ASuggestedBoundaryforHeisenberg suncertaintyprinciple

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1 ASuggestedBoundayfoHeisenbeg sunetaintypinile Esen aade Haug Nowegian Univesity of Life Sienes Januay 9, 08 Abstat In this ae we ae ombining Heisenbeg s unetainty inile with Haug s suggested maimum veloity fo anything with est-mass; see [,, 3, 4]. This leads to a suggested eat bounday ondition on Heisenbeg s unetainty inile. The unetainty in osition at the otential maimum momentum fo subatomi atiles as deived fom the maimum veloity is half of the Plank length. Pehas Einstein was ight afte all when he stated, od does not lay die. O at least the die may have a stite bounday on ossible outomes than we have eviously thought. We also show how this suggested bounday ondition seems to make big onsistent with Heisenbeg s unetainty inile. We obtain a mathematial eession fo big that is fully in line with emiial obsevations. Hoefully ou analysis an be a small ste in bette undestanding Heisenbeg s unetainty inile and its inteetations and, by etension, the boade imliations fo the uantum wold. Key wods: Heisenbeg s unetainty inile, maimum veloity of matte, oint atile, bounday ondition, big, Plank mass atile, Plank length, edued Comton wavelength. Intodution Haug [,, 3, 4] has eently intodued a new maimum veloity fo subatomi atiles (anything with mass) that is just below the seed of light. The fomula is given by v ma = whee is the edued Comton wavelength of the atile we ae tying to aeleate and is the Plank length [5]. This fomula an be deived fom seial elativity by simly assuming the maimum feueny one an have is the Plank feueny, o that the shotest wavelength ossible is the Plank length. We will also get the same fomula if we assume that the ultimate fundamentaatile has a satial dimension eual to and is always taveling at the seed of light, a model outlined by [6, ]. This maimum veloity uts an ue bounday ondition on the kineti enegy, the momentum, and the elativisti mass, as well as on the elativisti Dole shift in elation to subatomi atiles. Basially, no fundamentaatile an attain a elativisti mass highe than the Plank mass, and the shotest edued Comton wavelength we an obseve fom length ontation is the Plank length. In addition, the maimum feueny is limited to the Plank feueny. Hee we will ombine this euation with Heisenbeg s unetainty inile. Heisenbeg s Unetainty Pinile in Relation to Maimum Momentum Heisenbeg s unetainty inile [8] isgivenby l () () esenhaug@ma.om. Thanks to Vitoia Tees fo heling me edit this manusit. Also thanks to Alan Lewis, Daniel Du y, aue, and AvT fo useful tis on how to do high eision alulations. Thanks to Mike MCulloh fo vey useful omments on the fist daft of this ae. See also Kennad [9], who was the fist to ove this moden ineuality based on the wok of Heisenbeg.

2 whee is onsideed to be the unetainty in the osition, is the unetainty in the momentum, and is the edued Plank onstant. Haug [] has shown that the maimum momentum fo a fundamentaatile likely is given by ma = s ma = ma = s mvma v ma mv ma l! mv ma ma = m ma = m! l l (3) Based on this we an find a lowe bounday in the unetainty of the osition,, fo of any fundamental atile when assuming that is limited to the maimum momentum fo the subatomi atile in uestion. Fom this we get m mv ma v ma m v ma l m l (4) and sine the Plank mass an be witten as m =, we an ewite this as l l (5) Fo any known fundamentaatile, >> l l.thisgivesus so we an use the fist tem of a seies eansion: l l (6) and when >> we have a vey good aoimation by (7) In othe wods, the maimum unetainty in the osition of any fundamental subatomi atile (when assuming is eual to the maimum momentum of the atile) is half the Plank length. This lies in

