_{1}

^{*}

Reciprocity may be understood as relation of action and reaction in the sense of
*Hegel’s* philosophical definition. Quoting
*Kant*, freedom and ethical necessities are reciprocally limited. In this contribution, a more mathematical than philosophical reflection about reciprocity as an ever-present dual property of everything was given. As a crystallographer, the author is familiar with the action of
*Fourier* transforms and the relation between a crystal lattice and its reciprocal lattice, already pointing to the duality between particles and waves. A generalization of the reciprocity term was stimulated by results of the famous
*Information Relativity* (IR) theory of
*Suleiman* with its proven physical manifestation of matter-wave duality, compared to the set-theoretical E-
*Infinity* theory developed by
*El Naschie*, where the zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the empty set boundary surrounding the pre-particle. Expectedly, the most irrational number
of the golden mean is involved in these thoughts, because this number is intimately connected with its inverse. An important role plays further
*Hardy’s* maximum quantum entanglement probability as the fifth power of
φ and its connection to the dark matter. Remembering, the eleven dimensions in
*Witten’s* M-theory may be decomposed into the
*Lucas *number L
_{5} = 11 =
φ
^{−5} –
φ
^{5}. Reciprocity is indeed omnipresent in our world as piloting waves that accompany all observable earthen and cosmic matter. As a side effect of the IR theory some fundamental constants such as the gyromagnetic factor of the electron,
*Sommerfeld’s* fine-structure constant as well as the charge of the electron must be marginally changed caused by altered relativistic corrections. Consequences also arise for our vision about the evolution of life and consciousness.

Recently, the author reported on a reciprocity relation between the mass constituents of the universe and Hardy’s maximum quantum entanglement probability of two quantum particles [

The following chapter deals with the Fourier transform and the reciprocal lattice of crystallography that describes results of diffraction on crystals as a special kind of particle-wave duality. The reciprocity relation between boundary and enclosed area of a circle is shortly treated, followed by the reciprocity property of the golden ratio in a separate chapter. Then the fractal Cantorian set theory of El Naschie [_{e}) as well as electron pairing in superconductors [_{e} value, concerning the relativistic shift correction. Related to it, the charge of the electron has to be changed, too. In addition, a corrected version of the Niehaus EZBW (extended Zitterbewegung may serve as a probabilistic model for the electron [

The particle-wave duality is also essential in the quantum information theory, where the unit of information is given by the quantum bit (qubit) coined by Schumacher [

S U ( 2 ) ≅ H 1 ∋ Q = z 1 | Y e s 〉 + z 2 | N o 〉

where z_{1} and z_{2} are complex numbers, and z 1 2 + z 2 2 = 1 [

The diffraction of X-rays of sufficient wavelength on crystals leads to a characteristic diffraction pattern, where the electron density ρ ( r ) of a crystal structure is transformed to a reciprocal lattice, weighted with intensities I ( h ) ∝ | F ( h ) | 2 , where F ( h ) represents structure amplitudes (Fourier coefficients) according to the transform

F ( h ) = ∫ V ρ ( r ) e 2 π i h r d V . (1)

The vectors r = x a + y b + z c and h = h a * + k b * + l c * are position vectors of the crystal lattice respectively the reciprocal lattice of a diffraction pattern. The lattice parameters a, b, c respectively a^{*}, b^{*}, c^{*} are the repeat units along the lattice axes. The reader may follow a reciprocal lattice exercise in more detail, given as a lecture in [

The inverse Fourier transform, using the structure amplitudes as coefficients, delivers the electron density ρ ( r ) of the crystal

ρ ( r ) = ∫ V * F ( h ) e − 2 π i h r d V * , (2)

where V * ^{ }is the reciprocal volume.

In this way, a crystal structure can be completely solved by means of a diffraction experiment.

However, because only intensities I(h) are measured, phases of the structure amplitudes are lost and must be recovered by elaborated crystallographic methods.

