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On the Fundamental Constants of Nature Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 June 2020 doi:10.20944/preprints202006.0017.v1 On the Fundamental Constants of Nature David Humpherys [email protected] https://orcid.org/0000-0001-7375-7897 Abstract Planck units of length, mass, and time are fundamental constants of nature. Tra- ditional constants including Planck’s constant, the gravitational constant, the elementary charge, and many others are comprised of these three fundamental units. Physics equations are functions in which maximum potentials defined by the Planck units are reduced by one or more proportionality operators, producing observed quantities of natural phenomena. Natural symmetries constrain the relationships between length, mass, and time, yielding the physical dynamics of momentum, action, force, and energy. The Planck units quantify mechanical, gravitational, and electromagnetic properties of the universe and offer a com- mon language for interpreting the standard model interactions. Units associated with the electromagnetic interaction are translated into units of length, mass, and time, including the coulomb, ampere, volt, tesla, henry, weber, farad, ohm, and siemen. Keywords Planck units · natural units · fundamental constants · physical constants · Planck’s constant · gravitational constant · particle mechanics 1 Introduction At the close of the 19th century, Max Planck unveiled the constant of proportionality that bears his name today [1–3]. Planck’s discovery ushered in a new era of quantum physics and ~ became ubiquitous in equations describing the physical universe on small scales. At the same time, Planck showed that combining ~, G, and c in the right proportions creates natural quantities of length, mass, and time. More than a century later, the system of derived Planck units—featuring enigmatic ratios of the same three constants—has expanded deeper into the fields of quantum mechanics, thermodynamics, electromagnetism, and gravity [4–9]. Despite sharing the same building blocks, these formulas offer little insight into the physical meaning of ~ and G, or the Planck units they define. A deeper examination of the physical constants and the entire system of Planck units reveals a beautifully simple structure that warrants re-evaluation of what is fundamental. The opportunity eluding Planck at the time, and many others since, is recognizing that the © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 June 2020 doi:10.20944/preprints202006.0017.v1 2 fundamental ingredients of the universe are measured in units of length, mass, and time— not in quantities of Planck’s constant or the Gravitational constant. From the relationships between these three fundamental units emerge the symmetries on which modern day phys- ical theories are built. The fundamental Planck units not only explain the entire system of Planck units more simply and elegantly; they redefine the most important physical constants and equations in ways that expand our understanding of the natural universe. Planck’s constant and the gravitational constant can be stated in Planck units as ~ = lPmPc (1) l G = P c2: (2) mP Similarities between the two constants are immediately apparent in these simple forms, most notably in the relationship between length and mass. The product form of the rela- tionship found in Planck’s constant is ideal for determining the extensive properties of ele- mentary particles, whereas the quotient form of the gravitational constant is better suited for calculating the intensive properties of large gravitational bodies. Equations 1 and 2 can be derived from the equations of Planck length and Planck mass, shown in appendix A. The composition of Planck’s constant and the gravitational constant are demonstrated in table 1 using CODATA values. Table 1: Planck’s constant and the gravitational constant are comprised of fundamental units of length, mass, and time. Constant Unit Value −35 ~ = lP 1:616255 × 10 m −8 × mP 2:176434 × 10 kg × c 299; 792; 458 m=s = 1:054572 × 10−34 kgm2=s2 −35 G = lP 1:616255 × 10 m −8 ÷ mP 2:176434 × 10 kg × c 299; 792; 458 m=s × c 299; 792; 458 m=s = 6:67430 × 10−11 m3=kgs2 The knowledge that ~ and G are comprised of more fundamental quantities is a basis for reinterpreting physical laws explained by these two constants. For all the equations that rely on Planck’s constant and the gravitational constant, it is in the relationships between a handful of Planck units that we obtain definitive predictions about the world. Table 2 gives an extensive list of physical constants restated in fundamental quantities of length, mass, and time. Each constant is defined in elementary form using the three fun- damental Planck units, and additional forms are given in units of Planck energy, Planck momentum, and the speed of light. