Three Wave Hypothesis, Gear Model and the Rest Mass

Transcription

1 Three We Hypothesis, Ger Model nd the est Mss M. I. Snduk School of Engineering, Fculty of Engineering nd Physicl Science, OA, Uniersity of Surrey, Guildford Surrey GU 7XH, UK Abstrct: Three We Hypothesis (TWH) is of reltiistic quntum foundtion. The formultion of TWH hs perfect similrities with beel ger model. The rest mss is considered within TWH nd s consequence of similrity, between TWH nd the ger model, it is found tht rest mss frequency is relted the frequency of touch point of the ger in reltie to the lrge wheel. In regrding this frequency insted of the smll wheel frequency, the ger model leds to fourector representtion of prticle. Keywords: Three we hypothesis, We function, four ector, est mss, Ger, e roglie we. PAS:03.65, 03.75, 04.80

2 . Introduction Since the erly dys of quntum mechnics, nd fter de roglie's we hypothesis, mny physicists (in ddition to de roglie) tried to consider prticle within the frme of we phenomen. e roglie We hs been inestigted experimentlly wheres the theories of we origin of prticle were fcing hrd obstructions. In the second hlf of the lst century mny reserchers he been considered two or three wes representtions [,, 3, 4, 5]. In ddition to de roglie we there is ompton we; others postulted third we. One of those hypotheticl works is Three We Hypothesis (TWH) of Horodecki [5, 6]. It is reltiistic-quntum model; nd reltes mssie prticle to we phenomen. The hypothesis ssumed dul we in ddition to ompton nd de roglie wes. So fr there is no experimentl eidence confirming TWH. Howeer, The TWH shows tht: -There re two dispersie, wes (de roglie nd dul), nd trnsformed ompton we. -The recognized reltiistic quntities by the lb obserer (,,, the o o ngulr frequency nd rdius, for trnsformed nd rest stet relted to ompton we) re relted to the product of dispersie prmeters. 3-There re two reltiistic representtions [5, 6]. The first is in reltie to de roglie we: ( ) ± + (-), o nd the other, to dul we: ( ) ± o β (-). The single ngulr frequency formul of TWH is [7]: ( ) ± ( ) T (-3). T

3 Where,,, re de roglie frequency, dul we frequency, nd the trnsformed ompton frequency respectiely. The sme form my be obtined for rdius representtion. The rtio of we prmeters is: T (-4) β Where,, β re de roglie welength, dul welength, nd the rtio of elocity of prticle to the elocity of light ( β ) respectiely. The we c elocities re: c (-5) A work in 007 showed perfect similrity between TWH formultion nd ger model [7]. This mechnicl model is irtul ger model (VGM). Howeer, TWH is of reltiistic nd quntum bse, nd is combintion of them. Tht combintion (system) hs perfect similrity with ger model. The present work nd with id of the similrity of formultion, tries to find reltionship between VGM nd specil reltiity... Ger Model A preious work [7] exhibited perfect similrities between the system of equtions of the TWH nd clssicl ger of two perpendiculr wheels (s tht is shown Fig.-). This is irtul ger model (VGM), nd it is pure mthemticl model. It hs clssicl feture but it is not clssicl mterilistic model. The ger model is chrcterized by [7]: - The totl ngulr frequency of the perpendiculr wheels: ( + ) ± (-6). Similr formul for the rdius ( ) cn be obtined. - The ger rtio: (-7). 3

4 3- The elocities: (-8) (-9),, nd where,,,, nd (-0), nd re the first, second nd the totl ngulr frequencies, the first, second, rdii of the wheels, liner elocity, then the ger Fig. (-) The ger. rtio respectiely. The limit of is. epresenting ger model in terms of wheel frequencies only (without the totl frequency), one my obtin [7]: ( ) ± ( ) (-). Sme form my be obtined for rdius representtion... The similrity The similrity between the two models TWH nd VGM is quite obious in: - tio similrity, Eqs. (-4) nd (-7). - Velocities similrity, Eqs. (-5) nd (-9). 3- Single system representtion, Eqs. (-3) nd (-)... The correspondences The similrity my led to the following correspondences: 4

