International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 DOI: 10.5923/j.ijtmp.20211101.03
Measurement Quantization Describes the Physical Constants
Jody A. Geiger
1Department of Research, Informativity Institute, Chicago, IL, USA
Abstract It has been a long-standing goal in physics to present a physical model that may be used to describe and correlate the physical constants. We demonstrate, this is achieved by describing phenomena in terms of Planck Units and introducing a new concept, counts of Planck Units. Thus, we express the existing laws of classical mechanics in terms of units and counts of units to demonstrate that the physical constants may be expressed using only these terms. But this is not just a nomenclature substitution. With this approach we demonstrate that the constants and the laws of nature may be described with just the count terms or just the dimensional unit terms. Moreover, we demonstrate that there are three frames of reference important to observation. And with these principles we resolve the relation of the physical constants. And we resolve the SI values for the physical constants. Notably, we resolve the relation between gravitation and electromagnetism. Keywords Measurement Quantization, Physical Constants, Unification, Fine Structure Constant, Electric Constant, Magnetic Constant, Planck’s Constant, Gravitational Constant, Elementary Charge
ground state orbital a0 and mass of an electron me – both 1. Introduction measures from the 2018 CODATA – we resolve fundamental length l . And we continue with the resolution of We present expressions, their calculation and the f the gravitational constant G, Planck’s reduced constant ħ corresponding CODATA [1,2] values in Table 1. and those values typically resolved with ΛCDM. The Calculations start with measurement of the magnetic electromagnetic constants involve several concepts and will constant. Along with defined values, this provides a be discussed later. CODATA value for the fine structure constant Notably, the fundamental expression is provided without 7.2973525693 10-3 which may be considered a physically explanation. The difference between the Planck and significant guide for the remainder of the calculations. The electromagnetic descriptions of the fine structure constant count distance n =84.6005456998 corresponding to Lr are not discussed. Our goal, initially, is to demonstrate blackbody radiation may be resolved with twelve digits of the approach. The formulation, physical significance and physical significance knowing its approximate count n =84 Lr explanation of each expression is the purpose of this paper. of l . The value of l is not needed in that the value of f f Of the many descriptions of phenomena, it may come as a n is a mathematical property of discrete counts. The Lr surprise that there are few expressions that describe discrete product, Q n is calculated using the Pythagorean Theorem L Lr behavior as a count of some fundamental measure [4]. Q =(1+n 2)1/2-n . Such that Q +n describes the L Lr Lr L Lr Perhaps one of the first and most notable is Planck’s hypotenuse of a right-angle triangle of sides n =1 and some Lr expression for energy, E=nhv. That said, the property of count n of the reference l , then Q n =0.49998253642 and Lr f L Lr discreteness exists with respect to several phenomena (i.e., with this we can resolve θ kg m s-1. θ also describes the si si those radii that identify orbits where there is a highest mean angle of polarization with respect to the plane of entangled probability of electrons being fundamental to atomic theory). X-Rays [3] and has no units when describing properties of In that we describe phenomena mathematically in relative the universe. With θ , the defined value for c and the si terms, it follows that the property of discreteness carried fundamental expression l m =2θ t – resolved from Planck’s f f si f within such expressions is disguised beneath the Unit expressions [6] – we resolve fundamental mass m f macroscopic definitions that make up much of classical and the Planck form of the inverse fine structure constant mechanics. α −1. Using Planck’s expression along with measures for the p In this paper, we reduce the classical nomenclature to a
* Corresponding author: more fundamental set of terms that incorporates a description [email protected] (Jody A. Geiger) of discreteness. We accomplish this by taking the existing Received: Dec. 29, 2020; Accepted: Jan. 16, 2021; Published: Jan. 25, 2021 classical nomenclature and incorporating the concept of Published online at http://journal.sapub.org/ijtmp counts of fundamental measures to accommodate the
30 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
possibility of discrete measure. However, measure is not to and then apply that nomenclature to existing phenomena to be assumed countable or discrete. As such, we refrain from learn if there exists a physically significant correspondence. introducing biases, provide an accommodating nomenclature Table 1. CODATA and MQ expressions for the physical constants
Values Expressions CODATA [1] MQ -1 * -1 θsi=(1/84)((1/2QLnLrαc) + 137)) 3.26239 kg m s 3.2623903039 kg m s -8 † -8 mf=2θsi/c 2.176434 10 kg 2.1764325398 10 kg −1 ‡ αp =84θsi - RND(42θsi) 137.04077 137.04078553 -35 † -35 lf= mea0αp/mf 1.616255 10 m 1.6161999121 10 m -44 † -44 tf=lf/c 5.391247 10 s 5.3910626133 10 s 2 -11 3 -1 -2 † -11 3 -1 -2 G=c lf/mf 6.67408 10 m kg s 6.6740779430 10 m kg s 2 2 -34 ‡ -34 ħ: (ħ/θsilf) -2(ħ/θsilf)=(1/nLr) 1.054571817 10 Js 1.0545719462 10 Js § Ωdkm=(θsi-2)/(θsi+2) (dark energy) 68.3% 68.362416104% § Ωobs=4/(θsi+2) 31.6% 31.637583896% § Ωvis= Ωobs/2θsi 4.8% 4.8488348953% § Ωuobs= Ωobs-Ωvis (dark matter) 26.8% 26.788749000%
* Shwartz and Harris measure of theta corresponding to the signal and idler of entangled X-rays [2]. † 2010 CODATA Recommended values [2]. ‡ 2018 CODATA Recommended values [1]. § Cosmological parameters as published by the Planck Collaboration [4,5]. We begin with three notions: Heisenberg’s uncertainty focuses on correlating this to expressions that describe principle, the universality of the speed of light, and the nature. expression for the escape velocity from a gravitating mass. Moving forward, we describe how discrete measure skews Each describes a bound to measure, respectively a lower the measure of length, an effect like that of Special Relativity bound, an upper bound, and a gravitational bound, the latter (SR). Not accounting for this effect reduces the precision of being needed to incorporate the mass bound with respect to expressions, especially those that include Planck’s constant. the prior two. Using the new nomenclature, we identify three It is for this reason that the Planck Units have largely been properties of measure: discreteness, countability, and the considered coincidental and without physical significance. relationship between the three frames of reference. After Once completing the expressions for the fundamental resolving minimum count values for length, mass, and measures, their relationship, and a quantum interpretation of time, we then resolve physically significant values for gravity, we commence Section 3 describing the fundamental the fundamental measures, matching values in the 2010 constants resolving their values and physical significance CODATA [2] to six digits. Importantly, we learn that using only the MQ nomenclature (i.e., lf, mf, tf, and θsi). It is measure with respect to the observer is discrete, whereas here that we part with the self-referencing definitions that measure with respect to the universe is non-discrete. This have deadlocked modern theory. Specifically, we redefine difference allows us to resolve the constants and the laws of the physical constants not as functions of one another (i.e., 2 nature. ɛ0=1/μ0c ) [1], but as functions of the fundamental measures. We identify this presentation as the Informativity Several examples include elementary charge, the electric and approach – a term that describes the application of magnetic constants, Coulomb’s constant, the fine structure measurement quantization (MQ) to the description of constant, and the gravitational constant. phenomena. The nomenclature we call MQ. There are With new definitions for gravitation and several papers that apply MQ to describe phenomena in electromagnetism written in a shared and physically distinct disciplines such as: quantum mechanics, classical mechanics nomenclature, we establish a physical reference with which (including gravity, optics, motion, electromagnetism, to resolve what differentiates them. There are five relativity), and cosmology [7–11]. Nevertheless, a discussion expressions that describe their difference: two describe an of the physical constants is prerequisite to a thorough observational skew in measure, one describes the fine understanding of MQ. For that purpose, the first half of this structure constant as a count while another describes paper is a review of concepts established in prior papers elementary charge using only fundamental units. The final [10,11]. term – a mathematical constant – describes what separates Foremost, we introduce a consequence of discrete length; the energy of a particle from a wave. Although described discrete units of length limit the precision with which objects entirely as a function of mathematical constants, the can be measured relatively. Importantly, the property of particle/wave duality is difficult to physically ascertain. The discreteness is not only intrinsic to measure but also to the correlation, we admit, lacks the purity of classical concepts laws that describe what we measure. The methods section such as distance, velocity, and elapsed time.
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 31
From a broader perspective, the correspondence between mass, and time with values in SI units. They are widely geometry and physical expression becomes even more recognized today as Planck Units. Notably, Planck Units important. Its correspondence arises in so many expressions differ from Stoney units by a factor of α1/2 as a result of their 2 that we feel compelled to identify such descriptions as transformation αħc↔e /4πε0. consistent with the phrase, ‘the metric approach’, short Unfortunately, a clear physical correlation between the for geometric or a consequence of the geometry of a Planck Units and observed phenomena did not exist. phenomenon. We do not mean to emphasize the Expressions using Planck Units corresponded to mathematical properties of this correspondence as to say that measurements of three digits at best. Moreover, the values such properties follow the same consistency as that of SR. In for length, mass, and time were too small (e.g., the Planck this light, we present physical constants, such as the fine time) or too large (e.g., the Planck mass) to correspond to the structure constant and Planck’s reduced constant as counts of phenomena being measured. Over time, the Planck Units one fundamental measure. Importantly, the metric approach were largely relegated to the status of a legitimate discipline is not distinct from classical mechanics. Nevertheless, many without a known physical significance. This said, Planck of the physical constants are described as counts of one Units are still taught and used in specific branches of modern fundamental constant to be discussed at the outset in Section theory (i.e., superstring theory and supergravity) because of 2.5. their consistency regarding many phenomena. Finally, using MQ to describe the physical constants In the century since, we find ourselves still divided by the resolves several discrepancies between classical theory and physical constants, which are so commonly used in classical measurement. For one, the precision limits of Planck’s unit mechanics and the corresponding Planck descriptions, expressions are resolved. Furthermore, disagreement which in rare but specific cases carry a count term thereby between Planck’s expression for the ground state orbital of recognizing the countability of phenomena. The most an atom and that of electromagnetic theory is resolved. notable and well-understood example relates to Planck’s Disagreement between Newton’s expression for gravitation initial observations of blackbody radiation whereby he and an MQ description of quantum gravity is resolved. published his expression for energy, E=nhv, n representing Issues with singularities in classical theory are resolved. the count term for Planck’s ‘quantum of action’ [6]. The physical significance of the fine structure constant To break the deadlock, we skip forward to the present and is resolved. Physically independent definitions of the ask an interesting but seemingly straight-forward question. Is electromagnetic constants are resolved. A shared physical it the phenomenon or the measure of the phenomenon that is foundation for the unification of gravity and quantized? electromagnetism is resolved. The gravitational constant is The question is interesting as the phenomenon of resolved as a function of the magnetic constant to eleven quantization has always been regarded as quantum both significant digits. Additionally, several notable insights physically and in measure. To explore this further, we afforded by MQ are presented. Most importantly though, the consider that we have discovered a box of pencils. First, we solutions do not just provide six to eleven-digit ask how we know they are pencils? The only answer to this is correspondence to measurement, but a comprehensive that there exists a reference pencil against which we have physical description using the most fundamental tenants of identified the phenomenon of a pencil and labelled it as such. classical theory. We recognize that pencils are physically divisible, but for this thought experiment, we also recognize that the measure 1.1. Theoretical Landscape of a pencil is bounded and as such indivisible. The first observations regarding a formalism of physically To test our conjecture, we take the pencils from the box significant units were published by George Stoney in 1881 and place them on the desk. Our objective is to measure the with respect to experiments concerning electric charge [12]. phenomenon that is “pencil”. Having completed this There did not exist a specific nomenclature with which to measure, we divide the pencils into two equal stacks and conveniently describe the phenomena. Thus, Stoney derived measure again. Unfortunately, we are unable to evenly new units of length, mass, and time normalized to the divide the stack. We theorize that one stack has an even existing constants G, c, and e. These units later became count of pencils and the other odd. To test the conjecture, we known as Stoney units. However, little more was discovered proceed to divide each stack again. The process is a success for the two decades that followed. with an even count stack but cannot be achieved with an odd In 1899, discrete phenomena became important. It was count stack. The experiment may be repeated with the same then that Max Planck submitted his paper regarding result; the odd count stack cannot be divided. Why, because observations of quantization with respect to blackbody there exists no definition for half a reference and this is the radiation [6]. Moreover, he resolved a new constant of nature, physical significance of a quantized phenomenon. which he later identified as a ‘quantum of action’. Today, We could look at other means of measure and perhaps this is known as Planck’s constant and is denoted with the achieve some form of a division with respect to a different symbol h. A factor of this behavior also appeared as h/2π, dimension, but if our definition of ‘pencil’ is indeed natural, later to be assigned the symbol ħ. With an understanding of c, that being the most fundamental of measures, then it is not G, and ħ, Planck was able to derive expressions for length, possible to measure a fractional count of the reference
32 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
