ABSTRACT

The paper concerns the fundamental constants of physics. The fine structure constant is inferred first with the help of a generalized Fibonacci sequence. Next this generalized sequence is also implemented in order to show the interconnection between various fundamental constants. The analysis of an extended definition of Fibonacci sequence reveals that the fundamental constants take the meaning of archetypal templates that model the physical appearance of the observable Universe. Once more appears in this conceptual frame the natural and serendipitous link between quantum and relativistic theories.

Keywords:

Fibonacci Sequence, Fundamental Constants

1. Introduction

The standard definition of Fibonacci sequence is

$F=1,1,2,3,5,8,13,\cdots$ (1.1)

quoted in the following as $F=F\left(1.1\right)$ to emphasize the first two terms of the sequence. The first two numbers are crucial in determining all terms of the sequence, being by definition each term sum of the two preceding ones. This sequence was originally aimed to calculate the reproduction rate of rabbits under appropriate fertility hypotheses: the first two terms are required equal to one to signify the first couple of rabbits. If each couple becomes fertile at the end of the first month and gives birth to a new couple when the second month is completed, then the meaning of each term of the sequence is the number of fertile couples per month. From a mere mathematical point of view, it is well known why the Fibonacci numbers are related to the so-called “golden ratio”. In principle, however, no conceptual reason requires input values just equal to 1 if the sequence is not essentially restricted to the original purpose to which it was early aimed. In other words, nothing hinders to generalize this sequence in order to describe a wider range of natural phenomena simply changing the first two terms. Here is proposed the generalization of $F\left(1.1\right)$ putting

$F^{*}=F^{*}\left(f_0,f_1\right),$ (1.2)

where $f_0$ and $f_1$ are two arbitrary numbers in principle not necessarily integers; this affects all successive terms. Clearly, the standard Fibonacci sequence F is uniquely defined; instead the mathematical implications of the modified sequence $F^{\mathrm{<mo>\; *\; </mo>}}$, and thus their related physical meaning as well, just depending on how are defined the initial terms. Defining indeed

$F^{*}=f_0,f_1,\left(f_1+f_0\right),\left(2f_1+f_0\right),\left(3f_1+2f_0\right),\left(5f_1+3f_0\right),\left(8f_1+5f_0\right),\cdots$ (1.3)

i.e.

$F_1^{*}=f_0,F_2^{*}=f_1,F_4^{*}=2f_1+f_0,F_n^{*}=F_n\left(f_0,f_1\right)=f_1{F}_{n-1}+f_0{F}_{n-2},$ (1.4)

the last equation evidences that now the standard values $F_n$ still appear in $F_n^{\mathrm{<mo>\; *\; </mo>}}$ as a linear combination of $F_{n-1}$ and $F_{n-2}$ with coefficients $f_0$ and $f_1$. As it actually means that the $\left(n-1\right)$ -th and $\left(n-2\right)$ -th terms of the sequence (1.2) are all multiplied by $f_1$ and $f_0$ respectively, (1.4) reads identically

$F_n^{*}=F_{n-1}^{\left(1\right)}+F_{n-2}^{\left(0\right)},F_n^{\left(1\right)}=f_1{F}_{n-1},F_{n-2}^{\left(0\right)}=f_0{F}_{n-2},n>2$ (1.5)

the inequality reminds that the standard coefficients of (1.3) appear indeed for $n>2$, where are in fact calculable all $F_n^{\mathrm{<mo>\; *\; </mo>}}$ as sum of two previous $F^{\left(0\right)}$ and $F^{\left(1\right)}$ now defined by the arbitrary $f_0$ and $f_1$. The recursive rule is therefore unchanged. Since the single terms of the new sequence $F_n^{\mathrm{<mo>\; *\; </mo>}}$ are actually linear combinations of two terms with different n of the standard sequence $F_n$, reasonably each addend of the former could have in principle its own physical meaning likewise as the number calculated in the latter. It is therefore reasonable the guess why the aforesaid ability of F in describing the natural phenomena should still hold and be even enhanced by the additional freedom degrees inherent $F^{\mathrm{<mo>\; *\; </mo>}}$. This makes significant the appropriate definition of selected specific terms $F_1^{\mathrm{<mo>\; *\; </mo>}}$ and $F_2^{\mathrm{<mo>\; *\; </mo>}}$ suitable to provide information of interest about specific physical effects, likewise as each $F_n$ yields the number of couples of rabbits reproduced in the n-th generations from the initial couple. To exemplify this point return back to the original Fibonacci sequence, where each term represents the result of all previous reproduction steps. There is no way to introduce in this (1.1) the chance that either partner generated from the initial couple, fertile by definition, could actually be sterile; in other words, all successive generations of rabbits are assumed healthy and fertile exactly as the former one. The common sense suggests however that in practice, for any biological reason, this is not realistic; is missing in the early Fibonacci calculation any reference to possible discontinuity caused by an external action affecting the reproduction regularity of rabbits. From a physical point of view this chance could instead take place for example during the evolution of a system formed by an increasing number of particles from its initial configuration, e.g. early number of allowed states, to its current configuration after an appropriate number of intermediate steps; thus the evolutionary implication inherent the Fibonacci sequence can be advantageously modified and tailored to specific physical systems through the first two terms, as suggested by (1.4). In principle the deterministic character of (1.1) is inconsistent with the chance of describing quantum phenomena; also, the early F necessarily concerns phenomena describable via pure numbers. Instead the addends of $F_n^{\mathrm{<mo>\; *\; </mo>}}$, as proposed here, skip this requirement, i.e. the various terms are required to be neither integers nor dimensionless only; is crucial in this respect the numerical and dimensional choice of the coefficients $f_0$ and $f_1$. In practice, the first few terms of $F^{\mathrm{<mo>\; *\; </mo>}}$ read for example

$\begin{array}{l}{F}_3^{*}=2f_1+f_0,F_7^{*}=13f_1+8f_0,\\ F_{15}^{*}=610f_1+377f_0,F_{20}^{*}=6765f_1+4181f_0;\end{array}$ (1.6)

i.e. the coefficients that multiply $f_1$ and $f_2$ are just the respective numbers $F_{n-1}$ and $F_{n-2}$ of the standard sequence. So it seems worth investigating the chances opened by the kind of generalization prospected by (1.4), in particular the choice of the terms $F_n^{\mathrm{<mo>\; *\; </mo>}}$ pertinent to the specific physical problem of interest. For example, it is possible to multiply $F_n^{\mathrm{<mo>\; *\; </mo>}}$ by an arbitrary dimensional factor $k_c$ in order that (1.6) reads

$\begin{array}{l}{E}_3^{*}=\left(2f_1+f_0\right)k_c,E_7^{*}=\left(13f_1+8f_0\right)k_c,\\ E_{15}^{*}=\left(610f_1+377f_0\right)k_c,E_{20}^{*}=\left(6765f_1+4181f_0\right)k_c;\end{array}$ (1.7)

the notation exemplifies the particular case where

$k_c=\text{unit}\text{energy}.$ (1.8)

So the terms of (1.6) and (1.7) are numerically identical although the latter represents a sequence of energies expressed in energy units corresponding to $k_c$, whereas the pure numbers of the standard Fibonacci sequence turn into energies directly related to the evolution of a physical system: the steps that describe the total number of the n-th generation of rabbits turn into the allowed energies of the system at various time steps. In the following we implement (1.6) for simplicity of notation, subtending however that when necessary the numerical terms of the generalized sequence $F_n^{\mathrm{<mo>\; *\; </mo>}}$ represent identically the physical meaning of the sequence

$F_n^{\mathrm{<mo>\; *\; </mo>}}\equiv \frac{E_n^{\mathrm{<mo>\; *\; </mo>}}}{k_c}\mathrm{.}$ (1.9)

So (1.4) will be also implemented in a form immediately consequent putting

$f_0={\zeta }^{″}f,f_1={\zeta }^{\prime }f;$ (1.10)

owing to (1.5) it is possible to write (1.4) as exemplified in (1.6)

$F_n^{*}=\left({\zeta }^{\prime }{F}_{n-1}+{\zeta }^{″}{F}_{n-2}\right)f,f=f\left(\zeta \right),$ (1.11)

where $\zeta$ along with ${\zeta }^{\prime }$ and ${\zeta }^{″}$ are arbitrary parameters. Moreover, without loss of generality, it is sensible the chance of regarding the n-th terms of $F_n^{\mathrm{<mo>\; *\; </mo>}}$ via the definitions