5 - ( ) (-). Then, consequently (since ) one cn sy:,, nd (-3). - + (-4), + nd where, nd re the totl ngulr frequency nd the rdii. 3- c (-5), + then: +, nd (-6), + where,,,, re phse elocity of dul we, phse elocity of de roglie we, elocity of first wheel, elocity of second wheel nd the prticle elocity respectiely. It is esy to find tht: c (-7). + The spce of this VGM is three dimensionl Euclidin spce. Since TWH is of quntum nd reltiistic bse nd this model exhibit similrity with VGM, so the ger model my led to gie explntions for both specil reltiity nd quntum mechnics. Using VGM, the present work, tries to go in reerse direction to obtin the specil reltiity nd quntum foundtions.. The rest ompton frequency According to TWH, ompton reltie frequency is: ( ) (-) 5

6 It is relted to system of two wes. The rest ompton frequency, Eqs. (-) nd (-), one cn find: ( ) ] o [ (-) This is relted to system of two dispersie wes s well; but one of them ( ) is reltie to the other. Using the ger model similrity shows tht: The quntity x ( ) x (-3) hs no nlogy in ger system. The ( ) frequency of first wheel reltie to the second wheel, or: is represented the (-4) my represent the frequency of the touch point of the two wheels. Eq. (-3) becomes: x (-5) So, one cn sy tht: o x (-6). The reltie ger For lb obserer the recognized quntities (with correspondent's fundmentl ger) re (Eq.(-)): ( ), ompton frequency, nd πc λ de roglie frequency. Wheres, hs no experimentl eidence. So let, us try to reformult the system without. The new system is in term of, nd ny other obserble quntity. In ddition to Eq. (-) one my get mny different forms for the reltionship between nd. Using Eqs. (-4) nd (-), on gets: + x (-6), 6

7 Eq.(-6) represents new ger or reltie ger. The new ger depicted in Fig.-, nd its ger rtio (of Eq.(-6)) is: x (-0). x Where x,, x, nd, re the first, second nd the totl ngulr frequencies, the first,second, rdii of the wheels, then the ger rtio, respectiely. The prmeters (,,,, x x Fig (-) eltie ger ). The new ger rtio (Eq. (-0)) ger rtio ( ) is: re functions of the independent ger wheels prmeters in terms of the independent x x (-). From Eqs. (-) nd (-) one obtins: or x (-) + x (-3) Owing the considertion of rtio of fundmentl ger (Eq.(-7)), nd reltie considertion s well, so cn be represented s: 7

8 (-4), where:, nd (-5). Eq. (-3) cn be rewritten s: x (-6), nd, for the rdii: x (-7). Let: (-8). Γ Then x Γ (-9) x Γ (-0) Where,Γ re reltie elocity of first wheel, second wheel nd constnt, quntity (independent of the elocities) respectiely. From Eqs. (-7), (-9) nd (-0), it is esy to find tht: Γ Γ x (-A), nd for rdius: Γ x Γ x (-). 8

9 These forms re inrint forms (dot product of two 4-ectors), nd in this cse the nd look s reltiistic inrint quntities. The min reson behind this x x result is considering the system without nd... Similrity nd correspondences In compression Eqs. (-) with specil reltiity representtions, one finds: x o nd x o (-) where o, o re the proper quntities. Owing to:,, nd Γ c (-3). - Unrecognizing of nd formulte the system without it, nd -regrding the rtio with the reltie ger spce; Minkowski spce is obtined. So VGM my led to the reltiistic formultion for prticle of reltie energy ( h ). These reltiistic formultions does not exhibit ny feture of ger system. 4. onclusion The reltie considertion for the ger is equilent to the obsertion process. The obserble entity is prticle of rest mss. Accordingly, there is no ger model cn be obsered. Tht system is irtul ger model. eferences. s, S.N.: e roglie we nd ompton we. Phys. Lett. 0 A, (984).. Mukhopdhyy, P.: A correltion between the ompton welength nd the de roglie welength. Phys Lett. 4A, 79-8 (986). 3. ELAZ,.: On e broglie we nd ompton wes of Mssie Prticles. Phys. Lett. 09A, 7-8 (985). 4. Kostro, L.: A three-we model of the elementry prticle. Phys Lett. 07 A, (985). 5. Horodecki,.: e broglie we nd its dul we. Phys. Lett. 87A, (98). 6. Horodecki,.: Superluminl singulr dul we. Lett. Noo imento 38, (983). 7. Snduk, M. I.: oes the Three We Hypothesis Imply Hidden Structure? Apeiron, 4, No. 3-5 (007). Ailble i 9

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