phenomenon. Consistently, we find our efforts foiled such correlation, geometry invites further consideration. that the measure of the last pencil ends up in one stack or the A second and equally important issue relates to the other. existing classical nomenclature with which we describe There does exist one remaining concern. Thus far, we nature (i.e., length, mass, time, energy, charge). Modern present only the notion that we cannot measure a target nomenclature does not easily accommodate descriptions of smaller than a natural unit. Nevertheless, can a target be discrete phenomena. Yes, there exists a means with which smaller than a natural unit? Particles are in fact smaller than to resolve or at least conjecture discrete values associated the Planck mass. Indeed, we may certainly describe a length with a phenomenon, but a nomenclature that includes an smaller than the Planck length. Therefore, we ask the reader independent set of discrete terms separate from the reference to entertain the idea that what exists and what is measured measures may be more successful. are physically different and that difference describes a To succeed in this endeavor requires new tools with physically important property of nature. which to describe measure. In addition to resolving an This property will be resolved in its entirety but doing so understanding of the measurement discrepancy presented requires a careful presentation of physical clues, one built above, we need an expression that correlates the three upon the next. With that in mind, we begin Section 2. measures—lf, mf and tf—without inclusion of the physical constants. We must identify the properties of measure. Moreover, we must understand why those properties exist 2. Methods and under what circumstances they are immutable or skewed. 2.1. Considerations for a New Approach Note that working with dimensionless count terms also Before we express the physical constants, we must resolve carries limitations [13,14] or at least physically significant values for the fundamental measures. Historically, these rules of use. Specifically, they present an inability to: have been described using Planck expressions [6]. For resolve a physical quantity if there are more than three evaluations, we used the 2010 CODATA for comparison of dependent variables, most calculations [2]. Once we have resolved the properties derive a logarithmic or exponential relation, of measure, it will be better understood why the 2010 resolve whether a term involves derivatives, methods used to resolve Planck Units are physically more distinguish a scalar from a vector, and significant. Planck’s expressions [2] are verify dimensions given two or more dimensionless 1/2 terms. G l 1.616199 1035 m , (1) p 3 The first three are restrictions on use, but in no way lessen c the physical significance of MQ descriptions. Yes, use of the 1/2 dimensionally correlated count terms of MQ are restricted to G t 5.39106 1044 s , (2) basic operations: addition, subtraction, multiplication and p 5 c division. Nonetheless, this is rarely an issue with respect to 1/2 describing most classical phenomena. c 8 mp 2.1765110 kg . (3) The latter two limitations are cause for concern especially G when working with dimensionless values such as the fine While the expressions serve as a reasonably accurate structure constant. Fortunately, the count terms used in MQ guide, they will not suffice for our purposes. For instance, if differ from the traditional definition of a dimensionless value; we present the expression for Planck time such that the each count is dimensionally bound to a measure: nL to length, remaining values are supplied using the 2010 CODATA, we nM to mass and nT to time. Moreover, unlike a dimensionless 2 resolve a value for G such that value, MQ count terms may not be combined (i.e., nLnM≠n ). 2 Finally, each count term is, in definition, correlated to its 25 5.39106 1044 299792458 5 dimensional counterpart: l=n l , m=n m , and t=n t . While t pc L f M f T f G , (4) 34 attention must be given to avoid expressions that are 1.054571817 10 dimensionally ambiguous, rarely do the issues typical of dimensionless values become physically significant in MQ. G 6.67385 1011 m 3 kg 1 s 2 . (5) Similarly, with respect to length, then 2.2. Physical Significance of Measure 2 3 −11 3 −1 −2 G=lp c /ħ=6.67385×10 m kg s and with respect to Before we begin, we must distinguish the fundamental 2 −11 3 −1 −2 mass G=ħc/mp =6.67431×10 m kg s . All three measures of MQ from those of Planck. A subscript p is used values disagree with the 2010 CODATA value for to specify Planck units, whereas a subscript f is used for the −11 3 −1 −2 G=6.67408×10 m kg s . Is this a misunderstood fundamental measures, specifically, lf for length, mf for mass geometry, new physics, or inaccuracies in measurement and tf for time. In that we have not resolved the fundamental precision? Perhaps, but also consider that ((6.67431 + measures, we use Planck Units as a guide. The arguments 6.67385)/2 = 6.67408)×10−11 m3 kg−1 s−2. Considering a 6σ and expressions are to be considered as such until the
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 33
properties of measure and the values of the fundamental the observer and the particle. We have measures are resolved. 2 Beginning with our understanding of light and 2 nlLp n l n m . (11) Heisenberg’s expression for uncertainty [15,16], we resolve Lr p M p nt 2 both counts and values for each measure. The speed of light Tp is described as a count nL of length units lp divided by a count With these constraints, it follows that the minimum count nT of time units tp, then c=nLlp/nTtp such that values at the threshold ħ/2 correspond to a minimum distance n l and a momentum comprising a minimum mass n m , a nnLT . (6) Lr p M p minimum length nLlp and a minimum time nTtp. Replacing Using c=lp/tp and Planck’s expressions for length and mass, the value of ħ with the result from Eq. (8), we then have we also resolve that the product of their squares is 2 nL l p l p m p c G 2 ()n l n m , (12) 22 Lr p M p lmpp, (7) nT t p2 t p G cc32 2 2nLr n M n L n T . (13) lmpp clm . (8) pp t Identifying two additional conditions, we may constrain p the expression sufficiently to resolve the count values for Finally, using Heisenberg’s expression [16] to describe each dimension. We begin with a description of G using the the uncertainty associated with the position σX and expression for escape velocity. momentum σ of a particle, P 1/2 1/2 2GM 2GnMf m v , (14) XP , (9) r n l 2 Lf
we can resolve physically significant values for nL, nM, and Such that v=nLlf/nTtf, given that nL=nT=1 (Eq. 6) and nT. We begin by clarifying how we intend to use nM=1/2 (Eq. B7), we resolve that Heisenberg’s expression to achieve our goal. This involves 22 nL l f2 Gn M m f identifying the physical properties of uncertainty we intend , (15) to isolate. 22 ntTf nlLf The uncertainty principle asserts a limit to the precision 3 3 3 3 with which certain canonically conjugate pairs of particle nL l f l f1 l p t p properties can be known. However, this differs from our goal G . (16) 2 2 2mm 3 of resolving the certain minimum measurements of a particle mf n T t f t ffp t p at the threshold, ħ/2. Therefore, we introduce a special case To resolve the second condition, we return to the of the use of variances. expression for escape velocity, again reducing the expression Although the expression for variance is usually written to Planck units and/or counts of those units. Such that r=nLrlp to describe the certain properties of many targets, we and M=n m and where we consider G at the bound v=c, then modify this usage to describe the certain properties of M p 1/2 many measurements whereby the measurement, whether 2GM v , (17) applicable or physically significant, is uncertain. With this, r we then consider the solution for only the minimum count 3 values for length, mass, and time such that the conjugate pair 2 ltpp c2 nm, (18) is equal to the threshold at ħ/2; that is, n l3 m Mp Lr pt p p NN 2 2 XXPP nn 2 . (19) ii . (10) Lr M ii11 NN2 Given 2nLrnMnL=nT (Eq. 13) and nL=nT (Eq. 6), then To the extent that the minimal count N is reducible to a 21nnLr M . (20) certain measure describing a single particle, we consider Then, as expected, with nLr=2nM (Eq. 19), we find measures at N=1. The variance terms for position and momentum reduce such that there is a certain length 2 2nnMM 1, (21) 2 1/2 l=((Xi−푋 ) /1) corresponding to the variance in X and a 2 1 certain momentum mv=((P −푃 )2/1)1/2 corresponding to the n , (22) i M 4 variance in P. We write each term in the MQ nomenclature, i.e., l=nLrlp and mv=ml/t=nMmp(nLlp/nTtp). Note also that the 1 nM . (23) count nL for the change in velocity is distinct from the 2 position count nLr, the latter describing the distance between
34 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
We may continue the reduction given nL=nT (Eq. 6) and We find again that this case is the same as the first. Thus, 2nLrnMnL=nT (Eq. 13), whence we obtain we can conclude that measure is physically significant only 1 if a whole unit count of the reference is made. This may be 2n n n , (24) summarized with the following observations: Lr2 L T O2: Fundamental measures are discrete and countable. nT nLr 1 . (25) O3: Fundamental measures length and time each define a nL reference.
Along with nL=nT (Eq. 6) and such that nL and nLr describe We single out fundamental mass as exempt from this the phenomenon of length, then analysis. Mass is a consequence of our description of length and time. It is not a physically significant minimum measure. nLr n L n T 1. (26) By example, one may resolve an expression for length Thus, we recognize with the observation that starting with the expression for time. This arises in all
O1: There are physically significant fundamental units of physical descriptions of either dimension by definition of measure: length, mass, and time. their measure (i.e., divide lp by c to get tp). Conversely, one may not resolve a value for length or time starting with the That is, there is a physically significant lower threshold to 3 2 measure as described by the resolved counts. The measures expression for mass. The realization that G=lf /tf mf is a do not imply that a phenomenon may not be less than a consequence of the observation that the measure of G is coincident with this relation. To use that realization to minimum. Rather, a length or elapsed time less than lp and tp establish physical significance is circular. may not be measured with greater precision. Notably, mp is a composite of the length and time, an important count but not 2.4. Measurement Frameworks a minimum measure. Moreover, the above calculations do not imply that measure is discrete or countable. Resolving As established to this point, we recognize that measure is a these properties requires further analysis. property of references. With respect to this observation, we can then consider that the universe may be described as a 2.3. Discreteness of Measure space, time and mass. Locations in that space represent We now entertain measures larger than the bounds places of observing mass in elapsed time. And with respect identified in the prior section. Again, as before, we describe to every place the visible motion may not exceed c. measure as a count of some fundamental unit of measure, in In that the rate of visible motion from all places is defined this case, a count of the fundamental unit of length. We also by the maximum c, we also recognize that the classical expand our analysis to include macroscopic measures (i.e., definition of a universe as a physically significant frame can have no external reference. Importantly, we then observe that any distance greater than the reference lp). By example, measure with respect to the universe (i.e. with respect to the consider two sticks, one a length of 5.00 lp and the other a space) must be non-discrete. length of 5.25 lp. The difference may then be described as The observation brings to our attention a big picture view 5.25lllppp 5.00 0.25 . (27) of measure, non-discrete with respect to the universe yet discrete relatively between objects. It is for this reason we Is the result measurable? No. As resolved above, any describe measure with respect to the universe using a count of the fundamental measure less than 1 cannot be self-defining frame of reference. We describe measure measured. Therefore, with respect to the Heisenberg relatively between phenomena using a self-referencing uncertainty principle, the gravitational constant, the speed of frame of reference. Distinguishing the properties of measure light, and the expression for the escape velocity, this and how we describe measure enables a clearer description difference cannot be measured. This is also to say that all of phenomena. macroscopic measures may be observed only as a whole unit In working through various examples, we demonstrate count of the reference measure. that it is the difference between these two frames of reference While the presentation is extendable, let us clarify with that give rise to many, if not all, of the constants and laws of another length difference, two sticks such that one is 10.25 l p nature. If it were possible to reference points external to the and the other is 5.00 l , p universe, there would exist no differential between two 10.25lp 5.00 l p 5.25 l p . (28) frameworks and many of the observed behaviors of nature would not exist. With these observations, we observe that The difference here is physically significant and not discrete. To verify this statement though would also require O4: Measure with respect to the observer is discrete. that the result be distinguishable from a whole unit count O5: Measure with respect to the universe is non-discrete. equal to five units of the reference. We compare the result To demonstrate these observations with a mathematical with a count of 5 lp, description applicable to an observable phenomenon, we 5.25lllppp 5.00 0.25 . (29) propose an experiment that may be described by each of three frameworks. A frame describing measure with respect
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 35
to the universe carries the property of non-discreteness. The ascertain that the angular measure at each point is either remaining two carry the property of discreteness. The along a line or at 90° except for those points along a experiment also abides by two design prerequisites. We hypotenuse. The design, as such, does not require that we introduce no additional measures, such as angles, and at introduce angular measure. Moreover, as all prerequisites are every instant in time, the observer must have access to all agreed prior to setup, the experiment does not initially available information. incorporate time. Using the standard understanding of a Cartesian Note that there are two discrete frameworks, one in which coordinate system, we illustrate the three frameworks in Fig. A certifies the length 퐴 퐵 (the Reference Framework) and 1. With respect to the different origins of information (i.e., the other in which C certifies the length 퐵 퐶 (the the frameworks), we then recognize the differences in the Measurement Framework). The Target Framework contains discreteness of measure. The three frameworks are: both A and C for which the unknown length 퐴 퐶 is a member. Reference Framework—This is the framework of the In this way, all information in the system is defined with only observer where properties of the reference 퐴 퐵 are observed. the presence of members A and C. With respect to the standard understanding, this framework Using the Pythagorean Theorem such that 퐴 퐵 =퐵 퐶 , we differs only in that measure is a count function of discrete recognize that 퐴 퐶 ≈1.414. However, with respect to the length measures equal to one. observer only a discrete reference count may be measured Measurement Framework—This framework shares of 퐴 퐶 . It is with this conflict that we conclude that properties with the Reference Framework. It is characterized the difference 1.414−1.000=0.414 describes a physically as some known count of the reference describing where significant property of the universe. In the section that count properties of the reference 퐵 퐶 are observed. follows, we show that this difference is the phenomenon of Target Framework—This framework is characterized gravity. by the property of measure of non-discreteness, that being the framework of the universe that contains the phenomenon 2.5. Gravity 퐴 퐶 . Having resolved that measure has a lower threshold and is discrete and countable, we now address the physical significance of a phenomenon with respect to the discrete and non-discrete frames. The three frameworks described in Section 2 are represented in Fig. 2. Side a is always the reference count 1. Side b is some known count of the reference. The hypothenuse of the right-angle triangle Side c is then resolved using the Pythagorean Theorem.
