Does Heisenberg s Uncertainty Collapse at the Planck Scale? Heisenberg s Uncertainty Principle Becomes the Certainty Principle

Transcription

1 Does Heisenberg s Unertainty Collapse at the Plank Sale? Heisenberg s Unertainty Priniple Beomes the Certainty Priniple Espen Gaarder Haug Norwegian Uniersity of Life Sienes June 7, 08 Abstrat In this paper we show that Heisenberg s unertainty priniple, ombined with key priniples from Max Plank and Einstein, indiates that unertainty ollapses at the Plank sale. In essene we suggest that the unertainty priniple beomes the ertainty priniple at the Plank sale. This an be used to find the rest-mass formula for elementary partiles onsistent with what is already known. If this interpretation is orret, it means that Einstein s intuition that God Does Not Throw Die with the Unierse ould also be orret. We interpret this to mean that Einstein did not beliee the world was ruled by strange unertainty phenomena at the deeper leel, and we will laim that this leel is the Plank sale, where all unertainty seems to ollapse. The bad news is that this new-found ertainty an only an last for one Plank seond! We are also uestioning, without oming to a onlusion, if this ould hae impliations for Bell s theorem and hidden ariable theories. Key words: Heisenberg s unertainty priniple, Plank length, Plank partile, Plank momentum. The Three Giants In 899, Max Plank introdued what he alled natural units, namely the Plank length, the Plank mass, the Plank seond, and the Plank energy [, ]. He deried these fundamental units from Newton s graitational onstant [3], the speed of light, and the Plank onstant. In 905, Albert Einstein introdued speial relatiity theory [4]. In 97, Heisenberg [5] introdued his unertainty priniple. The Heisenberg unertainty priniple is one of the ornerstones in uantum mehanis. The Plank onstant is also a key here. Howeer, the Plank length, the Plank mass, the Plank energy, and the Plank time hae neer been really understood or diretly linked to a onsistent uantum theory. Albert Einstein is, of ourse, also one of the founders of uantum theory, in partiular with his insight on the photoeletri e et. Howeer, he was ery skeptial on muh of what followed in uantum physis, espeially in relation to strange unertainty phenomena. It was not neessarily the ase that he did not beliee in suh models, but he felt that the theories did not apture the fuliture of reality. Einstein is famous for his statement, see [6] God Does Not Throw Die with the Unierse From the deriations and logial reasoning that we are working with here, it looks like Einstein was right, een though many hae maintained that he was wrong on this point. We will use onepts from speial relatiity theory, Max Plank, and Heisenberg, and we find that the unifiation of these three Giants of Physis seems to lead to a breakdown of unertainty at the Plank sale. Further, this an be used to derie well-known formulas for the rest-mass of partiles. Does Unertainty Collapse at the Plank Sale? Heisenberg s unertainty priniple is gien by p x ~ Lloyd Motz, while working at the Rutherford Laboratory in 96, [7, 8, 9] suggested that there was probably a ery fundamentaartile with a mass eual to the Plank mass; today it is known as the Plank mass partile. The momentum of a Plank mass partile must also follow the relatiisti mass rule and be espenhaug@ma.om. Thanks to Vitoria Teres for heing me edit this manusript and thanks to Douglas G. Danforth for finding an important typo.

