are related to basic concepts of physics and arithmetic.

2.2 A Natural Unit of Proper Time

For each particle with rest mass \(m_0\), a natural unit of proper time \(T_0\) is uniquely provided by de Broglie’s periodic phenomenon. The latter postulates the existence of a proper frequency, or de Broglie frequency \(\nu _0\) [3] defined by the particle’s rest mass \(m_0\) via a proportionality factor comprised of constants of nature:

$$T_0 = \nu _0^{-1} = (h/c^2)(m_0^{-1}).$$
(1)

The role of (1) in defining the proper time of a free \(m_0\)-particle is discussed in [4] and [6]; a somewhat related remark about preceding the notion of time by the notion of a periodic process can be found in Einstein’s paper [7].

The accuracy of the time unit \(T_0\) is determined by the accuracy of the particle’s rest mass \(m_0\), as well as by the accuracy of the respective fundamental constants of nature. As such, the latter define (or, equivalently, are defined by) the fundamental properties of spacetime. Suppose that there is no particle whose rest mass \(m_0\) is determined with the same accuracy as the constants of nature. This would mean that no unit of proper time, i.e., no time could be determined with the accuracy of the constants of nature, and, consequently, the constants of nature would not be functioning properly as such. This argument suggests the existence of a particle with rest mass \(m_0\) whose accuracy matches the accuracy of a constant of nature, which implies that it (the respective \(m_0\)-particle) is in this sense a fundamental particle. And indeed, the rest masses of fundamental particles are in the list of the constants of nature.

2.3 De Broglie Proper Time and Connection to Natural Numbers

The necessity of counting time units presupposes a connection with numbers. It is worth calling to mind Dedekind’s [8] consideration that (emphasis added) “... the whole arithmetic is a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already formed individual to the consecutive new one to be formed”.

Let us follow how the ‘simplest arithmetic act’ is manifested by de Broglie’s periodic phenomenon (1). Accounting for the necessity of particle’s temporal beginning (start), the periodicity postulated by (1) may equally mean [6] a reproduction of the \(T_0\)-unit with de Broglie frequency \(\nu _0\). Then, the uninterrupted, on-going proper time of a free particle is started by one \(T_0\)-unit at the start \(\text {S}\) and continued by the ‘simplest arithmetic act’ of consequent, continuous adding of \(T_0\)-units, which is equivalent to the generation of a sequence \(T_0\), \(T_0T_0\), ... , \(iT_0\), ... . Supposing the strict identity of the \(T_0\)-units, the sequence is isomorphic to the sequence of natural numbers (positive integers) 1, 2, ... , defined in accordance with ordinal arithmetic developed by Peano. In particular, this means that every natural number i has a natural number successor SUC(i), and SUC(i) is \(i +1\). The time, which is defined by the number of counts of de Broglie units \(T_0\) determines a particle’s de Broglie proper time.

Fig. 1
figure 1

De Broglie time \(t_n\)

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It will be supposed that reproduction of a \(T_0\)-unit corresponds to the present instant \(\text{P}={\text{P}}_1\), which remains corresponding to one, \({\text{P}}_1 \leftrightarrow 1\) (this may as well be considered as a ‘numbers-based’ definition of the present—more discussion is postponed to Sect. 5.2). Then, at each reproduction of the present \(T_0\)-unit, the ‘previous present’ \(T_0\)-unit is added to the past. This brings the idea of uninterrupted, ongoing proper time as a continuous shifting of one (present) \(T_0\)-unit to form what is defined as the ‘past’. Since at the moment of temporal beginning, the present \({\text{P}}_1\) coincides with the start \(\text {S}\), the start can be labelled by one: \(\text {S} = {\text{S}}_1\) (see Fig. 1). The next act of reproduction means that the start does not correspond to the present anymore, but is shifted by one \(T_0\)-unit to the past. Since the present and the start are the only identifiable instants of time, the \({\text{P}}_1 \leftrightarrow 1\) condition implies that the shifting renumbers the start, \({\text{S}}_1 \rightarrow {\text{S}}_2\). Further, each consecutive act of reproduction of a \(T_0\)-unit at the present uniquely corresponds to a consecutive renumbering of starts as \(\text {S}_{i-1} \rightarrow \text {S}_i\). Therefore, the sequence \({\text{S}}_1\), \({\text{S}}_2\), ... , uniquely corresponds to the sequence \(1T_0\), \(2T_0\), ... , isomorphic to the sequence of natural numbers 1, 2, ... . Thus, we arrive at the idea of considering the particle’s ongoing proper time as its ageing time. This ascribes the following meaning to particle’s ‘ageing’: a consecutive reproduction of \(T_0\)-units at the present instant is equivalent to the respective shifting of the particle’s temporal beginning to the past. The number n of consecutive reproductions of a \(T_0\)-unit uniquely defines an interval of the particle’s de Broglie proper time \({\text{P}}_1 \text {S}_n=t_n\) (see Fig. 1):

$${\text{P}}_1 {\text{S}}_{n} \equiv t_n = nT_0.$$
(2)

It should be noted that the de Broglie proper time has no relation to the proper time defined in the STR, which refers to an externally defined reference frame. The continuity of a free particle’s existence is equivalent to its continuous ‘ageing’ as a consecutive increasing of its interval of proper time as \({\text{P}}_1{\text{S}}_n \rightarrow {\text{P}}_1{\text{S}}_{n+1}\), which itself is isomorphic to the ongoing sequence of natural numbers SUC(n) \(= n + 1\). Such an uninterrupted binding of the present, throughout the past, to the start denotes the free particle’s uninterrupted ‘history’.

3 Proper Space

3.1 Basic Assumptions

By intuition, it seems obvious to define particle’s spatial coordinates with respect to an external reference frame. However, as already discussed, any externally imposed determination would violate the rigid condition of a completely isolated entity. A possibility is suggested to avoid this inconsistency by assuming that an interval of the free particle’s proper space, or its ‘proper distance’Footnote 1, can be defined based on the relation of the particle’s proper time \(t_n\) to numbers. The particle’s proper distance thus introduced has to retain consistency with \(t_n\) defined by (2), which implies:

  • The existence of a natural unit of an \(m_0\)-particle’s proper distance that satisfies the same requirements as for the de Broglie unit of proper time \(T_0\) (1), namely, to be determined by nothing else but the particle’s rest mass and fundamental constants of nature;

  • For a given interval of the particle’s proper time \(t_n\)(2) that elapsed at the present moment \({\text{P}}_1\), the particle’s proper space has to be associated with the same present moment \({\text{P}}_1\) and the same sequence of natural numbers 1, 2, ... , n that defines \(t_n\);

  • It is possible to define (construct) the set of intervals of a free particle’s proper space by constructing the set of positive and negative integers from the set of natural numbers belonging to the \(t_n\)-defining sequence 1, 2, ... , n.

In what follows we will define the set of intervals of proper space along with related proper velocities based on a realization of these requirements.