Applying a transform such as the Fourier transform one goes from the object space to the image space or reciprocal space. If the original variable would be the time, then the transformed reciprocal variable would be a frequency, exemplified by the Laplace transform of electrical decay processes. Going from the object space to the reciprocal space one may heretically ask, what could be the Fourier transform of the entire cosmos, delivering an inverse universe or whatever else?

Quoting the references [

A = π = 4 ∫ 0 1 1 − x 2 d x , (3)

where π is Archimedes’ constant, the well-known circle constant. One obtains the circumference C by using the reciprocal of the integrand

C = 2 π = 4 ∫ 0 1 1 1 − x 2 d x . (4)

This connection between the boundary and the enclosed area is of fundamental importance. It may be thought of as a geometrical analog to the more general matter-wave duality that is being treated below. Besides, Archimedes’ constant π and the golden ratio φ as the fractal numerical dominators of our existence show an intimate numerical connection, and we may ask, in which manner nature makes use of this [

In 2014 Mushkolaj [_{c} functions of the form

T c ( collision ) ∝ ( M 1 ⋅ M 2 ⋅ Δ x 2 ) − 1 , T c ( spring ) ∝ M 1 ⋅ M 2 ⋅ Δ x 2 (5a, b)

where M_{1}, M_{2} are the colliding or by a spring connected masses, and Δx is the distance between atoms respectively the spring stretch length in Hooke’s region.

If we associate the atomic collision model with particles and the spring model with waves, we are faced with a reciprocity relation between the two excitation variants and again with the duality between particles and waves in a special form.

The golden mean or golden ratio φ is an omnipresent number in nature, found in the architecture of living creatures as well as human buildings, music, finance, medicine, philosophy, and of course in physics and mathematics [

φ = 5 − 1 2 = 1 1 + 1 1 + 1 1 + ⋯ (6)

It impressively underlines the fractal character of this number. Most obviously, the golden mean mediates stability of a system, because only “particles” as the center of gravity of vibrations with most irrational winding survive. Important relations involving φ are summarized below. However, to prevent confusion, in textbooks of mathematics the reciprocal value for φ is frequently used.

φ = 5 − 1 2 = 0.618033988 ⋯ , φ − 1 = 1 + φ = 5 + 1 2 = 1.618033988 ⋯ (7a, 7b)

φ 2 = 1 − φ = 0.381966011 ⋯ , φ − 2 = 2 + φ = 2.618033988 ⋯ (8a, 8b)

( φ 2 + 1 ) − 1 = ( φ − 2 + 1 ) / 5 (9a)

or equivalently ( φ − 2 + 1 ) − 1 = ( φ 2 + 1 ) / 5 (9b)

φ 5 = 1 − φ 1 + φ φ 2 = 0.090169943 ⋯ (10)

φ 5 2 + 5 2 φ 2 = 1 (11)

φ 3 + 2 φ 2 = 1 (12)

Hardy’s maximum quantum entanglement probability of two quantum particles [_{τ} as entanglement variable, running from not entangled states to completely entangled ones, is given by

P = p τ 2 1 − p τ 1 + p τ (13)

This function, displayed in

The probability function according to Equation (13) can be recast to an adapted distribution by means of a varied Fisher transformation (

z = 5 2 ln ( 1 + p τ 1 − p τ ) = 5 ⋅ artanh ( p τ ) (14)

where the pre-factor was chosen as a = 5 2 = φ + 1 2 . Then one gets for f (z)

f ( z ) = ( exp ( z a ) − 1 exp ( z a ) + 1 ) 2 exp ( z a ) − 1 (15)

A comparison is made between the curves displayed in

Moreover, in a subsequent contribution a geometric analog to Hardy’s probability function will be presented with significance, besides geometry, to crystallography, electrostatic, botany, coding theory and other disciplines.

Infinite continued fraction representations of φ^{5} and its inverse yield

φ 5 = 1 11 + 1 11 + 1 11 + ⋯ (16)

φ − 5 = 11 + φ 5 = 11 + 1 11 + 1 11 + 1 11 + ⋯ (17)

We notice that L_{5} = 11 is a Lucas number. It results from the definition

L n = φ − n + ( − φ ) n (18)

The L_{n} number series was named after the French mathematician François Édouard Anatole Lucas (1842-1891).