$\begin{array}{l}f\left(\zeta \right)={\sigma }_1{\alpha }^{\circ }+{\sigma }_2{\alpha }^{\circ 2}+\cdots ,\\ {\zeta }^{\prime }=1+{{\sigma }^{\prime }}_1{\alpha }^{\circ }+{{\sigma }^{\prime }}_2{\alpha }^{\circ 2}+\cdots ,\\ {\zeta }^{″}=1+{{\sigma }^{″}}_1{\alpha }^{\circ }+{{\sigma }^{″}}_2{\alpha }^{\circ 2}+\cdots \mathrm{,}\end{array}$ (1.12)

i.e. replacing the first two numbers of the standard sequence (1.1) with the series expansions (1.12) defining the parameters ${\zeta }^{\prime }$ and ${\zeta }^{″}$, which just for this reason are both defined with 1 as zero order approximation value. The new information introduced via (1.12) is that now appears a unique parameter ${\alpha }^{\circ }$ as a function of which is in principle calculable each $F_n^{\mathrm{<mo>\; *\; </mo>}}$ via suitable values of the various coefficients ${{\sigma }^{\prime }}_j$ and ${{\sigma }^{″}}_j$, by definition uniquely fixed once for all and focused on the physical problem of interest. Note that if ${{\sigma }^{\prime }}_j$ and ${{\sigma }^{″}}_j$ are defined as

${\displaystyle \underset{j}{\sum }{{\sigma }^{\prime }}_j{a}^{\circ j}}\ge 0,{\displaystyle \underset{j}{\sum }{{\sigma }^{″}}_j{a}^{\circ j}}\ge 0,{\displaystyle \underset{j}{\sum }{\sigma }_j{a}^{\circ j}}=k_c\mathrm{,}$

then, whatever the specific value of $a^{\circ }$ might be, (1.11) reduces to either (1.6) or (1.7) depending on whether $k_c=1$ or for example $k_c=unitenergy$ ; clearly the latter chance implies regarding ${\sigma }_j$ as dimensional factors with physical dimensions of energy according to (1.9).

So the positions (1.12) seem reasonable to define (1.11) consistently with particular cases of standard sequence (1.1) or extended sequence (1.6) or even dimensional extended sequence (1.7).

Also, ${\sigma }_j=0$ imply $f\left(\zeta \right)=0$ and $F_n^{*}=0$, which means that for ${\alpha }^{\circ }=0$ the terms $F_{n>2}^{*}=0$ are identically null. This makes sense because (1.11) and (1.12) hold for $n>2$ ; so, with reference to the standard (1.1), $F_1^{*}=1$ and $F_2^{*}=1$ are enough to return back to the case of an initial couple of infertile rabbits.

Nevertheless, depending on the specific physical problem, it is possible that the general (1.11) admits useful simplifications; putting for example ${\zeta }^{\prime }={\zeta }^{″}$, one finds

$F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>}}=\left(F_{n-1}+F_{n-2}\right){\zeta }^{\prime }f\mathrm{.}$ (1.13)

This particular case is interesting because $F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}\approx F_{n-1}+F_{n-2}$ is compliant just with the original Fibonacci intuition; indeed, recalling (1.9), ${\zeta }^{\prime }f$ is the general form of the factor $k_c$, which in turn can become itself mere dimensional factor. Furthermore, it is also possible that the series expansions (1.12) can be truncated to the respective first order approximation terms; if so, then (1.11)

$F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}\approx \left(F_{n-1}+F_{n-2}\right)\left(1+{{\sigma }^{\prime }}_1{\alpha }^{\circ }+{{\sigma }^{\prime }}_2{\alpha }^{\circ 2}+\cdots \right)\left({\sigma }_1{\alpha }^{\circ }+{\sigma }_2{\alpha }^{\circ 2}+\cdots \right)$ (1.14)

reduces to

$F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}\approx \left(F_{n-1}+F_{n-2}\right)\left(1+{{\sigma }^{\prime }}_1{\alpha }^{\circ }\right){\sigma }_1{\alpha }^{\circ }\mathrm{.}$ (1.15)

Finally it is clear that just for this reason, despite all manipulations so far introduced, holds anyway

$F_n^{***}>0$ (1.16)

this condition is necessary in order that $F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}$ and the initial $F_n$ is conceptually consistent. It is not enough that both are defined by the same sum rule of successive terms, it is also necessary that all terms are in both cases still positive although numerically different. If so, then (1.16) can be legitimately regarded as generalized Fibonacci sequence even for ${\alpha }^{\circ }<0$. Note that (1.13) is simply a particular case of (1.11) but in principle it does not contain any approximation, as instead (1.15) does. Thus these last equations must be motivated by and justified for the specific physical problem of interest case by case. Nevertheless both contain the essential feature that substantiates the aforesaid premise of the present paper, i.e.: 1) the physical dimensions of the parameter ${\alpha }^{\circ }$, and thus of ${\sigma }^{\prime }$ too, to fit the specific physical problem and 2) the actual chance of generalizing (1.1) through appropriate definitions of both parameters. Remains still evident the link between (1.15) and (1.1), which is identically reproduced putting in particular ${\alpha }^{\circ }=1$ and ${{\sigma }^{\prime }}_1={\sigma }_1=0$. It is clear however that (1.15) does not require all higher order approximation coefficients ${{\sigma }^{\prime }}_{j>1}=0$, as it concerns the first order of approximation only. So, despite the mere numerical character of the approximation (1.15), any deviation from these boundary conditions is itself a sensible generalization of (1.1). If $F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}$ is known for any n, (1.15) can be trivially solved with respect to ${\alpha }^{\circ }$ ; i.e.

${\alpha }_{\pm }^o=\frac{-{\sigma }_1\pm \sqrt{{\sigma }_1^2+4{{\sigma }^{\prime }}_1{\sigma }_{1}R}}{2{{\sigma }^{\prime }}_1{\sigma }_1},R=\frac{F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}}{F_{n-1}+F_{n-2}}\mathrm{,}$ (1.17)

where ${\sigma }_1>0$ and $R>0$ by definition whereas ${{\sigma }^{\prime }}_1$ can be in principle positive or negative.

The aim of the present paper is to show what has to do $F^{\mathrm{<mo>\; *\; </mo>}}$ with the fine structure constant $\alpha$, whose acknowledged numerical value is

$\alpha =\mathrm{0.007297352664.}$ (1.18)

As concerns this number, recall that implementing

$e=4.8025\times {10}^{-10}\text{e}\text{.s}\text{.u}\mathrm{.},G=6.67430{10}^{-8}{\text{cm}}^3\cdot {\text{g}}^{-1}\cdot {\text{s}}^{-2}$

one finds that [1]

$e=k_g\alpha G,k_g=0.986{\text{g}}^{3/2}\cdot \text{s}\cdot {\text{cm}}^{-3/2}i\mathrm{.}e\mathrm{.}\frac{eG}{\hslash c}\approx 1.01{\text{cm}}^{3/2}\cdot {\text{g}}^{-3/2}\cdot {\text{s}}^{-1},$ (1.19)

being $k_g$ dimensional proportionality constant. The fact that this constant is numerically very close to 1 cannot be accidental, rather it shows that reasonably $\alpha$ provides a direct physical link between e and G. The explanation and related implications have been concerned in [1] ; however here it is worth emphasizing that even this relationship must be someway inferable in a general conceptual frame aimed to understand profoundly the physical meaning of $\alpha$ and to calculate its numerical value.

The standard Fibonacci sequence is, per se, a mere recursive list of pure numbers; nevertheless it actually appears in several events and objects occurring in Nature [2] [3] [4] [5]. For example it is known that it describes the shape of nautilus shell and spirals during hurricane formation, undergoing cancer cell division and filament profile during spiral galaxy formation, chicken egg contour and curls of numerous insects [6] [7] [8], to name just a few. In principle nothing requires the correspondence between sequence of pure numbers and natural occurrences quantifiable through lengths, energies and momenta; as a matter of fact, however, the spirals consisting of arcs of circles inscribed into contiguous squares with size ratios progressively increasing in agreement with the Fibonacci numbers effectively overlap the observed shapes of galaxy arms and insect curls. The chance of examining these dimensionless ratios is just an example of how even pure numbers can take the worth of observable physical evidence. A huge literature is known about how and why this sequence is surprisingly suitable in describing natural phenomena [9]. The chance of acknowledging the existence of well-defined fingerprints while observing largest and smallest natural phenomena cannot be accidental; rather it suggests that, although surprisingly, a mere sequence of pure numbers becomes the key criterion underlying occurrences usually measured through dimensional quantities like lengths or times or velocities and explained through familiar concepts of physics like minimum energy or maximum probability.