3.2 Proper Space and Its Connection to the Integers

An obvious option for defining a natural unit of proper space consistent with the suggested requirements is offered by multiplying the de Broglie time unit \(T_0\) by c as the only fundamental constant of nature that possesses the distance over time dimension. Since the \(m_0\)-particle’s \(T_0\) (1) is defined by the coefficient that includes \(c^2/h\), this presupposes, first, separating \(c^2\) from h and then passing from \(c^2\) to c, the latter implying a \(\pm c\) choice of sign. As a result, the natural unit of proper space is

$$\sqrt{c^2}T_0 = \pm c T_0 = \pm \frac{ h}{m_0c},$$
(3)

its value being the Compton wavelength \(\lambda _C\). It seems reasonable to connect the opposite signs in (3) with opposite directions of a distance unit. With respect to numbers, this relates to the appearance of positive and negative integers. Then, a particular integer \(n_x\), positive or negative, multiplied by the distance unit value \(\lambda _C=cT_0\) may be associated with positive or negative directions of a particular interval of proper space, or proper distance \(x_n\) as

$$x_n = n_xcT_0=n_x \lambda _C.$$
(4)

The introduction of two opposite directions of proper space of a free particle, which is based exclusively on particle’s rest mass and numbers, consistently avoids any relation to an external reference frame, as distinct from the left-moving and right-moving solutions of wave propagation in the SRT, which relates to some externally defined coordinate system. The reason for the notations \(x_n\) and \(n_x\) is to relate them to a spatial coordinate, as well as to stress their relation to (origination from) the elapsed interval of the proper time \(t_n\) (see Fig. 2.) To define the proper space \(n_x\) has to take all values of the set of integers that can be formed from the \(t_n\)-defining sequence 1, 2, ... , n. According to a constructive definition (see, for instance, [9]) the set of integers is formed from a given set of natural numbers by ordered pairs (a, b), where \(a,b \in \{1, 2, \ldots , n;\, a \ne b\}\) as a result of subtracting b from a; this yields \(n_x \in \{-n+1, \ldots, -2, -1, 1, 2, \ldots , n-1\}\). The reason for choosing this constructive approach as the definition of the set of integers is its consistency with physical phenomena. Note that the value \(n_x = \pm n\) yielding \(x_n = \pm ncT_0\) does not appear. For \(n_x >0\), the number of all possible pairs (a, b) that yield a particular \(n_x\) is \(n-n_x\).

3.3 Proper Velocity

Since all intervals of proper space \(x_n\) originate from the same elapsed interval of proper time \(t_n\) (2), it is feasible to explicitly include the latter in (4). After multiplying and dividing by n the right-hand-side of (4) and accounting for (2), one arrives at the expression

$$x_n = \frac{n_x}{n}ct_n .$$
(5)

Since \(x_n\) is proportional to \(t_n\), it is appropriate to associate the proportionality coefficient with a definition of proper velocity (see Fig. 2):

$$v_{xn} = \frac{n_x}{n}c.$$
(6)

As can be seen, the proper velocity’s value is straightforwardly defined as a fraction of c; since \(x_n=\pm ncT_0\) does not appear (Sect. 3.2), c is the ultimate upper limit of \(v_{xn}\). According to the \(n_x\)-values that can be constructed for a given \(t_n\)-defining n-value in (2), the proper velocities vary from \(v_{xn} = \pm c/n\) when \(n_x = \pm 1\) to \(v_{xn} = \pm c(n - 1)/n\) when \(n_x = \pm (n - 1)\). As follows from (4)–(6), for n running from 1 to infinity, the normalized over c, or fractional proper velocity \(v_{xn}/c = n_x/n\) runs through all rational fractions \(n_x/n \in \{(-n+1)/n, \ldots, -2/n, -1/n, 1/n, 2/n, \ldots, (n-1)/n\}\). Therefore, there is a correspondence between the construction of integers and rational fractions from the natural numbers on the one hand, and the respective proper distances and proper velocities on the other hand.

It is obvious that the meaning of a free particle’s proper velocity thus introduced differs essentially from the conventional relative velocity referred to an external reference frame. In a particle’s proper spacetime, the proper velocity and proper distance are defined conjointly as related to the same interval of De Broglie proper time \(t_n\); some examples can be seen in Fig. 2. The proper velocity has nothing to do with the speed of covering a distance—it may rather be understood as the speed of ‘producing’ (appearing, forming) a respective proper distance

As one may notice, for fixed \(t_n\) the connection between a particle’s proper distance and its proper velocity \(v_{xn} = t_n^{-1}x_n\) has the form of Hubble’s law, with \(t_n^{-1}\) bearing the meaning of the ‘present Hubble constant’. This analogy does not seem accidental for a completely isolated entity that falls under the definition of an isolated universe possessing a temporal beginning. The proper time \(t_n\) (2) would then acquire the meaning of the ‘present age’ of the particle’s ‘universe’.

3.4 Geometrical Properties

As far as the geometrical properties of a free particle’s proper space are concerned, the following aspects will be addressed: (i) connection of the interval of a particle’s proper space to the segment of a straight line, and (ii) indications that proper space has to be three-dimensional. The segments of a straight line \(\text {L}\) are subject to the fundamental axiom (of the relationship between algebra and geometry), which predicates a one-to-one correspondence between the points of a segment and the real numbers \(\mathbb {R}\). In order to assure the continuity of the straight line, the latter contains infinitely many points that correspond to no rational number: ‘the straight line \(\text {L}\) is infinitely richer in point-individuals than the domain \(\mathbb {Q}\) of rational numbers in number-individuals’ [8], and thus a straight line possesses a correspondence to numbers if and only if the irrational numbers \(\mathbb {I}\) are included. But irrational numbers are not expressed by any ratio of integer numbers, the relation of which to proper time, proper distance, and proper velocity has been outlined in the preceding discussion. The question arises if there could be anything that points beyond \(\mathbb {Q}\) in the search for the correspondence of the interval of the free particle’s proper space to numbers.

Suppose that the interval of proper space \(x_n\) (4) corresponds to a segment of the straight line \(\text {L}\). Then, it has to be defined by the segment’s ends \({\text{P}}_1\) and \({\text{P}}_1^{\prime }\); we will denote \(x_n\) as \({\text{P}}_1 {\text{P}}_1 ^{\prime }\). Since \({\text{P}}_1\) and \({\text{P}}_1^{\prime }\) represent the same particle, they have to be in principle indistinguishable, therefore the \(x_n\) direction (sign) cannot change under the inversion of the segment’s ends \({\text{P}}_1 \leftrightarrow {\text{P}}_1^{\prime }\), which, however, does not hold if a directed \(x_n\) behaves as a polar vector. This means that \(x_n\), and therefore also the directed distance unit, has to behave as an axial vector that changes its direction at inversion. Such behaviour assumes a property related to rotation and therefore demands going beyond the one-dimensional space of a distance, i.e., demands involving two more spatial dimensions related to opposite rotations. This requirement implies that the particle’s proper space has to be three-dimensional.