Figure 1. Count of distance measures along segment 퐴 퐶 Although each framework is described with respect to the observer’s point-of-view, we also recognize the different properties of measure associated with each framework. With these constraints, we now address how information regarding the count of length measures in the Measurement Framework is obtained by the observer relative to the
Reference Framework. Moreover, we make this presentation to establish values for the fundamental measures. As such, Figure 2. Count of distance measures between an observer and target we will no longer use Planck Units, instead proceeding with Importantly, as a reference, Side a is prerequisite to any the terms lf, mf and tf. count of the reference along Side b to resolve Side c. Consider a system of grid points separated with a fixed Assigning a count other than 1 to the reference would count of lf along the shortest axis. Specifically, there must be introduce a factor representation (i.e., a=2) of the reference enough points to form a square such that the length of each for all sides concealing the discrete count properties of the hypotenuse of the square is also equal. To set up the grid described phenomenon. Hence, initially, we propose that a laser pulse rangefinder is used at 1/2 each point along with the time-of-flight principle to ascertain 2 c1 nLb . (30) a match to the prescribed requirements. In this way, we
36 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
3 We conjecture that any non-integer count QL of the mass accretion is written such that Macr=θsi mf/2tf ([7], Eqs. reference along the unknown Side c relates to a change in 135 and 136) then θsi is dimensionless, having no units at all −1 distance. We may describe this as repulsion when rounding (note: Macr is a rate kg s ). Likewise, as expressed in the up or attraction when rounding down. Notably, for all fundamental expression lfmf=2θsitf ([7], Eq. 47), θsi has units −1 solutions, QL is less than half as evident by its largest value kg m s . As demonstrated in Eq. (B7), θsi has units of radians. ~0.414 at Sides a=b=1 and therefore attractive. Moreover, Each measure of θsi is physically significant and corresponds because Side c always rounds down, we find that nLr=nLb for to the measurement data to six significant digits. all observations. Thus, for each count nT of fundamental time So, why does this constant differ from the other constants tf, the model describes a count of lf that is closer by that we are so familiar with? In part, because the other constants are each a composite of this constant and in 2 1/2 QL n Lc n Lb 1 n Lb n Lb . (31) part because this constant is a composite of all three dimensions θsi=lfmf/2tf. The units carried by θsi depend on the Because the measure of Side c always rounds down, phenomenon and the selected frame. Described with respect moving forward, we replace the term n with n . We also Lb Lr to the Measurement Frame, θsi usually carries the units of identify nLr as the ‘observed measure count’. With the loss of momentum. Described with respect to the Target Frame of the remainder Q relative to the whole unit count is Q /n , L L Lr the universe, θsi carries no units. For specific descriptions we now have an important dimensionless ratio that describes with respect to electromagnetic phenomena, θsi carries the gravity. units of radians. Examples are presented throughout the We may express this ratio in meters per second squared paper, but for nearly all cases, θsi is defined with respect −2 2 −1 (m s ) by multiplying by lf for meters and dividing by tf for to either the Measurement (kg m s ) or the Target seconds squared. This describes the loss of distance at the (dimensionless) frame. maximum rate of one sampling every tf seconds per second, Ql 2.6. Fundamental Measures Lf . (32) 2 ntLr f With a quantum definition for gravity, we can now resolve When compared with a classical description, we notice the simplest relation that describes the fundamental measures. now that the quantity is scaled. Hence, we introduce the This approach is sensitive to the skewing effects of discrete scaling constant S, multiplying by c/S to resolve. Notably, c measure and as such we cannot use the measure of ħ, a describes the rate of increasing space relative to observers in quantum property resolved where the effects described by all spaces as identified with respect to the classical the Informativity differential (Appendix A) are significant. description of the universe. In the following, we will learn Conversely, use of the measures of c and G are acceptable. that the scaling constant S is fundamental to the relation that Note also that the units for θsi are kilogram meters per second. As described in Appendix C, we then have: describes the three measures. Such that r=nLrlf and c=lf/tf, then 2G 2 6.67408 1011 3.26239 l si 1.61620 1035 m , 232 f 33 QL l fc QQcc Q L l f c c (299792458) LL , (33) 2 (35) ntLr f S nLr t f S n Lr l f t f S rS l 11 3 f 2Gsi 2 6.67408 10 3.26239 44 Q c G t f 5.39106 10 s , L . (34) c c44 (299792458) rS 2 r (36) The expression describes gravity as the difference 3 c 2si 2 3.26239 8 between the non-discrete measure with respect to the mtff 2.17643 10 kg . (37) universe and the discrete measure of the observer. When G c 299792458 2 compared with Newton’s expression G/r , we see a distance We may approach a solution to the fundamental between the two curves that is immeasurable, beyond the expression—the simplest expression that relates the three sixth digit of precision for all distances greater than 2,247 lf. measures—in two ways. One is we may replace G given that 3 The curves differ by QLnLr, which describes a skewing of G=c tf/mf (Eq. 16); the other is we may solve for G using the measure due to the discreteness of measure, an effect we expressions for lf and mf, then set them equal and reduce. refer to as the Informativity differential. As derived in That is, Appendix A, QLnLr approaches 1/2 with increasing distance. nL l f n M m f 2 si n T t f , (38) In Appendix B, we replace S with θsi because of its correlation in value to the signal and idler polarization angle l m 2 t . (39) with respect to the plane of X-rays at maximum quantum f f si f entanglement [3]. Notably, the term is not a radian for all Here, all counts in Eq. (38) are notably equal to a value of contexts, but the value of θsi=3.26239 is constant for all one. This differs from their minimum values as well as the physical contexts. For instance, when the expression for count for mass, nM=1/2 (Eq. 23). Explicitly, the fundamental
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 37
expression is not a description of the lower count bound of Heisenberg uncertainty principle. Importantly, the Planck each dimension. Moreover, in MQ, we often ignore the discrepancies observed in Eqs. (1)–(3) with respect to the Informativity differential and instead replace QLnLr with its 2010 CODATA are reduced to the sixth significant digit (see macroscopic limit of ½ as described in Appendix A. The Table 2). more precise expression, which we refer to as the expanded form, is Table 2. Planck’s expression calculated with quantum ħ and macroscopic ħf values for Planck’s constant
sit f Informativity Length Mass θ (radians) Time (s) lmff . (40) differential (m) (kg) QnL Lr Planck’s Reduced 1.616228× 2.17647 5.39116× 3.26250 As such, many MQ expressions are affected by the Constant ħ 10−35 ×10−8 10−44 Informativity differential, each having expanded Planck’s 1.616200× 2.17643 5.39106× counterparts. Although the calculation does involve several Fundamental 3.26239 10−35 ×10−8 10−44 steps, it is required when describing quantum phenomena, Constant ħf especially phenomena less than 2,247lf. 2010 CODATA 1.616199× 2.17651 5.39106× −35 −8 −44 Estimates [2] 10 ×10 10 2.7. Physically Significant Discrepancies with ħ We mention that the small rounding effect that occurs in Expressions that use measures, both macroscopic and −35 the length result (0.0000006×10 m) is subsequently quantum, have limited precision because of the Informativity amplified in the mass. Had the rounding gone the other way, differential. In this section, we explore those effects as they differences with the CODATA would not exist. That said, apply to the measures of G and ħ. This was demonstrated in neither case displays a seventh-digit physical significance (Eqs. 4 and 5) where the resolution of the gravitational and as such should not be considered. There exists no constant using Planck’s expression for time presented a physically significant difference between the MQ discrepancy in the fourth significant digit, a value of description and the measurement data. 2.4×10−15 with respect to the 2010 CODATA. Because the With Planck’s reduced constant adjusted for the effects of measure of G is a property of macroscopic phenomena and ħ the Informativity differential, we may apply the value to a measure of quantum phenomena, it is necessary to resolve expressions that include macroscopically measured terms. the effects of the Informativity differential to present a value For instance, the value of θ as described in Appendix B of ħ as it would appear if measured macroscopically. We will si using G and c may now be presented using ħ (see listing in call this ħ . This, in turn, would be suitable when measured in f f Table 3). Each value precisely matches the Shwartz and expressions that include measures resolved macroscopically. 3 2 Harris measures [3]. Given c /G=ħ/lf (Eq. 1) and the fundamental expression θsi=lfmf/2tf (Eq. 39), we resolve ħf with respect to Table 3. Predicted radian measures of the k vectors of the pump, signal, 3 and idler for the maximally entangled state at the degenerate frequency of macroscopic measures G=c (tf/mf) (Eq. 16), then X-rays using Planck’s fundamental constant ħf 3 lf m f l fc l f ff θp θs θi si , (41) 2 (ħf/2lf)−π π−(ħf/2lf) π−(ħf/2lf) 2tff 2 G 2l 2 l π−θ f Max (0.1208) (−0.1208) (−0.1208)
34 2π−(ħf/2lf) (ħf/2lf) (ħf/2lf) 2 l 1.05457 10J s . (42) θMax f si f (3.02079) (3.26239) (3.26239) The approach physically validates our understanding of Likewise, we may expand our understanding of the the derivation of lim f(Q n )=1/2 (Appendix A), which nLr→∞ L Lr relationship between G and ħ with the following correlation. had we instead used the expanded form of the Informativity We start with the fundamental expression l m =2θ t (Eq. 39), differential (Eq. 38), would then yield ħ=θ l /Q n . f f si f si f L Lr then The result describes Planck’s reduced constant at the macroscopic limit, although the measure of ħf at any distance 2sitm f l f f , (43) greater than 2,247l will reasonably approximate the limit. f 2 Conversely, at the quantum distance 84.9764lf 4sitm f 2 sil f f , (44) (Appendix D)—that distance corresponding to the measure 2 of blackbody radiation—and ħ=θsilf/QLnLr with QLnLr = 4sitm f f f , (45) 0.499983, then the value of ħ is as we recognize it today. We identify the 84.9764lf distance as the blackbody t f 4 c3 2 c 3 , (46) demarcation. si f m f Note that ħf is a function of only θsi and lf when accounting for the contraction of length associated with discrete measure. 4cG 23 . (47) The approach changes our understanding of the physical si f significance of ħ, now being a count property of the Here, all the terms are macroscopic, and hence we have
38 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
appropriately replaced ħ with ħf. We then move the terms we 1.616200 1035 2.17643 10 8 find in Planck’s expression for length (Eq. 1) to the right, 2 1 3.26239 , (55) leaving the remaining terms for the fundamental length (Eq. 5.39106 1044 35) to appear on the left. This brings to our attention that it is the lack of the Informativity differential that limits Planck’s 6.52478 6.52478 kg m s1. (56) expressions to three digits of precision. Having both a physically significant description of the fundamental length Finally, we recognize that the quantum approach to and accounting for the skewing effects arising from discrete describing gravity also allows for a calculation of the measure, we bring the two expressions together thus gravitational constant using only the measure of light (i.e., resolving the measurement discrepancy found in G as lf =tfc) and θsi. The approach again corresponds to the 2010 presented in Eqs. (4) and (5). Specifically, we have CODATA to six significant digits. 2 2 4G QL n Lc n Lb 1 n Lb n Lb , (57) si , (48) 3 f c 34 2 34 QL 1 (6.18735 10 ) 6.18735 10 22 , (58) 4G f G si , (49) 8.08100 1036 cc63 Q c3 8.08100 1036 (299792458) 3 1/2 L 2G f G si . (50) rsi 1 3.26239 . (59) 33 cc 6.67408 1011 m kg12 s- In the same way, we can take this expression and using ħf, Similar examples extend to electromagnetic phenomena. Planck’s reduced constant adjusted for the Informativity 3 1/2 −35 The effects of the Informativity differential with respect to differential, solve. However, (ħG/c ) =1.61623×10 m those constants will be discussed in the section to follow. To incorrectly resolves the measured value. Using ħf, the summarize these results, we present in Table 4 the expression is now mathematically equivalent to six digits, Informativity differential with respect to three physical 11 2 6.67408 10 3.26239 constants. We recall that the value for θsi comes from the Shwartz and Harris experiments, not from the CODATA, 2997924583 , (51) which presently does not recognize this value. 1/2 34 11 1.05454 10 6.67408 10 Table 4. Physical constants calculated with quantum ħ and macroscopic ħf values for Planck’s constant 3 299792458 Planck’s Physical Constants reduced 35 35 1/3 3 2 1.61620 10 1.61620 10 . (52) constant c=(4Gθsi/ħ) θsi=ħ/2lf G=c lf /ħ −11 Returning to Eqs. (4) and (5), and replacing ħ with the ħ 299788980 3.26250 6.67385×10 −11 distance sensitive measure adjusted for the Informativity ħf 299792458 3.26239 6.67408×10 2010 differential, ħf, the discrepancy with the 2010 CODATA for 299792458 3.26239 6.67408×10−11 3 2 −11 CODATA [2] the gravitational constant G=c lp /ħ=6.67385×10 is also resolved, specifically
3 2 3. Results 32 299792458 1.616200 1035 c l f G 34 . (53) 3.1. Fine Structure Constant f 1.05454 10 11 3 -1 -2 Considering the new descriptions offered by MQ, we 6.67408 10 m kg s present four expressions that describe the fine structure constant α. Concepts from MQ are used to resolve an Moreover, replacing ħ with ħf properly accounts for the skewing effects of the discrete measure as applies to Swartz understanding of each. More importantly, we present a singular physical description of their differences— that is, and Harris’s measure of θsi (Appendix B), how the distortion of measure explains the difference in 34 f 1.05454 10 3.26239 kg m s1 . (54) value between each expression. si 35 2l f 2 1.616199 10 Before we begin, we note that counts of θsi are central to the presentation. Specifically, the count factor 42 of θsi The dimensional homogeneity problem is also solved. determines the value of the fundamental fine structure From the fundamental expression, 2θsi=lfmf/tf, we find a constant αf and is physically correlated to the charge mathematical correspondence with the 2010 CODATA coupling demarcation, a distance associated with α and values [2], described as a count of lf. The two terms—lf and θsi—are
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 39
proportional as described by the count values of each term demonstration. and related by the fundamental expression. Given the We shall next discuss the metric and Informativity minimum count terms are nL=nT=1 and the corresponding differentials and their relation to each of the measurement count term for mass nM=1/2, (Eqs. 23, 26), then the minimum frameworks. We do not address the change in the value of α count of θsi with respect to the Reference Frame is obtained with respect to increasing energy as described in QED. from the fundamental expression, Nonetheless, this presentation does address the ground state of α; a description that incorporates high-energy phenomena nnLM 1 1/ 2 1 n . (60) is to be a topic of further research. Also, we note that αħ is not si nT 12 physically interesting because a coupling of the Informativity The physical significance of counts and their relation to differential to the measure of a phenomenon that already frames of reference are best understood with respect to the accommodates the effects of this skew is duplicative. Given unity expression described in ([8], Eq. 111) of a “Quantum that ħf differs from ħ precisely by the Informativity Model of Gravity Unifies Relativistic Effects …” as differential, the calculation presents an opportunity to published in Journal of High Energy Physics, Gravitation demonstrate two means of applying this effect, each and Cosmology; resulting in the same value. The expression and value for each of the four descriptions are: 2 1/3 2 1 t f n 42 137.020 , (62) Lm 1 . (61) f si lmff nLc 1 mae 0 p 137.041, (63) As is true with the fundamental expression, the mlff combination of terms lfmf/tf has no units, defined with respect 1 c to the frame of the universe. The unity expression describes 4π 137.036 , (64) c 0 2 the dimensional measures of the prior counts with respect to e the expansion of the universe, yet notably excludes the factor 1 f c 1/2, the constant of proportionality in the fundamental 4π 0 137.031 . (65) e2 expression lfmf/2=θsitf which correlates the dimensional terms. When working with the nondimensional expressions To explain their relationship, we begin with αf and then of MQ, how counts apply to specific phenomena must be demonstrate how each of the remaining expressions validated by the physical and value correlations of the differ. Two distinct measurement-skewing effects must be resulting description, the difference in this case being a considered. The metric differential is notably different than description with respect to the self-defining frame of the that of the Informativity differential; the latter describes the universe, as opposed to the self-referencing frame of the skew in measure arising from the discreteness of measure observer. and is defined with respect to the self-referencing frame of With respect to the charge coupling demarcation a the observer, that is, between phenomena in the universe. non-discrete distance of nLr=276/θsi=84.6005 corresponds to The metric differential Δf describes the shift in measure that a count of nθ=RND(84.9764/2)=42 in the Measurement exists between the discrete and non-discrete frames. The Frame. We present the MQ expression for the inverse of the function RND to be used below means to round to the nearest −1 fundamental fine structure constant as αf =42θsi. However, whole-unit value (glossary). And the count nθ is 42, as θsi is defined with respect to the Target Frame and as such is discussed above, corresponding to the measure of the charge dimensionless. A second description of α was discovered by coupling demarcation. Planck in his work with the Planck units. He observed that α Beginning with a general expression for the metric could be described as a function of the electron mass me differential, then and the radial distance to the first ground state orbit a0 (i.e., 2 2 f n nn si RND si , (66) the Bohr radius 4πɛ0ħ /mec ). We identify his description with the designation αp. A third description follows from 42 RND 42 . (67) electromagnetism, which also serves as the CODATA f si si −1 definition for α. We identify this description as αc. A fourth To resolve a Planck form of the expression αp , we start −1 expression follows from MQ, modifies the CODATA with αf and adjust; that is, we add the metric differential definition such that Planck’s reduced constant ħ is adjusted Δf. The addition accounts for the physically significant for the Informativity differential, ħf with respect to a difference between the discrete and non-discrete frames of macroscopic distance. We identify this description as αħ. reference. There are other descriptions, such as α=Z G /4 written as the 0 0 11+ 42 42 RND 42 , (68) impedance and conductance of a free vacuum and the p f f si si si product of the Bohr radius α=r /r such that r =ħ/m c. e Q Q e 1 84 RND 42 137.041. (69) The first four descriptions though will suffice for our p si si
