ASuggestedBoundaryforHeisenberg suncertaintyprinciple

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1 ASuggestedBoundayfoHeisenbeg suncetaintypincile Esen aade Haug Nowegian Univesity of Life Sciences Januay, 07 Abstact In this ae we ae combining Heisenbeg s uncetainty incile with Haug s suggested maimum velocity fo anything with est-mass; see [,, 3]. This leads to a suggested eact bounday condition on Heisenbeg s uncetainty incile. The uncetainty in osition at the otential maimum momentum fo subatomic aticles (as deived fom the maimum velocity) is half of the Planck length. Pehas Einstein was ight afte all when he stated, od does not lay dice. O at least the dice may have a sticte bounday on ossible outcomes than we have eviously thought. We also show how this suggested bounday condition seems to make big consistent with Heisenbeg s uncetainty incile. We obtain a mathematical eession fo big that is fully in line with emiical obsevations. Hoefully ou analysis can be a small ste in bette undestanding Heisenbeg s uncetainty incile and its inteetations and by etension, the boade imlications fo the uantum wold. Key wods: Heisenbeg s uncetainty incile, maimum velocity matte, oint aticle, bounday condition, big, Planck mass aticle, Planck length, educed Comton wavelength. Intoduction Haug [,, 3] has ecently intoduced a new maimum velocity fo subatomic aticles (anything with mass) that is just below the seed of light. The fomula is given by v ma = c whee is the educed Comton wavelength of the aticle we ae tying to acceleate and is the Planck length [4]. This fomula can fo eamle be deived fom secial elativity by simly assuming that the maimum feuency one can have is the Planck feuency c, o that the shotest wavelength ossible is the Planck length. We will also obtain the same fomula if we assume that the ultimate fundamentaaticle has a satial dimension eual to and always is taveling at the seed of light, a model outlined by [, 5]. This maimum velocity uts an ue bounday condition on the kinetic enegy, the momentum, and the elativistic mass, as well as on the elativistic Dole shift in elation to subatomic aticles. Basically, no fundamentaaticle can attain a elativistic mass highe than the Planck mass, and the shotest educed Comton wavelength we can obseve fom length contaction is the Planck length. In addition, the maimum feuency is limited to the Planck feuency. Hee we will combine this euation with Heisenbeg s uncetainty incile. Heisenbeg s Uncetainty Pincile in Relation to Maimum Momentum Heisenbeg s uncetainty incile [7] isgivenby esenhaug@mac.com. Thanks to Victoia Teces fo heling me edit this manuscit. Also thanks to Alan Lewis, Daniel Du y, aue, and AvT fo useful tis on how to do high ecision calculations. Thanks to Mike McCulloch fo vey useful comments on the fist daft of this ae. See also Kennad [8] whowasthefistto ove thismodenineualitybasedonthewokofheisenbeg. l () ()

2 whee is consideed to be the uncetainty in the osition, is the uncetainty in the momentum, and is the educed Planck constant. Haug [] has suggested that the maimum momentum fo a fundamentaaticle likely is given by ma = s ma = ma = s mvma v ma c mv ma l! c c mv ma c ma = mc ma = m c c c! l l (3) Based on this we can find a lowe bounday in the uncetainty of the osition,, fo of any fundamental aticle when assuming the is limited to the maimum momentum fo the subatomic aticle in uestion. Fom this we get m c mv ma v ma c m v ma l m c l (4) and since the Planck mass can be witten as m =, we can ewite this as c c c l l (5) Fo any known fundamentaaticle, >>, so we can use the fist tem of a seies eansion: l l.thisgivesus l l (6) and when >>, we have a vey good aoimation by (7) In othe wods, the maimum uncetainty in the osition of any fundamental subatomic aticle (when assuming is eual to the maimum momentum of the aticle) is half the Planck length. This lies in