The assumed rotation-related property suggests passing in (1) to a cyclic de Broglie frequency \(\omega _0\) and in (3) to a ‘reduced’ Compton wavelength as a distance unit, which is supposed to behave as an axial vector. Though these considerations will be kept in mind, in what follows the ‘non-cyclic’ units based on (1) and (3) will be used when it does not influence the analysis.

Let us return to the correspondence of an interval of particle’s proper space to a straight line, which assumes that irrational numbers have to be related to the entity’s proper spacetime, or its free motion. But if irrational numbers do not correspond to proper velocity, since the latter by definition corresponds to rational numbers, in what way might irrational numbers be connected to free motion? What remains is the aforementioned necessity of axial vector symmetry, which assumes a rotation-related property. Though the latter lacks any constructive definition, its geometrical description should be continuous, in the sense that there should be no ‘empty cuts’ in its correspondence to the points of a segment of a straight line. In other words, there may exist a correspondence of the state of free motion, including the rotational property, to all real (rational \(\mathbb {Q}\) and irrational \(\mathbb {I}\)) numbers, that is, a correspondence to all points of an interval of particle’s proper space to a segment of a straight line.

4 Proper Spacetime

4.1 Basic Assumptions

Introducing the interval of proper space \(x_n\) = \({\text{P}}_1 {\text{P}}_1^{\prime }\) means that the moment of a particle’s present existence (Sect. 2.1) must be related simultaneously both to \({\text{P}}_1\) and \({\text{P}}_{{\text{1}}}^{\prime}\). This can be regarded as ‘splitting’ the present \({\text{P}}_1\) into a ‘dual-present’, \({\text{P}}_1 \rightarrow {\text{P}}_{{\text{1}}} {\text{P}}_{{\text{1}}}^{\prime }\) (see Fig. 2), which means that along with de Broglie time \(t_n = {\text{P}}_1{\text{S}}_n\) (2), another interval of proper time \({t_{n^{\prime}}} = {\text{P}}_1^{\prime } {\text{S}}_{n}\) has to appear. Thus, the present \({\text{P}}_1\) is connected to the start \(\text {S}_n\) by two intervals of proper time \({\text{P}}_1{\text{S}}_n\) and \({\text{P}}_{{\text{1}}}^{\prime } {\text{S}}_n\). As both \(t_n\) and \({t_n^{\prime}}\) relate to the same present and the same start, they must be determined by the same sequence of numbers 1, 2, ..., n. Consequently, the only possible expression for \({t_n^{\prime}}\) is \({t_n^{\prime}}= {nT_0^{\prime}}\), in which the time unit \(T_0^{\prime}\) differs from the de Broglie time unit \(T_0\).

Fig. 2
figure 2

An example of how to resolve de Broglie time \(t_n\) (2), \(n = 5\), over spatial and temporal coordinates in dimensionless units \(^{DL}t_n = n\delta _0\), \(^{DL}x_n = n_x\delta _0\), \(^{DL}{t_n^{\prime}} = n\delta _0^{\prime}\), where \({\delta _0^{\prime}}= \delta _0(1-n_x^2/n^2)^{1/2}\) ; (a) \(n_x = 2\), proper velocity (6) \(v_{xn} = 0.4c\); (b) \(n_x = 4\), \(v_{xn} = 0.8c\)

Full size image

From a geometrical viewpoint, introducing proper distance means that a two-point present-past connection \({\text{P}}_1{\text{S}}_n\) is now replaced by a three-point present-past connection \({\text{P}}_1 {\text{P}}_1^{\prime } {\text{S}}_n\). (It is noteworthy that three is the only number of points that coincides with the number of all possible straight line segments connecting the points). It will be assumed that along with \(x_n\) (see Sect. 3.4), also \(t_n\) and \(t_n^{\prime}\) can be considered to correspond to the respective straight-line segments. Their expression via numbers would be fully consistent if all three segments are dimensionless. In what follows, after addressing the mentioned requirements, the connection between the squared values of \(x_n\), \(t_n\), and \(t_n^{\prime}\) is proposed, and their transformation to the energy-momentum form is analysed.

4.2 Dimensionless Unit of a Particle’s Spacetime

The possibility of defining a dimensionless unit, which is common for proper time and proper space appears to be essential since any dimension externally imposed on a free entity does not seem to be strictly consistent with its completely isolated existence. The question is how to reduce the respective units defined by (1) and (3) to one common dimensionless unit of space and time? To follow the ‘everything from a particle’s rest mass’ strategy exploited so far such a unit must be defined by nothing else than the particle’s rest mass \(m_0\) and the constants of nature. For this purpose, it is necessary to use a fundamental phenomenon, other than de Broglie’s, that connects \(m_0\) to a particle’s proper quantity that possesses a time or space dimension; then, accounting for a factor c, the respective ratios of (1) or (3) to that quantity would provide the sought-after dimensionless unit. A phenomenon, which is defined by constants of nature as fundamental as h and c, is gravitation, providing the expression for an \(m_0\)-particle’s gravitational radius \(R_0\) (see, for instance, [1])

$$R_0 = \frac{2Gm_0}{c^2} ,$$
(7)

which includes the gravitational constant G. By dividing \(\lambda _C\) (3) by \(R_0\) and \(T_0\) (1) by \(R_0c^{-1}\), we obtain a dimensionless unit, which is the same for the \(m_0\)-particle’s proper space and proper time:

$$\frac{\lambda _C}{R_0} = \frac{T_0}{R_0c^{-1}} = \frac{hc}{2G}m^{-2}_0.$$
(8)

It is meaningful that this unit is just a product or ratio of all three fundamental constants of nature, namely of the general theory of relativity (GTR), QM, and the STR, with \(m_0^2\) as the scaling factor (the relation of the latter to gravitational and inertial mass is addressed in Sect. 6).

Slightly modifying (8) allows us to use the notation (\(\hbar c/G)^{1/2}=m_P\), where \(m_P\) is the so-called ’Planck mass’, and to introduce the dimensionless spacetime unit as

$$\delta _0 = \frac{\hbar c}{G} m_0^{-2} \equiv \left( \frac{m_P}{m_0} \right) ^2 .$$
(9)

Passing to a dimensionless unit defines the scale of the \(m_0\)-particle’s proper spacetime just by its characteristic number, thus, for an electron \(\delta _0 \simeq 5.708 \times 10^{44}\). To return to the standard time (2) in seconds, one must pass from \(\delta _0\) to \(T_0\) given by (1); as follows from (8) and (9), this means multiplying \(\delta _0\) by \(2\pi G m_0/c^3\).