40 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
The value is identical to the value resolved with Planck’s 1 1 1 1 c p p1 2Q L n Lr 2 Q L n Lr p , (74) expression to the precision of θsi, i.e., six digits. Conversely, descriptions of αc and αħ differ from that of αp 1 c2Qn L Lr 84si RND 42 si 137.036 . (75) by the Informativity differential. To proceed, we must know the non-discrete count nLr of lf associated with the charge We repeat this process once again to resolve the value one coupling of a cesium atom absorbing a photon, namely, the would find when using the Informativity differential adjusted charge coupling demarcation, described here; value for Planck’s constant, ħf, only our base measure is not −1 −1 αp , but now αc . Specifically nLr 276 / si 84.6005 . (70) 1 We can then resolve the Informativity differential as C c12 Qn L Lr , (76) 1/2 2 11 , (77) QL n Lr n Lr1 n Lr n Lr 0.499983 . (71) c C 1 1 1 The differential skewing of measure between the Planck c c1 2Qn L Lr 137.031. (78) and electromagnetic expressions Δ(P-C) due to MQ is again Each of the values match the corresponding 2018 the differential between α −1 and α −1. We multiply the p c CODATA values to the same precision as the measure of θsi, Informativity differential by two to resolve the skew with i.e., six digits. Also, of interest is the difference between the respect to the self-defining frame of the universe, not the calculated values with respect to the modern and MQ radial description respective of an observer/target relation. expressions. While MQ calculations are constrained to six Then digits of physical significance —the precision with which 1 12 Qn . (72) we can measure θsi—on comparing the values of the PC p L Lr resulting calculations, we find that the difference between th Notably, the Informativity differential is a contraction the modern and MQ expressions are consistent to the 10 effect (i.e., like gravity). Subtracting two differentials 2QLnLr significant digit corresponding to the precision of the −1 −1 of αp from αp (i.e., 1−2QLnLr), then modern measurement (see Table 5). The consistency of the difference emphasizes a correlation that extends in 11 cp PC , (73) parallel between the MQ expressions and the physical measurements. Table 5. Modern and MQ expressions for the inverse fine structure constant, their values and difference
Values Expressions for α CODATA MQ Difference Differential −1 αf =42θsi - 137.02038 - - −1 2 −1 αħ =4πɛ0ħfc/e 137.03123 137.03123 0.0000088261 αc (1-2QLnLr) −1 2 −1 αc =4πɛ0ħc/e 137.03600 137.03600 0.0000088263 αp (1-2QLnLr) −1 αp = mea0/mflf 137.04077 137.04077 0.0000088263 42θsi−RND(42θsi)
Moreover, we now have one physical approach to describe This approach has changed. Now, the elementary charge, all expressions. The difference between them is a function of Planck’s constant h, and the speed of light in vacuum c are the differential. The approach supports the position that the defined values, leaving the magnetic constant as a measured expressions are not in error but are a physically significant value that determines the value of ɛ0. The magnetic constant, consequence of MQ relative to the measurement distance. as before, is a function of α. This is most relevant in the long-standing discrepancy With the expressions presented, we may approach between the Planck and electromagnetic interpretations. definitions for the electromagnetic constants anew. For one, There has been no physically correlated explanation for their we may replace Planck’s reduced constant with the difference to date. We also draw attention to the metric following expression (Eq. 54) given QLnLr=1/2, differential and its physical significance when describing sil f differences in measure between the two frames of reference. f . (79) QnL Lr 3.2. Electromagnetic Constants Next, we may replace α with αc reducing the description of Until May 20, 2019, the value of the elementary charge e ɛ0 to a function of θsi, c, and e. Although ɛ0 is defined, the had been defined as an exact number of Coulombs [17]. This determination of ɛ0 follows as a function of e, fundamental gave a specific value for the electric constant ɛ0 as a function units, and mathematical constants. With QLnLr=0.499983 at of the magnetic constant μ0, which in turn follows from the charge coupling demarcation, then the elementary charge and the fine structure constant α.
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 41
2 2 2 discrete and non-discrete frames, i.e., 42θsi-RND(42θsi). As 1 eeQnL Lre , (80) such, we may describe the differential b-d of e as an equality 0 2 2hl c 4π c 4π c f 0c Cf Cfsfi Cf with the quantization ratio b/d. Collectively, the two define the fundamental elementary charge which when multiplied Qn 2 84 RND 42 e2 L Lr si si by the metric differential Δ between the frames – the 0 fr 2cπsil f . (81) product being a function of rq and b-d (i.e., b/d=b-d) – give us e in the local frame. To do so, we leverage the known 8.85419 1012 Fm 1 p value of the discrete difference d=θsi at the demarcation 42θsi. 2 Given that μ0=1/ɛ0c and c=lf/tf, we also resolve two more Notably, the demarcation counts for all phenomena round to constants. The magnetic constant, for instance, is 42. 2 112πsit f d , (82) b . (87) 0 2 2 2 d 1 0c QnL Lr 84si RND 42 si e With this we isolate and resolve the fundamental value of 6 1 0 1.25664 10 Hm . (83) ef as a function of b. Keep in mind, charge is not and may not be known as a count of θ nor is it known in terms of the Coulomb’s constant k is si e fundamental measures. As such, there exist no dimensionally 1 2πsil f c homogeneous precedent to validate our expression. To k , (84) e 4π 2 2 compensate, we express all measures in their fundamental 0 4πQnL Lr 84si RND 42 si e form, lf, mf, tf, θsi, and ħ, replacing dimensional homogeneity sil f c with physical homogeneity. k e 2 Having physically correlated each measure, then the base 2Q n 84 RND 42 e2 . (85) LLr sis i b is the elementary charge ef as a function of the fundamental 8.98755 109 Nm2 C -2 mass f(mf) relative to its quantum of angular momentum ħf. We begin by mapping each description to θsi. For instance, New to modern theory, θsi is also the radial rate of the corresponding momentum is θsi=(1/2)lfmf/tf, indicating expansion defined with respect to the universe (i.e., not per that we should divide ħf /2. Moreover, we observe that the Mpc) and an angular measure (in value) corresponding to phenomenon of elementary charge presents itself at the the plane of polarization for maximally quantum-entangled upper bound c. Therefore, we resolve the upper count bound X-rays in specific Bell states [3]. As such, we have expanded of mf in relation to the count of tf as the mass frequency our physical definition of electromagnetic theory to include bound. As described in Appendix E and the third paper ([9], both quantum and cosmological phenomena. Appendix 5.3), then Although the electric and magnetic constants have been 33 reduced, in part, as a function of the metric and Informativity mf m f l f m f differential, elementary charge remains problematic in that a MRRb f() R si (88) tf t f2 t f known description of e does not exist as a count of θsi. That is, 2 the non-discrete frame of the universe provides a geometry 42 of only counts and mathematical constants. Nevertheless, lf m f l f m f nM m b f() R n Lr l f n Lr (89) elementary charge is a multi-dimensional measure in the 2 2t 2t f f discrete frame of the observer. For this reason, there exists no 2 mathematical counterpart. We are forced to describe e with a Hence, f(mf)=mf relative to θsi=lfmf/2tf (i.e., the remaining terms). With both mapped to θsi, then the physically discrete physical approach and correlate that to the 2 non-discrete frame of the universe with the metric homogenous expression is b=mf /(ħf/2). With this description differential. resolved with respect to the macroscopic measures of G and Before we begin, we note briefly that a second way to c, we then use ħf=2θsilf to reduce. Thus, describe the metric differential is as a ratio of counts of θsi. mmff ee . (90) Specifically, we introduce the quantization ratio, taking the f f f /2 sil f non-discrete product nθθsi and dividing by its discrete product, The approach confirms the identification of θsi as the divisor/difference d. We remove θsi from our definition of nsi rq ; (86) the base b and account for the squared relation of mf at the RNDnsi bound, 2 Its physical significance is discussed extensively in the m f sections to follow. We may now resolve an expression for be f . (91) −1 l f elementary charge. Recall in Eq. (69), that αp was resolved with respect to a differential (i.e., a difference) between the We now solve for the fundamental elementary charge in
42 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
terms of the fundamental units, 19 ec 1.60217 10 Coulombs . (102) 22 ef m f e f m f With a description of the elementary charge comprising , (92) si fundamental measures, we describe the electric and magnetic sill f f constants. Such that lfmf=2θsitf and ec=2QLnLrep=2QLnLrΔfref, 2 2 2 then ef m f si e f m f l f si , (93) 2 2 QnLLr 84si RND 42 si 2 l f si 19 e , (103) e 1.60513 10 . (94) 0 c f 2 2πsilc f m f si 1 2 2 2 With respect to units, there is no convention in describing QnL Lr 84si RND 42 si l f si fr 2Qn , phenomena relative to different frames. The issue becomes 0 LLr 2 2πsilcf m 1 more complex with elementary charge, now a presentation of f si the geometry used to describe the fundamental fine structure (104) constant. It is conjectured that charge may have a geometric 4 2 2 QnL Lr 84si RND 42 sil f c fr origin, a function of m kg-2, but more research would be , (105) 0 2 needed to fully resolve the physical significance of this 4πm fs i 1 description. We continue by correlating this description to the To be discussed in detail in the section on unification, Measurement Frame by applying the metric differential. This we may replace the metric differentials with gamma γ. The is described as a product of the quantization ratio between effects described by γ are geometric, a function of the the two frames. Note, the differential is an offset of one point-of-view of the observer and not intrinsic to the relative to the demarcation. The differential should have described phenomenon. For this reason, it is physically significant to separate these characteristics. been applied to ef, but we are forced to apply it to ep, making this an approximation. That said, a solution is presented in 2 21si Section 3.7 that resolves the true value. For now, we describe γ , (106) 4 2 this as nθ-1 such that nθ=42. QnL Lr 84si RND 42 si fr n 1 Making the substitution then, si 0.998194 . (95) fr 2 RNDn 1 c l f si 8.85413 10121 Fm . (107) 0 2πγ m Taking the product, we resolve the Planck equivalent of f 2 elementary charge. With μ0=1/ɛ0c , then 2 2 n 1si l f si 1 4πm f si 1 ep fre f . (96) .(108) RNDn 1 2 0 2 4 4 2 si mf si 1 0c QnL Lr 84si RND 42 sil f c fr We present the Informativity differential relative to the 2πγ m f demarcation count (both the charge coupling and blackbody 1.25665 1061 H m . (109) 0 4 l demarcations product the same six-digit value) and resolve c f the differential between the Planck equivalent ep and the Coulomb’s constant is then CODATA form of the elementary charge ec; 2 1 m f si 1 2 1/2 k , (110) QL n Lr n Lr1 n Lr n Lr 0.499983 , (97) e 4π 4 2 2 0 QnLrL 84si RND 42 si l f c fr
ep 12 QLn Lr . (98) γ m f PC k 8.98761109 Nm 2 C 2 . (111) e 2 2c l f Subtracting two differentials 2QLnLr of ep from ep (i.e., 1-2QLnLr), then The value of each constant is compared with the 2018 CODATA values in Table 6. eec p PC , (99) Notably, there is a skew of 0.55 in the sixth digit of e from ecp e ep 1 2QL n Lr 2 QLL n r ep , (100) that found in the CODATA. This stems from the application of the differential to ep instead of ef. Moreover, there is a 2 n 1 l six-digit precision constraint in the measure of θsi that is si f si ec 2QnL Lr . (101) amplified in ɛ and k . We will address this by resolving a RND n 1 2 0 e si m f si 1 more precise value for γ in Section 3.7.
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Table 6. Electromagnetic constants as a function of the fundamental measures and appoximated γ
Values Physical Constants 2 −2 e (C) ɛ0 (F m−1) μ0 (H m−1) ke (N m C ) MQ 1.60217×10−19 8.85413×10−12 1.25665×10−6 8.98761×109 2018 CODATA [1] 1.60218×10−19 8.85419×10−12 1.25664×10−6 8.98755×109
To explore the physical meaning of these expressions 2π h . (118) further, we modify the definition of μ0 to incorporate h as a 0 25 f part of the expression. First, recall that the blackbody l f c demarcation (Appendix D) is a function of the Informativity With γ incorporating several effects, each a function differential, which may be used to solve for the demarcation relative to the observer, then the discrete properties of at 84.9764lf. With this solution, we resolve first the metric Planck’s constant hf, the quantum of action, are directly differential Δfr=0.998194 and then the Informativity proportional to that of μ0. differential QLnLr=0.499986. Given ħf=2θsilf, then 2 3.3. Properties of the Atom 1 21 si h . (112) 0 2 4 25 2 f When working with the MQ nomenclature, we may more 0c QnL Lr 84si RND 42 sil f c fr easily recognize the permissible properties of phenomena. The expression hints at magnetic phenomena being a For instance, we may ask what an MQ description of an discrete count of Planck’s constant, a quantum of action. elementary charge looks like to understand atomic structure. Does this tell us more about its discrete properties? Perhaps, With the observation of charge appearing in nature as a but we must refine this understanding to improve the discrete count of fundamental units, we may then look to the physical correlation. The focus here is on π, which was lost component terms to see if they vary and, if so, what other values of e are permitted. From Eq. (101) and given with the introduction of hf. To understand the physical 2 2 2 importance of π, observe the expressions below; they θsi /mf =c /4, then describe the relationship between the energy of a 2 n 1 1 l f si fundamental unit of mass E and the energy of a photon E . e 2Qn si , (119) m l c LLr 2 RNDn 1 si 1 m From the fundamental expression lfmf=2θsitf and with si f ħf=2θsilf (our comparison is with the measure of mass), then 2 QnLLr n 1si c l f nMf sic e . (120) E mcc22 n si , (113) c mM 2 RNDn 1si si 1 QL n Lr c Q L n Lr We observe that all values are constant. Subsequently, sil f 1 nM2π sic El nMMMhv n2π v n 2π , (114) given that elementary charge is measured only as a QL n Lr t f QLLn r whole-unit count of e, we find that charge must be a whole-unit count of the observed phenomenon. Importantly, E 11nMf sic/ Q L n Lr m , (115) the component terms that describe e are physical and El2π n M sic / Q L n Lr 2π numerical constants. To imply that e could take on a fractional value would require that one or more of the EElm 2π . (116) fundamental measures—lf or θsi or c—was not fundamental. These expressions describe the role of π between The description does not accommodate fractional charges descriptions of particle and wave phenomena. That is, the inferred with respect to quarks, leaving the conjecture that numerical constant that divides them is 2π. Returning to the charge is a physically measurable property of quarks Planck modified definition for μ0,we find that it is more unsupported. fundamental to retain π. This is evident when we observe that O6: Charge is not a physically measurable property of the difference between Eqs. (108) and (112) is quarks.