3 3 stong ontast to standad hysis, whee thee is basially no bounday on the maimum momentum a fundamentaatile an ahieve as long as it is below infinity. Theefoe, in the standad theoy thee is no limit on how lose an be, elative to zeo. As [0] eently has shown, this leads to absud ossibilities fo elativisti mass, kineti enegy, and momentum. Unde the standad theoy, an eleton ould attain a elativisti mass eual to the est-mass of the Moon, the Eath, the Sun, and even the entie obsevable univese while still taveling below the seed of light. In the new theoy esented by Haug no fundamentaatile an attain a elativisti mass lage than the Plank mass. Unde ou new inteetation of Heisenbeg s inile thee is an eat ue limit on the momentum eual to the Plank momentum, and it is idential fo all subatomi fundamental atiles. Natually this will only hold tue beause thei maimum veloities ae not the same and ae deendent on thei edued Comton wavelengths. Ou theoy gives an eat limit on how lose v an get to. Fo eamle, fo an eleton this maimum veloity is s l v ma = (8) e This is the same maimum veloity as given by [, ]. These alulations euie vey high eision and wee alulated in Mathematia. In ou view, one ossible inteetation is that the edued Comton wavelength of the eleton is ontated down to the Plank length at this maimum veloity, as disussed by [7]. In this ase, we annot laim that the eleton is at an eat oint loation 0, simly beause it is not a oint atile. The edued Comton wavelength is, in ou view, the distane fom ente to ente between two indivisible atiles that make u the eleton, taveling bak and foth ounte-stiking. When they ae ultimately omessed (due to length ontation of the void in between the indivisibles making u the fundamentaatile), the atiles must lie side by side. The edued Comton wavelength is now. And ou best estimate of whee the eleton is now would be half the Plank length, that is to say, in the middle of its ontated edued Comton wavelength. Heisenbeg s unetainty inile ombined with ou maimum veloity fomula ossibly indiates that thee an be no oint atiles. Altenatively, one an just inteet this as if thee is a known maimum momentum fo a fundamentaatile, then this must be the maimum unetainty in momentum and then thee must be a limitation on how low the unetainty in loation an be, taking the Heisenbeg inile into aount. Based on this maimum veloity Haug laims that the Plank-mass atile and the Plank length ae the same and is invaiant as seen fom any efeene fame. This an only hold tue if the Plank mass only lasts fo an instant. The Plank mass an be seen as the ollision of two light atiles, and theefoe onstitutes the tuning oint of light. When a hoton hanges dietion by 80 degees (baksatteing) does it not, at the vey tuning oint, stand still fo an instant? The onet of the ollision of two hotons eating matte was fist suggested by Beit and Wheele 934, see []. Imliations fom light olliding with light have eently eeived ineased attention, see fo eamle [, 3, 4]. The shotest we an have in elation to a given momentum is l, whihagainanbeusedtofind the maimum veloity fo any subatomi atile. We used seveal di eent set-us in Mathematia; hee is one of them: N[St[ (6699 0^( 4))^/( ^( 9))^], 50], whee ^( 4) is the Plank length and ^( 9) is the edued Comton wavelength of the eleton. An altenative way to wite it is: N[St[ (SetPeision[ ^( 35))^, 50]/(SetPeision[ ^( 3))^, 50]], 50].

4 4 mv v v v v v v v + l ale l m lm l v ale l v ale v v l ale l ale ale l + l v v l v (9) + This is the maimum unetainty in veloity fo a subatomi atile with known mass o known edued Comton wavelength. A Taylo seies eansion gives v ale + l + 3 l and the maimum veloity fomula suggested by Haug is l v ma = l + l l l In both fomulas we get a highly auate esult by using the fist tem of the Taylo eansion and we see they ae the same. We ae not the only ones to suggest an absolute minimum unetainty in the osition of any atile, suh as an eleton. Adle and Santiago [5] have, based on assumed gavitational inteation of the hoton and the atile being obseved, modified the unetainty inile with an additional tem. By doing this they find a minimum unetainty in the osition that is not fa fom ou edition. The stength in ou esult is that no additional tems in the Heisenbeg inile ae needed to get a minimum unetainty in the osition of any atile, and theeby also a maimum limit in the unetainty of the momentum. 3 Time and Enegy In tems of time and enegy, Heisenbeg s unetainty inile an be witten as t E () Haug [] has shown that the maimum kineti enegy of a fundamentaatile with edued Comton wavelength of is given by (0) ()