2 p p = mp =0 () Haug [0,, ] has shown that the maximum eloity any partile with rest-mass likely an attain is r max = where is the redued Compton waelength of the partile and is the Plank length. Only for the Plank mass partile does =, and only for the Plank mass partile do we hae maximum eloity of zero: s max = l p () =0 (3) This remarkably also means the Plank mass partile always has zero momentum. Howeer it is normally assumed the Plank mass momentum is p p = m p. This we do not disagree with, but we think the orret interpretation is that there is a rest-mass momentum eual to m when a partile stands still, and that the Plank mass partile always stand still when it exist, and only hae rest-mass momentum. It also only hae rest-mass energy and no kineti energy. No partile with rest mass an anyway moe at the speed of light, as one mistakenly ould suspet is indiated by p p = m p. A rest mass partile moing at the speed of light would mean infinite kineti energy, whih is impossible. The key is that the Plank mass partile must stand still we will laim that the Plank mass partile only last for one Plank seond. We also get a hint about the lifetime of a Plank partile from the Plank aeleration, a p = m/s. The Plank aeleration is assumed to be the maximum possible aeleration by seerahysiists; see [4, 5], for example. The eloity of a partile that undergoes Plank aeleration will atually reah the speed of light within one Plank seond: a pt p = =. Howeer, we know that nothing with rest-mass an trael at the speed of light, so no normal partile an undergo Plank aeleration if the shortest possible aeleration time interal is the Plank seond. The solution is simple. The Plank aeleration is an internal aeleration inside the Plank partile that within one Plank seond turns the Plank mass partile into pure energy. This also explains why the Plank momentum is so speial, namely always m p, unlike for any other partiles, whih an take a wide range of eloities and therefore a wide range of momentums. All other known partiles hae maximum eloities extremely lose to that of the speed of light, but these far exeed what an be ahieed at the Large Hadron Collider today, making empirial work di ult. This maximum eloity and iew that the Plank mass partile must stand absolutely still mean that the Lorentz symmetry must be broken at the Plank sale, something that also is predited by seeral uantum graity theories [3]. All other partiles an show a wide range of momentum, beause they an hae signifiant ariations in their eloity and therefore, they also hae unertainty in their momentum. As long as we assume that the Plank partile has a known momentum of m p, thenwefindthat p p x ~ m p x ~ ~ m p x ~ m p x We know that the mass of any elementary partile an be written in the form m = ~ and sine the redued Compton waelength of the Plank mass partile is = then we must hae (4) (5) m p = ~ Inputting formula 6 into formula 4 and soling with respet to x we get (6) ~ ~ x x (7)

3 3 This gies two important insights. Many (perhaps een most) physiists are of the opinion that the Plank length is the minimum distane we an measure, and from the analysis aboe, this would mean that we annot hae an unertainty smaller than the Plank length. This has seeral important impliations; for example, it means the speed limit of just <annotholdfor anything with rest-mass, as the highest relatiisti mass must now be the Plank mass; see [6] for detailed disussion on this point. Instead, we get the exat speed limit gien by Haug s maximum speed limit for anything with rest-mass. This speed limit is, for any known obsered partile, ery lose to the speed of light, exept for the speial ase of the Plank partile where it surprisingly is zero. Beause there is no unertainty in the Plank momentum due to there being no unertainty in the eloity of the Plank partile, we will laim there is no longer an unertainty in its position. This orresponds well with the points aboe, where we hae shown that the Plank partile must stand absolutely still, but only for one Plank seond. So, its position is simply the shortest possible distane we an measure, een hypothetially, whih is the Plank length. That is to say, only for the Plank mass partile an we know the momentum and the position at the same time and, in fat, we only need one of them and then we an dedue the other one. This means that at the Plank sale Heisenberg s unertainty priniple breaks down and beomes the ertainty priniple as reently showed in uantum mehanial alulation in [7]. This seems to also indiate that the minimum unertainty in position for all other partiles seems to be limited by the Plank length x minimum = (8) To reiterate, we laim all unertainty will likely disappear at the Plank sale, but this world is ertain for only one Plank seond. The Plank momentum is linked to the speed of light and no mass an moe at the speed of light. Howeer, a Plank partile an and must dissole into pure energy within one Plank seond. Still, for all non-plank partiles we hae p x ~ m x ~ (9) If we now assume we know the rest-mass of the partile in uestion, an eletron, for example, then the unertainty in momentum must ome from the unertainty in the eloity. This means we hae m ~ x (0) Now if we set x to what we know is the minimum possible unertainty in it, namely the Plank length, and we know the rest-mass of the partile, then it is een more lear that what is ausing the unertainty in the momentum is the unertainty in the eloity: m ~ ~ ~ () Soled with respet to this gies