Since a dimensionless unit is determined by the ratio of the particle’s proper distance unit over its singularity size, it should be large enough to avoid distortion by gravity to substantiate the requirement of correspondence to a straight line (Sect. 3.4). Therefore, the necessary condition for the rest mass \(m_0\) of a particle (a fundamental one, see Sect. 2.2) is

$$\frac{m_P}{m_0} \gg 1 ,$$
(10)

which is well fulfilled for fundamental particles. The fact that for \(m_P\) comparable to \(m_0\) one would have difficulties with correspondence to a straight-line geometry, and thus with the introduction of a flat particle’s proper space, may provide an argument for explaining a somewhat puzzling question (see, for instance, a remark in [10]): why is the Planck mass so huge when compared to the masses of fundamental particles, whereas Planck units of time and length are so small when compared to the respective de Broglie time period and Compton wavelength?

4.3 Basic Equation

The equivalence of positive and negative \(x_n\) values implies that the sought-after connection between \(t_n\), \(x_n\), and \({t_n^{\prime}}\) (Sect. 4.1) must include the squared values of spatial intervals, and thus also of the time intervals, \(t_n^2\) being an ultimate invariant at given n, all possible combinations of \(x_n\) and \({t_n^{\prime}}\) will be defined in a unique way for a fixed \(t_n^2\) by supposing, in Euclidean space, the orthogonality of \(x_n\) and \({t_n^{\prime}}\), which appear at the resolution of \(t_n\) into the respective orthogonal projections (see Fig. 2). This allows us to introduce the set of the respective orthogonal spatial \(x_n\) and temporal \({t_n^{\prime}}\) Euclidean projections of the de Broglie time \(t_n\).

Passing to dimensionless spacetime coordinates becomes possible by applying the dimensionless unit (9) to the \(m_0\)-particle’s \(t_n\) and \(x_n\) defined by  (2) and (4). This yields their respective dimensionless analogues \(^{DL}t_n = n\delta _0\) and \(^{DL}x_n = n_x\delta _0\); obviously, \(|^{DL}x_n|<^{DL}t_n\) since \(|n_x|<n\). The condition \({t^{\prime}}_n = {nT_0^{\prime}}\) (Sect. 4.1) means that \({T_0^{\prime}}<T_0\), which yields \(^{DL}{t_n^{\prime}} = {n\delta _0^{\prime}}\) with \({\delta _0^{\prime}} < \delta _0\) (see Fig. 2). Considering \(^{DL}x_n\) and \(^{DL}{t_n^{\prime}}\) as the Euclidean projections of dimensionless de Broglie time \(^{DL}t_n\) yields

$$\begin{aligned} (^{DL}t_n)^2 = (^{DL}x_n)^2 + (^{DL}{t_n^{\prime}})^2 , \end{aligned}$$
(11)

in this way providing a framework of a Euclidean space. The entire set of \((^{DL}x_n, ^{DL}{t_n^{\prime}})\)-coordinates may be considered as forming the proper spacetime of a free, \(t_n\)-‘aged’ particle (see the remark in Sect. 3.4 regarding the necessity of two more spatial dimensions). It is notable that \(^{DL}{t_n^{\prime}}\) and \(^{DL}x_n\) appear from \(^{DL}t_n\) as the only possible alternatives generated either by a different (diminished) number \(|n_x|\) of the same unit \(\delta _0\), or by the same number n of a different (diminished) unit \({\delta _0^{\prime}}\), as follows from (11), \({\delta _0^{\prime}}=\delta _0 (1-n_x^2/n^2)^{1/2}\).

The basic equation (11) allows one to consider the de Broglie time as a vector \(\mathbf {t_n}(^{DL}x_n,^{DL}{t_n^{\prime}})\), \(|\mathbf {t_n}| = ^{DL}t_n\) in \((^{DL}x_n,^{DL}{t_n^{\prime}})\)-space. To follow explicitly the connection to numbers, let us expand \(\mathbf {t_n}\) over the orthonormalized unit vectors \(\varvec{\chi }=\mathbf {x_n}/(n_x\delta _0)\), \(|\mathbf {x_n}|=^{DL}x_n\), and \(\varvec{\tau^{\prime}}=\mathbf {{t_n^{\prime}}}/({n\delta _0^{\prime}})\), \(|\mathbf {{t_n}^{\prime}}|=^{DL}{t_n^{\prime}}\), \(|\varvec{\chi }|=|\varvec{\tau^{\prime}}|=1\), which form the \(\{\varvec{\chi },{\varvec{\tau }^{\prime}}\}\)-basis. We get:

$${\mathbf{t}}_{{\mathbf{n}}} = \;^{{DL}} x_{n} \chi + ^{{DL}} t^{\prime}_{n} \tau ^{\prime} = ({\mathbf{t}}_{{\mathbf{n}}} \chi )\chi {\text{ + }}({\mathbf{t}}_{{\mathbf{n}}} {\mathbf{\tau ^{\prime}}}){\mathbf{\tau ^{\prime}}}.$$
(12)

Expressing the respective projections of \({\mathbf{t_n}}\) onto \({{\varvec{\chi }}}\) and \({{\varvec{\tau }^{\prime}}}\) via \(^{DL}t_n\) we get \(^{DL}x_n = C_x ^{DL}t_n\) and \(^{DL}{t^{\prime}}_n = C_{{t^{\prime}}}^{DL}t_n\), where the distance-connected projection coefficient \(C_x = (n_x/n) = v_{xn}/c\) is just the proper velocity (6) normalized by c, while the \(^{DL}{t_n^{\prime}}\)-connected projection coefficient is \(C_{{t^{\prime}}} = (1-n_x^2/n^2)^{1/2} = (1 - v_{xn}^2/c^2)^{1/2}\). As can be seen, provided \(C_x^2+C_{{t^{\prime}}}^2=1\), \(C_x^2\) and \(C_{{t^{\prime}}}^2\) acquire the meaning of the partition coefficients for resolving the de Broglie time over the respective orthogonal space-like and time-like coordinates of a particle’s proper spacetime. Passing to the n-independent unit vector \({\varvec{ \tau _0}}={\mathbf{t_n}}/nT_0\) with \(|{\varvec{\tau _0}}|=1\) and expanding \({\varvec{\tau _0}}\) over \({\varvec{\chi }}\) and \({\varvec{\tau }^{\prime}}\) yields

$$\begin{aligned} {\varvec{\tau }_\mathbf{0}} = {(\varvec{\tau }_\mathbf{0} \varvec{\chi }) \varvec{\chi }} + {(\varvec{\tau }_\mathbf{0} \varvec{\tau^{\prime}})\varvec{\tau^{\prime}}} = C_x \varvec{\chi }+ C_{{t^{\prime}}} {\varvec{\tau }^{\prime}}, \end{aligned}$$
(13)

which means that \(C_x = {\varvec{\tau }_\mathbf{0} \varvec{\chi }}\) and \(C_{{t^{\prime}}} = {\varvec{\tau }_\mathbf{0} \varvec{\tau^{\prime}}}\) determine the \({\varvec{\tau }_\mathbf{0 }}\)-orientation \({\varvec{\tau }_\mathbf{0 }}(\theta )\) in the \(\{ {\varvec{\chi }, \varvec{\tau }^{\prime} }\}\)-basis as \(C_{x}=\cos \theta\) and \(C_{{t^{\prime}}}=\sin \theta\).