2πmf l fc4π si t fc2 π22 si l f π f h f . (117) In a similar fashion, such that me=nMmf, (i.e., nM is not a physically significant count, but is constant) Planck’s Such that (2θsilf)=ħf, it is more fundamental to replace hf expression for the fine structure constant may be arranged as and then ħ, which would then place us back to where we started. That is, electromagnetism is best described as a 1 mae 0 P 84si RND 42 si , (121) wave phenomenon (epitomized by π) in a classical spacetime mlff using the same fundamental measures used to describe gravitational curvature. l f a0 84si RND 42 si . (122) That said, we also observe that nM
44 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
Consequently, the ground state orbital a0 of the electron A difference between the correlated terms that describes must exist precisely as described. one or more other phenomena. Consider now the application of the MQ nomenclature—a O7: The ground state orbital of an atom is invariant with set of physically distinct terms—to our descriptions of respect to the fundamental length lf and the count nM of mf of gravitation and electromagnetic phenomena. Given that the an electron (i.e., lf/nM). electric and magnetic constants are inversely proportional up There are no variable terms in the description. Importantly, to the square of the speed of light (i.e., ɛ =1/μ c2), we we find that the Informativity differential, applicable to terms 0 0 consider only the relationship between G and ɛ . We reduce in the numerator and denominator, cancels out such that it is 0 the expression for G (Eq. 34) and compare it with the also not a part of the description. Thus, we would expect expression for ɛ (Eq. 105), arranged such that the differentials are a function of the relative distance of the 0 dimensional terms fall to the end. observer, not an intrinsic property of the atom. 3 34 Qrc QL l f1 Q L n Lr l f 3.4. Unification G L n l , (126) 33Lr f si tt si One of the greatest endeavors of the modern era has been si f f to provide a physically significant and meaningful 42 tf Q L n Lr l f l f c unification of gravity with electromagnetism. We present G , (127) l m Q n3 m that this endeavor is challenging in that there is no clear f f L Lrt f f roadmap as to what constitutes unification. For instance: i) 4 2 2 Should one present a one-for-one match between strings as QnL Lr 84si RND 42 si fr l f c . (128) described in String Theory with respect to each of these 0 2 m phenomena? ii) Would this be recognized as the most 4πsi 1 f satisfactory solution? iii) Would a correlation between two The dimensional terms for the electric constant are distinct field expressions constitute a better unification? iv) precisely those that describe gravitation. To complete their What about a classical approach using only the laws of correlation, we make a final substitution, motion? 4 2 Moreover, let us consider the existence of a match. In that QnL Lr 84si RND 42 si fr each phenomenon is different, there would exist a physical 0 G . (129) 4π 1 2 differential. What differential—additional constants and si geometry—would be acceptable? One must bear in mind that the description of G is a Let us entertain what may be considered a step towards distance-sensitive property of the observer, not an intrinsic unification by presenting an example of what is not property of gravitation. Moreover, note that QLnLr=0.499983 unification to help clarify the definition. Consider the at 42θsi and the differential Δfr=0.998158 both describe expression for the product of the electric and magnetic observational phenomena not intrinsic to the compared constants and multiply both sides by G, phenomena. The physical significance of that which G describes their difference excludes the observer’s relative 00G . (123) motion, and the distance is independent of the c2 measurement-skewing effects between the inertial frame and Granted, such a coupling of fields is nonsensible, but our the observed phenomenon. goal is to then reduce and demonstrate a fundamental We now consider each of these ‘other phenomena’ that 3 expression that masquerades as unification. With G=c tf/mf, distinguish gravitation from electromagnetism, as follows. replace G on the right-side, thus solving for G. Two measurement-distortion phenomena: the metric 3 and Informativity differentials 1 c tlff G , (124) 00 2 The Informativity differential is described by QLnLr at 42θsi c mmff and the metric differential is described by Δfr. Each term describes a relative skew in measure. Importantly, the l f G . (125) Informativity differential addresses the skew in lf with m f 00 respect to the self-referencing frame. The metric differential addresses the skew in θ with respect to the self-defining Why does this fail to demonstrate unification? Among si frame. other things, the expression fails to provide term descriptions that can be defined independent of the unified phenomena. First frame correlation: the metric differential associated With this example, we identify unification as being with the fine structure constant nomenclature of physically distinct terms that are Given that the inverse fundamental fine structure constant independent of the correlated terms, is 42θsi, we then apply the metric differential to resolve the A definition of each correlated term comprising distinct Planck equivalent as observed in the Measurement Frame. terms and mathematical constants, and The expression is a function of the count of the base measure
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 45
θ corresponding to the charge coupling demarcation. 2 si m f 84si RND 42 si . (130) Em 2π0 . (139) l f Second frame correlation: the metric differential Although γ is a necessary part of the calculation, we associated with elementary charge consider it an external consequence of the geometry between The terms below describe the metric differential the observer, the target, and the universe. When resolving associated with elementary charge. The expression may be the properties, the overall geometry is important to the considered a mathematical constant correlating the measure calculation, but not relevant to the intrinsic properties of the of e between the discrete and non-discrete frames; phenomenon. Consequently, we consider the above energy 1 expression a physically correlated function of the electric . (131) constant and fundamental measures, the remainder γ being 2 si 1 geometric relative to our point-of-view. Notably, the extra 2 fundamental units mf /lf are precisely the base relative to θsi One particle/wave correlation: as a function of energy 2 used to describe e. For instance, substituting mf /lf with Eq. The last term, found in the denominator, is 2π. As (94), then expressed in Eq. (116), the term may be described as the ratio 2 of the energy of a fundamental unit of mass m with respect to si f Eemf 2π0 . (140) that of a photon; si 1 As such we find the product of energy and charge to El 2π = . (132) describe one revolution and the electric constant, the E m remaining terms a function of the observer’s point-of-view. Collectively, the five expressions—all of which comprise mathematical constants—describe differences that 3.5. Demarcations and Fundamental Constants distinguish the electric constant from gravitational curvature. MQ allows us to use the physical correlation we have With γ representing those terms that describe the skew in made between counts of θsi and that of α to do the same for measure and geometries external to the intrinsic properties of Planck’s reduced constant. We clarify that one must choose the two phenomena, to resolve values with the demarcation most appropriate to the phenomenon being described. We will use a discrete 21 2 γ si , (133) approach to the fine structure constant and then a second 4 2 QnL Lr 84si RND 42 si fr approach to resolve Planck’s reduced constant. The fine structure constant is defined against the then the correlation of G to ɛ is −1 0 Target Frame as αf =42θsi=137.020. When compared to 11 the Measurement Frame, the value of α corresponds to 0 , (134) G 2π the nearest whole unit count RND(nθθsi)=137. Hence, the quantization ratio is rq=137.020/137=1.00015. We express G 2π0 . (135) this as As expected, the correlation follows the same form as for n r si . (141) energy which carries no geometric component γ, E =2πE . q l m RNDnsi We advance one more expression with respect to energy. Arranging Eqs. (115) and (134) with both equaling 1/2π, we The ratio is a numerical description of the relationship then set them equal yielding between the non-discrete and discrete frames. Quantization ratios may also be defined by the inverse, but this relation has E m 0 . (136) proved most useful. EGl Thus, the gravitational constant corresponds to the energy of a photon as the electric constant does to the energy of mf, with γ describing the four additional geometries not intrinsic to the phenomena. We may also describe the energy Em of mf. Given El=2πθsic/QLnLr (Eq. 114) and θsi=QLnLrlfmf/tf, then 2 tmff EEE0 , (137) m l l 0 3 G l f 22 2π c tf m f Q L n Lr l f m f2πc t f m f E si , (138) m 0033 QL n Lrllff t f Q L n Lr Figure 3. Diagram of quantization ratio terms
46 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
In that we are working with counts, which are with a value of 1 (Graph 1; also listed in Table 7). Points are nondimensional, the approach is both universal and resolved such that y=1 for the quantization ratio at the applicable to all dimensions (i.e., θsi, lf). Importantly, the beginning, end, and middle of the sequence. Notably, what is approach allows us to resolve α. However, before we begin, being counted – θsi, lf, widgets – is irrelevant and affects only we briefly define some terminology (see Fig. 3). the magnitude of the quantization ratios along the y-axis. Demark—Given a plot of quantization ratios, there is a The resolution of the midpoint is a function of the x-axis repeating pattern. Demark identifies the first point in each count quantization, that is entirely a function of counting. pattern such that y=1, with the average of a discrete set of points immediately to the left y<1 and the average of a discrete set of points immediately to the right y>1. Sequence—A plot of quantization ratios comprising points including both beginning and end demarks. Series—The set of sequences for which the beginning demarks share identical quantization ratios. A series may be distinguished as having a base demark nθ with repeating demarks such that each demark in the series is a whole unit count of the base (i.e., nθ: 42, 84, 126, …, 1050). Before we begin, we emphasize that quantization ratios are a function of discrete counts nθ of θsi, but we use those counts to resolve α with respect to the charge coupling Graph 1. Plot of Quantization Ratios Describing the Charge Coupling Demarcation demarcation, as a non−discrete multiplier nL of lf. A physical value is not needed, but we will need to map the pattern to a Table 7. Metric Approach to Planck’s Reduced Constant physical phenomenon, and we achieve this as a count of Calculate Values some measure. The two measures – θsi and lf – are correlated nθθsi↔nLlf Start Midpoint End with respect to their count values such that nθ=RND(nL/2) as described in Eqs. (60) and (61). Moreover, resolving the Dataset 82.148363623 84.600553570 87.052743541 midpoint—as is required for resolving the demarcation Such that n θ =276, we find the charge coupling associated with α —produces the same value with any Lr si demarcation is n =276/θ =84.6005. sequence to a considerable precision. Lr si Conversely, we do not have an expression for Planck’s Moreover, sequences are a function of the separation of reduced constant with respect to the Target Frame. We can data points along the x-axis (i.e., their quantization which is resolve its demarcation distance with Eq. (70). Knowing θsi described by rq=nθθsi/RND(nθθsi), Eq. 86). Graph 1 is and lf, then displayed with non-discrete x-axis values nθ incremented by 2 0.1 in separation. A 0.2 separation produces sequences that sil f are half in length along the x-axis. A 0.3 separation produces () 2 l , (142) si f n a line that connects the upper left point of each sequence Lr with its lower right point. Each may be used to obtain the 2 2 1 same result, although larger separations of the data points 2 , (143) become increasingly difficult to resolve. Importantly, the sill f si f 84.9764 quantization separation is what produces the physically significant pattern that describes the charge coupling 1.05457 1034 Js . (144) demarcation. Note also that different graphing programs will render We will look at this more closely later with greater differently. For instance, online tools such as desmos.com precision. But for now, we note that we can also write Eq. will not connect all the data points left to right as a (143) as a function. Setting x=ħf/θsilf and y=1/nLr., we then continuous plot. Conversely, MS Excel does. have Given the charge coupling demarcation is associated with x2 y2 2 x 0 . (145) a count of nθ=RND(nLr/2)=42, we may resolve the mid-point The solution as displayed in Fig. 4 for ħ/θ l falls on the of that sequence near nθ=42 and then scale the count of lf si f with respect to the constant of proportionality. Or we may point (2.000069857, 0.000139718) on the right parabola. resolve the non-discrete mid-point of the sequence near The axis of symmetry for both parabolas fall parallel to the x-axis. The expression can also be reduced. With ħ =2θ l nθ=84.9764 (Eq. 70) (i.e., any sequence may be used). To f si f avoid scaling, we proceed with the latter. The demarcation and ħ=θsilf/QLnLr, then count may be resolved relative to the midpoint of the second 1 2 x . (146) full sequence displayed in Graph I. l Q n Both the demarks and the halfway point fall on the y-axis si f f L Lr
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 47
expansion of the universe in relation to the fundamental measures, each expression as described by the Pythagorean Theorem.
3.6. Metric Approach to Series of Sequences As demonstrated with both the fine structure constant and Planck’s reduced constant, a metric approach can be used to identify physically significant values. In turn, those values describe physical characteristics of our universe, such as the quantum of action h and the ground state orbital of an electron a0. Moreover, there exists a physical correlation between the approach (i.e., frames of reference) and what we measure; in the case of ħ, we have an 11σ correspondence to its presently measured value. While the application of counts of fundamental measures to the description of Figure 4. Skew in spatial measurement as a count of lf identifies the blackbody demarcation constants is new, we may at least consider what additional properties may be deduced. Substituting 1/QLnLr for ħ/θsilf, then Table 8. Repeating quantization ratio demarks for counts nθ up to 64 2 2 1 2 1 Whole unit Counts n Corresponding with Repeating . (147) n θ Lr Quantization Ratios QL n Lr Q L n Lr 84.976 4 3 6 Thus, the x-axis describes the skew in measure between 4 8 12 16 20 24 28 32 36 40 the non-discrete frame of the universe and the discrete frame 7 14 21 of the observer, also known as the Informativity differential. The effect is geometric, independent of the rate of expansion 11 22 33 44 15 30 45 60 75 90 105 (i.e., 2θsi, which is defined relative to the universe). The 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 y-axis describes the blackbody demarcation. In that y is a 19 function of x (i.e., f(x)=(x2−2x)1/2), it follows that the 323 342 361 380 399 418 437 456 … 627 demarcation distance is also independent of the 23 46 69 92 115 138 161 184 207 230 253 276 299 322 expansion—that is, independent of all system parameters 26 52 particular to our universe. Both axes are dimensionless 27 54 81 108 135 counts of lf. 31 62 93 Although the expression is initially expressed as a 34 68 102 136 170 204 function of Planck’s reduced constant, it can just as well be 35 70 expressed as a function of θsi or several other physical 39 78 constants. The relation does not describe the constants that 41 82 make up the expression but describes the skewing of length 84 126 168 210 252 294 336 378 420 462 504 546 588 630 42 relative to the Informativity differential. 672 714 756 798 840 882 924 966 … 1050 While the vertex of the right parabola corresponds to the 49 98 147 blackbody demarcation, of interest is the vertex of the left 50 100 150 200 parabola (positive in the x- and y-axes by a miniscule value). 53 106 159 212 265 Is there a physical significance to this second property and will it be instrumental to understanding virtual particles? 58 116 122 183 244 305 366 427 488 549 610 671 732 793 854 915 Do the two demarcations provide insight into the energy 61 jumps associated with electrons? At this moment, we have 976 1037 1098 1159 1220 1281 … 6893 established a new understanding of quantum phenomena, but 64 128 how to translate these MQ descriptions is physically unclear. Consider, for instance, not a specific count series (i.e., 42, As a final note, Eq. (147) may be reduced given that 84, 126, ..., 42n, ...), but all count series of θ . Among them, n =84.9764 to resolve a blackbody demarcation of si Lr we look for repeating patterns in the quantization ratios. If 2 QnL Lr the ratio values repeated indefinitely or were not otherwise 21QnL Lr , (148) 84.9764 constrained, they would be uninformative. However, there 2 are constraints to the relationships that may exist. That is, the QL 21 Q L n Lr . (149) quantization ratios associated with each demark repeat up to With this, we find that the Informativity differential is also a certain point. The corresponding series for the first 64 a unity expression, just as Eq. (61), and describes the counts are listed in Table 8.
