ANewBoundayfoHeisenbeg sunceainypincile ood Bye o he Poin Paicle Hyohesis? Esen aade Haug Nowegian Univesiy of Life Sciences Januay 5, 07 Absac In his ae we ae combining Heisenbeg s unceainy incile wih Haug s new insigh on he maimum velociy fo anyhing wih es-mass; see [,, 3]. This leads o a new and eac bounday condiion on Heisenbeg s unceainy incile. The unceainy in osiion a he oenial maimum momenum fo subaomic aicles as deived fom he maimum velociy is half of he Planck lengh. Pehas Einsein was igh afe all when he saed, od does no lay dice. O a leas he dice may have a sice bounday on ossible oucomes han we have eviously hough. We also show how his new bounday condiion seems o make big consisen wih Heisenbeg s unceainy incile. We obain a mahemaical eession fo big ha is fully in line wih emiical obsevaions. Hoefully ou analysis can be a small se in bee undesanding Heisenbeg s unceainy incile and is ineeaions and by eension, he boade imlicaions fo he uanum wold. Key wods: Heisenbeg s unceainy incile, maimum velociy mae, oin aicle, bounday condiion, big, Planck mass aicle, Planck lengh, educed Comon wavelengh. Inoducion Haug [,, 3] has ecenly inoduced a new maimum velociy fo subaomic aicles (anyhing wih mass) ha is jus below he seed of ligh. The fomula is given by v ma = c whee is he educed Comon wavelengh of he aicle we ae ying o acceleae and is he Planck lengh [4]. This maimum velociy us an ue bounday condiion on he kineic enegy, he momenum, and he elaivisic mass, as well as on he elaivisic Dole shif in elaion o subaomic aicles. Basically, no fundamenaaicle can aain a elaivisic mass highe han he Planck mass, and he shoes educed Comon wavelengh we can obseve fom lengh conacion is he Planck lengh. In addiion, he maimum feuency is limied o he Planck feuency, he Planck aicle mass is invaian, and so is he Planck lengh (when elaed o he educed Comon wavelengh). Hee we will combine his euaion wih Heisenbeg s unceainy incile. Heisenbeg s Unceainy Pincile in elaion o Maimum Momenum Heisenbeg s unceainy incile [5] isgivenby () whee is consideed o be he unceainy in he osiion, is he unceainy in he momenum, and is he educed Planck consan. e-mail esenhaug@mac.com. Thanks o Vicoia Teces fo heling me edi his manusci. Also hanks o Alan Lewis, Daniel Du y, aue and AvT fo useful is on how o do high ecision calculaions. See also Kennad [6] hawashefiso ove hismodenineualiybasedonhewokofheisenbeg. l ()
Haug [] has shown ha he maimum momenum fo a fundamenaaicle likely is given by ma = s ma = ma = mvma v ma c mv ma l! c c mv ma ma = mc ma = m c c +c c l l (3) Based on his we can find a lowe bounday in he unceainy of he osiion,, fo of any fundamenal aicle when assuming he is limied o he maimum momenum fo he subaomic aicle in uesion. Fom his we ge m c mv ma v ma c m v ma l m c l (4) and since he Planck mass can be wien as m =, we can ewie his as c c c l l (5) Fo any known fundamenaaicle, >> l l.thisgivesus so we can use he fis em of a seies eansion: l l (6) and when >> we have a vey good aoimaion by (7) In ohe wods, he maimum unceainy in he osiion of any fundamenal subaomic aicle (when assuming is eual o he maimum momenum of he aicle) is half he Planck lengh. This lies in song conas o sandad hysics, whee hee is basically no bounday on he maimum momenum a fundamenaaicle can achieve as long as i is below infinie. Theefoe, in he sandad heoy hee
3 is no limi on how close can be, elaive o zeo. As [7] ecenly has shown, his leads o absud ossibiliies fo elaivisic mass, kineic enegy, and momenum. Unde he sandad heoy, an elecon could aain a elaivisic mass eual o he es-mass of he Moon, he Eah, he Sun and even he whole obsevable univese while sill aveling below he seed of ligh. In he new heoy esened by Haug no fundamenaaicle can aain a elaivisic mass lage han he Planck mass. Unde ou new ineeaion of Heisenbeg s incile hee is an eac ue limi on he momenum, and i is idenical fo all subaomic fundamenaaicles. Naually his will only hold ue because hei maimum velociy is no he same and is deenden on hei educed Comon wavelengh. Ou heoy gives an eac limi on how close v can ge o c. Fo eamle, fo an