In other words, here is the first leading idea of the present paper: it seems sensible to expect that the numbers (1.1) are actually a sort of core boundary condition that controls even the fundamental constants of Nature.

In this way, regardless of how these constants or their combinations govern fields or interactions, these numbers can be identified as templates to which conform from time to time any physical observable; the measure units and their physical dimensions become therefore simply factors to turn the experimental observation into quantifiable data and predicting ability of standard scientific research.

Here is thus the second leading idea of this paper: to demonstrate the existence of hidden templates to which Nature conforms through its fundamental constants.

In this respect the indirect correspondence between pure numbers and natural events seems a limiting restriction to the actual worth of (1.1) and its full implementability; nevertheless it suggests the realistic chance of describing observable phenomena directly through measurable parameters someway related to (1.4), which requires in turn to convert the pure numbers of the sequence (1.1) into dimensional quantities like (1.9) with specific physical meaning. The basic step in this respect is the idea of extending the early Fibonacci sequence in order to calculate both dimensionless and dimensional constants of the Nature, like $\alpha$ or proton to electron mass ratio $r_{pe}$ or Avogadro number $N_A$ on the one hand and electron charge e or Boltzmann constant $K_B$ or G on the other hand.

In fact, this will be done implementing appropriately the coefficients $f_0$ and $f_1$.

To understand this point in agreement with (1.7) and (1.9), note that according to (1.5) $F^{\mathrm{<mo>\; *\; </mo>}}$ is sum of two sequences obtained multiplying (1.1) a first time by $f_0$ and a second time by $f_1$ and next collecting the particular terms $f_1{F}_{n-1}$ and $f_0{F}_{n-2}$ as in (1.4). Analogously multiply all terms of F a first time by an arbitrary dimensional factor $k_a{f}_1$, so that all $F_n$ turn into $k_a{f}_1{F}_n$, and next by another arbitrary dimensional factor $k_b{f}_0$, so that all $F_n$ turn now into $k_b{f}_0{F}_n$ ; the two sequences obtained in this way consist of terms with well defined physical dimensions, in general different when $k_a\ne k_b$. If for example the dimensional factor $k_a$ is defined as cm⁻¹∙g⁻¹∙s⁻¹, then the $f_1{k}_a$ means $f_1$ per unit length, unit mass and unit time. If more realistically $k_a$ is unit energy and $k_b$ unit momentum, then the sequence (1.1) early introduced to describe uniquely the population increment of rabbits now generates two separate sequences concerning possible time evolutions of energy and momentum of a physical system; note however that nothing changes from a mere numerical point of view, i.e. energy and momentum sequences are still defined by (1.1). Eventually, multiply again these sequences by ${f^{\prime }}_0$ and ${f^{\prime }}_1$ whose physical dimensions are defined by $k_c/k_a$ and $k_c/k_b$ as follows

$\begin{array}{l}{F^{\prime }}_n^{\ast }={f^{\prime }}_0{F^{\prime }}_{n-2}+{f^{\prime }}_1{F^{\prime }}_{n-1},{f^{\prime }}_0=f_0\frac{k_c}{k_a},{f^{\prime }}_1=f_1\frac{k_c}{k_b},\\ {F^{\prime }}_{n-2}=F_{n-2}{k}_a,{F^{\prime }}_{n-1}=F_{n-1}{k}_b\end{array}$ (1.20)

or identically

$\begin{array}{l}{F^{″}}_n^{\ast }={f^{″}}_0{F^{″}}_{n-2}+{f^{″}}_1{F^{″}}_{n-1},{f^{″}}_0=f_0{k}_c{k}_a,{f^{″}}_1=f_1{k}_c{k}_b,\\ {F^{″}}_{n-2}=\frac{F_{n-2}}{k_a},{F^{″}}_{n-1}=\frac{F_{n-1}}{k_b},\end{array}$ (1.21)

being $k_c$ a third dimensional factor. So the addends ${f^{\prime }}_0{F^{\prime }}_{n-2}$ and ${f^{\prime }}_1{F^{\prime }}_{n-1}$ have both physical dimension $k_c$ and therefore can be summed up likewise in (1.5), which yields indeed ${F^{\prime }}_n^{\ast }=k_c{F}_n^{\ast }$ in this example. Since all factors just introduced are in principle arbitrary, but actually defined here for convenience unitary with appropriate physical dimensions, in fact (1.20) and (1.21) keep identically the numerical features of (1.7) although describing through $k_a$ and $k_b$, for example, unit energy and unit momentum. The primed notations emphasize that ${F^{\prime }}_n^{\ast }$ at the left hand side and right hand side of (1.20) are numerically identical to $F_n^{\ast }$ of (1.4), being in turn ${f^{\prime }}_0$ and ${f^{\prime }}_1$ numerically identical to $f_0$ and to $f_1$ as well; so the numerical values of the early addends $f_0{F}_{n-2}$ and $f_1{F}_{n-1}$ of $F_n^{\ast }$ are now directly referable to different physical quantities concerning the physical features of any system. The same considerations hold for (1.21).

In fact the current literature about the physical and biological implications of the standard sequence (1.1) concerns essentially contour profiling, shape matching and considerations on the importance of the golden ratio inherent its recursive rule only. All of this seems astonishing on the one hand, but also reductive on the other hand; the sequence of numbers (1.1) represents a preferential principle of Nature, quantitative formulas should be inferable likewise as from any physical law. The present paper proposes a mathematical model to overcome the gap between quantitative physical data, i.e. the acknowledged values of some physical constants, and results predictable via (1.3) through direct and detailed calculations assigning specific physical meaning to f₀ and f₁.

At this point however the true challenge is to demonstrate how far an appropriate choice of dimensional units really allows describing in fact a physical system; the goal is to implement explicitly ${F^{\prime }}_n^{\ast }$ and ${F^{″}}_n^{\ast }$ without changing the numerical value of $F_n^{\ast }$, anyway governed by the Fibonacci recursive rule. The chance of obtaining physical information must still hold while preserving the numerical values of two generalized Fibonacci sequences merged together.

The Section 2 introduces some preliminary hints about the fine structure α; this allows to calculate a first approximation numerical value of this constant in the Section 3. The Section 4 shows that with the help of the extended Fibonacci sequence the numerical value of α can be calculated with improved approximation, one order of magnitude better. The Section 5 shows how to reveal via (1.3) with the help of (1.20) and (1.21) the subtle link between various fundamental constants, thus clarifying the concept of template itself. The importance of (1.20) and (1.21) will appear in the Section 5: owing to the numerical coincidence of ${f^{\prime }}_0$ and ${f^{\prime }}_1$ with $f_0$ and $f_1$, and thus that of ${F^{\prime }}_n^{\ast }$ with $F_n^{\ast }$, the calculations will be carried out implementing (1.4) although ${f^{\prime }}_0{F^{\prime }}_{n-2}$ and ${f^{\prime }}_1{F^{\prime }}_{n-1}$ will actually concern dimensional physical quantities. Accordingly, the paper consists in fact of two parts: the former one describes a “standard” approach based on (1.17) to infer $\alpha$, the latter one clarifies how actually the fundamental constants of Nature conform themselves to hidden templates compliant with a unique recursive rule, still that guessed by Fibonacci.

The text is organized with the main intention of making the exposition as self-contained as possible. For this reason also a few basic concepts of classical electromagnetism are shortly quoted below.

