Generalized Fibonacci Sequence: Possible Template for the Constants of Nature

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 ap-pearance of the observable Universe. Once more appears in this conceptual frame the natural and serendipitous link between quantum and relativistic theories.


Introduction
The standard definition of Fibonacci sequence is ever, 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 ( ) where 0 f and 1 f 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 , and thus their related physical meaning as well, just depending on how are defined the initial terms. Defining indeed 8 5 , F now defined by the arbitrary 0 f and 1 f . The recursive rule is therefore unchanged. Since the single terms of the new sequence * n F are actually linear combinations of two terms with different n of the standard sequence n F , 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 . This makes significant the appropriate definition of selected specific terms * 1 F and * 2 F suitable to provide information of interest about specific physical effects, likewise as each n F 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 * n F , 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 0 f and 1 f . In practice, the first few terms of * F read for example  (1.6) i.e. the coefficients that multiply 1 f and 2 f are just the respective numbers , the notation exemplifies the particular case where unit energy.
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 c k , 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 * n F represent identically the physical meaning of the sequence  [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 im-  (1.4). Analogously multiply all terms of F a first time by an arbitrary dimensional factor 1 a k f , so that all n F turn into 1 a n k f F , and next by another arbitrary dimensional factor 0 b k f , so that all n F turn now into 0 b n k f F ; the two sequences obtained in this way consist of terms with well defined physical dimensions, in general different when a b k k ≠ . If for example the dimensional factor a k is defined as cm −1 •g −1 •s −1 , then the 1 a f k means 1 f per unit length, unit mass and unit time. If more realistically a k is unit energy and b k 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 0 f ′ and 1 f ′ whose physical dimensions are defined by c a k k and c b k k as follows   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 Journal of Applied Mathematics and Physics 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 f 0 and f 1 .
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 n F * ′ and n F * ′′ without changing the numerical value of n F * , 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; f ′ with 0 f and 1 f , and thus that of n F * ′ with n F * , the calculations will be carried out implementing (1.4) although 0 2 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 α , 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.

The Fine Structure Constant: Preliminary Considerations
Before obtaining an approximate numerical estimate of the dimensionless constant α 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 where c ξ  is the proportionality constant linking r εδ to α . Owing to this physical meaning of α , try to identify an energy ε and length r δ considering a bound system of charges. In particular it is sensible to think r εδ 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 Journal of Applied Mathematics and Physics where e m 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 , p x n t δ δ δεδ which in turn is a corollary itself of an operative definition 2 G c  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 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 ( ) Ze e defining α . This kind of interaction appears explicitly because it also follows that ( ) the systematic presence of Z n α suggests that just Z α is the key quantity to understand α . Despite (2.2) include neither spin/orbit nor spin-spin electron/nucleus interaction or the Lamb shift, conceptually crucial even for 1 Z = 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 ε and r δ of (2.1). The factor 1/2 is the fingerprint of the quantum uncertainty: although m and nucleus are r δ apart, the Coulomb-like form requires the total uncertainty range of the charge e − with respect to Ze that indeed is 2 r δ i.e. the total radial delocalization range of the electron around the nucleus. Of course r δ 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 , , is trivially the classical kinetic energy of a free particle of mass equal to the reduced mass m moving at radial velocity r v . 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 2 mc via Z n α , because now 4 e of (2.2) becomes hidden in 2 α of (2.6).
However, although ε is even expressed as classical kinetic energy of reduced mass m moving at velocity r v , the quantization is still evident through (2.7).
So, merging (2.14) and (2.15), from and owing to (2.6) Note now that if the electric and magnetic fields of electromagnetic waves are orthogonal, then in fact numerically EH = × E H 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 wave propagating within C V . 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 α : given ε , the greater the numerical value of α , 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 H ; the proportionality constant between the energies involves α . 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 1 .  n, 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 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 velocity 4 . Since    In effect, this result is nothing else that the deflection angle δφ of a photon moving in the gravity field of m at distance  , 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 phys-ics 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 r Φ acting on the electron in the field of nucleus as Φ is no longer required to be   2  3  3  1  2  3  3  3  3  3 , ; in this way the first addend B ε of  is the Bohr zero order term, the second addend yields the electron correlation terms as a function of α for a number of electrons 1 N ≥ , which is certainly possible in principle determining appropriately the series coefficients j ζ of (2.37). In particular the third addend 5 α ∝ is the fingerprint of the Lamb energy due to the "vacuum" polarization.
Even for 1 N = 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.  , . Of course the same holds considering ( ) ( ) 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

First Approximate Calculation of α
where clearly r a is the radial acceleration of m due to the force r Φ . So comparing with (2.22) This result can be calculated recalling that in [15] it has been found that the upper limit of high atomic number heavy nuclei is max 118 Z = . Then The deviation of α 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 car- here instead no change of c and C λ , 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 α is next carried out implementing the Fibonacci sequence.

The Generalized Fibonacci Sequence
Often an approximation is useful to check immediately and easily whether or This last case is interesting as i.e. if 1 1 σ ′ = , then o α + and o α − take probabilistic meaning whose sum yields the certainty. This means that *** n F has probabilistic meaning, as it can take here two values depending on whether α  takes either value α +  or α −  .
Clearly ( so that the solution of (4.4) with respect to a  reads ( ) Since by definition two known addends in other words, o α ± replace the factor ( ) 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 α . 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). now the deviation from (1.18) is one order of magnitude better than that of (3.4). Apart from the enhanced numerical agreement of this result, a crucial question arises now: is it possible to improve further the accuracy of the calculations? From the mere numerical point of view, certainly the implementation of (1.14) instead of (1.15) affects the final approximation inherent the value (4.9); so, it is in principle reasonable to expect that the calculation of o α ± including in (1.17) higher order terms prospected by (1.14) would bring to a result even better than (4.9).
The calculations proposed in this section aimed merely to provide a first answer to the challenge proposed at the end of the Section 1, i.e.: the Fibonacci sequence appears in fact adequate to calculate an acceptable value of α if appropriately implemented. Nevertheless the next section will describe in this respect a much better and far reaching approach by following a completely different strategy.

Calculation of Fundamental Constants via (1.4)
Consider again (1.4) with the purpose of correlating the fundamental constants of nature, which are now assumed all known: this section aims indeed to find the possible interconnection between these constants.   with other fundamental constants of nature, exploit the quantities of Table 2   To identify significant templates that govern physical constants and/or their combinations is in principle a difficult task. The fact that the values calculable via (5.4), (5.5) and (5.6) are actually abstract numbers, does not indicate "a priori" which specific physical properties they refer to; only "a posteriori" one acknowledges, by comparison like that of (5.4) and (5.9), which allowed physical constants and properties tentatively found are really correlatable. Moreover are crucial in this respect also the possible physical corollaries like (5.7) contextual to the numerical comparisons. Additional matches are shown in the following.
Consider the next template numbers (5.5) to find their possible corresponding fundamental constants and note that Last but not the least: the few remarks in Section 2, although very shortly sketched, are enough to show once again in a straightforward and elementary way the intimate link between relativity and quantum physics.

Conclusion
The sequence (1.1) is defined by its own sum rule of terms only, the sequence with j L s = + . So, the quantization implies itself the chance of the spin. Actually the existence of such half-integer angular momentum is shown starting from first principles in other papers, while being in fact j L s = ± [12] [16].
These short remarks aim merely to sketch the chance of justifying (2.2) and (5.8) via (2.3) only, in order to make the present paper as self contained as possible.