4.4 Proper Energy and Momentum

Let us transform equation (11) to an invariant quantity—the energy. Coming back to de Broglie unit (1) and accounting for (2) and (4), Eq. (11) can be written as

$$(nT_0 )^2 = (n_xT_0)^2 + (nT_0^{\prime})^2 .$$
(14)

In order to cancel the flow of time and to pass to energy-related quantities, let us multiply the terms of (14) by \(h^2/(T_0T_0^{\prime}n)^2\). We get:

$$\begin{aligned} (h\nu _0^{\prime})^2=\frac{n_x^2}{n^2}(h\nu _0^{\prime})^2+(h\nu _0)^2=\frac{v_{xn}^2}{c^2}(h\nu _0^{\prime})^2+(m_0c^2)^2, \end{aligned}$$
(15)

where \({\nu _0^{\prime}}=1/T_0^{\prime}\). As can be seen, we have arrived at the ultimately invariant quantity of the energy-momentum relation in the STR—the particle’s squared rest energy (\(h\nu _0)^2=(m_0c^2)^2\), which has been transformed from the particle’s proper temporal coordinate term \(({t_n^{\prime}})^2=(nT_0^{\prime})^2\) of (14) or (11). This relates the \(m_0\)-particle’s rest energy \(m_0c^2\) to the temporal coordinate \({t_n^{\prime}}\) of its proper spacetime. Let us focus on the first right-hand-side term, which is transformed from the spatial coordinate \(x_n^2 = (n_xT_0)^2\) in (14) or (11). As is seen from (14) and (15) it takes the form of the squared relativistic linear momentum term of energy-momentum relation in the STR:

$$\begin{aligned} \frac{n_x^2}{n^2}\left( h\nu _0^{\prime}\right) ^2 =m_0^2c^4 \left( \frac{n_x^2}{n^2}\right) \left( 1-\frac{n_x^2}{n^2}\right) ^{-1}=c^2v_{xn}^2m_0^2 \left( 1-\frac{v_{xn}^2}{c^2} \right) ^{-1}\equiv p_{xn}^2c^2 , \end{aligned}$$
(16)

where a conventional notation \(p_{xn}^2 = m_0^2v_{xn}^2/(1-v_{xn}^2/c^2)\) is used for the respective proper quantity, which is expressed by numbers. Following the analogy of (15) to the STR energy-momentum formalism, \(p_{xn}^2c^2\) represents the difference between the squared values of a particle’s full proper energy and its rest energy. As determined by the proper velocity referred to as the speed of ‘producing’ proper space (Sect. 3.3), \(p_{xn}^2c^2\) may be considered as the ’spatial’ component of proper energy necessary for ’producing’ a particular proper space interval \((x_n)^2\).

Let us determine the total \(p_n^2c^2\) value, which is summed over all proper distances, that is, over all \(n_x\) for a fixed n defining an elapsed de Broglie time \(t_n\). According to the construction of the set of integers from pairs of natural numbers that belong to the \(t_n\)-defining sequence 1, 2, ..., n (Sect. 3.2), it is necessary to perform a weighted summation of (16) accounting for the weight factor. For \(n_x >0\) the weight factor has to be taken as the number \(n - n_x\) of all possible pairs \(a, b \in [1, n]\) that yield a particular \(n_x\) for a fixed n, which is normalized by the sum of such possibilities over all \(n_x\) from 1 to \(n - 1\), or \(n(n-1)/2\). As a result, after accounting for \(n_x < 0\), we get

$$\begin{aligned} p_n^2c^2=\frac{4m_0^2c^4}{n(n-1)}\sum _{n_x=1}^{n-1}\frac{n_x^2}{n_x+n} . \end{aligned}$$
(17)

As can be seen, the major contribution to \(p_n^2c^2\) is made by the terms with large \(n_x\)-values comparable to n, which correspond to large proper distances comparable to \(ct_n\).

It is of decisive importance to check whether the particle’s \(p_n^2c^2\) converges to a finite value for growing n since it would be difficult to imagine a divergence of proper energy with growing n, which means with evolving de Broglie time \(t_n\), for an isolated particle. And indeed, by finding the asymptotic limit of (17) as \(n \rightarrow \infty\), one gets the convergence of \(p_n^2c^2\) to a finite value:

$$\begin{aligned} p^2c^2=\lim _{n\rightarrow \infty }p_n^2c^2=2\left( 2 \ln 2-1 \right) m_0^2c^4 \simeq 0.7726m_0^2c^4 . \end{aligned}$$
(18)

which is determined exclusively by the rest mass, as should be expected for a free \(m_0\)-particle; note that it is distributed (delocalized) within large proper distances of the order of \(ct_n\). Accounting for Eq. (15) and its connection to (11) and (14), the squared full energy of a particle’s spacetime acquires a value determined exclusively by its rest mass, being the sum of a spatial component (linear momentum term) (18) and a temporal component (rest energy term) \((m_0c^2)^2\). Then, the full energy of the \(m_0\)-particle’s proper spacetime expressed in terms of the momentum, or the \(m_0\)-particle’s Hamiltonian [1] will possess a numerical value \(\text {H} = m_0c^2\sqrt{2(2\ln 2-1) + 1}\). The assumption that two other spatial dimensions (see Sect. 3.4) yield the same contribution to the full proper energy as given by (18) would increase the respective term of the Hamiltonian by a factor of 3.

5 Wave-Like Properties

5.1 The Phase of the Wave

The periodic nature of de Broglie time \(t_n\) (1) presupposes that the connection between proper space and proper time should exhibit wave-like properties. The very nature of a wave process requires this connection to be linear. For this purpose, let us express \(nT_0^{\prime} = {t_n^{\prime}}\) in (14) as

$${t^{\prime}}_n = t_n\left( 1-\frac{n_x^2}{n^2} \right) ^{1/2} = t_n \left( 1-\frac{v_{xn}^2}{c^2}\right) ^{1/2} ,$$
(19)

then multiply and divide the obtained expression by \((1-v_{xn}^2/c^2)^{1/2}\). After explicitly including \(v_{xn} = x_n/t_n\) in the numerator, we arrive at

$$\begin{aligned} {t^{\prime}}_n = \frac{ t_n - x_nv_{xn}/c^2}{\sqrt{1 -v_{xn}^2/c^2}}. \end{aligned}$$
(20)

As can be seen, for a particular \(v_{xn}/c = n_x/n\), time \(t_n\) and distance \(x_n\) enter the right-hand-side of (20) as a linear combination, thus providing a general condition for the argument of a wave-like process (see also [6]). Namely, the proper spatial coordinate \(x_n\) scaled by \(c^2/v_{xn}\) is linearly connected to the de Broglie time \(t_n\) in such a ‘phase-matching’ way that their difference, scaled by a factor \((1- v^2_{xn}/c^2)^{1/2}\), remains equal to the temporal coordinate \({t^{\prime}}_n\) of particle’s proper spacetime.