2. The Fine Structure Constant: Preliminary Considerations

Before obtaining an approximate numerical estimate of the dimensionless constant $\alpha$ that characterizes the electromagnetic interaction, introduce first a few well-known concepts indicating how to start from first principles. Consider preliminarily the possibility of writing

$\alpha =\frac{e^2}{\hslash c}=\xi \frac{\epsilon \delta r}{\hslash c},$ (2.1)

where $\xi /\hslash c$ is the proportionality constant linking $\epsilon \delta r$ to $\alpha$. Owing to this physical meaning of $\alpha$, try to identify an energy $\epsilon$ and length $\delta r$ considering a bound system of charges. In particular it is sensible to think $\epsilon \delta r$ at the right hand side as product of hydrogen-like atom properties, whose analytical expressions are known and simple. As are useful short reminds about outcomes of elementary wave mechanics for a charged reduced mass m in a central field of nuclear charge $Ze$, introduce

$\epsilon =-\frac{Z^2{e}^{4}m}{2n^2{\hslash }^2},\delta r=\frac{n^2{\hslash }^2}{Zm{e}^2},m=\frac{m_{e}A}{m_e+A},$ (2.2)

where $m_e$ and A are for example electron reduced and nucleus masses. In several papers, e.g. [10] [11], these formulas have been found without solving any wave function, but as straightforward consequence of the statistical formulation of the quantum uncertainty only

$\delta p\delta x=n\hslash =\delta \epsilon \delta t,$ (2.3)

which in turn is a corollary itself of an operative definition $\hslash G/c^2$ of space time [11] in the frame of an evolutionary quantum model of Universe [12]. Anyway, multiplying side by side the first two (2.2) one finds

$\epsilon \delta r=-\frac{\left(Ze\right)e}{2},$ (2.4)

whence $\left|\xi \right|=2/Z$ so that

$\alpha =\frac{2}{Z}\frac{\epsilon \delta r}{\hslash c}.$ (2.5)

The remarkable simplicity of (2.2) is the reason of having introduced just the non-relativistic hydrogen-like atom, and not for example a complex many electron atoms: the latter implies complex many electron correlations and thus the lack of analytical formulas able to include various forms of interaction, the former is implemented as electromagnetic interaction between two charges only compliant with $\left|\left(Ze\right)e\right|$ defining $\alpha$. This kind of interaction appears explicitly because it also follows that

$\begin{array}{l}\left|\epsilon \right|=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^{2}m{c}^2=\frac{\left(Ze\right)e}{2\delta r},\\ \delta r=\frac{n\hslash }{mc}\left(\frac{n}{\alpha Z}\right)=n{\lambda }_C\left(\frac{n}{\alpha Z}\right),\frac{\hslash }{mc}={\lambda }_C;\end{array}$ (2.6)

the systematic presence of $\alpha Z/n$ suggests that just $\alpha Z$ is the key quantity to understand $\alpha$. Despite (2.2) include neither spin/orbit nor spin-spin electron/nucleus interaction or the Lamb shift, conceptually crucial even for $Z=1$ although in fact numerically negligible, valuable information is in fact hidden in these equations; this suggests implementing (2.2) as an acceptable basis in defining both $\epsilon$ and $\delta r$ of (2.1). The factor 1/2 is the fingerprint of the quantum uncertainty: although m and nucleus are $\delta r$ apart, the Coulomb-like form requires the total uncertainty range of the charge $-e$ with respect to $Ze$ that indeed is $2\delta r$ i.e. the total radial delocalization range of the electron around the nucleus. Of course $\delta r$ is itself arbitrary and unknowable because n is arbitrary [1] ; thus the physical content of (2.6) is well beyond that of the mere Coulomb law. Moreover it is easy to realize with the help of (2.3) that

$\frac{\delta r}{\delta t}=\frac{n{\lambda }_C}{\delta t}\frac{n}{\alpha Z}=\frac{{\lambda }_C\delta \epsilon }{\hslash }\frac{n}{\alpha Z}=\frac{\delta \epsilon }{mc}\frac{n}{\alpha Z}=v_r,v_r=\frac{\delta r}{\delta t},$ (2.7)

being $v_r$ the average radial velocity of m. So, owing to (2.6),

$\frac{\alpha Z}{n}=\frac{\delta \epsilon /m{v}_r}{c}=\frac{\delta v_r}{c}$ (2.8)

having defined $\delta v_r$ as $\delta \epsilon /m{v}_r$ by dimensional reasons; indeed

$\delta \epsilon =m{v}_r\delta v_r=m\delta \left(v_r^2\right)/2i.e.\epsilon =m{v}_r^2/2+const$ (2.9)

is trivially the classical kinetic energy of a free particle of mass equal to the reduced mass m moving at radial velocity $v_r$. However the non-trivial consequence of this short analysis is that the early Bohr energy can be expressed not only via Coulomb energy, as it is obvious, but also via an expression not explicitly referred to charge interaction; rather it is simply related to the rest mass $m{c}^2$ via $\alpha Z/n$, because now $e^4$ of (2.2) becomes hidden in ${\alpha }^2$ of (2.6). However, although $\epsilon$ is even expressed as classical kinetic energy of reduced mass m moving at velocity $v_r$, the quantization is still evident through (2.7).

At this point, before implementing these results to evaluate $\alpha$, note that just (2.6) suggest further chances to manipulate the early Bohr formulas to highlight explicitly the electromagnetic character of the Coulomb interaction via the electromagnetic field associated to a radial electromagnetic wave: if it is true that magnetic field H is generated by moving charges, then this H and the electric field E itself should be someway both hidden in (2.2). Also, the force $\Phi$ acting on m due to these fields should also be contextually inferable. To show these points note first of all that trivial manipulations of (2.2) yield

$\begin{array}{l}\left|\epsilon \right|=\frac{Z^2{e}^5{m}^2{\hslash }^2}{2n^2{\hslash }^{4}em}=\frac{E{\hslash }^{2}c}{2emc}=\frac{E\hslash {\lambda }_{C}c}{2e}=\frac{E{\mu }_B}{\alpha },\\ E=\frac{Z^2{e}^5{m}^2}{n^2{\hslash }^4}={\left(\frac{\alpha Z}{n}\right)}^2\frac{e}{{\lambda }_C^2},{\mu }_B=\frac{e{\lambda }_C}{2}\end{array}$ (2.10)

and also

$\begin{array}{l}\frac{e{\lambda }_C}{{\mu }_B}=2,\left|\epsilon \right|=\frac{{\mu }_B}{e{\lambda }_C}{\left(\frac{\alpha Z}{n}\right)}^{2}m{c}^2=\frac{\left(Ze\right)e}{2\delta r}=H{\mu }_B,\\ H=\frac{m{c}^2}{e{\lambda }_C}{\left(\frac{\alpha Z}{n}\right)}^2=\frac{Ze}{{\lambda }_C\delta r},\end{array}$ (2.11)

which define the moduli E and H of electric and magnetic fields inside the atom; contextually is even defined the Bohr magneton ${\mu }_B$. In fact (2.11) suggest what is well known, i.e. that charges randomly moving around the nucleus induce H, which however at the atom scale is quantized itself. Multiplying side by side (2.10) and (2.11), implement thus

${\epsilon }^2=\frac{EH{\mu }_B^2}{\alpha }\mathrm{.}$ (2.12)

Moreover, since

${\mu }_B^2=\frac{e^2{\lambda }_C^2}{4}=\frac{e^2{\lambda }_C^3}{4{\lambda }_C}=\frac{3}{16\pi }\frac{e^2}{{\lambda }_C}{V}_C=\frac{3}{16\pi }\eta V_C,V_C=\frac{4\pi }{3}{\lambda }_C^3,\eta =\frac{e^2}{{\lambda }_C},$ (2.13)

it follows then

${\epsilon }^2=\frac{3}{16\pi }\frac{EH{V}_C\eta }{\alpha };$ (2.14)

hence, according to the first (2.6),

$\frac{{\epsilon }^2}{\eta }=\frac{Z^2{e}^4}{4\delta r^2}\frac{{\lambda }_C}{e^2}=\frac{Z}{2}\frac{{\lambda }_C}{\delta r}\frac{Z{e}^2}{2\delta r}=\frac{Z}{2}\frac{{\lambda }_C}{\delta r}\left|\epsilon \right|.$ (2.15)

So, merging (2.14) and (2.15), from

$\frac{{\lambda }_C}{\delta r}\left|\epsilon \right|=\frac{3}{8\pi \alpha Z}EH{V}_C$ (2.16)

and owing to (2.6)