elecon his maimum velociy is v ma = c s l e c 0.9999999999999999999999999999999999999999999994 (8) This is he same maimum velociy as given by [, ]. These calculaions euie vey high ecision and wee calculaed in Mahemaica. In ou view, he igh ineeaion is mos likely ha he educed Comon wavelengh of he elecon is conaced down o he Planck lengh a his maimum velociy, as discussed by [8]. In his case, we canno claim ha he elecon is a an eac oin locaion 0, simly because i is no a oin aicle. The educed Comon wavelengh is, in ou view, he disance fom cene o cene beween wo indivisible aicles ha make u he elecon, aveling back and foh coune-siking. When hey ae ulimaely comessed (due o lengh conacion of he void in beween he indivisibles making u he fundamenaaicle), he aicles mus lie side by side. The educed Comon wavelengh is now. And ou bes esimae of whee he elecon is now would be half he Planck lengh, ha is o say, in he middle of is conaced educed Comon wavelengh. Heisenbeg s unceainy incile combined wih ou maimum velociy fomula ossibly indicaes ha hee can be no oin aicles. In he secial case of a Planck mass aicle, we find ha =. This may sound damaic, bu he coec ineeaion is simly ha he momenum of a Planck-mass aicle always is zeo, since he Planck mass aicle is sanding sill as obseved fom any efeence fame; see []. I would also claim ha Heisenbeg s unceainy incile may no be ideally suied fo descibing he siuaion fo any aicle ha is meely sanding sill (a es). Again he shoes we can have in elaion o a momenum is l, whichagaincanbeusedofind he maimum momenum fo any subaomic aicle. mv v c v v v c v c l l m lm v l v c This leads o a uadaic euaion wih a negaive and osiive soluion fo v, whee only he osiive soluion seems o make acical sense 3 l,namelyhav = c. This gives us he maimum l momenum fo any subaomic aicle eual o ma = m c m c. l c We used seveal di een se-us in Mahemaica; hee is one of hem: N[S[ (6699 0^( 4))^/(386593 0^( 9))^], 50], whee 6699 0^( 4) is he Planck lengh and 386593 0^( 9) is he educed Comon wavelengh of he elecon. An alenaive way o wie i is: N[S[ (SePecision[.6699 0^( 35))^, 50]/(SePecision[3.86593 0^( 3))^, 50]], 50]. 3 O he minus soluion could be ineeed as a aicle aveling in he oosie diecion of he lus soluion. v c v c (9)
4 We ae no he only ones o sugges an absolue minimum unceainy in he osiion of any aicle, such as an elecon. Adle and Saniago [9] have, based on assumed gaviaional ineacion of he hoon and he aicle being obseved, modified he unceainy incile wih an addiional em. By doing his hey find a minimum unceainy in he osiion ha is no fa fom ou edicion. The sengh in ou esul is ha no addiional ems in he Heisenbeg incile ae needed o ge a minimum unceainy in he osiion of any aicle, and heeby also a maimum limi in he unceainy of he momenum. 3 Time and Enegy Heisenbeg s unceainy incile in ems of ime and enegy can be wien as E (0) Haug [] has shown ha he maimum kineic enegy of a fundamenaaicle wih educed Comon wavelengh of is given by E k,ma = E k,ma = E k,ma = s s E k,ma = mc mc v ma c mc c mc c mc c c mc l! l! mc mc E k,ma = mc mc E k,ma = c c E k,ma = c c E k,ma = c c c () We can use his esul in Heisenbeg s ime enegy unceainy ineualiy euaion E c c c () and when >>, we have a vey good aoimaion by Which is half a Planck second. I is woh menioning ha he half Planck second and half Planck lengh found as bounday condiions hee ae eacly he same as he esuls we obained when looking a he Loenz ansfomaion in he limi of he maimum velociy of mass [0]. c (3)