Many researchers have found the golden ratio to be important by trying to uncover secrets of the universe and its mass respectively energy distribution [

If one deals with exponential functions, the author has learnt from Sherbon [

exp ( W ( z ) ) = W ( z ) / z (19)

Especially is

Ω = W ( 1 ) = exp ( − W ( 1 ) ) = 0.5671432904 ⋯ (20)

W ( 2 ) = 1 2 exp ( − W ( 2 ) ) (21)

and

W ( 1 2 ) = 2 ⋅ exp ( − W ( − 1 2 ) ) (22)

The quoted publication of Sherbon [

Often you wonder why our world is what it is. Fundamental numbers such as the golden ration φ, the circle constant π as well as Sommerfeld’s fine-structure constant α and their obvious similarities play an important role. Some approximations should illustrate it. So one can connect the number π with the reciprocal of the Sommerfeld constant α − 1 ≈ 137 [^{5} [

π ≈ 3 + 16 137 − 24 = 3.1415929 ⋯ (23)

π − 3 π = 0.04507 ⋯ ≈ φ 5 2 = 0.04508 ⋯ (24)

3 π = 0.9549296 ⋯ ≈ 5 2 φ 2 = 0.954915 ⋯ (25)

α − 1 = 137 + 2 5 φ 5 = 137.0360 ⋯ (26)

These approximations, believed to be accidental by others, may find now and then application in the following discussions.

In a previous publication the author drew attention to the numerical similarity between the golden mean and the Madelung constant [

The Madelung constant α_{2D} was iteratively determined with very high precision by Triebl [

α 2 D -NaCl = 1.615542626711299 ⋯ (27)

The α_{2D} value is very close to the quotient of two Fibonacci numbers, 21/13 = 1.615385, ... and can be adapted to φ^{−1} by only slight distortion of the square net along the two dimensions or by involving the third one to allow a quite flat curvature [

The difference to the inverse of the golden mean φ^{−1} is only marginal and gives

φ − 1 − α 2 D -NaCl = 0.002491362 ⋯ (28)

This almost numerical equality was applied to Villata’s lattice universe [

1 + α 2 D ≈ 1 + φ − 1 = φ − 2 (29)

This relation leads the author to a proposal for the golden mean based calculation of the mass constituents of the universe [

A certain reciprocity property may be suggested for a two dimensional rocksalt-type matter-antimatter lattice independent of whether it is a real possibility. The question is whether a conditionally flat lattice universe with a Madelung constant of φ^{−1} would guarantee sufficient stability to exist over long periods of time.

El Naschie’s E-infinity (ε^{∞}) theory [_{Q} is symbolized by the bi-dimension of the zero set, while the guiding wave W_{Q} surrounding the quantum particle is given by the bi-dimension of the empty set according to

dim ( X ) = ( n , d c ( n ) ) (30)

where n is the Urysohn-Menger topological dimension [

d c ( n ) = ( φ − 1 ) n − 1 (31)

represents the Hausdorff dimension [

It results for P_{Q} dim ( P Q ) = ( 0 , φ ) (32)

respectively for W_{Q} dim ( W Q ) = ( − 1 , φ 2 ) (33)

By using these dimensions a probabilistic quantum entanglement calculation [_{M}, dark matter e_{DM}, entire dark constituents e_{ED}, and pure dark energy e_{PD} as follows

e M = 1 2 1 − φ 1 + φ φ 2 = φ 5 2 = 0.04508497 (34)

e E D = 1 − e M = 5 2 φ 2 = 0.9549150 (35)

e D M = 3 2 φ 4 = 0.218847 (36)

e P D = 2 φ − 1 2 = 0.736068 (37)

e M + e D M + e P D = 1 (38)

Recasting the matter amounts into a suitable form,

e M = 1 10 5 φ 5 , e D M = 1 10 ( 5 φ 5 ) − 1 = 0.2218 (39)

a reciprocity relation was confirmed between e_{M} and e_{DM} giving a persuasive equation for the pure dark energy [

e P D = 1 − 1 10 ( 5 φ 5 + ( 5 φ 5 ) − 1 ) = 0.7331 ( 73.31 % ) (40)

Such quantum entanglement based coincidence means that the constituents of the cosmos should not be considered independent of each other, which was confirmed by the IR theory.