5 5 E k,ma = E k,ma = E k,ma = s s E k,ma = m m v ma m m m m l! l! m m E k,ma = m m E k,ma = E k,ma = E k,ma = (3) We an use this esult in Heisenbeg s time enegy unetainty ineuality euation t E t t t (4) and when >>, we have a vey good aoimation by t Whih is half a Plank seond. It is woth mentioning that the half Plank seond and half Plank length found as bounday onditions hee ae eatly the same as the esults we obtained when looking at the Loentz tansfomation in the limit of the maimum veloity of mass [6]. (5) 4 Big and Heisenbeg s Unetainty Pinile As shown in [3], the maimum veloity an also be witten as l v ma = = m (6) whee is Newton s gavitational onstant [7] andm is the mass of a fundamentaatile. It is imotant to undestand m in this ontet is not just any mass; this mass must have a edued Comton wavelength. In othe wods, it is the mass of fundamentaatiles. Based on this obsevation, we an assess whethe o not we an use this in ombination with Heisenbeg s unetainty inile to deive a theoetial value of big. We ae not the fist to suggest that Heisenbeg s unetainty inile ould be elated to Newtonian gavity. MCulloh [8] has shown that Newton s gavity fomula basially an be deived fom Heisenbeg s unetainty inile. Howeve, he has not shown how big also an be deived fom it.

6 6 We ould also say that this is just anothe way to show the maimum veloity fo matte may be onsistent with Heisenbeg s unetainty inile, although this should not be onsideed as evidene that we will get big fom Heisenbeg s unetainty inile. We have mv ma v ma m v ma m m m m m 4 m m l 3 4 m m 4 4m v ma v ma m m 3 l m m l l (7) To wite the gavitational onstant as = 3 has aleady been suggested by Haug [9, 0] in ode to simlify a seies of eessions in Newtonian and Einsteinian gavity end esults. It has also been deived by dimensional analysis [3] and used to simlify the euation fom of the Plank units. Futhe, Haug has suggested that the Plank length (at least in a thought eeiment) an be found indeendent of based on the maimum veloity fomula. Sine v ma hee is a funtion of the univesal onstants, and one ould ty to ague that this is a evidene must be a funtion of and and and not that is a funtion of. In othe wods, that must be a univesal onstant and is just a deived onstant. Howeve, the beauty of the maimum veloity fomula is that and anel out and we ae left with that v ma only is a funtion of, and the edued Comton wavelength of the atile in uestion,, andnotof and. It is woth ointing out that the edued Comton wavelength of an eleton an be found eeimentally, omletely indeendent of any knowledge of, see[]. To find one needs the edued Comton wavelength that an be found totally indeendent on as well at the maimum veloity fo an eleton, v ma. This maimum veloity has to be found eeimentally. This maimum veloity fo an eleton is vey lose to, but still highe than the veloities one oeates with at the Lage Hadon Collide. Howeve, the fat that something is edited but not yet found, is not a su ient agument fo ejeting a theoy outight. Ou fomula fo big gives the same value as the gavitational onstant, as is known fom eeiments, it an atually be alibated to the eeiments. Thee is still onsideable unetainty about the eat measuement of the gavitational onstant. Eeimentally, substantiaogess has been made in eent yeas based on vaious methods. See, fo eamle, [, 3, 4, 5, 6]. In the fomula esented hee, the unetainty lies in the eat value of the Plank length, as well as in ; theseedoflight =