The phase of such a wave-like process is obtained after multiplying (20) by the cyclic de Broglie frequency \(2\pi / T_0=\omega _0\), which yields

$$\begin{aligned} \omega _0 {t^{\prime}}_n = \frac{ \omega _0(t_n - x_nv_{xn}/c^2)}{\sqrt{1 -v_{xn}^2/c^2}}=\omega _0^{\prime}t_n-k_{xn}x_n, \end{aligned}$$
(21)

where \(\omega _0/(1-v_{xn}^2/c^2)^{1/2}=\omega _0^{\prime}=2\pi /T_0^{\prime}\), while the factor

$$\begin{aligned} V_{xn}=c^2/v_{xn}=(n/n_x)c \end{aligned}$$
(22)

can be identified as a proper phase velocity. Then, \(\omega _0^{\prime}/V_{xn}=k_{xn}\) acquires the meaning of the proper wave number, the respective proper de Broglie wavelength being

$$\begin{aligned} \Lambda _{xn}=2\pi /k_{xn}=\left[ h/(m_0v_{xn})\right] \left( 1-v_{xn}^2/c^2\right) ^{1/2}. \end{aligned}$$
(23)

The comparison of (23) and (16) immediately reveals the connection of \(k_{xn}\) to the proper linear momentum \(p_{xn} = \hbar k_{xn}\); it is meaningful that this appears just as a result of transforming (11) to the expressions containing the respective recognizable quantities. Though (21) and (23) have the form of a Lorentz transformation in the STR and of the wavelength of the de Broglie matter wave, respectively, they possess a different meaning since the spatial and temporal coordinates are defined as proper, in the particle’s proper spacetime, without any relation to an external reference frame.

Let us follow the spreading of maximal value of proper distance \(|x_n^{max}| = (n - 1)cT_0\) with ongoing de Broglie time \(t_n \rightarrow t_{n+1}\), or \({\text{P}}_1{\text{S}}_n \rightarrow {\text{P}}_1{\text{S}}_{n+1}\) (Sect. 2.3) in the wave-like process (20)–(23). For a three-dimensional particle’s proper space (Sect. 3.4) the wave-like process may be considered as expanding of the radius of ‘wave front’ \(r_{n-1} = |x_n^{max}|\) as \(r_{n-1} \rightarrow r_n\), or \((n - 1)cT_0 \rightarrow n(cT_0)\). As discussed in Sect. 2.3, the flow of \(t_n\) corresponds to a consequent periodic reproduction, at the present instant \({\text{P}}_1\), of a \(T_0\)-unit of proper time, and consequently, of a \(cT_0\)-unit of proper space. In relation to integers, this corresponds to the periodic reproduction, at the present, of \(n_x = 1\). It can be seen that extending of \(r_n\) as \((n - 1)cT_0 \rightarrow ncT_0\) corresponding to \(S_{n-1} \rightarrow S_n\) takes place at the present; indeed, \(n_x = 1\) corresponds to the proper velocity \(v_{xn} = c/n\) (6), and consequently, to the proper phase velocity \(V_{xn} = nc\) (22). Therefore the time interval taken for the wave front to expand equals \(r_n/V_{xn} = T_0\), which means that the expansion occurs at the present. Along with that, the number \(n-n_x\) of all possible pairs \(a, b \in [1, n]\) that yield a particular positive value \(n_x\) is increased from \(n-n_x\) to \((n+1)-n_x\), or by 1. And since each such increasing occurs at the present, the entire proper space belongs to the present, in agreement with the assumption in Sect. 3.1.

5.2 Duality and Indistinguishability

Let us address the feasibility of combining the wave-like property of an \(m_0\)-entity’s proper spacetime with its particle-related property, which means treating the wave-particle duality, within the numbers-based approach. The main issue is to preserve the feature of a ‘point-like’ particle while introducing the proper distance that is connected with proper time in a wave-like way. It is rather clear that possessing only a temporal proper coordinate \({t_n^{\prime}}\) while \(x_n\) remains zero is consistent with regarding the \(m_0\)-entity as ‘purely’ a particle, which agrees with the correspondence of the \((^{DL}{t^{\prime}}_n)^2\) term in (11) to the rest energy term \((h\nu _0)^2 = m_0^2c^4\) in (15). Note that this fits the orthogonality condition of proper temporal and spatial coordinates considered in Sect. 4. In a simplified way, one may say that according to Eq. (20) the \(m_0\)-entity is identified with a particle by \({t^{\prime}}_n\) in the left-hand-side and with a wave (wave’s phase) by the right-hand-side, the equality sign standing for the equivalence of wave- and particle-like properties.

Conformity of the temporal coordinate \({t_n^{\prime}}\) (14) with a ‘point-like’ particle’s property means that, since \({t_n^{\prime}}\) is defined by a sequence 1, 2, ... , n, which is counting the \(T_0^{\prime}\)-units, a unit of proper distance has to remain zero in each count. But a zero distance unit contradicts its definition (3) as a Compton wavelength \(\lambda _C\), or, more precisely, as (see Sect. 3.4). This contradiction may be resolved by supposing that consists of two one-half units . Then, the particle-related property associated with \({t_n^{\prime}}\) would have been provided by a summed-to-zero combination \(\pm [(\frac{1}{2})\hbar /(m_0c)- (\frac{1}{2})\hbar /m_0c]\) whereas the wave-related property associated with \(x_n\) is provided by a summed-to-a-Compton-unit combination \(\pm [(\frac{1}{2})\hbar /(m_0c)+ (\frac{1}{2})\hbar /m_0c]\).