3 3 stong contast to standad hysics, whee thee is basically no bounday on the maimum momentum a fundamentaaticle can achieve as long as it is below infinite. Theefoe, in the standad theoy thee is no limit on how close can be, elative to zeo. As [9] ecently has shown, this leads to absud ossibilities fo elativistic mass, kinetic enegy, and momentum. Unde the standad theoy, an electon could attain a elativistic mass eual to the est-mass of the Moon, the Eath, the Sun, and even the whole obsevable univese while still taveling below the seed of light. In the new theoy esented by Haug no fundamentaaticle can attain a elativistic mass lage than the Planck mass. Unde ou new inteetation of Heisenbeg s incile thee is an eact ue limit on the momentum eual to the Planck momentum, and it is identical fo all subatomic fundamentaaticles. Natually this will only hold tue because thei maimum velocity is not the same and is deendent on thei educed Comton wavelength. Ou theoy gives an eact limit on how close v can get to c. Togive an eamle, fo an electon this maimum velocity is v ma = c s l e c (8) This is the same maimum velocity as given by [, ]. These calculations euie vey high ecision and wee calculated in Mathematica. In ou view, one ossible inteetation is that the educed Comton wavelength of the electon is contacted down to the Planck length at this maimum velocity, as discussed by [6]. In this case, we cannot claim that the electon is at an eact oint location 0, simly because it is not a oint aticle. The educed Comton wavelength is, in ou view, the distance fom cente to cente between two indivisible aticles that make u the electon, taveling back and foth counte-stiking. When they ae ultimately comessed (due to length contaction of the void in between the indivisibles making u the fundamentaaticle), the aticles must lie side by side. The educed Comton wavelength is now. And ou best estimate of whee the electon is now would be half the Planck length, that is to say, in the middle of its contacted educed Comton wavelength. Heisenbeg s uncetainty incile combined with ou maimum velocity fomula ossibly indicates that thee can be no oint aticles. Altenatively, one can just inteet this as if thee is a known maimum momentum fo a fundamentaaticle. Then this must be the maimum uncetainty in momentum and fom the Heisenbeg incile thee must be a limitation on how low the uncetainty in that location can be. In the secial case of a Planck mass aticle, we find that =. This may sound damatic, but one ossible inteetation is simly that the momentum of a Planck mass aticle always is zeo, since the Planck mass aticle is standing still as obseved fom any efeence fame; see []. I would also claim that Heisenbeg s uncetainty incile may not be ideally suited fo descibing the situation fo any aticle that is meely standing still (at est). Based on this maimum velocity Haug claims that the Planck mass aticle and the Planck length is the same and is invaiant as seen fom any efeence fame. This can only hold tue of the Planck mass only lasts fo an instant. The Planck mass can be seen as the collision of two light aticles, and theefoe constitutes the tuning oint of light. When a hoton tuns diection by 80 degees (backscatteing) does it not, at the vey tuning oint, stand still fo an instant? The collision of two hotons to ceate matte was fist suggested by Beit and Wheele 934; see [0]. The imlications dawn fom the instances whee light is colliding with light have ecently eceived inceased attention; see [,, 3], fo eamle. Again, the shotest we can have in elation to a momentum is l, whichcanbeusedtofindthe maimum momentum fo any subatomic aticle. We used seveal di eent set-us in Mathematica; hee is one of them: N[St[ (6699 0^( 4))^/( ^( 9))^], 50], whee ^( 4) is the Planck length and ^( 9) is the educed Comton wavelength of the electon. An altenative way to wite it is: N[St[ (SetPecision[ ^( 35))^, 50]/(SetPecision[ ^( 3))^, 50]], 50].