$\frac{\alpha Z}{n^2}=\frac{{\lambda }_C}{\delta r}$ (2.17)

one finds

$\left|\epsilon \right|=\frac{3}{8\pi }{\left(\frac{n}{\alpha Z}\right)}^{2}EH{V}_C,\frac{EH}{\left|\epsilon \right|}={\left(\frac{\alpha Z}{n}\right)}^2\frac{2}{{\lambda }_C^3}.$ (2.18)

Note now that if the electric and magnetic fields of electromagnetic waves are orthogonal, then in fact numerically $EH=\left|E\times H\right|$ while remaining true of course that the modulus at right hand side is related to cross vector product that points towards the propagation direction of the wave. In this sense a running plane wave is also described by and equivalent to the quantized energy in the volume $V_C$ of hydrogenlike atom; i.e. photons generated by electron decay between discrete energy levels propagate as steady e.m. waves. In other words the energy density field HE of e.m. wave in $V_C$ is quantized, whereas photons are just the energy quanta of field corresponding to

$\delta \left|\epsilon \right|=\delta \left|HE{V}_C\right|$ (2.19)

between n-th allowed levels. Note in this respect that the second couple of (2.3) reads

$\delta \left|\epsilon \right|=n\hslash /\delta t=n\hslash \omega ,$ (2.20)

in agreement with Planck, whereas the first couple reads $\delta p=nh/2\pi \delta x$. Then

$2\pi \delta x=n\lambda \text{if}p=p_0+\frac{h}{\lambda }$ (2.21)

according to De Broglie wave definition of momentum with $p_0$ arbitrary constant.

So the uncertainty requires on the one hand the corpuscular nature of the charge e, see Equations (2.9) and (2.2) itself, but also wavelike energy and momentum on the other hand, see oscillating fields E and H that describe an e.m. wave propagating within $V_C$. So the wave confined in this volume cannot be radiated without changing n, whereas photons can be however emitted/absorbed compatibly with the Pauli principle. Obviously analogous conclusion holds for any quantum system of charges bound with interaction strength constant $\alpha$ : given $\left|\epsilon \right|$, the greater the numerical value of $\alpha$, the greater the energy density EH. In conclusion, at the left hand side of (2.19) still appears the well known quantized Bohr energy, at the right hand side appears the modulus EH of the Poynting vector $E\times H$ ; the proportionality constant between the energies involves $\alpha$. As the last (2.11) shows that eH has physical dimensions of force, the question at this point is whether this force can be guessed and inferred itself straightforwardly from the basic Equations (2.2). While (2.2) are well known, they have marked the birth of the old quantum mechanics, nothing in principle excludes the chance of defining also a further equation alternative to (2.4); note indeed that appears at left hand side of (2.16) the ratio $\left|\epsilon \right|/\delta r$. So, owing to the definitions of ${\lambda }_C$ and ${\mu }_B$, write with the help of second (2.2)

${\Phi }_r=\frac{\left|\epsilon \right|}{\delta r}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{m^2{c}^3}{n\hslash }=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{m{c}^2}{n{\lambda }_C}=m{a}_r,a_r=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{c^2}{n{\lambda }_C},$ (2.22)

where ${\Phi }_r$ has in fact physical dimensions of force; in effect this force appears reasonably expressed as m times a new amount $a_r$ having physical dimensions of acceleration, quantized itself too. If so, then the force here acknowledged acting on m in the field of nucleus is the radial component of a more general quantized electromagnetic interaction force of modulus $\Phi =\left|\Phi \right|$ between nucleus and electron at Bohr radial distance $\delta r=\left|\delta r\right|$ around it, thus delocalized in an ideal sphere of diameter $2\delta r$ centered on the nucleus. This explains why $a_r$ results defined via $c^2$ over n times the length $2{\lambda }_C$ compliant with the quantum uncertainty. Replacing in (2.16), one finds therefore owing to (2.17)

${\Phi }_r\delta r=\frac{3}{8\pi {\alpha }^2}EH{V}_C.$ (2.23)

The fact that at the left hand side appears ${\Phi }_r\delta r$ suggests putting

$\Phi \cdot \delta r=\left(\Phi \delta r\right)\cos \phi ,{\Phi }_r=\left|\Phi \right|\cos \phi ,$ (2.24)

being $\phi$ an appropriate angle. Before concerning in some more detail (2.23) and $\phi$ of (2.24), note that the definitions (2.10) and (2.11) allow inferring immediately from the radial force component ${\Phi }_r$ further well-known concepts that confirm the validity of the present considerations. Trivial manipulations of (2.22) yield with the help of (2.10)

$\begin{array}{l}{\Phi }_r=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{m{c}^{2}e}{2n{\mu }_B}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{m^2{c}^{2}e}{2mn{\mu }_B}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{{\left(mc\right)}^{2}c\gamma }{n{\mu }_B}\\ =\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{{\hslash }^{2}c\gamma }{n{\lambda }_C^2{\mu }_B},\\ \gamma =\frac{e}{2mc},\end{array}$ (2.25)

whence

$\begin{array}{l}{\Phi }_r=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{{\left(mc\right)}^2\gamma c}{n{\mu }_B}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{M^{2}c}{n{\lambda }_C^{2}j\left(j+1\right)}\frac{\gamma }{{\mu }_B},\\ M^2={\hslash }^{2}j\left(j+1\right),\frac{e{\lambda }_C}{{\mu }_B}=2\end{array}$ (2.26)

once having introduced the square angular momentum $M^2$ of m around the nucleus as a function of $j=l\pm s$ ; then it is possible to write

${\Phi }_r={\left(\frac{\alpha Z}{n}\right)}^3\frac{{\left|M\right|}^{3}c{\gamma }^2}{nj\left(j+1\right){\lambda }_C^2{\mu }_B^2},\frac{1}{2}=\gamma \frac{\left|M\right|}{{\mu }_B}$ (2.27)

and thus, recalling the last (2.11) and that ${\mu }_B/\gamma =\hslash$,

${\Phi }_r={\left(\frac{\alpha Z}{n}\right)}^3\frac{\hslash c\sqrt{j\left(j+1\right)}}{n{\lambda }_C^2}.$ (2.28)

As concerns the second (2.27), especially interesting is the particular case where $\left|M\right|=\left|L+S\right|$ reduces to the angular momentum of spin $\left|S\right|$ of m only, in which case the second (2.27) yields the so-called Land? factor of m; this equation reads indeed

$2=\frac{1}{\gamma }\frac{{\mu }_B}{\left|S\right|},\frac{1}{\gamma }=\frac{2mc}{e},$ (2.29)

Note however that usually $\gamma$ and ${\mu }_B$ are expressed through the rest mass $m_e$ corresponding to m rather than through the reduced m itself. For example, in the case of a hydrogen-like atom with one electron, (2.10) and (2.25) yield

$\begin{array}{l}\frac{1}{\gamma }=\frac{2mc}{e}=\frac{2\left(m/m_e\right)m_{e}c}{e}=\frac{m}{m_e}\frac{1}{{\gamma }_e},\\ {\mu }_B=\frac{e\hslash }{2mc}=\frac{e\hslash m_e}{2m_{e}mc}={\mu }_{eB}\frac{m_e}{m},\frac{m_e}{m}=\frac{m_e+A}{A};\end{array}$

hence replacing $\gamma$ and ${\mu }_B$ so far defined with ${\gamma }_e$ and ${\mu }_{eB}$, (2.29) turns into

$2=\frac{1}{\gamma }\frac{{\mu }_B}{\left|S\right|}=\frac{m}{m_e}\frac{1}{{\gamma }_e}\frac{m_e}{m}\frac{{\mu }_{eB}}{\left|S\right|}$

so that

$2=\frac{1}{{\gamma }_e}\frac{{\mu }_{eB}}{\left|S\right|},\frac{1}{{\gamma }_e}=\frac{2m_{e}c}{e},$

as expected in non-relativistic approach. Moreover, since from (2.6)

$\frac{1}{{\lambda }_C}{\left(\frac{\alpha Z}{n}\right)}^2=\frac{He}{m{c}^2},$

it also follows

$\frac{m{c}^2}{{\lambda }_{C}H}{\left(\frac{\alpha Z}{n}\right)}^2=e,\gamma =\frac{1}{2}\frac{c}{{\lambda }_{C}H}{\left(\frac{\alpha Z}{n}\right)}^2=\frac{2\pi \nu }{H},\nu =\frac{c}{2{\lambda }_C}{\left(\frac{\alpha Z}{n}\right)}^2,$ (2.30)

being clearly $\nu$ a frequency. Thus, results defined

$\omega =H\gamma =\frac{eH}{2mc},$

being $\nu =H\gamma /2\pi$ the Larmor frequency.