48 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
A quantization ratio rq is calculated such that We focus on the metric approach to consider whether both rq=nθθsi/RND(nθθsi). By example, consider the fine structure the geometry and the counts of fundamental measures constant, which we have shown to be physically correlated are important when describing observed phenomena. For with a count of 42. The quantization ratio is then instance, consider a mass divided into three equal parts; the physical properties of the parts are affected by the division, 42si 137.020 rq 1.00015 . (150) many of those properties being a straight-forward division by RND 42si 137 three. For example, the effects of gravitation from each part Carrying the operation out for each count … 1, 2, 3, 4, 5 … are now one-third of the original whole. we can then compare the quantization ratios for matching From another perspective, SR may be viewed as a values. In the case of the 42 series, there are matching ratios geometric phenomenon that is consistent with certain numerical properties. That is, there are specific at nθ equal to 42, 84, 126, 168 … 42n … 1050. There are no more matching values above or below a series. This series consequences to the observation of length, mass, and time is physically correlated to the fine structure constant, to relative to the numerical increase or decrease in velocity Planck’s constant, to the blackbody demarcation, and by between the observer and target. For this reason, an means of these constants to other phenomena. investigation of the permitted counts associated with the With respect to these results, we pose several questions. description of phenomena is important. Why are all discovered phenomena associated with the 3.7. Extending Precision of the Physical Constants 42 count series? Using the metric approach, we were able to resolve values Such that the count nθ=42 identifies the series that of several physical constants. For example, we resolved the corresponds to the fine structure constant and our fine structure constant as a count of θsi. Unfortunately, θsi is understanding of the quantum of action, we find also that the constrained to six digits of physical significance. Here, we first count value identifies the x-axis value associated with reverse the calculation resolving a more precise value of θsi the charge coupling demarcation. Moreover, we find that, as a measure of the magnetic constant and the charge adjusted for measurement skewing described by the metric coupling demarcation ‘distance’ in the Target Frame, and Informativity differential, a count of 42 θsi may be used n 276 . (151) to identify the ground state orbital a0 of an electron. With Lr si these and other physical correlations, we may inquire if there The expression is an essential observation that can be is something unique to this series that all values thus far are validated for any given count nθ and corresponding θsi such physically correlated with only this series. -1 that rq=1, nLrθsi=276. The value corresponds to αc defined Are there physical constants that are independent of the with respect to the Target Frame. With this we may resolve fundamental measures? the immeasurable distance associated with α. Had the rate of expansion for our universe been something 1/2 Q1 n2 n . (152) other than 2θsi, would that change some or all properties of L Lr Lr the universe? Recall that quantization ratios are a function of 1/2 fundamental measures and counts. Count properties such as 2 2 QLsi si si n Lr n Lr si . (153) the charge coupling demarcation and the Informativity differential are geometric, independent of the rate of 1/2 2 2 expansion. For this reason, there exists a path to realize that si 276 276 some physical constants, and as such some phenomena, QL . (154) although they may differ in value, will exist regardless of the si system parameters of our universe. Using the 2018 CODATA value for αc=7.2973525693 -3 Are discrete systems the cause of breaks in physical 10 [1] then, symmetry? 1 2QnL Lr 84si RND 42s i , (155) Importantly, we note that a non-discrete universe would c be symmetrically equal. However, a non-discrete universe with no external reference creates a discrete internal frame 1 QnL Lr , (156) of reference. This leads to asymmetries; for instance, 2c 84si RND 42 si comparing counts of sequences starting with odd x-values 2 (there are 21) to those starting with even x-values (there are 7) 2*276 si . (157) c 12/ of the first 64 whole-unit values (see Table 8). It is 2 2 si276 276 84 si RND42 si conjectured that this lack of symmetry in discrete systems is the source of physical variations we observe in nature. 3.26239030395 kg m s1 . (158) O8: A universe with no external frame of reference will not si be symmetrical in all aspects. Notably, a recent study resolves α to 81 parts per trillion,
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 49
-1 α =137.035999206(11) [18] with respect to the recoil 1.0545718177 1034 Js . (168) velocity of a rubidium atom that absorbs a photon. This measure demonstrated a strong disagreement with calculated The value of αc from Eq. (156) is defined as a function of ħ, -11 values for αp (a difference of 1.28 10 greater than the which makes this last calculation significant only for -1 presently accepted CODATA value of α =137.035999084 measured values of α not a function of ħ (i.e., αc=cμ0/2RK (21) [1]) suggesting that measurements of the cesium atom such that μ0 is measured directly). Moreover, to calculate a demonstrate a stronger correlation to the fine structure distance sensitive value that accounts for the Informativity constant. Morel, L., Yao, Z., Cladé, P. et al. Determination of differential we use the expanded form. We may, for instance, the fine-structure constant with an accuracy of 81 parts per resolve ħf at the upper count bound. This value is trillion. Nature 588, 61–65 (2020). approximately accurate for any macroscopic measurement. https://doi.org/10.1038/s41586-020-2964-7. sil f The charge coupling demarcation associated with the fine , (169) structure constant is then, QnL Lr 34 nLr 276 / si 84.6005456998. (159) f 2sil f 1.05453498445 10 Js . (170) 33 nlLr f 1.36731436664 10 m . (160) There is always a series of measures that underpin each calculation. Our initial measure of θsi comes from the 2018 The corresponding Informativity differential is at CODATA definition of αc which depends on the measure of QnL Lr 0.49999563388 . (161) μ0. The electric constant is also a function of μ0 with ħ, e, and c being defined. As such, we cannot use our more refined In addition to the quantum entanglement correlation, we values to then calculate elementary charge, the electric or now offer this second approach to the measurement of θsi. magnetic constants. Doing so would create a loop. But we Using the expression for mf from Eq. (37), then can improve our electromagnetic calculations. 2 Recall that we needed to apply the differential after m si 2.1764325398 108 kg . (162) f c resolving an expression for the metric differential. This approximation is wholly contained within γ. With our new The remaining two measures, l and t are functions of G f f expressions for the electromagnetic constants, we can isolate constrained to six digits. However, we may use Planck’s gamma with respect to the measure of each constant formula for the fine structure constant (Eq. 63) to resolve lf. −1 independently. We must also solve αp as a function of the new θsi. 1 1 γ=2G π0 1.19967279331. (171) p 84si RND 42 si 137.040785532 . (163) 0 2 Now with mf and αp as a function of the new value for θsi γ Gc 1.19967279331 . (172) and with which the remaining values are measured, then 2π
maep0 35 γ 2kGe 1.19967279331. (173) l f 1.61619991203 10 m . (164) m f Notably, the value of γ is the same for all three measurements. Such that the most appropriate measurement We use α over α in recognition that Planck’s expression p c is used for each solution, then should not be adjusted for the Informativity differential as G demonstrated in Eq. (69). In turn, using the defined value for = 8.85418781280 1012 Fm1. (174) the speed of light leads to a value for fundamental time. 0 2πγ l f 44 2πγ 6 -1 t f 5.39106426132 10 s . (165) 0 1.2566370621 10 H m . (175) c Gc2 In that the value of c is defined we have interpreted this to γ 9 2 2 have no effect on the precision of the result. Continuing, the ke 8.9875517923 10 Nm C . (176) 2G gravitational constant when measured macroscopically is Table 9. Electromagnetic constants as a function of the fundamental l f G c2 6.67407794280 10 11 m 3 kg 1 s 2 . (166) measures and γ m f Physical Values Constants 2 −2 Moreover, the value of ħ such that all values derive from ɛ0 (F m−1) μ0 (H m−1) ke (N m C ) the measure of μ , the blackbody demarcation n and c is 8.85418781280 1.2566370621× 8.987551792 0 Lr MQ ×10−12 10−6 26×109 2 2 1 2018 8.8541878128(13) 1.25663706212 8.987551792 2 , (167) −12 −6 9 CODATA ×10 (19)×10 3(14)×10 sill f si f 84.976352961
50 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
We compare these results in Table 9. energy of light reflects Einstein’s equation E=mc2. Such that There are many values that may be described as a El=2πEm (Eq. 184) and Em=2θsic (Eq. 182), then combination of these, now with more precise results. For 2 2πlmff instance, the ΛCDM distributions are now resolvable ([11], Em4π cc 2π 2 . (186) l si2 f Eqs. 82-86) to twelve digits of physical significance, t f 2 2 si 68.3624161047% , (177) Importantly, the value of 2π is what distinguishes the dkm 2 energy of mass m from that of light. si 2 f Consider now the phenomenon of force. This relation, 4 31.6375838953% , (178) resolved earlier in the paper, describes how gravitation and obs 2 electromagnetism are correlated. si 2 G 2π0 . (187) obs vis 4.84883489523% , (179) 2si Once again, the two phenomena are separated in value by 2π. Gamma, incidentally, is used to indicate four geometries uobs obs vis 26.7887490001% . (180) instrumental to the calculation, but not representative of the Lastly, some values in this paper quote the 2010 intrinsic properties of either phenomenon. CODATA results. There have been changes in the 2018 Consider now the CMB power spectrum. As presented in measure of G reflecting new measurement techniques that Appendix F, we observe that the x-axis coordinate of the are more subject to the effects of the Informativity peak of each curve is distinguished as a function of π. differential. That is, there are calculated values, some terms 2 2 measured macroscopically while others measured quantumly. Ω : πy, si , (188) dkm 2 As such, we have endeavored to use measures that are least si 2 affected by the Informativity differential. π5 4 Ω :,y , (189) 3.8. Particles vs. Waves obs 2*9 2 si 2 We have provided expressions that correlate particle and wave phenomena. Invariably, they differ by a constant, 2π. 42 Ω : π3 y, si , (190) What does this describe? Why are there so many phenomena uobs 3 that differ by this value? And what is the physical si 2 si significance of this difference? The answers to these 2 1 questions are at the heart of particle/wave duality. They arise Ω :9π2 y, si , (191) v 3 from a geometry that underlies the universe. si 2 si To better grasp the scope of physical phenomena that differ by this value we will review. Firstly, consider energy, 5 2 that is the energy of a fundamental unit of mass m and a Ωvis : π y, . (192) f 3 2 corresponding quantum of light ([7], Eqs. 48-51). si si This is not unexpected with electromagnetic descriptions, 22si sic E mcc n n , (181) but we have demonstrated that these descriptions are m MQ nc M Q n L Lr L Lr temporal in origin. Moreover, there is a relativistic offset. Applying the offset to each x-value, Em 2 sic , (182) 2/3 1 π ElnM hf v h ft f , (183) , (193) si E 2 c 2sil f f 1 we account for the skewing effects of measure between m si . (184) El h f/ t f hff22π π the earliest and present epochs ([11], Eq. 90). Let us now consider what a particle is in terms of energy. Their difference is 2π. Moreover, n is not a whole unit The energy of a fundamental unit of mass is Em=2θsic. value, at least not in terms of counts. Written using Planck’s Notably, such that 2θsi is the rate of universal expansion HU expression E=nhv, we find that n is 1/2π when describing and c is the velocity of all points relative to observers at the mass. Therefore 2π appears in expressions describing visible bound – a system perimeter, that spacetime where electromagnetic phenomena. there is no information beyond the bound, we find that Em is hv the product of the expansion parameter and the perimeter El f 1 Em hf v . (185) velocity. 2π 2π 2π EH c . (194) Note also that expansion of the expression to describe the mU
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1/2 1/2 What, then, is mass? What is the relation between mf and a 33 2 tfcc2 n M t f universe? Is there a greater physical significance to π? c nm , (196) Mfr m n l f Lr f n 21M . (197) nLr We recognized that the observer and the target cannot both occupy the same space at the same time. Thus, the count value for nLr must be greater than 0. Moreover, nLr must be greater than two times nM, nM also greater than 0. To correlate this to relativity, the measurement distortion expressions ([8], Eqs. 31-34) also described by SR are 1/2 2 nLm Figure 5. Arc length of a circle of radius lf, subtending angle θ radians tt1 , (198) ol2 nLc That is, we bring together a suite of phenomena each which carry the value of π across multiple disciplines: energy, 1/2 n2 gravitation, electromagnetism, cosmology, and epochs. We ll1 Lm , (199) ol2 correlate them, demonstrating that it is π which stands nLc between them. Yet, we also describe π as a geometry 1/2 reflective of a description of a circle. As described in Fig. 5, 2 nLm such that the circumference of a circle is C=2πr, it is that mmol/1. (200) n2 geometry which describes this difference as a ratio, the Lc circumference divided by the radius. Moreover, given the The counts correspond to their classical counterparts such radius of a circle equal to lf, we find that the radian measure that θ corresponds to half of a quantum of energy ħ/2, again h si 22 2 2 divided by 2π (i.e., C/r). We typically refer to this as its nLm nLm l f t f v angular momentum. From each of these observations, we . (201) n22tl n c may observe that Lc ff Lc