Importantly, if one compares the results given here with the following ones of the information relativity (IR) theory, then El Naschie’s set theoretical approach is restricted to v → c , whereas the more general IR theory delivers results for the recession velocity β = v c in the hole range 0 ≤ β ≤ 1 (c is the speed of light).

Many formal explanations or physical constructs that bothered long time the world of physics are overcome by the new exciting Information Relativity theory, developed by Suleiman [_{M} of a moving body with velocity v and rest density ρ_{o}

e M = 1 2 ρ v 2 = 1 2 ρ o c 2 1 − β 1 + β β 2 = e o 1 − β 1 + β β 2 , (41)

where β = v c is the recession velocity respectively e o = 1 2 ρ o c 2 .

The matter energy density reached its maximum at a recession velocity of β = φ . Replacement of this special value in Equation (41) gives

( e M ) max = e o 1 − φ 1 + φ φ 2 = e o φ 5 = e o ⋅ 0.09016994 ⋯ (42)

Remembering, φ^{5} represents Hardy’s quantum probability at the maximum. This result was commented by the author in a publication before mentioned [

Suleiman aptly characterized the behavior at the critical point β_{cr} = φ as phase criticality at cosmic scale [

e D M e o = 2 β 3 1 + β (43)

The relations are depicted in

φ 3 + 2 φ 2 = 0.236067976 ⋯ + 0.763932023 ⋯ = 1 (44)

The difference gives 2 φ 2 − φ 3 ≈ ( 2 − 1 ) 4 π .

The case, where according to the Information Relativity theory of Suleiman [_{6} = 18 (see Chapter 5 and

e M = e D M = L 6 − 1 = 1 18 = 0.055555 ⋯ = ( φ 6 + φ − 6 ) − 1 ≈ 0.055728089 ⋯ = φ 6 (45)

Furthermore, if the recession velocity at β_{eq} = 1/3 is mirrored at β_{cr} = φ, it resulted β_{mir} = 0.9027. In its vicinity at β = 0.89297 the matter energy density would be exactly φ^{5}/2 = 0.04508497∙∙∙ respectively the dual dark component 0.7523∙∙∙ ≈ 0.763932∙∙∙ = 2φ^{2} (

It approximately indicated a situation that is elaborated for v → c by means of the fractal set theory summarized before in Chapter 8.

In

Suleiman’s IR theory validates once more the importance of the golden mean in solving physical phenomena. Reciprocity is given by the proposed duality between particle and wave.

As was demonstrated by Suleiman (

The electron, considered as center of compacted information, still keeps its secrets, but not for long. Whereas the hydrogen atom problem was just solved by Suleiman [

l l 0 = 1 + β 1 − β (46)

other constructs like the electron spin [

The g_{e} factor of the electron, conceived as a classical charged particle, is determined by the relation

μ → = g e μ B S → h ¯ , μ B = e 2 m (47)

where μ → is the observable magnetic moment, μ B is the Bohr magneton, and S → is the spin of the electron, e respectively m are charge respectively mass of the electron, and h ¯ is the reduced Planck constant.

However, the spin as half-integer quantum number of the electron was introduced without any physical justification [

Remembering that the “anomalous part” of the gyromagnetic factor Δ g e was recently given by a simple and solely golden mean representation with sufficient accuracy [

Δ g e = ln ( 1 + φ 6 24 ) = 0.002319312 ⋯ (48)

while a series expansion yields a value more accurate up to the tenth decimal place

g e = 2 + φ 6 24 − 1 2 ( φ 6 24 ) 2 − 1 4 ( φ 6 24 ) 3 = 2.002319304 ⋯ (49)