4 4 + apple l p apple apple apple l p l p + l p () + This is basially the same deriation as gien by Haug in a working paper preiously [6]. What is new in this paper is that we are showing how Heisenberg s unertainty priniple likely leads to a breakdown of unertainty at the Plank sale. 3 Future Researh: Bell s Theorem Seeral researhers hae pointed out that by impliitly assuming alossible Bell measurements our simultaneously, then alroofs of Bell s Theorem [8] iolate Heisenberg s unertainty priniple [9]. We wonder what it ould mean for the interpretation of Bell s Theorem if the Heisenberg unertainly priniple breaks down at the Plank sale and we then go from unertainty to ertainty (determinism). If Heisenberg?s unertainty priniple breaks down at the Plank sale, ould this open up the possibility of hidden ariables as suggested by Einstein, Podolsk and Rosen in 935? See [0]. Cloer, as ited aboe, laims that Only time-independent lassial loal hidden ariable theories are forbidden by iolations of the original Bell ineualities; time-dependent uantum loal hidden ariable theories an satisfy this new bound and agree with experiment. Further Cloer interestingly states [] By impliitly assuming that all measurements our simultaneously, Bell s Theorem only applied to loal theories that iolated Heisenberg s Unertainty Priniple. We are urrently studying more about Bell s theorem and hidden ariable ideas. Although it is too early to draw any onlusions at this point, we enourage others to see if the extended ersion of Heisenberg s unertainty priniple presented in this paper an proide further insights here. 4 Conlusion In this paper, we hae shown that Heisenberg s unertainty priniple likely ollapses to a ertainty priniple at the Plank sale. This indiates that Einstein was right when he laimed God Does Not Throw Die. The Plank mass partile is uniue and is the only partile that has a known momentum eual to p = m. There is likely no room for unertainty in the eloity of a Plank mass partile, simply beause it is at absolute rest, een as obsered aross di erent referene frames. This hypothesis is supported by uantum mehanial deriations and is ready for ealuation by other researhers. Referenes [] M. Plank. Naturlishe Masseinheiten. Der Königlih Preussishen Akademie Der Wissenshaften, p. 479., 899. [] M. Plank. Vorlesungen über die Theorie der Wärmestrahlung. Leipzig: J.A. Barth, p. 63, see also the English translation The Theory of Radiation (959) Doer, 906. [3] I. Newton. Philosophiae Naturalis Prinipia Mathematia. London,686. [4] A. Einstein. On the eletrodynamis of moing bodies. Annalen der Physik, English translation by George Barker Je ery 93, (7),905.

5 5 [5] W. Heisenberg. Über den anshaulihen inhalt der uantentheoretishen kinematik und mehanik. Zeitshrift für Physik, (43):7 98,97. [6] W. Hermanns. Einstein and the Poet. Branden Books, 03. [7] L. Motz. Gauge inariane and the struture of harged partiles. Il Nuoo Cimento, 6(4), 96. [8] L. Motz. A graitational theory of the mu meson and leptons in general. Rutherford Obseratory, Columbia Uniersity, 966. [9] L. Motz. The uantization of mass. Rutherford Obseratory, Columbia Uniersity, 97. [0] E. G. Haug. The graitational onstant and the Plank units: A simplifiation of the uantum realm. Physis Essays Vol 9, No 4, 06. [] E. G. Haug. The ultimate limits of the relatiisti roket euation: The Plank photon roket. Ata Astronautia, 36,07. [] E. G. Haug. Can the Plank length be found independent of big G? Applied Physis Researh, 9(6),07. [3] F. Kislat and H. Krawzynski. Plank-sale onstraints on anisotropi lorentz and pt inariane iolations from optiaolarization measurements. Physial Reiew D, 95, 07. [4] G. Sarpetta. Letter Nuoo Cimento,, 5, 984. [5] D. F. Falla and P. T. Landsberg. Blak holes and limits on some physial uantities. European Journal of Physis, 5,994. [6] E. G. Haug. Does Heisenberg s unertainty priniple predit a maximum eloity for anything with rest-mass below the speed of light? Working Paper Norwegian Uniersity of Life Sienes, 08. [7] E. G. Haug. Reisiting the deriation of Heisenberg s unertainty priniple: The ollapse of unertainty at the Plank sale. ixra:805.09, 08. [8] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physis,,964. [9] M. Cloer. Bell s theorem: A ritiue. arxi:uant-ph/05006, 005. [0] A. Einstein, B. Podolsky, and N. Rosen. Can uantum mehanial desription of physial reality be onsidered omplete? Phys. Re., (47): ,935. [] M. Cloer. Bell s theorem: A new deriation that preseres Heisenberg and loality. arxi:uant-ph/ , 004.