Furthermore, comparing last (2.30) and first (2.6) one finds $\left|\epsilon \right|=\nu {\lambda }_{C}mc=\left(\nu h\right)\left({\lambda }_C/h\right)mc$, where $\nu {\lambda }_C$ has physical dimensions of velocity; so being ${\lambda }_C=h/mc$ it follows that by definition $\left|\epsilon \right|=h\nu$ whereas ${\lambda }_C/h=p$ by dimensional reasons, which are the old Planck and De Broglie relationships already identified while commenting (2.18). Note that all relations are quantized, being inferred from early (2.2) only, i.e. without any consideration about the classical definitions of angular and magnetic momenta. Moreover, dividing side by side (2.22) and (2.11), one finds

$\frac{{\Phi }_r}{H}=\frac{1}{2}\left(\frac{\alpha Z}{n}\right)\frac{e}{n};$ (2.31)

so according to (2.30)

$\frac{n}{\alpha Z}\frac{v_{\lambda }}{c}=\frac{1}{2}\left(\frac{\alpha Z}{n}\right),v_{\lambda }=\nu {\lambda }_C$

and thus replacing in (2.31)

${\Phi }_r=H\frac{v_{\lambda }e}{c},v_{\lambda }=\frac{{\lambda }_C\nu }{\alpha Z},$

being $v_{\lambda }$ the modulus of velocity $v_{\lambda }$ calculated via Compton length of m times the frequency defined in (2.30). The vector equation corresponding to this scalar relationship can be nothing else but

$\Phi =e\frac{v_{\lambda }}{c}\times H$

as stated before about (2.18), the magnetic field $H$ of the wave along with the electric field $E$, must be orthogonal to its direction propagation velocity $v_{\lambda }$. Hence, $v_{\lambda }\times H=v_{\lambda }Hn$, with the unit vector $n$ that defines the propagation direction of a transversal e.m. plane wave made by orthogonal magnetic and electric oscillating fields, whose quantization implies the concept of photon.

Nevertheless this elementary approach has a further interesting implication. Define the function

$G{\Phi }_r=\frac{1}{2}G{\left(\frac{\alpha Z}{n}\right)}^3\frac{m^2{c}^3}{n\hslash }=X{c}^4,$ (2.32)

where G is the gravity constant and thus X a dimensionless function to be found; indeed the physical dimensions of the product G times force are velocity⁴. Since

$X=G\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{m^2}{n\hslash c},$

then

$X=G\frac{m^2}{{ϵ}_0\lambda },{ϵ}_0\lambda =n\frac{\hslash c}{\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3}$

being $\lambda$ an arbitrary length and ${ϵ}_0$ an energy defined as

${ϵ}_0=\frac{n\hslash {\omega }_0}{\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3},{\omega }_0=\frac{c}{\lambda }.$

So

${ϵ}_0=G\frac{m_1{m}_2}{\lambda },m_1{m}_2=\frac{m^2}{X}.$ (2.33)

Note that being $\lambda$ arbitrary, are also arbitrary $m_1{m}_2$ and thus $m_1$ and $m_2$ themselves. The interesting fact is that even the gravitational law appears to be nested in a natural way in the present elementary approach.

This result supports the validity of (1.19). Eventually calculate also

$\frac{G}{{\Phi }_r}=\frac{G}{m{a}_r}={\left(\frac{\mathcal{l}}{m}\right)}^2,$

where $\mathcal{l}$ is another arbitrary length; the last equality in inferred by dimensional reasons. Thus, owing to the second (2.22),

$\mathcal{l}={\left(\frac{mG}{a_r}\right)}^{1/2},a_r=\frac{mG}{{\mathcal{l}}^2}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{c^2}{n{\lambda }_C}$

that defines $\mathcal{l}$. Consider now the particular value ${\mathcal{l}}_{bh}$ of the length $\mathcal{l}$ defined as follows

${\mathcal{l}}_{bh}=\frac{2mG}{c^2},$ (2.34)

which implies

$\frac{2mG}{c^2}={\left(\frac{mG}{a_r}\right)}^{1/2},\frac{4{\left(mG\right)}^2}{c^4}=\frac{mG}{a_r},a_r^{\left(bh\right)}=\frac{c^4}{4mG}=\frac{1}{2}\frac{c^2}{{\mathcal{l}}_{bh}};$

the form of the particular acceleration $a_r^{\left(bh\right)}$ is the same as that reported in (2.22) for $a_r$. Note that

$a_r^{\left(bh\right)}\frac{\mathcal{l}}{c^2}=\frac{\mathcal{l}{c}^2}{4mG}=\frac{1}{\delta \varphi }$ (2.35)

the last equality is obvious, thinking that in general $v^2/r$ is the classical centripetal force acting on a point mass traveling along a circular path at tangential speed v. So $\mathcal{l}\delta \varphi$ is the length of an arc of circumference defined by the angle $\delta \varphi$ traveled by the point mass. The fact that here appears in particular $v\equiv c$ means that this elementary interpretation holds for a photon. Thus

$\delta \varphi =\frac{4mG}{\mathcal{l}{c}^2}$ (2.36)

In effect, this result is nothing else that the deflection angle $\delta \varphi$ of a photon moving in the gravity field of m at distance $\mathcal{l}$, which corresponds to the famous light deflection of light beam traveling in a curved space time; the first order approximation of Einstein approach corresponds here to having approximated via (2.35) the actual photon path just along a circular arc.

This short and elementary discussion has shown that basic concepts of classical electromagnetism are hidden in the simple Bohr results (2.2), in turn inferable as straightforward and serendipitous corollaries of (2.3). Moreover this result shows once more what has been emphasized several times in previous papers [10] [11] [12] [13], i.e. the intimate and natural merging of quantum physics and relativity.

It is useful therefore to show shortly here how all results hitherto inferred are related to classical physics on the one hand and relativistic physics on the other hand; of course here this last aspect of the problem is shortened as much as possible, it has been more thoroughly examined in [11] [12]. Nevertheless, simple and short final remarks allow extending these non-relativistic concepts simply implied by the early Bohr approach. Define indeed more in general ${\Phi }_r$ acting on the electron in the field of nucleus as $\Phi$ is no longer required to be radial component as prospected by the second (2.24), whereas the electron delocalization around the nucleus is described by $\delta r\cos \phi$. Then assuming $\phi =\phi \left(\alpha \right)$, as it is natural owing to (2.24) and (2.22), and expanding in series $\cos \phi$ around $\phi =0$ one finds recalling (2.22)

$\mathcal{E}={\epsilon }_B\left(1+{\zeta }_1\alpha +{\zeta }_2{\alpha }^2+\cdots \right)={\epsilon }_B+\cdots ,{\epsilon }_B=\Phi \delta r,{\zeta }_j={\zeta }_j\left(Z,A,m,N\right),$ (2.37)

being clearly ${\zeta }_j$ the coefficients of series expansion; in this way the early Bohr energy ${\epsilon }_B$ introduced in (2.2) and so far implemented results to be in fact the first order term only of a more complex energy $\mathcal{E}$ in which are hidden further forms of interaction. This latter statememt can be explicitly evidenced writing