O9: Phenomena come in pairs separated in value by 2π. And using the escape velocity expression, then In all cases, we find it an inevitable conclusion that energy, 3 2 2Gmf2 Gn M m f2 t f c mass, and force each present themselves in pairs, partner v nMf m , (202) phenomena separated by a geometry of 2π, C/r. lf n Lr l f n Lr l f m f 3 32 3.9. Singularities 2ntMfc 2nnc 2 c v2 MM , (203) Singularities in modern theory are encountered in nLr l f n Lrc n Lr situations such as General Relativity (GR) [19] when used to v2 n describe phenomena at the extreme of the measurement 2 M 1. (204) 2 domain (i.e., the center of a black hole or the universe as a c nLr quantum singularity). Their mathematical and physical equivalence are then Notably, MQ is a discrete nomenclature, a physically demonstrated by their combination. Notably, the significant description of phenomena as counts of three presentation is intentionally simplistic. A more exhaustive fundamental measures, l , m and t . In that there are f f f presentation is available in the second paper, ‘Measurement no fractional counts of physical significance in the Quantization Unifies Relativistic Effects …’ [8]. Measurement Frame, there are no opportunities for singularities. The value of any count in an expression starts n2 n Lm 2 M . (205) with one and has an upper bound which does not exceed the 2 nLc nLr Planck frequency, 1/tf. A demonstration of the issue as occurs in GR may be We find then, no opportunity for a speed parameter better understood in analysis of the expression for escape in relativistic expressions to present singularities. All velocity ve. Consider the velocity bound such that v=c. Then descriptions of phenomena must satisfy nLr≥1, nM≥1 and 1/2 nT≥1 such that all counts are whole unit. The root cause of 2GM singularities arises as a by-product of a non-discrete ve , (195) r nomenclature when describing phenomena. MQ recognizes the physical significance of discrete measure and
52 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
respectively modifies modern nomenclature with a more just beginning? We should consider the latter. precise terminology. Physically significant bounds are then Finally, the attentive reader will note that there is no intent more easily recognized. This physically significant rule set to push geometry into the spotlight. The correlation of so does not allow singularities and preserves an understanding many phenomena to numerical qualities is not our focus. of the properties that underlie what is being described. We emphasize, as has long been the tradition of science, that finding the best math that describes the world around us is our greatest endeavor. That more of nature appears 4. Discussion geometric is irrelevant. In this paper, we use MQ to resolve physically significant descriptions of the constants of nature. We developed a 5. Glossary of Terms foundation for their origin—a differential between the discrete framework of the observer and the non-discrete Blackbody demarcation framework of the universe—and various relations, variations That distance at 84.9764lf, (Appendix D) corresponding to of the fundamental expression. Importantly, we complete a the measure of blackbody radiation, the value of picture of what were formally unanswered questions: Why QLnLr=0.499996 and the value of ħ as we recognize it today. do the physical constants exist? Why do the laws of nature Charge coupling demarcation exist? Why is the universe symmetrical and where is it That distance at 84.6005, corresponding to the measure of nonsymmetrical? Why does the use of classical mechanics the fine structure constant and the value of Q n =0.499996. present singularities? L Lr While these questions were at some point intangible, we Fundamental expression now look to new questions. Why is the blackbody The simplest expression that relates the three measures, demarcation at 42θsi? Why is the fine structure constant a length lf, mass mf, and time tf. The expression is lfmf=2θsitf function at 42θsi? Why is Planck’s expression for α a such that the value of θsi is 3.26239. differential of the fine structure constant, a function of 42θsi? Informativity And finally, why is the ground state orbital of an atom a A field of science that recognizes the physical significance function of 42θ ? si of nature as a consequence of mathematical form. We use A review of Table 8 offers some possibilities. If starting Measurement Quantization (MQ) as an approach to describe values are conjectured to be based on orbitals for the atom the discrete properties of nature revealing how nature and (which then need adjustment for the effects of force), it mathematical form are correlated. This is achieved with a follows that the universe might have corresponded to any nomenclature of counts of physically significant count series, for example, 41θ or 39θ . However, these si si fundamental units of measure applied to the existing series afford only two orbitals for atoms. It may also be classical laws of modern theory. conjectured that there is no preferential series. The universe Notably, MQ is just one approach. Other approaches are could have just as easily corresponded to the 19θ or 61θ si si likely to arise specific to quantum theory, information theory count series. This leads to the question, why are there so and high energy physics. Regardless of the approach, we many odd count series compared with even, 21 to 7 for the identify the science of Informativity as any discipline that first 64 whole-unit counts? Is this lack of symmetry at the recognizes the constants and laws of nature as a consequence root of the matter/anti-matter differential? These questions of mathematical form. are purely speculative, intended to spark interest for further investigations. Conversely, the presentation addresses only Informativity differential verifiable results with straight-forward physical correlations. A skewing of the measure of length because of the Among other discoveries, the greatest challenge we discreteness of measure. The effect is geometric and may be 2 1/2 have encountered is a better understanding of unification. described as a count of lf such that QLnLr=((1+nLr ) -nLr)nLr. We may agree that unification involves a description of QL is the non-discrete count of lf that is lost at each count of tf each phenomenon correlated with an equality. In that the with respect to elapsed time. This effect is known as gravity, phenomena are physically distinct, there will be some count although the differential itself is also recognized as a of additional phenomena that distinguish them. Nonetheless, skewing of measure in a fashion like relativity, except with as demonstrated in Eqs. (105)–(108), it is easy to correlate respect to distance, not motion. phenomena and identify other phenomena that distinguish Metric differential them. Thus, a more fundamental definition is needed. Follows the same geometric displacement described by Perhaps unification requires a shared fundamental the Informativity differential but is calculated as a difference nomenclature and physically significant distinct terms between a discrete and non-discrete count n of a fundamental independent of the correlated terms. Perhaps some element measure. of naturalness and/or elegance is prerequisite to the solution. Are these the ground rules for unification? If agreed, then RND have we completed unification of the four forces or are we The term RND is used often in MQ and meant to describe
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a physical process whereby the measure of an object at a as some known count of the reference length measure. non-discrete distance appears to be at the nearest count of Target framework—This framework is characterized by a fundamental measure. Thus, if a calculated value has the property of measure of non-discreteness, that being the remainder less than one half, the lower count is measured. If framework of the universe that contains the phenomenon. It a calculated value has a remainder greater than one half, shares those properties of the self-defining frame of reference the upper count is measured. A calculated value with a but in relation to a local phenomenon. remainder equal to one half has an uncertain count value, Self-referencing frame of reference—A system of either above or below the non-discrete value. Counts apply to geometric axes anchored with respect to objects within the all dimensions, but with respect to distance all calculated universe to which measurements of size, position or motion values are less than one half and as such round down. can be made. One may assign expressions comprising Self-referencing physically significant terms. However, those terms are An expression defined with respect to the observer’s defined with respect to other terms (i.e., ħ, ɛ0, μ0, α, lf, mf, tf) inertial frame of reference. such that there exists no external anchor with which to resolve any term independently. Self-defining Self-defining frame of reference—A system of geometric An expression defined with respect to the universe as a axes anchored with respect to the universe to which frame of reference. measurements of size, position or motion can be made. One System parameters may assign expressions comprising physically significant Any constant value associated with a self-defining terms. Those terms, also known as system parameters, are defined numerically with respect to properties of the expression (i.e., θsi). universe (i.e., rate of universal expansion 2θ , age of the MQ nomenclature si universe AU), but have no relation with respect to phenomena A nomenclature of all fundamental units of measure—lf,, external to the universe. mf , tf, and θsi—which are discrete, countable, and may be used exclusively to describe all observed phenomena in the universe. The acronym MQ stands for measurement ACKNOWLEDGMENTS quantization. The nomenclature is applied to the well-tested We thank Edanz Group (www.edanzediting.com/ac) for and strongly supported laws of classical mechanics. editing a draft of this manuscript. Quantization ratio The ratio is a numerical representation of the relation between the discrete and non-discrete frameworks, a Appendices function of the count n of a fundamental measure, typically APPENDIX A: INFORMATIVITY DIFFERENTIAL QLnLr θsi or lf. The ratio nθsi/RND(nθsi) is defined with RND(*) ([11], Appendix A) denoting the value of the argument rounded to the nearest whole-unit. In analysis of Heisenberg’s uncertainty principle, we Demark—Given a plot of quantization ratios, there is a resolve properties of measure demonstrating discreteness, repeating pattern. Demark identifies the first point in each countability and in relation to three frames of reference. pattern such that y=1, with the average of a discrete set of Notably, the physical significance of discrete measure also points immediately to the left y<1 and the average of a demonstrates that there is a skew between such a description discrete set of points immediately to the right y>1. and that of an expression not constrained by a whole-unit Sequence—A plot of quantization ratios comprising count of fundamental measures. We find that difference best points including both beginning and end demarks. described by QLnLr and refer to this term as the Informativity Series—The set of sequences for which the beginning differential. Resolving the limits to QLnLr is valuable when demarks share identical quantization ratios. The series may working with MQ expressions. be distinguished as having a base demark B with repeating We approach this goal by recognizing that the product of demarks such that each demark in the series is a whole unit QLnLr (Eqs. 30 and 31) is count n of the base (i.e., nB: 42, 84, 126, …, 1050). 2 QL n Lr1 n Lb n Lb n Lb . (A1) Frameworks Reference framework—This is the framework of the Recall that there are three frames involved in a description observer. With respect to the traditional understanding, this of measure, such that Side a is the reference count nLa, Side b framework differs only in that measure is a count function of is some count nLb of that reference and Side c is the measured discrete length measures equal to one. It shares those count, such that nLc=nLr+QL and nLb=nLr. Such that we wish properties of the self-referencing frame of reference but in to describe Side c, we drop the nLb term and adopt the term relation to the inertial frame of the observer. nLr in that r is more commonly associated with the measure Measurement framework—This framework shares of distance. properties with the Reference Framework. It is characterized We verify that nLb=nLr such that the highest value for QL is
54 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
2 0.5 nLb=1 where (1+1 ) –1≈0.414 and the ‘observed’ distance We will now describe a second dimensional quality of θsi, of Side c – a count nLc of lf – is always rounded to the nearest a numerical correlation in its value as a momentum with integer value. That value is equal to the count nLr. It is respect to the angle of polarization of X-rays when satisfying QL=√2-1 at its highest and quickly approaches 0 with a maximally entangled Bell state. Correlated to the lattice increasing nLr. vector 퐺 , two vectors are described: one, a polarization of the electric field in the scattering plane and two, the 2 Q n1 n n n . (A2) polarization of the electric field orthogonal to the scattering L Lr Lr Lr Lr plane. This new approach allows us to offer a new expression At the lower limit nLr=1, then limr=1f(QLnLr) =√2−1. for θsi such that the two vectors differ in sign, a division of Conversely, dividing by nLr, adding nLr, squaring, the magnitude of the coordinate components of the pump 2 subtracting nLr , and dividing by 2, we find that the upper wavevector. Thus, the x and y components are respectively limit is the arccosine and arcsine of the lattice vector and may be 2 described as (−npxcos(θ), npysin(θ)). QL 1 QnL Lr . (A3) With this we resolve the magnitude of each vector, the 22 pump, signal and idler. We denote the vectors (a function of In analysis of this expression, we recognize with the pump frequency or the phase matching properties of the increasing nLr that QL decreases to 0. The left term drops out nonlinear optical crystal) with the symbol n and distinguish such that the Informativity differential QLnLr approaches 0.5. the signals with the subscript p, s or i. To identify the At 2,247 lf the value of QLnLr rounds to 0.5 to six significant coordinate axes, we add an x or y. 4 digits, with no difference in the ninth digit at 10 lf. G arccosx , (B1) APPENDIX B: MEASUREMENT OF θsi ([11], Appendix n n n B) sx ix px In addition to a physical correlation of θ with measure of Gy si arcsin, (B2) the magnetic constant (Eq. 157), we have correlated θ in si nnniy iy py value with respect to X-rays in maximally entangled states as described by Bell. Measurements were taken by Shwartz and The component vectors yields are then Harris and published in their 2011 paper [3]. Within it, they also offer a model that describes their results.
cosnnsx n ix n px, sin iy n iy n py G , (B3)
nsxcos n ix cos n px cos , n iy sin n iy sin n py sin G , (B4)
nsxcos , n sy sin n ix cos , n iy sin n px cos , n py sin G , (B5)
nsxcos , n sy sin n ix cos , n iy sin n px cos , n py sin G . (B6) We complete the reduction in three steps. First, we move the pump coordinate to the right alongside the lattice vector. We then take the angular difference of the y-component to make the sine positive. And finally, we match the form found in the x-component.