This result may be compared to the high accuracy of the best known experimental value for g_{e} determined as one-electron cyclotron transition for an electron trapped in an electrostatic quadrupol potential (Penning trap) [

g e = 2.00231930436182 ( 52 ) (50)

In a subsequently presented seminal idea of He et al. [

g e 2 = φ ˜ ( 1 + φ ˜ ) (51)

with the value φ ˜ = 0.6190713336307 ( 34 ) as He-Chengtian average [_{e}/2. One can calculate φ ˜ by a very simple formula, which resembles the representation for φ (Equation 7(a)) and delivers exactly the given value

φ ˜ = 1 2 ( 5 + ( g e + 2 ) ( g e − 2 ) − 1 ) = 1 2 ( 1 + g e 2 − 1 ) (52)

and for the IR corrected value of g_{e} = 2.0023190900 (see Chapter 11)

φ ˜ = 0.619071237 ⋯ (53)

Using this formula, the gyromagnetic factor resulted simply as function of α/π [

g e ≈ 2 1 + v K 2 π c = 2 1 + α π = 2.00232147 (54)

giving φ ˜ = 0.619072302 ⋯ (55)

where v_{K} is the Klizing speed and c the speed of light.

The latest released values of Sommerfeld’s fine-structure constant α [

α = 0.0072973525693 ( 11 ) (56)

α − 1 = 137.035999084 ( 21 ) (57)

An approximation using the α/π series expansion yielded

g e ≈ 2 1 + α π − 1 2 ( α π ) 2 = 2.002318778 ⋯ (58)

or g e ≈ 2 1 + ln ( 1 + α π ) = 2.002318782 ⋯ (59)

and further φ ˜ = 1 2 ( 5 + 4 ⋅ ln ( 1 + α π ) − 1 ) = 1 2 ( 5 + ln ( 1 + α π ) 4 − 1 ) = 0.619071099 ⋯ (60)

The deviation between this a bit underdetermined φ ˜ value and the newly relativistic corrected one is in the seventh decimal place as well as the corresponding g-factors. It is hoped that precisely re-determined experimental factors may lessen these deviations further.

One may ask, what the infinitely continued fraction representation of φ ˜ would result in. We can write similar to the golden mean [

φ ˜ = 5 + δ − 1 2 = 1 1 − δ 1 + 1 1 − δ 1 + 1 1 − δ 1 + ⋯ (61)

The calculation with δ 1 = 0.00374774 ≈ 1 266. 6 ¯ ≈ δ φ ≈ φ 5 4 ! yielded φ ˜ = 0.619071096 . Indeed, the number 266. is very interesting. Division of this number by integers frequently delivers numbers with repeating decimals, exemplified by 266. 6 ¯ / 24 = 11. 1 ¯ . If one associates this number with rounds, then one would need 27 ones to complete 20-times the full 360 degrees extent.

With an assumed involvement of the fifth power of the golden mean in the continued fraction representation one may speak of a nested golden mean representation. This result supports once more the fractal-deterministic approach chosen for the physics of the electron beyond the ad hoc half-spin assumption, characterizing the electron as complexly nested resonating entity. An alternating approach for the gyromagnetic factor is given in Appendix II.

The calculation of the electron’s gyromagnetic factor is the prime example for application of the QED. However, a cascade of Feynman diagram calculation must be done to determine the pre-factors of systematic perturbative expansions in powers of α/π [_{c} = cyclotron frequency, B = magnetic field strength in Tesla) was performed using the familiar relativistic factor γ. However, γ should be replaced by the mass transformation according to the IR theory [

m m 0 = 1 − β 1 + β (62)

For the classical case the corrected frequency ω_{c} is

ω c = ω 0 ( 1 − E n m c 2 ) (63)

where the energy E_{n} of the nth quantum state of a harmonic cyclotron oscillator is given as

E n = ( n + 1 2 ) h ¯ ω c (64)

The classical relativistic shift δ in the cyclotron frequency per energy quantum was approximated by the level spacing of the harmonic oscillator giving [

δ = − h ¯ ω c 2 / ( m c 2 ) (65)