$\mathcal{E}={\epsilon }_B+\left({\zeta }_1\alpha +{\zeta }_2{\alpha }^2+{{\zeta }^{\prime }}_3{\alpha }^3+\cdots \right){\epsilon }_B+{{\zeta }^{″}}_3{\alpha }^3{\epsilon }_B+\cdots ,{\zeta }_3={{\zeta }^{\prime }}_3+{{\zeta }^{″}}_3;$ (2.38)

in this way the first addend ${\epsilon }_B$ of $\mathcal{E}$ is the Bohr zero order term, the second addend yields the electron correlation terms as a function of $\alpha$ for a number of electrons $N\ge 1$, which is certainly possible in principle determining appropriately the series coefficients ${\zeta }_j$ of (2.37). In particular the third addend $\propto {\alpha }^5$ is the fingerprint of the Lamb energy due to the “vacuum” polarization. Even for $N=1$ appear thus expectable in principle the electron spin-orbit and spin-spin interactions missing in (QQQ). These higher order effects have been considered in [12] [13] simply starting from the statistical formulation of quantum uncertainty, but will not be considered here for brevity. However it is enough noticing that rewriting (2.3) as $\delta \epsilon =v\delta p$, with velocity modulus defined by $v=\delta x/\delta t$, and multiplying both sides by $\epsilon$, one finds $\epsilon \delta \epsilon =\epsilon v\delta p$. To shorten as much as possible this discussion, put now by definition $p=\sigma \epsilon v$, where $\sigma$ is a proportionality constant; this position is purposeful to obtain $\epsilon \delta \epsilon =\sigma \epsilon v\delta \left(\epsilon v\right)$ that reads then

$\frac{1}{2}\delta \left({\epsilon }^2\right)=\frac{1}{2}\sigma \delta {\left(\epsilon v\right)}^2,\sigma =constant\mathrm{.}$

Hence $\delta \left({\epsilon }^2\right)=\delta \sigma {\left(\epsilon v\right)}^2$ yields identically

$\delta \left({\epsilon }^2\right)=\delta \left(\sigma {\left(\epsilon v\right)}^2+const\right)\mathrm{,}$ (2.39)

because of course $\delta const=0$ by definition. So the last equation reads

${\epsilon }^2=\sigma {\left(p/\sigma \right)}^2+const\mathrm{.}$

Since by dimensional reasons $\sigma =velocit{y}^{-2}$, this constant velocity can be nothing else but $\sigma =c^{-2}$. So

${\epsilon }^2={\left(pc\right)}^2+const,p=\epsilon \frac{v}{c^2}\mathrm{.}$ (2.40)

Moreover implement the boundary condition for $v\to 0$ to find const, which has physical dimensions of square energy: as

$\underset{v\to 0}{\lim }\frac{p}{v}=m=\frac{{\epsilon }_0}{c^2},$

then (2.39) reads

${\epsilon }_0^2={\left(m{c}^2\right)}^2,\underset{v\to 0}{\lim }{\epsilon }^2=const={\epsilon }_0^2={\left(m{c}^2\right)}^2.$ (2.41)

Clearly (2.41) and (2.40) imply the Lorentz factor $\sqrt{1-v^2/c^2}$. Calculate with the help of (2.2) and (2.40) via (2.3), then

${\left(\delta \epsilon \right)}^2-{\left(\delta pc\right)}^2=\frac{n^2{\hslash }^2}{{\left(\delta t\right)}^2}-\frac{n^2{\hslash }^2{c}^2}{\delta x^2}={\left(n\hslash \right)}^2\frac{{\left(\delta x\right)}^2-{\left(\delta ct\right)}^2}{{\left(\delta x\delta t\right)}^2}\mathrm{<mo>\; ;\; </mo>}$

if one implements arbitrary ranges $\delta \epsilon ={\epsilon }_2-{\epsilon }_1$ and $\delta p=p_2-p_1$ at the left hand side with $\epsilon$ and p fulfilling (2.40), it follows that the ratio at the right hand side must be invariant. This means

${\left(\delta x\right)}^2-{\left(c\delta t\right)}^2=inv,\delta x\delta t=inv\mathrm{.}$

Of course the same holds considering ${\left(\delta pc\right)}^2-{\left(\delta \epsilon \right)}^2$ that would yield ${\left(c\delta t\right)}^2-{\left(\delta x\right)}^2=inv$. These invariant results are actually the signature of the quantum uncertainty; they hold not only for a free particle, but also for a particle of reduced mass m subjected to e.m. interaction. It is known that the 4D invariant interval, here symbolized by $\delta x^2$ only instead of $\delta x_i^2\mathrm{,}i=1,2,3$ for 3 space coordinates, is the conceptual basis of special relativity [14].

Finally implement Equation (2.8) dividing both sides by $c^2\delta t$ and write according to (2.7)

$\frac{\delta \epsilon /m{c}^2}{v_r\delta t}=-\frac{\delta \varphi /c^2}{\delta r}=-\frac{\delta v_r}{\delta t},\varphi =\frac{\epsilon }{m},$ (2.42)

so that, replacing $\epsilon =\hslash \omega$ to describe in particular a photon in the field $\varphi$, one finds the relationship $\delta \left(\hslash \omega /m{c}^2\right)=\delta \varphi /c^2$ ; thus the respective energy changes yield

$\frac{\delta \omega }{{\omega }_0}=\frac{\delta \varphi }{c^2},{\omega }_0=\frac{m{c}^2}{\hslash }.$ (2.43)

Being $\delta \omega$ arbitrary, define in general $\delta \omega ={\omega }_1-{\omega }_0$. Note that (2.42) has the form of one dimensional space gradient component $\delta \varphi /\delta r$ of the function $\varphi$ equal to velocity component change rate ${\stackrel{\dot{}}{v}}_r=\delta v_r/\delta t$, i.e. in the usual 3D form $\nabla \varphi =-\stackrel{\dot{}}{v}$, typical of the energy potential [14]. So, regarding the left hand side of (2.43) as the energy change of a photon in the field gradient $\varphi$, the result

$\frac{\delta \omega }{{\omega }_0}=\frac{{\varphi }_s-{\varphi }_o}{c^2}$

concerns energy loss $\delta \omega <0$ of a photon moving against the field strength gradient; the subscripts stand for source and observer respectively. This equation is in particular the red shift of a photon beam moving from the emission point of a source, ${\varphi }_s$, towards the observation point, ${\varphi }_o$, with decreased energy potential ${\varphi }_s<{\varphi }_o$.

These short considerations have evidenced the importance of the constant $\alpha$ in determining the physical properties of hydrogenlike atoms and the existence of quantized electromagnetic field required by the elementary Bohr formulas. Mostly important, this elementary analysis of the e.m. interaction shows that there is no conflict between QM and relativity; rather, trivial algebraic steps are enough to bridge both theories.

With this introductory background in mind, it is possible to tackle self-consistently in the next sections the problem of determining the numerical value of $\alpha$.

3. First Approximate Calculation of α

After having implemented (2.2) to emphasize a few basic concepts of electromagnetism, is reasonable the attempt to implement also ${\Phi }_r$ to get a preliminary estimate of $\alpha$. Note that the Equations (2.22) contain explicitly m, whose value determines whether these equations describe specifically hydrogenlike atoms or mesic atoms or even a proton/antiproton system; thus it is convenient in principle to rewrite (2.22) in dimensionless form

${\Phi }_r^{*}={\Phi }_r\frac{n\hslash }{m^2{c}^3}=\frac{{\Phi }_r}{2\pi }\frac{nh}{m^2{c}^3}=\frac{n{\Phi }_r}{\pi }\frac{{\lambda }_C}{2m{c}^2}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3$ (3.1)

in order that ${\Phi }_r^{\mathrm{<mo>\; *\; </mo>}}$ holds for any bound system governed by electromagnetic interaction with central charge $Ze$ regardless of the particular reduced mass m of the charge e. First of all (3.1) suggests identifying at the left hand side of the last equality the amount $\left({\Phi }_r/2\right)\left(h/m^2{c}^3\right)n$ such that

$\frac{n{\Phi }_r}{m{a}_r}=\frac{n{\Phi }_r}{{\Phi }_r}=n=\frac{\pi }{2}{\left(\frac{\alpha Z}{n}\right)}^3,{\Phi }_r=m{a}_r,a_r=\frac{2c^2}{{\lambda }_C},$ (3.2)

where clearly $a_r$ is the radial acceleration of m due to the force ${\Phi }_r$. So comparing with (2.22)

$a_r=\frac{2c^2}{{\lambda }_C}=\frac{1}{2}{\left(\frac{\alpha Z}{n}\right)}^3\frac{c^2}{n{\lambda }_C}=\pi {\left(\frac{\alpha Z}{n}\right)}^3\frac{c^2}{{\lambda }_C}$

i.e.