Let us follow the relation of the particle-related property to numbers. Introducing an \(m_0\)-entity’s de Broglie proper time \(t_n\) relies on the entity’s singular-present \({\text{P}}_1\) (Sect. 2), which it is reasonable to associate with a singular particle, while, when introducing proper distance, one necessarily introduces ‘dual-present’ \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) (Sect. 4.1, Fig. 1). The premise ‘singular-present–singular-particle’ implies the premise ‘dual-present–dual-particle’, which means that, at the present instant, the particle should be simultaneously identified with \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\). Indeed, if (i) a particle is associated with \({t_n^{\prime}}\) and (ii) identifying \({t_n^{\prime}}\) with in principle (see Sect. 3.4) indistinguishable \({\text{P}}_1\) or \({\text{P}}_1^{\prime}\) makes no sense, then \({t_n^{\prime}}\), and together with it, the particle, has to be identified with the ‘dual-present’ \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\). This can be associated with a particle’s delocalization because the particle’s ‘dual-present’ \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) is ascribed to the entire set of proper distances \(x_n = {\text{P}}_1 {\text{P}}_1 ^{\prime}\). The question arises how the particle’s ‘dual-present’ property agrees with the correspondence of its present existence to 1. Following the discussion in Sect. 2.3, a possible answer lies in tracing how exactly the ongoing proper time is ensured by continuous adding, at the present, of a \(T_0\)-unit to an already existing (and corresponding to the present) \(T_0\)-unit, ‘shifting’ the latter to the past. It is essential that such adding presumes a simultaneous (within \(T_0\)) existence, at the present, of both the added \(T_0\)-unit and the previous \(T_0\)-unit, thus explaining the very possibility of a ‘dual-present’ \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\). One may notice that the same relates to constructing the, isomorphic to the ongoing time flow, sequence of natural numbers by a continuous, uninterrupted addition of 1 in \(\text {SUC}(i) = i + 1; i \in [1, n-1]\). Here, a unique symmetry occurs for \(i = 1\) when passing from ‘1’ to ‘1’ + ‘1’, which presumes the simultaneous existence of (consequently, relation of the present instant to) ‘1’ and ‘1’ \(+\) ‘1’. Indeed, it is not possible to distinguish whether the present corresponds to ‘1’ or to ‘1’ + ‘1’: if the present only corresponds to ‘1’, there is no ongoing sequence of natural numbers, therefore no time flow, whereas if it only corresponds to ‘1’ + ‘1’, it does not distinguish an individual entity. In other words, because of ongoing time, the premise “‘1’ corresponds to a singular-present” equivalently means that “‘1’ and ‘1’ \(+\) ‘1’ correspond to a dual-present”. This justifies relating the particle’s existence both to the single-present \({\text{P}}_1\) and to the dual-present \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\), which, in fact, justifies the very idea that a free \(m_0\)-particle’s proper space relates to the present.

Relation to numbers makes it possible to numerically assess the partition of an \(m_0\)-entity’s particle-like and wave-like properties just by the respective values of the coefficients introduced by (13): \(C_{{t^{\prime}}}^2 = 1-(v_{xn}/c)^2 = 1-(n_x/n)^2\) and \(C_x^2 = (v_{xn}/c)^2 = (n_x/n)^2\). In particular, the condition to be fully particle-like, that is, to possess only a temporal proper coordinate \({t^{\prime}}_n\), would mean that \(n_x = 0\), and, therefore, \(C_{{t^{\prime}}}^2 = 1\) and \({t^{\prime}}_n = t_n\). Since \(n_x = 0\) does not appear in the construction of integers from pairs of natural numbers (a, b) where \(a,b \in \{1, 2, ... , n ; a \ne b\}\) (see Sect. 3.2), the smallest \(n_x\)-value is 1, which brings the Compton wavelength (3) as the minimal ‘size’. It may be of interest to recall the STR notion of ‘event’ [1], which would break the condition of a fully isolated entity ‘at the moment it occurs’. Following the discussion in Sect. 2.3, this means that, at the entity’s present moment, the ongoing \(t_n\)-defining sequence 1, 2, ... , n ‘collapses’ into \(n =1, \text {S}_n \rightarrow {\text{S}}_1\). This can be considered as a ‘collapse’ of the dual-present \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) into a singular-present \({\text{P}}_1 \leftrightarrow {\text{S}}_1\), which means the \(m_0\)-entity’s ‘collapse’ into a particle with the ‘size’ of a distance unit. If the particle can be considered free after the event, \({\text{S}}_1\) would identify a new temporal beginning and restart the numbering of \(T_0\)-units. Associating the \(m_0\)-entity with a ‘pure’ wave without any particle-related property would mean \(C_x^2 = 1\), and, therefore, \(n_x = n\), which, as already mentioned, is not possible to construct from pairs (a, b).

It is of evident interest to consider the possibility of attributing the ‘dual-present’ \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) to two particles. Let us begin by considering two identical (fundamental) \(m_0\)-particles, which are assumed to remain fully isolated after their common start (creation, temporal beginning), say, following a decay process. Since the present instant \({\text{P}}_1\) is common for both particles at the start, \({\text{S}}_1= {\text{P}}_1\), and the correspondence of \({\text{P}}_1\) to 1 should remain common for a strictly isolated two-particle system. The sequence 1, 2, ..., n numbering common \(T_0\)-units now relates to both particles providing their common interval of proper time \(t_n = nT_0\) from the common present \({\text{P}}_1\) to the common start \(\text {S}_n\). Let us suppose that, after elapsed \(t_n\), the particles are separated by a distance \(x_n = {\text{P}}_1 {\text{P}}_1^{\prime}\). The question is: under what conditions can \(x_n\) be considered proper for both particles? To comply with the reasoning in Sect. 3.4, the ultimate condition should be the possibility to identify the particles with in principle indistinguishable dual-present \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\). And this becomes possible if and only if the particles themselves are in principle indistinguishable, which is exactly the case for a fundamental particle. Then, the set of \((x_n, {t_n^{\prime}})\)-coordinates that define proper spacetime can be equivalently related to any of the particles, thus defining their common spacetime as proper. This means that the distance \(x_n = {\text{P}}_1 {\text{P}}_1^{\prime}\), which corresponds to an integer \(n_x\) (4), as well as the respective proper velocity \(v_{xn}\) (6), can be considered relative, while remaining proper. The same relates to the proper momentum \(p_{xn}~\) (16) supplying an argument for the ‘reality’ of the proper energy (17) and (18). Identifying the proper reference frame with any of the particles would mean identifying the \((x_n, {t_n^{\prime}})\)-coordinates with the ‘other’ particle in this reference frame. Since this means relating the temporal coordinate \({t_n^{\prime}}\) (19) to a particle that possesses (now also relative) velocity \(v_{xn}\), one arrives at the ‘time dilation’ phenomenon in the STR. As is understandable, there is no need for any clock synchronization hypothesis since two indistinguishably-identical \(m_0\)-particles represent two ideally-equal clocks (see the remark in [6]), which are inevitably synchronized at the common start, thus providing a common \(t_n\); the full equivalence of relating the system of reference to each of the particles is obvious. Regarding the relation to the foundations of QM, one may notice that while the temporal coordinate \({t^{\prime}}_n\) (19) is diminished with respect to the de Broglie time \(t_n\), the corresponding frequency \({\nu _0^{\prime}} =1/T_0^{\prime}\) is increased with respect to \(\nu _0\) in (1) as \({\nu _0^{\prime}} = \nu _0(1- n_x^2/n^2)^{-1/2}\) [see (14)]. As a consequence, the respective QM momentum \(h\nu _0^{\prime}/c\) also increases. The fact that correspondence of the present instant to 1 now relates to both particles is consistent with possessing the one-entity-property, which points to the particles’ entanglement. What is more, relating the opposite signs of one-half Compton units to the respective \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) now means relating the opposite signs of the factor \(\pm \frac{1}{2}\hbar\) to the respective particles. Then, if an ‘event’ (the ‘collapse’ of a common 1, 2, ..., n sequence into 1) occurs, at the present, breaking the condition of being free for one of the particles, it (the ’event’) in fact occurs at the dual-present \({\text{P}}_1\) and \({\text{P}}_1^{\prime}\) of two indistinguishable particles, and so it is, in principle, impossible to distinguish to which one the ‘event’ occurs. Since, at a common present instant, the particle-related time \({t_n^{\prime}}\) corresponding to the summed-to-zero combination \(\pm [ (\frac{1}{2})\hbar /(m_0c)-(\frac{1}{2})\hbar /(m_0c)]\) now relates to both particles, fixing, say, a factor \(+(\frac{1}{2})\hbar\) to one of the particles means simultaneous appearance of a factor \(-(\frac{1}{2})\hbar\) of the other particle at whatever proper distance \(x_n\).