4 4 mv v v c v v c v c l m lm v l v c l c v c v c (9) This leads to a uadatic euation with negative and ositive solutions fo v, whee only the ositive solution seems to make actical sense 3,namelythatv = c l. This is the maimum uncetainty in velocity fo a subatomic aticle with known mass o known educed Comton wavelength. This gives us the maimum momentum fo any subatomic aticle eual to ma = m c l.andwhen >>, this is aoimately eual to the Planck momentum, ma m c. We ae not the only ones to suggest an absolute minimum uncetainty in the osition of any aticle, such as an electon. Adle and Santiago [4] have, based on assumed gavitational inteaction of the hoton and the aticle being obseved, modified the uncetainty incile with an additional tem. By doing this they find a minimum uncetainty in the osition that is not fa fom ou ediction. The stength in ou esult is that no additional tems in the Heisenbeg incile ae needed to get a minimum uncetainty in the osition of any aticle, and theeby also a maimum limit in the uncetainty of the momentum. 3 Time and Enegy Heisenbeg s uncetainty incile in tems of time and enegy can be witten as t E (0) Haug [] has shown that the maimum kinetic enegy of a fundamentaaticle with educed Comton wavelength of is given by 3 O the negative solution could be inteeted as a aticle taveling in the oosite diection of the ositive solution.

5 5 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 this esult in Heisenbeg s time enegy uncetainty ineuality euation t E tc t t c c () and when >>, we have a vey good aoimation by t Which is half a Planck second. It is woth mentioning that the half Planck second and half Planck length found as bounday conditions hee ae eactly the same as the esults we obtained when looking at the Loentz tansfomation in the limit of the maimum velocity of mass [5]. c (3) 4 Big and Heisenbeg s Uncetainty Pincile As shown in [3], the maimum velocity can also be witten as l v ma = c = c m (4) c whee is Newton s gavitational constant [6] andm is the mass of a fundamentaaticle. It is imotant to undestand that m in this contet is not just any mass; this mass must have a educed Comton wavelength. In othe wods, it is the mass of fundamentaaticles. Based on this obsevation, we can assess whethe o not we can use this in combination with Heisenbeg s uncetainty incile to deive a theoetical value of big. We ae not the fist to suggest that Heisenbeg s uncetainty incile could be elated to Newtonian gavity. McCulloch [7] has shown that Newton s gavity fomula basically can be deived fom Heisenbeg s uncetainty incile. Howeve, he has not shown how big also can be deived fom it.

6 6 We could also say that this is just anothe way to show the maimum velocity fo matte may be consistent with Heisenbeg s uncetainty incile, although this should not be consideed as evidence that we will get big fom Heisenbeg s uncetainty 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 (5) To wite the gavitational constant as = c3 has aleady been suggested by Haug [8, 9] in ode to simlify a seies of eessions in Newton and Einstein gavity end esults. It has also been deived by dimensional analysis [3] and used to simlify the euation fom of the Planck units. Futhe, Haug has suggested that the Planck length (at least in a thought eeiment) can be found indeendent of based on the maimum velocity fomula. Since v ma hee is a function of the univesal constants, and c, one could ty to ague that this ovides evidence that must be a function of and and c and not that is a function of. In othe wods, must be a univesal constant and is just a deived constant. Howeve, the beauty of the maimum velocity fomula is that and cancel out and that we ae left with v ma only as a function of c, and the educed Comton wavelength of the aticle in uestion,, and not of and. It is woth ointing out that the educed Comton wavelength of an electon can be found comletely indeendent of any knowledge of ; see[0]. To find we need the educed Comton wavelength that can be found indeendently of, as well at the maimum velocity fo an electon, v ma. Thismaimumvelocityhas to be found eeimentally. This maimum velocity fo an electon is vey close to c, butitisstillhighe than the velocities that ae in oeation at LHC. Even so, the fact that something is edicted and not found yet is not a good enough agument to comletely eject a theoy yet. Ou fomula fo big gives the same value as the gavitational constant, as it is known fom eeiments, and it can actually be calibated to the eeiments. Thee is still consideable uncetainty about the eact measuement of the gavitational constant. Eeimentally, substantiaogess has been made in ecent yeas based on vaious methods. See [,, 3, 4, 5], fo eamle. In the fomula esented hee, the uncetainty lies in the eact value of the Planck length, as well as in ; theseedoflightc = iseactbydefinition. Atthe