For the IR theory one yields a much greater and positive shift because the cyclotron frequency yields now

ω c ≈ ω 0 ⋅ ( 1 + 2 ⋅ 2 E n m c 2 ) (66)

The relativistic shift δ is approximated by

δ ≈ d ω c d n = 2 ω 0 2 m c 2 ⋅ d E n d n = ω 0 2 m c 2 E n ⋅ h ¯ ω c = 2 ⋅ ω 0 2 E n m c 2 (67)

The gyromagnetic factor as g_{e}/2 can be determined from the observed eigenfrequencies [

g e 2 = 1 + ω ¯ a − ω ¯ m ω ¯ c − ω ¯ m (68)

where the ω ¯ values are marginally modified with respect to the free-space values, ω a is the anomalous frequency, and the spin frequency is ω s = ω c + ω a , ω ¯ z 2 2 ω ¯ c = ω ¯ m is the magnetron frequency, using the dip frequency ω_{z} in Hz [_{c} = 149.2 GHz, the classical relativistic shift is calculated to be δ = −2π∙182.1 Hz compared to the IR corrected one giving δ = +2π∙14.78 Mhz. One can estimate that g_{e} becomes noticeably smaller by a factor of approximately 1.0001, meaning a correction of g_{e} from the seventh decimal point downwards to about g e ≈ 2.00231909 ?

Now the scientific community is waiting for a most precise redetermination of the g-factor as well as the related Sommerfeld constant by experts [

Nature presents much more relationships to keep in mind, where the golden mean is involved, and superconductivity is no exception. However, we must reassess the theory considering the dark matter surrounding the moving electrons, which dive into the dark after marriage, or in other words, become superconducting under special conditions. Before a golden ratio in the spin dynamics of the quasi-one-dimensional Ising ferromagnet CoNb_{2}O_{8} was experimentally verified next a phase critical point by Coldea et al. [_{c} superconductors with the golden mean in the form of Hardy’s maximum quantum probability of two particles [

σ 0 ≈ 8 π φ 5 = 0.2293 (69)

Obviously, this optimum is again near a quantum critical point in the phase diagram. In addition, the relation of the Fermi speed to the Klitzing speed comes out as

v F v K ≈ 2 π φ 5 = 0.0571 (70)

Both relations document the fractal nature of the electronic response in superconductors. It was suggested recently that the same is true for conventional superconductors [_{c} superconductors [

Interestingly, some time ago the present author connected the optimum transition temperature T_{co} of high-T_{c} superconductors with a Fibonacci number f_{i}, proportional to a domain width, by the relation T_{co} = 12,000/f_{i} [

Quantum entanglement of two moving electrons may be influenced by local interaction of their interwoven dark matter surroundings, quoting the cogwheel picture of Suleiman [

Nature repeatedly applied its building plans, based on the hierarchical golden number system, from largest to smallest dimensions, from the cosmos to the smallest living cell. Inasmuch the golden ratio is involved, reciprocity is considered as a vital element of life. Recently, thoughts to the link between cosmology and biology are impressively formulated [

The evolution of life may take place similar to the statistical bootstrap model of colliding heavy particles, so the Hagedorn temperature T_{H} comes into play. I quote the formulation of Rafelski and Ericson [

This nesting looks like a Menger sponge [_{H} around ambient temperatures.

With respect to the entire energy density of φ^{2} at the phase critical point β = φ one may suggest formulating the Hagedorn temperature T_{H} proportional to the squared golden mean φ, where α ′ is formally the tension of a string.

T H a g ≈ φ 2 ⋅ 1 α ′ (71)

It remains to interpret the not liked string tension by a more appropriate thermodynamic quantity at ambient equilibrium conditions.

The duality between a compacted entity and its surrounding in general as well as the duality between a moving particle or body and the accompanying wave or reciprocity between matter and dark matter is the very spice of life. This was proven by the beautiful information relativity theory of Suleiman. Reciprocity is impressively formulated by the words of Wolfgang Pauli: “God made the bulk; surfaces were invented by the devil” (quoted from [

The author appreciated the critical reading of the manuscript by Prof. Ramzi Suleiman, University of Haifa, and the Triangle Research and Development Center (TCRD), who enriched the scientific community with his famous information relativity theory. The author is also grateful for the constructive criticism of a very creative reviewer.