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 June 2020 doi:10.20944/preprints202006.0017.v1 On the Fundamental Constants of Nature 3 Table 2: Traditional constants are composites of more fundamental quantities of length, mass, and time. The elementary form of these constants reveals physical meaning that was previously obfuscated by the composite values. Constant Sym Std form Elem form Alt forms MKS value EM value r ~G −35 Planck length lp [1] lp - 1:616255 × 10 m - c3 r ~c Planck mass m [1] m - 2:176434 × 10−8kg - p G p r ~G −44 Planck time tp [1] tp - 5:391247 × 10 s - c5 r 3 ~c lpmp kgm Planck momentum pp [4, 5] mpc 6:524784 s - G tp r 2 ~c5 lpmp 2 Planck energy E [4, 9] m c2 1; 956; 081; 000 kgm - p 2 p s2 G tp l2 m p p −34 kgm2 Reduced Planck const ~ - lp pp, Eptp 1:054571 × 10 s - tp 3 lp lp 3 Gravitational const G - c2 6:67430 × 10−11 m - 2 kgs2 mptp mp lp lp m Speed of light c - 299; 792; 458 s - tp tp p −44 −18 Planck charge qp 4π0~c [4, 8, 10] tp - 5:391247 × 10 s 1:876 × 10 C 4π~α p −45 −19 Elementary charge e; q [11] tp α - 4:605448 × 10 s 1:602 × 10 C µ0c 4 1 tp tp 4 Electric permittivity [11] 7:315968 × 10−63 s 8:854 × 10−12 F 0 2 3 2 kgm3 m µ0c 4πlpmp 4πppc 3 1 lpmp pp 3 2 Coulomb const k c2 1:087723 × 1061 kgm 8:988 × 109 Nm e 4 s4 C2 4π0 tp tp 4πα~ lpmp pp Magnetic permeability µ [11] 4π 4π 1:520851 × 1045 kgm 1:256 × 10−6 N 0 2 2 s2 A2 e c tp tp s 4 2 c lpmp Ep Ep 2 Voltage V [12] , 3:628253 × 1052 kgm 1:043 × 1027V p 3 s3 4πG0 tp tp qp r 6 4π0c tp qp tp 25 Current Ip [12] , 1 3:479 × 10 A G tp tp qp r 2 1 ~G lpmp 2 Inductance L [12] E 1; 956; 081; 000 kgm 1:616 × 10−42H p 7 2 p s2 4π0 c tp s 5 c mp pp Magnetic inductance B [12] 7:488021 × 1078 kg 2:152 × 1053T p 2 2 s2 4π0~G tp lptp Continued on next page Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 June 2020 doi:10.20944/preprints202006.0017.v1 4 Table 2 (continued) Constant Sym Std form Elem form Alt forms MKS value EM value 3 tp tp qp 3 Conductance G 4π c , 2:756147 × 10−53 s 0:03336S p 0 2 kgm2 lpmp Ep Ep 2 1 lpmp Ep Ep 2 Impedance Z [12] , 3:628253 × 1052 kgm 29:98Ω p 3 s3 4π0c tp tp qp 2 lpmp Ep 2 Impedance of vacuum Z µ c [11] 4π 4π 4:559397 × 1053 kgm 376:7Ω 0 0 3 s3 tp tp r 4 2 ~G tp tp 4 Capacitance C 4π [12] 1:485907 × 10−96 s 1:798326 × 10−45F p 0 3 2 kgm2 c lpmp Ep e2 Fine structure α [11] - - :0072973525693 - 4π0~c 2 2 α mec meα 1 Rydberg constant R1 [11] - 10973732 m - 2h 4πlpmp 2 2 1 meα lp 1 2 Rydberg energy hcR m α2c2 [11] m α2c2 2:179872 × 10−18 kgm - 1 e 2 e s2 2 2tp 2 2 2π~ πlpmp π 2 Magnetic flux quantum φ [11] p p E 7:193730 × 1010 kgm 2:067834 × 10−15Wb 0 2 p s2 2e αtp α 2 2π~ 2πmplp 2πEp 2 von Klitzing R [11] 3:124008 × 1055 kgm 25; 812:81Ω K 2 3 s3 e αtp αtp p 2 p 2e αtp α 2 Josephson K [11] 1:390100 × 10−11 s 4:835978 × 1014 Hz J 2 kgm2 V h πlpmp πEp p e~ α mp 2 −49 2 −24 J Bohr magneton µB [11] lp - 2:665808 × 10 m 9:274010 × 10 T 2me 2 me 6 G GF tp G Fermi coupling F [11] F 1:166379 × 10−5Ge=V2 - 3 3 9 3 3 3 ~ c lpmp lpEp 2 3 2e αtp αtp 3 Conductance quantum G [11] 6:402033 × 10−56 s 7:748092 × 10−5S 0 2 kgm2 2π~ πlpmp πEp lP lp −28 m Linear mass density - - 7:426161 × 10 kg - mP mp −43 Quantum constant lPmP - lpmp - 3:517672 × 10 kgm - 4 c mplp pp Ep 2 Force F , 1:210255 × 1044 kgm - p 2 s2 G tp tp lp r c7 lp c Acceleration a 5:560725 × 1051 m - p 2 s2 ~G tp tp α=Fine structure constant; A=ampere; C=coulomb; EP=Planck energy; F=farad; H=henry; kg=kilogram; lP=Planck length; m=meter; me=electron mass; mP=Planck mass; pP=Planck momentum; N=newton; s=second; tP=Planck time; S=siemens; T=tesla; W=weber; V=volt; Ω=ohm Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 June 2020 doi:10.20944/preprints202006.0017.v1 On the Fundamental Constants of Nature 5 2 Functions and Equations The benefits of replacing traditional constants with Planck units becomes clear in the refor- mulated versions of physical equations.
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