6 Concluding Remarks

Let us summarize the relation of the present approach to the foundations of the STR and QM. As has been proposed, proper spacetime may be ascribed to a free, nonzero-rest-mass particle based on a connection to natural numbers of de Broglie proper time defined by de Broglie’s periodic phenomenon. This, in a sense, conforms to the STR concept of proper time as read by the entity’s proper clock based on a periodic process; as stated in [7], it does not fall into circular reasoning “...if one puts the concept of periodic occurrence ahead of the concept of time”.

Yet introducing a free particle’s spatial coordinate as proper seems incompatible with the STR, according to which only the temporal coordinate may be considered as proper “...in a system of coordinates linked to the moving clocks” [1] (one may notice, however, that a system of coordinates that involves only time may seem not to be fully consistent with the generality of the spacetime concept in the STR). A numbers-based approach allows a constructive definition of a spatial coordinate as proper. Then, an interval of proper space must be connected to the particle’s temporal beginning not only by the de Broglie time interval \(t_n\), but also by another time interval \({t_n^{\prime}}\). The velocity-dependence of \({t_n^{\prime}}\) in the form of (19) suggests that \({t_n^{\prime}}\) acquires the property of the temporal coordinate in the STR; it is notable that the constant c appears as an ultimate velocity limit.

The ‘absolute’, existential meaning of the de Broglie time \(t_n\) as distinct from the temporal coordinate \({t_n^{\prime}}\) may add to the discussion on the inconsistency between the intuitive notion of time and the temporal component of four-dimensional relativistic spacetime. One may recall the analysis made by Gödel [11], who found a cosmological solution of the GTR, though for the special case of rotating matter, which, as he stated in [12], proves that relativistic time contradicts the very existence of ‘an objective lapse of time (whose essence is that only the present really exists)’ by demonstrating the ‘absurd’ possibility of ‘time travel’, say, to one’s ‘own past’.

Regarding the connection to the foundations of QM, a straightforward idea is to relate the de Broglie time \(t_n\) and the coordinate time \({t_n^{\prime}}\) to the respective wave-like and particle-like properties. Indeed, while the de Broglie time \(t_n\) enters the wave’s phase (20), the coordinate time \({t_n^{\prime}}\) is associated with possessing a particle-related property. Recalling the premise [2] of a particle’s interferences within its proper space may possibly suggest a way of experimentally verifying the features characteristic of the present approach. One possibility might be based on some version of a double-slit experiment: if the particle ‘sees’ the distance between the ‘slits’ as proper, the distance should not exceed the maximal distance value related to the particle’s ‘age’ \(t_n\), while the phase difference should include the same n for all spatial intervals \(x_n\).

The connection to numbers of a free, \(m_0\)-particle’s spacetime becomes complete provided the existence of a common dimensionless unit of time and space composed from the particle’s rest mast \(m_0\) and the constants of nature. It is meaningful that in such a unit (9) \(m_0^2\) appears as the product of two coefficients: the particle’s rest mass in (1), which determines the de Broglie frequency and, therefore, though designated as inertial mass, implies the basic quantum phenomenon, and the gravitational mass in (7). In general, these coefficients might have been totally different, and the very fact that they are reduced to the squared value \(m_0^2\) of coinciding ‘inertial-quantum’ and gravitational masses demonstrates their unification into an ‘inertial-quantum-gravitational’ mass that specifies the particle’s proper spacetime. And such a unification seems justified by the very circumstance that the factor \(\hbar c/G\) in (9), which scales \(m_0^2\) to the entity’s specific number, is composed from the three constants of nature, which are the fundamental constants of QM, the STR, and the GTR. It is notable that, if the existence of a dimensionless spacetime unit is considered as the starting point, the inclusion of \(m_0^2\) offers a sign choice for \(m_0\).

The most challenging issue is to comprehend the proper-space-related energy term in (18), which may be attributed to the ‘cost of forming space’. According to the suggested estimate, this energy is comparable with the particle’s rest energy \(m_0c^2\) and, as follows from (17), is mainly determined by large proper distances. A rather inspiring question is: can this energy be associated with dark energy of the universe as a whole? One could mention a purely formal argument: since a fully isolated entity possesses the features of an isolated universe, the considerations regarding the particle’s proper spacetime might be relevant. The applicability of the present reasoning to a system of indistinguishable particles with a common start may justify merging their proper spacetime into the proper spacetime of the universe as a whole (could it provide an argument for the hypothesis of a primordial particle as the end-constituent of matter?). One may notice that the proper-space-related particle energy is likely to fit such expected properties of the dark energy (see, for instance, [13]) as being recognized on the scale of large distances and being consistent with a homogeneous, observationally flat universe of very low density.

A direct involvement of numbers in defining a free \(m_0\)-particle’s proper spacetime forms the core of the suggested approach. The assumption that the formation of proper distances corresponds to the formation of positive and negative integers from natural numbers indicates that the de Broglie time interval defined by them is ‘prior’ to a spatial interval, just as natural numbers are prior to integers. Rational fractions may be related to proper velocity, fractional with respect to c. What is more, there is a symmetry-based argument (Sect. 3.4), which points to the necessity for space to be three-dimensional and possibly bears indications of irrational numbers. The very fact that a definition of a particle’s spacetime, which is fundamental for physics, is reduced to numbers, which form the basis of mathematics may contribute to the discussion of the ‘unexplainable role’ of mathematics in physics. A debatable epistemological issue concerns the ‘reality’ of numbers as mathematical objects (for an insight see, for instance, [14] and references therein): while the numbers are considered as not existing in spacetime, the usually agreed meaning of (physical) ‘reality’ would presume such existence. A possibility to comply with ‘reality’ would be to include the numbers into the definition of spacetime, and this is the core of the present approach. Accordingly, the proposition of the ‘reality’ of numbers should not mean that they exist in spacetime but that they constructively determine spacetime; in such a way, the ‘reality’ of numbers is reduced to the ‘reality’ of spacetime.