nsxcos ,n sy sin n ix cos , n iy sin n px cos2π , n py sin 2π G . (B7)
With this we resolve that θs=θi. Notably, we also see that idler are precisely a function of the measure at maximal the pump angle is θp=2π−θ. And such that the pump is split entanglement θMax. And by subtracting each angle from π evenly, we recognize that the momentum of the beam is (i.e., π−θp, π−θs, π−θi) we resolve line 1. divided. Thus, the angles of the k vectors with respect to the Table 10. Predicted radian measures of the k vectors of the pump, signal atomic pane must equal half the momentum of the entangled and idler for the maximally entangled polarization at the degenerate photons (i.e., S). Such that θsi=mflf/tf, then frequency of X-rays
3 θp θs θi lff c1 l 3 3 3 Smf . (B8) (lfc /2G)−π π −(lfc /2G) π−(lfc /2G) π−θ 22Gtf Max (0.1208) (−0.1208) (−0.1208) 2π−(l c3/2G) (l c3/2G) (l c3/2G) Referring to the Shwartz and Harris model, we recognize θ f f f Max (3.02079) (3.26239) (3.26239) that one additional data row may be resolved also using this approach. Using Eq. (B5) as described in line 2 of Table 10, In Table 11, we describe the Shwartz and Harris we recognize that the respective angles for the signal and projections with respect to two of five Bell states they
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identify as generating entangled photons. For consistency, Notably, we can also resolve the expression from Eq. (34), we also adopt their nomenclature, such that |H> is the but this assumes a physical correlation with respect to the polarization of the electric field of the X-ray in the scattering Newton description. Thus, we start with Eq. (33) and resolve 3 plane and |V> is the polarization orthogonal to the scattering that G=c tf/mf as a physically significant description with plane, which contains the incident k vector and the lattice k respect to our expression of quantum gravity. This may then vector 퐺 . be extended. Such that the Informativity differential lim f(Q n )=1/2, then Table 11. Angle setting in radians of the k vectors of the pump, signal, and nLr→∞ L Lr 33 3 idler for maximally entangled polarization states at the degenerate Q c Q c Q c tff m frequency [3] LLL , (C4) rsi n Lr l f si n Lr l f si m f t f Bell’s State θp θs θi 3 (|Hs, Vi > +|Vs, Hi >)/√2 0.1208 −0.1208 −0.1208 Q c Q m f LL G , (C5) 3.02079 3.26239 3.26239 r n l t si Lr f si f The MQ descriptions match the Shwartz and Harris r Q m f lG si L , (C6) measures to six digits, the extent with which each model has f 3 QcL ntLr si f physical significance. Notably, with respect to the nLr l f siQ m f m f measurement data, Shwartz and Harris note that the error in lL G G , (C7) f 32 measure is less than 2 micro-radians. QL cntLr si f c On top of the correlation already presented between 1 2sit f 2G gravitation and electromagnetism, the Shwartz and Harris lGsi . (C8) f 23 correlation continues to add to our physical understanding ccl f of θsi as having a distinct measurable value but differing For all macroscopic distance, the fundamental measures dimensional qualities as a function of the phenomenon being are measured. In this case, we find angular measure correlated 11 3 2Gsi 2 6.67408 10 3.26239 35 to the scalar constant S=lpc /2G, a composition of the l f 1.61620 10 m , c33(299792458) fundamental length lp, the speed of light c, and the gravitational constant G. By demonstration only, using the (C9) 2010 CODATA [2], the calculation matches the Shwartz and 11 l f 2Gsi 2 6.67408 10 3.26239 Harris results. t f c c44(299792458) , (C10) 3 35 3 lcp 1.616199 10 (299792458) S 5.391061 044 s 2G 2 6.67408 1011 . (B9) 3 2si c 2 3.26239 8 3.26239 kg m s1 mt 2.17643 10 kg . (C11) ffc G 299792458 As noted previously in this paper, newer approaches to the APPENDIX D: BLACKBODY DEMARCATION ([10], measure of G reflected in the 2014 and 2018 CODATA are Appendix B) affected significantly by the Informativity differential. For this reason, we consistently adopt comparisons with the 2010 The measure of Planck’s constant corresponds to a CODATA when discussing measures that include G. physical interaction at a specific distance. That distance is a count of l (Eq. 31) where n rounds to n and APPENDIX C: FUNDAMENTAL MEASURES ([10], Sec. f Lb Lr 3 3 3 3.4) Q rc c c l f L Q n l 3.26239 kg m s1 . si L Lr f We may resolve the fundamental measures using only our GGG 2 initial observations regarding an MQ description of quantum (D1) gravity and its relation to Newton’s expression for G. 2 3 We have substitute ħ/lf from Planck’s relation (Eq. 1). Such that G/r2=QLc /rθsi (Eq. 34), then factoring out the Thus Informativity differential limnLr→∞f(QLnLr)=1/2 (Appendix A) we resolve that 1/2 Q1 n2 n , (D2) 33 33 L Lr Lr 2 QQLLccllffcc G r nLr l f Q L n Lr . (C1) rsi si si2 si c3 3 Given that G=c t /m (Eq. 16), then siQ L n Lr l f Q L n Lr l f , (D3) f f G l2 33 f tlffc c , (C2) m 2 sill f si f f si n , (D4) Lr 1/2 l m 2 t . (C3) QL 2 f fsi f 1nnLr Lr We identify this as the fundamental expression.
56 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
1/2 2 41/2 2 sil f 2GM nLr n Lr n Lr , (D5) v , (E2) r l2 2 l 22 l n 2 1/2 1/2 2 4si f 2 si f si f Lr 4 32 n n n n , (D6) 2 QrLMcnM si 2n c Lr Lr Lr2 Lr c , (E3) rsi Q L n Lrc n Lr 22 2sill f si f n2 1, (D7) 2nM n Lr . (E4) Lr 2 Notably, the smallest count of mf with respect to lf may not -8 2 2 be less than the precision of the reference mf=2.17647x10 sil f 1 nl , (D8) kg. Then, Lr2 si f ( 2 l ) 1 2/sil f si f 8 2.17643 10 unitsmmff 1 unit 1 n . (E5) 7 Mb nLr 84.9764 . (D9) 1 unitlmff4.59468 10 units l f 1/
APPENDIX E: OBSERVABLE MASS BOUND ([9], Appx. Correlating both bounds the ratio is then 2 units of mf per 5.3) unit of lf such that 1/(1/mf). Thus, 2(1/(1/mf))=2mf. Moreover, In classical theory we recognize that no phenomenon may such that nMb and mf are equal in value without units, then the have a relative change in position greater than the speed of classical velocity bound is light. When we apply a MQ nomenclature to a description of nM relative motion, we would characterize such a change as an vmbccc2 f . (E6) nLr ‘observed measure count’ of lf per count of tf. Within the field of classical theory, such concepts are commonly The expansion of space HU=2θsi ([9], Eq. 27) is not recognized although the Planck Units have no known included in the expression. Given that HU is relative to the physical significance. diameter of the universe, divide by 2. Then, the radial With this presentation, we add to our understanding of expansion respective of orbital and escape velocity vb may be nature that the fundamental units are physically significant. written in two ways. The fundamental expression may be Counts of the fundamental measures are important and give used to convert between them. rise to the properties and characterization of observed 1 phenomena. It is with this understanding that we recognize vmb sic 2 f 204.054 km s , (E7) there also exists an upper ‘observed measure count’ of mf per vmc c 204.054 km s1 . (E8) count of tf. b f si We call the upper count bound of lf to tf the length Both θsi and our substitution of mf for nMb carry no units. frequency. We call the upper count bound of mf to tf the mass The expression describes the velocity bound corresponding frequency. Developing an expression that describes these to the upper count bound of m that may be discerned at a phenomena may at first seem straight-forward, but when f point in space. The corresponding mass bound is then vb accounting for the relative view of an observer in a spacetime, equal to the same as expressed with Newton’s expression. it becomes more complex. There are several questions that one can engage to untangle why. GMb f() R c 2m , (E9) Such that the count of mf has an upper count bound, what si f R happens when the observable mass exceeds this bound? Does that mean that the excess mass is invisible? And what 1 mf M222 Rcc 2 m 2 R 2 m , (E10) expressions describe a mass distribution that is uneven, for b f() R si f si f 3 G c t f instance, an observer in a galaxy? These questions are central to understanding dark matter and are discussed in this paper, 2 2 2mmff 2 ‘Measurement Quantization Describes Galactic Rotational M2 Rc m2 R , (E11) b f() R si f3 si l Velocities …’ [9]. For our needs, we will resolve only an c t f f expression describing observable mass. This will allow us to m22 m l m resolve the relation between mass and time at the bound c. 2 f f f f 3 MRRb f() R22 si si , (E12) Such that G=c tf/mf, then the escape velocity is equal to the lf2 t f l f classical velocity bound at c. 3 m f 32 MR . (E13) GM lf t f n M m f l f nnMM b f() R si v c . (E1) t f 32 Rttff mf n Lr l f n Lr n Lr We observe that mf in Eq. (E9) is a dimensionless The mass-to-length count bound with respect to the escape substitute for nMb. There are no units. But, in Eq. (E12) where 2 velocity is then R in meters cancels with lf in meters, we are left with mf
International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 57
and a single kilograms describing Mb-f(R). But in Eq. (E9) To break down the terms in the parenthesis, we note that we introduced the fundamental expression θsi=lfmf/2tf. mf=2θsi/c when defined with respect to the Target Frame Cancellations leave both R, tf and an additional mf follows c=nLulf/nTutf=(nLu/nTu)c. Thus, dimensionless. The result is kilograms, 2sin Tu 23 mu , (F5) 2 mmffkg c nLu Mb f() R2 si R m si R kg . (E14) lff m t But as there exists no reference for mu external to the universe, we recognize that m =1 in the Target Frame of the APPENDIX F: POWER SPECTRUM DISTRIBUTIONS u universe, a self-defining unity expression. We may then ([7], Sec. 3.11) organize terms on the right side of the equality such that Before we begin, there exists a significant question 2 n regarding the value of the cosmological constant. New si Lu , (F6) insights into its value may be resolved when considering c nTu Einstein’s observation that the speed of light is constant for nLu all observers. We extend this observation to recognize all mf . (F7) frames of reference for which an observer may consider. nTu It follows that the upper bound rate nL/nT of lf per tf at Now, with the fundamental expression describing the which two phenomena may regress with for all inertial relation between each dimension as a function of the frames, including those considered at the leading edge of a Measurement and Target Frames, we may reduce the terms system, must equal c. Moreover, the relation between length in the parenthesis of Eq. (F4) to resolve that and time in the Measurement Frame is described by the n t l fundamental expression. AU cTu f f n Tu 1 m m m m 1. (F8) fD f n l t f n f m 2 U Lu f f Lu f si c . (F1) m f Given an age of the universe equal to 13.799±0.021 billion years [5], then But, as we recognize, the speed of light is constant for all observers in consideration of all frames. Thus, in addition to DU2 si U 2 3.26239 13.799 90.035 bly , (F9) the observer’s Measurement Frame, light is also constant as DU 90.035 8 described with respect to the observer’s Target Frame. This m 2.1764 10 kg .(F10) f A c 13.799 299792458 implies that, U Notably, the same ‘unity’ arguments may be made with O10: Any measure other than that described by the fundamental expression would describe observers that respect to length and time, but unlike that made with respect observe a speed of light other than c. to mass, such that the relation between the remaining dimensions length and time (i.e. c=l /t ) is a known constant, We also recognize that the universe as defined by that f f the relation between time and mass as well as mass and space traveled by light since the Big Bang is physically length are not so easily measured. It is only because of the significant. This does not mean that matter occupies all space known value of c that this argument can be carried out. in the Target Frame of all observers. We recognize that the Moreover, note that the constancy of c for all observers presence of mass may differ from the space described by the is reflected in c=(D /A m ) (Eq. F10) such that D /A fundamental expression when defined relative to the Target U U f U U describes the rate of universal expansion. Thus, Frame and as such distinguish universal expansion (i.e. the expansion of the universe) from stellar expansion (i.e. the O11: Physical support for the constancy of light in all frames expansion of galaxies away from one another). exists only in a flat universe.
Replacing distance and elapsed time as used in the O12: The rate of universal expansion when defined with Measurement Frame with those terms corresponding to the respect to the system is constant, DU/AU=cmf=6.52478 ly/y. Target Frame nTtf=AU, nLlf=DU (i.e. the diameter DU and age With this, we may then approach a description of the AU of the universe), then universe as a function of the system volume (i.e., as described by the fundamental expression) and the critical ntnllmT f L f f f 2 si tntnl f T f L f , (F2) density defined with respect to that space. Such that V =(4/3)πR 3, G=c3t /m from (Eq. 16), R /A =θ c ([11], ntTfc U U f f U U si n l m 2 n t , (F3) Eq. 44), and where the critical density of a flat universe L f fnl si T f Lf ρc=Hf2/8πG [20] is a function of the Hubble frequency ([7], Eq. 60), then A c D mU 2 A . (F4) U f si U MVVMO U m U c obs , (F11) DU
58 Jody A. Geiger: Measurement Quantization Describes the Physical Constants
VU m A Usi (/) m f t f V U m t f 1 Ω Ω Ω 63.3624 4.84884 26.78876% . (F24) M 1 uobs obs vis dkm A(/) m t m A Usi f f f U si , (F12) VtU m f c 1 mRfU REFERENCES 3 tfc 4πRU t f l f MVM 11 [1] NIST: CODATA Recommended Values of the Fundamental dkm U mm R3 c obs m R t f U f U f , (F13) Physical Constants: 2018, (May, 2019), 2 https://physics.nist.gov/cuu/pdf/wall_2018.pdf. 4πRU M obs l f 1 c [2] P. Mohr, B. Taylor, and D. Newell: CODATA Recommend 3m f Values of the Fundamental Physical Constants: 2010, p. 3, 34H2π R 2 M l R 2 M l (2012), DOI: f U obs f2 U obs f http://dx.doi.org/10.1103/RevModPhys.84.1527. MHdkm 11 f 8πG32 mff Gm , (F14) [3] S. Shwartz, and S.E. Harris: Polarization Entangled Photons 2 at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), 1 RU M obs l f 1 arXiv: 1012.3499. 2 A 2Gm f U [4] Planck Collaboration, Planck 2013 Results. XVI. 2 Cosmologi-cal Parameters, Astronomy & Astrophysics, 571 R Mobs l f M obs l f 1 M U 1 22 c 1 (2013). dkm2 si A 22Gmff m G U , (F15) [5] Planck Collaboration, Planck Collaboration 2018 Results VI Mobs l f m f (2018), [arXiv:1807.06209]. 22c1 si 3 2m f c t f [6] M. Planck, Ü ber irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der 2 si Wissenschaften zu, Berlin, 5, p. 480 (1899). MMdkm obs 1. (F16) 2 [7] J.A. Geiger: Measurement Quantization Unites Classical and Organizing this expression in combination with the Quantum Physics, Journal of High Energy Physics observation that the sum of the dark and observable Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018). https://doi.org/10.4236/jhepgc.2018.42019. distributions must equal 1, then 2 [8] J.A. Geiger: Quantum Model of Gravity Unifies Relativistic 22Ωdkm Ω obs si , (F17) Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and ΩdkmΩ obs 1. (F18) Cosmology. Vol. 4, No. 4 (Oct 2018). https://doi.org/10.4236/jhepgc.2018.44038. We may combine the two expressions to resolve their distribution values. [9] J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, 2 2 Journal of High Energy Physics Gravitation and Cosmology. Ω si 68.3624% , (F19) dkm 2 Vol. 5, No. 2 (Apr 2019). si 2 https://doi.org/10.4236/jhepgc.2019.52028. 4 [10] J.A. Geiger: Physical Significance of Measure, Journal of Ωobs 31.6376% . (F20) 2 2 High Energy Physics Gravitation and Cosmology. Vol. 6, No. si 1 (Jan 2020). https://doi.org/10.4236/jhepgc.2020.61008. Finally, such that the leading edge of the universe is [11] J.A. Geiger: Measurement Quantization Describes History of expanding at the speed of light vU=c and such that 2θsic Universe – Quantum Inflation, Transition to Expansion, CMB defines the ratio of the observable relative to the visible Power Spectrum, Journal of High Energy Physics Gravitation Mobs/Mvis (i.e., two times the RU/AU=θsic referenced in the and Cosmology. Vol. 6, No. 2 (Apr 2020). first paragraph) we may use these constraints to resolve the https://doi.org/10.4236/jhepgc.2020.62015. remaining distributions. Note, Ωtot=1; as such, the term may [12] G. Stoney, On the Physical Units of Nature, Phil. Mag. 11. be dropped. 381 - 391 (1881).
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