7 7 is eat e definition. At the moment, the Plank length an only be found fom, butifwehadaess to muh moe advaned atile aeleatos than the Lage Hadon Collide, we ould eet to detet v ma and then bak the Plank length out fom thee. We laim that big is indeed a univesal onstant, but it is a omosite onstant that is deendent on thee even moe fundamental onstants, namely,,and. 5 Conlusion By ombining Heisenbeg s unetainty inile with the newly intodued maimum veloity on mass, we have shown that the smallest loation unetainty of a fundamentaatile is elated to half the Plank length, and that the shotest time inteval is elated to half the Plank time. This is the same finding as the one we obtained when we ombined this maimum veloity with the Loentz tansfomation [6]. Refeenes [] E.. Haug. The Plank mass atile finally disoveed! ood bye to the oint atile hyothesis! htt://via.og/abs/ , 06. [] E.. Haug. A new solution to Einstein s elativisti mass hallenge based on maimum feueny. htt://via.og/abs/ , 06. [3] E.. Haug. The gavitational onstant and the Plank units. A simlifiation of the uantum ealm. Physis Essays Vol 9, No 4, 06. [4] E.. Haug. The ultimate limits of the elativisti oket euation. The Plank hoton oket. Ata Astonautia, 36,07. [5] M. Plank. The Theoy of Radiation. Dove 959 tanslation, 906. [6] E.. Haug. Unified Revolution: New Fundamental Physis. Oslo,E..H.Publishing,04. [7] E.. Haug. Deiving the maimum veloity of matte fom the Plank length limit on length ontation. htt://via.og/abs/6.0358, 06. [8] W. Heisenbeg. Übe den anshaulihen inhalt de uantentheoetishen kinematik und mehanik. Zeitshift fü Physik, (43):7 98,97. [9] E. H. Kennad. Zu uantenmehanik einfahe bewegungstyen. Zeitshift fü Physik, (44):36 35, 97. [0] E.. Haug. Moden hysis inomlete absud elativisti mass inteetation. and the simle solution that saves Einstein s fomula. htt://via.og/abs/6.049, 06. []. Beit and J. A. Wheele. Collision of two light uanta. Physial Review, 46,934. [] B. King and C. H. Keitel. Photon hoton satteing in ollisions of intense lase ulses. New Jounal of Physis, 4,0. [3] O. J. Pike, F. Makenoth, E.. Hill, and R. S. J. A hoton hoton ollide in a vauum hohlaum. Natue Photonis, 8,04. [4] D. E. Chang, V. Vuletić, and M. D. Lukin. Quantum nonlinea otis hoton by hoton. Natue Photonis, 8,04. [5] R. J. Adle and D. I. Santiago. On gavity and the unetainty inile. Moden Physis Lettes A, 4. [6] E.. Haug. The Loentz tansfomation at the maimum veloity fo a mass. htt://via.og/abs/609.05, 06. [7] I. Newton. Philosohiae Natualis Piniia Mathematia. London,686. [8] M. E. MCulloh. avity fom the unetainty inile. Astohysis and Sae Siene, 349, 04.

8 8 [9] E.. Haug. Plank uantization of Newton and Einstein gavitation. Intenational Jounal of Astonomy and Astohysis, 6(),06. [0] E.. Haug. Newton and Einstein s gavity in a new esetive fo Plank masses and smalle sized objets. htt://via.og/abs/60.038, 06. [] S. Pasannakuma, S. Kishnaveni, and T. K. Umesh. Detemination of est mass enegy of the eleton by a Comton satteing eeiment. Euoean Jounal of Physis, 33(), 0. []. S. Bisnovatyi-Kogan. Cheking the vaiability of the gavitational onstant with binay ulsas. Intenational Jounal of Moden Physis D, 5(07),006. [3] B. File,. T. Foste, J. M. Muik, and M. A. Kasevih. Atom intefeomete measuement of the Newtonian onstant of gavity. Siene, 35, 007. [4] S. alli, A. Melhioi,. F. Smoot, and O. Zahn. Fom Cavendish to Plank: Constaining Newton s gavitational onstant with CMB temeatue and olaization anisotoy. Physial Review D, 80, 009. [5]. Rosi, F. Soentino, L. Caiauoti, M. Pevedelli, and. M. Tino. Peision measuement of the Newtonian gavitational onstant using old atoms. Natue, 50, 04. [6] S. Shlamminge. A fundamental onstants: A ool way to measue big. Natue, 50,04.