The author declares no conflicts of interest regarding the publication of this paper.

Otto, H.H. (2020) Reciprocity as an Ever-Present Dual Property of Everything. Journal of Modern Physics, 11, 98-121. https://doi.org/10.4236/jmp.2020.111007

About the Meaning of the Terms Reciprocity, Duality and Complementarity

These terms, omnipresent in many disciplines of science (physics, mathematics, philosophy, music, economy, etc.) can certainly have different meanings, even if they have something in common.

In mathematics, a reciprocal of a number is its multiplicative inverse, but an inverse is not necessarily a reciprocal. Reciprocity in the amounts of matter and dark matter is formulated according to Equation (39) of the main text. Important for mathematics and physics, a reciprocal vector system can be created by Fourier transformation. However, reciprocity in physics may have a more general meaning when describing a mutual dependence or influence.

Duality in mathematics can be demonstrated on platonic solids, also familiar for a crystallographer. The convex hull of the center points of each face of a starting polyhedron results in a dual polyhedron such that, for instance, the cube and the octahedron form a dual pair, but the tetrahedron is self-dual. In physics, the most prominent example for duality is that between matter and piloting wave in the sense of the De Brogly-Bohm approach [

Finally, the concept of complementarity in quantum physics has been formulated and coined by Bohr in his Como lecture of 1927, describing the familiar case of reciprocal uncertainty between position and momentum of an electron as conjugate variables [

In mathematics, a number and the complement to a number add up to a whole of some amount. If one performs the reciprocal of these numbers and renormalize the resulting values, then complement and primal number change their values [

Another approach for the gyromagnetic factor used the fifth power of φ ˜ with the value φ ˜ 5 = 0.09092922100312 . An approximation is the inverse Lucas number L_{5} = 11 as combination of two inversely related irrational numbers (see Equations (13) to (15))

( φ − 5 − φ 5 ) − 1 = ( 11 ) − 1 = 0.0909090 ⋯ (72)

However, physically more convenient is the expression

2 π ⋅ v K c = 0.0917012 ⋯

where v_{K} is the Klizing speed and c the speed of light. This term keeps no dimension, as required. If we are working with a speed, according to the IR theory the information offset has to be corrected. The speed transforms as

v v 0 = 1 + β (73)

combining the length transformation l l 0 = 1 + β 1 − β with the time transformation t t 0 = 1 1 − β , where β = v/c is the recession speed [

2 π v K c ( 1 + 2 π v K c ) ( 1 − 2 π v K c ) 5 = 2 π v K c ( 1 − ( 2 π v K c ) 2 ) 5 = 0.61907254 (74)

leading to g e 2 = 1.00116100597109 or g e = 2.00232201 (75)

Remarkably, this value is almost identical to the result of [

2 + φ 6 24 = 2.002322003 (76)

Only now we are allowed to associate the term 2 π v K c with Sommerfeld’s fine-structure constant α [

2 π v K c = 4 π α (77)

where α is a measure of the strength of interaction of an electron and a photon in the quantum electrodynamics theory (QED). The charge of the electron in QED (Lorentz-Heaviside) units has the numerical value of e = − 4 π α .

The accurate experimental value for the gyromagnetic constant could be attained from Equation (74) using an adapted fine-structure constant of

α ′ = 0.007297279955669 (78)

respectively α ′ − 1 = 137.037362698897742 (79)

where α ′ − α = 6.43 × 10 − 7 (80)

Tackling the problem of the not fully adapted accuracy in comparison to the experimental value, one can multiply the term under the fifth root of Equation (74) by a factor of 0.9999902180 or alternatively reduce the Klitzing speed by a factor of 0.99999004931863 respectively the charge of the electron by a factor of 0.99999502464694. This adjustment may result partly from a correction of g as well as α with respect to the IR theory, besides needed radiative corrections.