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The Large Numbers in a Quantized Universe

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The Large Numbers in a Quantized Universe Journal of Modern Physics, 2013, 4, 1647-1654 Published Online December 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.412205 The Large Numbers in a Quantized Universe Yan Ryazantsev* 52, 43 Dimitrova Street, Zagoryansky, Shchelkovo District, Moscow, Russia Email: [email protected] Received October 3, 2013; revised November 5, 2013; accepted November 29, 2013 Copyright © 2013 Yan Ryazantsev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The article relates to a decades-old problem of the mysterious coincidence between various large numbers of the mag- nitude ranging from 1040 to 10120 which sometimes appears in cosmology and quantum physics. Using well known clas- sical relations as well as the ideal Schwarzschild solution the exact relations of various large numbers, the fine structure constant and π were found. The new largest number law is claimed. The hypothetical approximations of the Hubble parameter—68.7457(82) km/s/Mpc, Hubble radius—14.2330(17) Gly, and some others were proposed. The exact formulae supporting P. Dirac’s large number hypothesis and H. Weyl’s proposition were found. It is shown that all major physical constants with the length dimension (from the Compton wave length of universe through the Planck and atomic scale up to the Hubble sphere radius) could be derived from each other, and the table of the specific conver- sion rules has been developed. The model shows that the Eddington-Weinberg relation can be transformed to precise identity. It is shown that both Bekenstein universal entropy bound and Bekenstein-Hawking Black Hole entropy bound are proportional to the largest number doubled. Keywords: Large Numbers Hypothesis; Hubble Sphere; Eddington Number; Cosmological Constant 1. Introduction 1 q2 fe 2 (2) The problem of Large Numbers dates back decades. The 4π0 r first problem statement and attempts at resolving it can 2 me be found in the studies by H. Weyl [1-3] and Sir A. fGg 2 (3) Eddington [4,5], who drew attention to the incredibly r large numbers of the dimensionless physical constants The second is the ratio of the universe radius to the found in the cosmology, electrodynamics and quantum classical electron radius: mechanics. The magnitude of these constants is so big Rc 20 40 80 120 40 10 , 10 , 10 , 10 as compared with the con- NDR 4.63 10 (4) rHr ventional mathematical constants, like π 3.14 and ee Euler’s constant e2.71 , that boggles imagination as where H is Hubble’s constant, re —classical electron P. Davies [6] noted. radius, R —Hubble sphere radius or radius of event The first of the two most popular dimensionless large horizon, rg —gravitational electron radius, fe —electros- numbers is the classical ratio between the gravitational tatic force between two electrons at a distance r, 0 — force and the electromagnetic force in any given distance, permittivity of vacuum, q —electron charge, f g — called by H. Weyl [2] “even more mysterious than the gravitational force between two electrons at a distance r, fine structure constant ”: G —Newton gravitational constant, me —electron mass. r r The proximity of magnitude of e and g values free 42 R r NDF 4.16 10 (1) e frgg led H. Weyl to the idea that the incredible weakness of were: gravitational interaction may be due to the ratio of the electron and the universe radiuses or to the total quantity *Yan Ryazantsev, Independent researcher. of particles in the universe—the Eddington number [5]. Open Access JMP 1648 Y. RYAZANTSEV We refer everybody interested in the history of stu- On the other hand, we know the formula for the dying the problem of large numbers to the reviews by S. electron gravitational radius that includes more or less Ray, U. Mukhopadhyay, P. P. Ghosh [7] and K. A. precisely measured physical values: Tomilin [8]. Gm The hypothesis by P.A.M. Dirac is one of the best e rg 2 (9) known hypotheses put forward to explain the problem of c large numbers [9-11]. He supposed that the reason for Therefore, the formula for the classical electron radius appearance of great magnitudes of dimensionless values that uses G can be easily obtained from (8) and (9): is their reliance on the equally large value, so-called NGm “cosmological time”. This led P. Dirac to the hypothesis DFe re 2 (10) of dependence of the gravitational constant and the c universe mass on time: We will not discuss now if the value of the classical 1 electron radius