5 4 Big and Heisenbeg s Unceainy Pincile As shown in [3], he maimum velociy can also be wien as l v ma = c = c m (4) c whee is Newon s gaviaional consan [] andm is he mass of a fundamenaaicle. I is imoan o undesand m in his cone is no jus any mass; his mass mus have a educed Comon wavelengh. In ohe wods, i is he mass of fundamenaaicles. Based on his obsevaion, we can assess whehe o no we can use his in combinaion wih Heisenbeg s unceainy incile o deive a heoeical value of big. We ae no he fis o sugges ha Heisenbeg s unceainy incile could be elaed o Newonian gaviy. McCulloch [] has shown ha Newon s gaviy fomula basically can be deived fom Heisenbeg s unceainy incile. Howeve, he has no shown how big also can be deived fom i. We could also say ha his is jus anohe way o show he maimum velociy fo mae may be consisen wih Heisenbeg s unceainy incile, alhough his should no be consideed as evidence ha we will ge big fom Heisenbeg s unceainy incile. We have mv ma v ma c m v ma m m c c m c m c m c 4 m c c m c c c 3 c 3 c 3 c 3 lc 3 c 4 m c m c 4 4m v ma cv ma m c cc c m c c c 3 l m c m l l 6.67384 0 (5) To wie he gaviaional consan as = c3 has aleady been suggesed by Haug [3, 4] in ode o simlify a seies of eessions in Newon and Einsein gaviy end esuls. I has also been deived by dimensional analysis [3] and used o simlify he euaion fom of he Planck unis. Fuhe, Haug has suggesed ha he Planck lengh (a leas in a hough eeimen) can be found indeenden of based on he maimum velociy fomula. This gives he same value as he gaviaional consan, as is known fom eeimens. Howeve, hee is sill consideable unceainy abou he eac measuemen of he gaviaional consan. Eeimenally, subsaniaogess has been made in ecen yeas based on vaious mehods. See, fo eamle,
6 [5, 6, 7, 8, 9]. In he fomula esened hee, he unceainy lies in he eac value of he Planck lengh, as well as in ; heseedoflighc =9979458iseacedefiniion. Ahemomen,he Planck lengh can only be found fom, bu if we had access o much moe advanced aicle acceleaos han he Lage Hadon Collide, we could eec o deec v ma and hen back he Planck lengh ou fom hee. We claim ha big is indeed a univesal consan, bu i is a comosie consan ha is deenden on hee even moe fundamenal consans, namely,,andc. 5 Conclusion By combining Heisenbeg s unceainy incile wih he newly inoduced maimum velociy on mass, we have shown ha he smalles locaion unceainy of a fundamenaaicle is elaed o half he Planck lengh, and ha he shoes ime ineval is elaed o half he Planck ime. This is he same finding as we obained when combining his maimum velociy wih he Loenz ansfomaion [0]. Refeences [] E.. Haug. The Planck mass aicle finally discoveed! ood bye o he oin aicle hyohesis! h://via.og/abs/607.0496, 06. [] E.. Haug. A new soluion o Einsein s elaivisic mass challenge based on maimum feuency. h://via.og/abs/609.0083, 06. [3] E.. Haug. The gaviaional consan and he Planck unis. A simlificaion of he uanum ealm. Physics Essays Vol 9, No 4, 06. [4] M. Planck. The Theoy of Radiaion. Dove 959 anslaion, 906. [5] W. Heisenbeg. Übe den anschaulichen inhal de uanenheoeischen kinemaik und mechanik. Zeischif fü Physik, (43):7 98,97. [6] E. H. Kennad. Zu uanenmechanik einfache bewegungsyen. Zeischif fü Physik, (44):36 35, 97. [7] E.. Haug. Moden hysics incomlee absud elaivisic mass ineeaion. and he simle soluion ha saves Einsein s fomula. h://via.og/abs/6.049, 06. [8] E.. Haug. Deiving he maimum velociy of mae fom he Planck lengh limi on lengh conacion. h://via.og/abs/6.0358, 06. [9] R. J. Adle and D. I. Saniago. On gaviy and he unceainy incile. Moden Physics Lees A, 4. [0] E.. Haug. The Loenz ansfomaion a he maimum velociy fo a mass. h://via.og/abs/609.05, 06. [] I. Newon. Philosohiae Naualis Pinciia Mahemaics. London,686. [] M. E. McCulloch. On gaviy and he unceainy incile. Asohysics and Sace Science, 349. [3] E.. Haug. Planck uanizaion of Newon and Einsein gaviaion. Inenaional Jounal of Asonomy and Asohysics, 6(),06. [4] E.. Haug. Newon and Einsein s gaviy in a new esecive fo Planck masses and smalle sized objecs. h://via.og/abs/60.038, 06. [5]. S. Bisnovayi-Kogan. Checking he vaiabiliy of he gaviaional consan wih binay ulsas. Inenaional Jounal of Moden Physics D, 5(07),006. [6] B. File,. T. Fose, J. M. Mcuik, and M. A. Kasevich. Aom inefeomee measuemen of he newonian consan of gaviy. Science, 35, 007. [7] S. alli, A. Melchioi,. F. Smoo, and O. Zahn. Fom Cavendish o Planck: Consaining Newon s gaviaional consan wih CMB emeaue and olaizaion anisooy. Physical Review D, 80, 009.
7 [8]. Rosi, F. Soenino, L. Cacciauoi, M. Pevedelli, and. M. Tino. Pecision measuemen of he Newonian gaviaional consan using cold aoms. Naue, 50, 04. [9] S. Schlamminge. A fundamenal consans: A cool way o measue big. Naue, 50,04.