7 7 moment, the Planck length can only be found fom, but if we had access to much moe advanced aticle acceleatos than the Lage Hadon Collide, we could eect to detect v ma and then back the Planck length out fom thee. We claim that big is indeed a univesal constant, but it is a comosite constant that is deendent on thee even moe fundamental constants, namely,,andc. 5 Conclusion By combining Heisenbeg s uncetainty incile with the newly intoduced maimum velocity on mass, we have shown that the smallest locational uncetainty of a fundamentaaticle is elated to half the Planck length, and that the shotest time inteval is elated to half the Planck time. This is the same finding that we obtained when combining this maimum velocity with the Loentz tansfomation [5]. Refeences [] E.. Haug. The Planck mass aticle finally discoveed! ood bye to the oint aticle hyothesis! htt://via.og/abs/ , 06. [] E.. Haug. A new solution to Einstein s elativistic mass challenge based on maimum feuency. htt://via.og/abs/ , 06. [3] E.. Haug. The gavitational constant and the Planck units: A simlification of the uantum ealm. Physics Essays Vol 9, No 4, 06. [4] M. Planck. The Theoy of Radiation. Dove 959 tanslation, 906. [5] E.. Haug. Unified Revolution: New Fundamental Physics. Oslo, E..H. Publishing, 04. [6] E.. Haug. Deiving the maimum velocity of matte fom the Planck length limit on length contaction. htt://via.og/abs/6.0358, 06. [7] W. Heisenbeg. Übe den anschaulichen inhalt de uantentheoetischen kinematik und mechanik. Zeitschift fü Physik, (43):7 98,97. [8] E. H. Kennad. Zu uantenmechanik einfache bewegungstyen. Zeitschift fü Physik, (44):36 35, 97. [9] E.. Haug. Moden hysics incomlete absud elativistic mass inteetation. and the simle solution that saves Einstein s fomula. htt://via.og/abs/6.049, 06. [0]. Beit and J. A. Wheele. Collision of two light uanta. Physical Review, 46,934. [] B. King and C. H. Keitel. Photon hoton scatteing in collisions of intense lase ulses. New Jounal of Physics, 4,0. [] O. J. Pike, F. Mackenoth, E.. Hill, and R. S. J. A hoton hoton collide in a vacuum hohlaum. Natue Photonics, 8,04. [3] D. E. Chang, V. Vuletić, and M. D. Lukin. Quantum nonlinea otics hoton by hoton. Natue Photonics, 8,04. [4] R. J. Adle and D. I. Santiago. On gavity and the uncetainty incile. Moden Physics Lettes A, 4. [5] E.. Haug. The Loentz tansfomation at the maimum velocity fo a mass. htt://via.og/abs/609.05, 06. [6] I. Newton. Philosohiae Natualis Pinciia Mathematica. London,686. [7] M. E. McCulloch. avity fom the uncetainty incile. Astohysics and Sace Science, 349, 04. [8] E.. Haug. Planck uantization of Newton and Einstein gavitation. Intenational Jounal of Astonomy and Astohysics, 6(),06. [9] E.. Haug. Newton and Einstein s gavity in a new esective fo Planck masses and smalle sized objects. htt://via.og/abs/60.038, 06.

8 8 [0] S. Pasannakuma, S. Kishnaveni, and T. K. Umesh. Detemination of est mass enegy of the electon by a Comton scatteing eeiment. Euoean Jounal of Physics, 33(), 0. []. S. Bisnovatyi-Kogan. Checking the vaiability of the gavitational constant with binay ulsas. Intenational Jounal of Moden Physics D, 5(07),006. [] B. File,. T. Foste, J. M. Mcuik, and M. A. Kasevich. Atom intefeomete measuement of the Newtonian constant of gavity. Science, 35, 007. [3] S. alli, A. Melchioi,. F. Smoot, and O. Zahn. Fom Cavendish to Planck: Constaining Newton s gavitational constant with CMB temeatue and olaization anisotoy. Physical Review D, 80, 009. [4]. Rosi, F. Soentino, L. Cacciauoti, M. Pevedelli, and. M. Tino. Pecision measuement of the Newtonian gavitational constant using cold atoms. Natue, 50, 04. [5] S. Schlamminge. A fundamental constant: A cool way to measue big. Natue, 50,04.