has any real physical significance. It is G (5) T enough that it is one of the energy status representations of an electron, a particle with the minimum self-energy 2 M T (6) among all charged particles. where M is the mass of universe, T is cosmological For transition to energy values, we will use a large time. mass number introduced by H.Weyl: The idea of time-varying constants was developed, in M E particular, by E. A. Milne [12]. To establish the ratios N eU (11) M mE and laws between modern values of the fundamental eU e 2 2 constants, we suggest studying values of large numbers where EMcU —the universe self-energy, Emcee at the current point of time, without taking into account h —the electron self-energy, —Compton wave their time derivative. P. Dirac’s assumption that New- U Mc ton’s gravitational constant and the mass of universe are not true constants but change over time gave rise to an h length of the universe, e —Compton wave length array of scientific discussions, experimental and theore- mc tical studies devoted to verification of the fundamental of the electron. constants in subsequent decades. No reliable proofs of By dividing (8) by (10), one can get the following variability of the physical constants were found yet. ratio: In this article, we will use the Hubble time, the RM2 parameter inverse to the Hubble constant, as approxi- (12) rNm mation of the cosmological time: eDFe 1 Thus, we obtain the precise correlation among the T (7) three large numbers out of (4), (11) and (12): H 2NNN (13) We referred to the simplest mathematics in narrating M DF DR the article; however, the results we obtained provide Besides the above large numbers, we will use a large quite good approximation to the most precise and energy number NW representing the ratio of the generally accepted values of physical parameters, such as universe’s self-energy EU and “the minimum vacuum me and re , taking into account their uncertainties. In energy” E as proposed by J. Casado [15]: particular, CODATA 2010 [13] as well as the measure- W ments of Mission Planck 2013 [14] were used. EU NW (14) EW 2. Revealing the Large Numbers Ratios hc where EH —quantum of energy with wave To discover the correlation between large numbers in our W 2 R epoch, let’s begin with the well known vacuum solution length 2πR , h —Planck’s constant; to the Einstein’s equations for spherically symmetric and We will also need another large number equal to the static universe. According to Schwarzschild’s solution, cube of N value: the universe’s radius R that coincides with the Black DR 3 Hole radius with the mass of M is determined by a R 3 NNUDR3 (15) well-known formula (Schwarzschild radius): re 2GM Introduction of a new designation for a large number R 2 (8) c NU is quite justified, because, as we will see below, this Open Access JMP Y. RYAZANTSEV 1649 number has a particular and substantial value. One of the Table 2. Calculated Large numbers values. simplest classical interpretations of this number: “the Parameter Value large number NU represents the total number of, say, N 4.16589 50 1042 ‘elementary clusters’ in the universe—the ratio between DF 40 volume VU of ball-like universe and the region Ve N DR 4.842 55 10 folded by sphere of radius r ”. Taking into account that 122 e N 1.135 39 10 this number has the greatest magnitude as compared with U 122 N 1.009 11 1081 other large numbers 10 , we suggest calling it “the M 121 Largest number”. NW 3.564 81 10 3. The Largest Number Law and Revealing the left-hand side, and the quantum ones in the right- of Dirac’s Proportionalities hand one the ratio (17) can be transformed into the Now let us use well known physical parameters with following relation: their uncertainties (see Table 1). The parameters listed in M 1 m n e (19) Table 1 enable us to calculate large numbers values (see x Rr22 e Table 2). To obtain NDF , we use (10): universe electron 1 rc2 The question is—what kind of a phisical law the last N e (16) DF equation represents? Gme In order to find out the answer we would like to We should pay an attention to the very close values of propose quite a simple classical model. We can apply the 121 NU and NW of the magnitude 10 . The ratio classical approach because we are dealing with constant between this two large numbers is macroscopic physical parameters and those ratios. Let us N consider a really large number of non-interacting quanta. n U 3.185(36) (17) x N All quanta are moving in all directions with a speed of W light, i.e. there are photons.
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