$n^4\frac{2}{\pi }={\left(\alpha Z\right)}^3,\alpha =\alpha \left(Z,n\right).$

In this way, $\alpha$ is a function of n and Z. To define uniquely the value of $\alpha Z$ is interesting to calculate this result in particular for the fundamental state $n=1$ and $Z=Z_{\max }$, which actually calculates the minimum value allowed for $\alpha$. Introducing thus the boundary condition $n=1$ and $Z=Z_{\max }$ that defines ${\Phi }_{\max }^{*}={\Phi }_r^{*}\left(n=1,Z=Z_{\max }\right)$, the last equation reads

$\alpha ={\left(\frac{2}{\pi }\right)}^{1/3}\frac{1}{Z_{\max }}.$ (3.3)

This result can be calculated recalling that in [15] it has been found that the upper limit of high atomic number heavy nuclei is $Z_{\max }=118$. Then

$\alpha ={\left(\frac{2}{\pi }\right)}^{1/3}\frac{1}{118}=0.0072902883,dev=-0.0968\%.$ (3.4)

The deviation of $\alpha$ calculated in this way from the true value (1.18) is due to the non-relativistic approximation of the Equations (2.2); even so, however, the approach hitherto followed appears basically correct.

The agreement provided by this kind of preliminary approach, although carried out through classical concepts of acceleration and force, explains the stability of the quantum system of interacting charges. The definition of acceleration in (3.2) reminds formally that $v^2/r$ of the circular motion of m tethered around a fixed point, which however before the birth of QM has posed the problem of explaining why there is no radiation loss in the case of a bound system of a charge m around a fixed nucleus. The reason is that classically, being in general $v=v\left(t\right)$ and $r=r\left(t\right)$, the emission of radiation is allowed to occur contextually to the spiral motion of the charge m approaching closer and closer to the nucleus till to the collapse of the system. However the definition (3.2) of $a_r$ prevents such behavior, because both c and ${\lambda }_C$ are constant quantities: here instead no change of c and ${\lambda }_C$, and then no collapse. Hence the atomic system does not radiate because m cannot spiral towards the nucleus; the Bohr postulate is actually the constancy of light speed in vacuum. Nevertheless the system can still emit discrete amounts of energy, we call them photons, changing n by integer amounts.

Anyway, a much more effective approach to obtain a better value of $\alpha$ is next carried out implementing the Fibonacci sequence.

4. The Generalized Fibonacci Sequence

Often an approximation is useful to check immediately and easily whether or not any reasoning or algebraic steps point to the right direction in calculating something, provided that however the basic requirements of the calculation model are still valid; for this reason this section starts just from (1.16) and (1.17), where ${\sigma }_1>0$ and $R>0$ by definition whereas ${{\sigma }^{\prime }}_1$ can be in principle positive or negative.

When switching from (1.1) to (1.3), two considerations arise about $f_0$ and $f_1$ replacing 1. The first problem concerns the chances $f_0>1$ and $f_1>1$ or $0<f_0<1$ and $0<f_1<1$, both possible in principle. The second problem is that just the most general meaning of (1.3) requires determining two initial unknowns, a sort of boundary condition without which the sequence is in fact undefined. These requirements are compliant with (1.10) regarding either ${{\sigma }^{\prime }}_j>0$ and ${{\sigma }^{″}}_j>0$ or ${{\sigma }^{\prime }}_j<0$ and ${{\sigma }^{″}}_j<0$ in the definitions (1.12). Actually both requirements are fulfilled in the simplified form (1.15) at the first order only, where ${{\sigma }^{\prime }}_1={{\sigma }^{″}}_1$ ; even neglecting the higher order coefficients, it is enough to regard either ${{\sigma }^{\prime }}_1>0$ or ${{\sigma }^{\prime }}_1<0$, in the following indicated with the notation $-\left|{{\sigma }^{\prime }}_1\right|$. For ${{\sigma }^{\prime }}_1>0$ holds the upper sign only, i.e. the first (1.17) reads

${\alpha }_{+}^o=\frac{-1+\sqrt{1+4{{\sigma }^{\prime }}_{1}R/{\sigma }_1}}{2{{\sigma }^{\prime }}_1},$ (4.1)

whereas for ${{\sigma }^{\prime }}_1<0$ both signs of the solutions are admissible but with the following condition

${\alpha }_{\pm }^o=\frac{-{\sigma }_1\pm \sqrt{{\sigma }_1^2-4\left|{{\sigma }^{\prime }}_1\right|{\sigma }_{1}R}}{-2\left|{{\sigma }^{\prime }}_1\right|{\sigma }_1}=\frac{1\mp \sqrt{1-4R\left|{{\sigma }^{\prime }}_1\right|/{\sigma }_1}}{2\left|{{\sigma }^{\prime }}_1\right|},4\left|{{\sigma }^{\prime }}_1\right|{\sigma }_{1}R\le 1.$ (4.2)

This last case is interesting as

${\alpha }_{+}^o+{\alpha }_{-}^o=\frac{1}{\left|{{\sigma }^{\prime }}_1\right|}$ (4.3)

i.e. if $\left|{{\sigma }^{\prime }}_1\right|=1$, then ${\alpha }_{+}^o$ and ${\alpha }_{-}^o$ take probabilistic meaning whose sum yields the certainty. This means that $F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}$ has probabilistic meaning, as it can take here two values depending on whether ${\alpha }^{\circ }$ takes either value ${\alpha }_{+}^{\circ }$ or ${\alpha }_{-}^{\circ }$. Clearly (1.17) admits in general several values of ${\alpha }^{\circ }$ depending on the number of series expansion terms in (1.14); yet it is reasonable to think that a probabilistic equation analogous to (4.3) still holds for the sum of all real solutions ${\alpha }_j^{\circ }$. Anyway, since $\left|{{\sigma }^{\prime }}_1\right|=1$ with ${{\sigma }^{\prime }}_1<0$ means ${{\sigma }^{\prime }}_1=-1$, now (1.15) reads

$\frac{F_n^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}}{{\sigma }_1}=\left(F_{n-1}+F_{n-2}\right)\left(1-{\alpha }^{\circ }\right){\alpha }^{\circ }\mathrm{.}$ (4.4)

As (3.4) implies that $\alpha$ must depend on $Z_{\max }^{-1}$, it is reasonable to expect also now

${\alpha }^{\circ }=\frac{const}{Z_{\max }},\frac{F^{\mathrm{<mo>\; *\; </mo>\; <mo>\; *\; </mo>\; <mo>\; *\; </mo>}}}{{\sigma }_1}\approx Z_{\max }$ (4.5)

so that the solution of (4.4) with respect to $a^{\circ }$ reads

${\alpha }_{\pm }^o=\frac{1\mp \sqrt{1-4Z_{\max }/\left(F_{n-1}+F_{n-2}\right)}}{2},4Z_{\max }\le F_{n-1}+F_{n-2}.$ (4.6)

Since by definition two known addends $F_{n-1}+F_{n-2}$ concur to the n-th term of the standard (1.1), which is thus known, this result is calculable as a function of $Z_{\max }$. If the present reasoning is correct, it should be true that

$\alpha =\frac{{\alpha }_{\pm }^o}{Z_{\max }}=\frac{1\mp \sqrt{1-4Z_{\max }/\left(F_{n-1}+F_{n-2}\right)}}{2Z_{\max }}\mathrm{<mo>\; ;\; </mo>}$ (4.7)

in other words, ${\alpha }_{\pm }^o$ replace the factor ${\left(2/\pi \right)}^{1/3}$ of (3.4). Yet the interesting fact is that now not only ${\alpha }_{\pm }^o$ depends itself upon $Z_{\max }$ but also that it can take two values. To assess this conclusion according to the condition (4.6), it is enough to put $Z_{\max }=118$, i.e. $4Z_{\max }=472$, and check which n of (1.1) fulfills

$\frac{1\pm \sqrt{1-472/\left(F_{n-1}+F_{n-2}\right)}}{2}\approx {\left(\frac{2}{\pi }\right)}^{1/3},F_{n-1}+F_{n-2}\ge 472$ (4.8)

if the reasoning is correct, then an appropriate value of n must exist such that either sign should provide a better approximation to the true value of $\alpha$. A few results for selected values of n are reported in following Table 1 that for convenience of comparison also reminds the reference value (4.8) early introduced in (3.3) to calculate (3.4).

${\left(\frac{2}{\pi }\right)}^{1/3}=0.86025\cdots$