A Different Derivation of the Calogero Conjecture

Transcription

1 1 Abstract: A Different Derivation of the Calogero Conjecture Ioannis Iraklis Haranas Physics and Astronoy Departent York University 314 A Petrie Science Building Toronto Ontario CANADA E ail: ioannis@yorku.ca In a study by F. Calogero [1] entitled Cosic origin of quantization an expression was derived for the variability of h with tie, and its consequences if any, of such an idea in cosology were exained. In this paper we will offer a different derivation of the Calogero conjecture based on a postulate concerning a variable speed of light, [] in conjuction with Weinberg s relationship for the ass of an eleentary particle. Introduction: Recently, Calogero studied and reconsidered the universal background noise which according to Nelsons s stohastic echanics [3,4,5] is the basis for the quantu behaviour. The position that Calogero took was that there ight be a physical origin for this noise. He iediately pointed his attention to the gravitational interactions with the distant asses in the universe. Therefore, he argued for the case that Planck s constant can be given by the following expression: 1/ [ R( ] 1/ 3/ (1) h α where α is a nuerical constant = 1, is the Newtonian gravitational constant, is the ass of the hydrogen ato which is considered as the basic ass unit in the universe, and finally R( represents the radius of the universe or alternatively: part of the universe which is accessible to gravitational interactions at tie t. [6]. We can obtain a rough estiate of the value of h if we just take into account that R( is the radius of the universe evaluated at the present tie t ie. R(t ) 1-8 c. Taking the value of q 1, then equation (1) gives the following value:[7] -6 h 6 1 c ()

2 We should ention that in (1) the only quantity that varies with tie is R(.[6] Next, let us assue that the expanssion rate dr(/dt of the radius of the universe R(, at any tie t, is equal to the speed of light c( at the sae tie t, and the relation can be written as follows: dr( c( = R & () t = (3) dt It should be noted that R( is not a physical length, but instead, it can be thought as a universal length of space itself with a present rate of expansion given by: dr t c = ( ) dt (4) Let us now consider Weinberg s relation for the ass of an eleentary particle π which is given by:[8] 1/ 3 h H π = (5) c where H is Hubble s constant, c is the speed of light, is the gravitational constant, and h is Planck s constant. If we now assue that c and H are functions of tie then solving for h we have: h 3 4π π = c( (6) H ( Substituiting now for c ( = R& () t and for at any tie t the expression for Hubble s constant naely: h () t R& H ( = and siplifying we obtain the final expression: R( 1/ ( ) 3 / [ R( )] 1/ π π t (7) We observe that (7) is the sae exprssion as the one derived by Calogero in [1] but differs by only the nuerical value of the constant α which, in our case, is just different than 1 and has the value of 6.8. In addition, we can now assue that the ass of the eleentary particle is the ass of the hydrogen. In [] eata obtains the value h 6 1 -

3 3 6 c -1. However, even if we take the typical value of an eleentary particle π 1-5 g we obtain: -7 h c (8) If we now use for the ass of the eleentary particle the ass of the hydrogen in our equation, we get a value for h which is greater by alost a factor of ten, ie: 5 h c (9) Next, let us deonstrate the analytical tie dependances of h for the different odel universes. For a universe with radiation where R( t 1/, then a universe with atter where R( t /3 and finally a De-Sitter exponentially expanding universe where R( e Ht we have respectively: h π h π h π 1/ 1/ 1/ 3 / 3 / 3 / [( H ) ] 1/ R t 1/ R 1/ e 3H Ro H t / 3 t 1/ / 3 1/ Taking (1 ) in the present era t = t = 1/ H = h -1 y = sec, and.5 h.85. [9] we obtain the following values of h rerspectively: (1) h h h c c c (11) Since we are now in the atter era, we can easily see that when taking into account the expansion of the universe we can write Calogero s result in ters of the average atter density, assuing that none of the other constants change in tie as follows: h (1) ρ 1/6 Upon expansion of the universe during which the ass density drops will ake the value of h grow. If we now put the ass of an eleentary particle π 1-5 g instead of the

4 4 ass of hydrogen and substituting for the average density in the universe ρ = 1-9 g/c 3 we obtain the following value for h: -7 h c (13) The fact is that not only the ass of hydrogen, but also the ass of an eleentary particle, gives us the value of h as derived by Calogero which could siply ean that any eleentary particle ight can also be considered as a basic ass unit in the universe which contributes to Calogero s origin of quantization. Conclusions A different derivation of Calogero s conjecture is given in this brief paper. The derivation is based on Weinberg s relation for ass of an eleentary particle, and also on the notion of a variable speed of light c(. A nuerical result of Planck s costant was obtained when the ass of the eleentary particle was substituted for the ass of hydrogen instead. Furtherore, three different expressions were obtained in which the tie dependance of h were obtained for three possible types of universes, and its value was estiated for the present era in the history of these universes. All values were within the actual value range of h that we know today. Finally an expession for h changing with atter density of the universe was obtained and showed that h changes as ρ -1/6. References: [1] F., Calogero, Cosic Origin of Quantization, Phys., Lett., A 8, 1997, [] I., I., Haranas, A Siple Calculation of a Possible Variation in the Speed of Light, Journal of Theoretics, Vol # 3, No, 1. [3] E., Nelson, Derivation of the Schrodinger equation fro Newtonian echanics, Phys. Rev., 15, 1966, [4] F.uerra, Structural Aspects of Stohastic Mechanics and Stohastic Field Theory, Phys., Rep., 77, 1981, [5] E., Nelson, Quantu Fluctuations,Princeton University Press, [6]., aeta, On the Cosological Iplications of the Calogero Conjecture, citeseer.nj.nec.co/cachedpage/431116/.

5 5 [7] As in Ref [6] [8] S., Weinberg, ravitation and Cosology, John Wiley, New York, 197. [9] P., J., E., Peebles Principles of Physical Cosology, Princeton University Press, 1993, p: 5, 1.