Generalized Uncertainty Principle, Extra-dimensions and Holography

Transcription

1 470 Brziin Journ of Physics, vo. 35, no. 2B, June, 2005 Generized Uncertinty Principe, Extr-dimensions nd Hoogrphy Fbio Scrdigi CENTRA - Deprtmento de Fisic, Instituto Superior Tecnico, Av. Rovisco Pis 1, isbo, Portug nd Roberto Csdio Diprtimento di Fisic, Università di Boogn nd I.N.F.N., Sezione di Boogn, vi Irnerio 46, Boogn, Ity Received on 23 December, 2004 We consider Uncertinty Principes which tke into ccount the roe of grvity nd the possibe existence of extr spti dimensions. Expicit expressions for such Generized Uncertinty Principes in 4 + n dimensions re given nd their hoogrphic properties investigted. In prticur, we show tht the predicted number of degrees of freedom encosed in given spti voume mtches the hoogrphic counting ony for one of the vibe generiztions nd without extr dimensions. 1 Introduction During the st yers mny efforts hve been devoted to crifying the roe pyed by the existence of extr spti dimensions in the theory of grvity [1, 2]. One of the most interesting predictions wn from the theory is tht there shoud be mesurbe devitions from the 1/r 2 w of Newtonin grvity t short nd perhps so t rge distnces. Such new ws of grvity woud impy modifictions of those Generized Uncertinty Principes GUP s designed to ccount for grvittion effects in the mesure of positions nd energies. On the other hnd, the hoogrphic principe is cimed to ppy to of the grvittion systems. The existence of GUP s stisfying the hoogrphy in four dimensions one of the min exmpes is due to Ng nd Vn Dm [3] ed us to expore the hoogrphic properties of the GUP s extended to the brne-word scenrios [4]. The resuts, t est for the exmpes we considered, re quite surprising. The expected hoogrphic scing indeed seems to hod ony in four dimensions, nd ony for the Ng nd vn Dm s GUP. When extr spti dimensions re dmitted, the hoogrphy is destroyed. This fct ows two different interprettions: either the hoogrphic principe is not univers nd does not ppy when extr dimensions re present; or, on the contrry, we tke seriousy the hoogrphic cim in ny number of dimensions, nd our resuts re therefore evidence ginst the existence of extr dimensions. The four-dimension Newton constnt is denoted by G N throughout the pper. 2 iner GUP in four dimensions from micro bck hoes In this Section we derive GUP vi micro bck hoe gednken experiment, foowing cosey the content of Ref. [5]. When we mesure position with precision of order x, we expect quntum fuctutions of the metric fied round the mesured position with energy mpitude E 2 x. 1 The Schwrzschid rdius ssocited with the energy E, R S = 2 G N E c 4, 2 fs we inside the interv x for prctic cses. However, if we wnted to improve the precision indefinitey, the fuctution E woud grow up nd the corresponding R S woud become rger nd rger, unti it reches the sme size s x. As it is we known, the critic ength is the Pnck ength, 1/2 GN x = R S x = p, 3 nd the ssocited energy is the Pnck energy ɛ p = 1 5 1/ p 2 G N If we tried to further decrese x, we shoud concentrte in tht region n energy greter thn the Pnck energy, nd this woud enrge further the Schwrzschid rdius R S, hiding c 3

2 Fbio Scrdigi nd Roberto Csdio 471 more nd more detis of the region beyond the event horizon of the micro hoe. The sitution cn be summrized by the inequities x 2 E for E < ɛ p 2 G N E c 4 for E > ɛ p. which, if combined inery, yied x 5 2 E + 2 G N E c 4. 6 This is generiztion of the uncertinty principe to cses in which grvity is importnt, i.e. to energies of the order of ɛ p. We note tht the minimum vue of x is reched for E min = ɛ p nd is given by x min = 2 p. 2.1 Hoogrphic properties In this section, we investigte the hoogrphic properties of the GUP proposed bove. We sh estimte the number of degrees of freedom nv contined in spti voume cube or hypercube of size. The hoogrphic principe cims tht nv sces s the re of the hyper-surfce encosing the given voume, tht is / p 2+n in 4 + n dimensions. For the four-dimension GUP considered in the previous section, Eq. 6, we find tht this scing does not occur. In fct, x min p nd cube of side contins number of degrees of freedom equ to 3 nv. 7 p We then concude tht this GUP, obtined by inery combining the quntum mechnic expression with grvittion bounds, does not impy the hoogrphic counting of degrees of freedom. 3 Ng nd Vn Dm GUP in four dimensions An interesting GUP tht stisfies the hoogrphic principe in four dimensions hs been proposed by Ng nd vn Dm [3], bsed on Wigner inequities bout distnce mesurements with cocks nd ight signs [6]. Suppose we wish to mesure distnce. Our mesuring device is composed of cock, photon detector nd photon gun. A mirror is pced t the distnce which we wnt to mesure nd m is the mss of the system cock + photon detector + photon gun. We c detector the whoe system nd et be its size. Obviousy, we suppose > r g 2 G N m c 2 = R S m, 8 which mens tht we re not using bck hoe s cock. Be x 1 the uncertinty in the position of the detector, then the uncertinty in the detector s veocity is v = 2 m x 1. 9 After the time T = 2 /c tken by ight to trve ong the cosed pth detector mirror detector, the uncertinty in the detector s position i.e. the uncertinty in the ctu ength of the segment hs become x tot = x 1 + T v = x 1 + T 2 m x We cn minimize x tot by suitby choosing x 1, 1/2 x tot T = 0 x 1 min =. 11 x 1 2 m Hence T x tot min = x 1 min + 2 m x 1 min 1/2 T = m Since T = 2 /c, we hve x tot min = 2 1/2 δ QM. 13 m c This is purey quntum mechnic resut obtined for the first time by Wigner in 1957 [6]. From Eq. 13, it seems tht we cn reduce the error x tot min s much s we wnt by choosing m very rge, since x tot min 0 for m. But, obviousy, here grvity enters the gme. In fct, Ng nd vn Dm hve so considered further source of error, grvittion error, besides the quntum mechnic one redy dessed. Suppose the cock hs spheric symmetry, with > r g. Then the error due to curvture cn be computed from the Schwrzschid metric surrounding the cock. The optic pth from r 0 > r g to generic point r > r 0 is given by see, for exmpe, Ref. [7] c t = r r 0 dρ 1 rg ρ = r r 0 + r g og r r g r 0 r g, 14 nd differs from the true spti ength r r 0. If we put = r 0, = r, the grvittion error on the mesure of is thus δ C = r g og r g r g r g og, 15 where the st estimte hods for > r g. If we mesure distnce 2, then the error due to curvture is δ C r g og 2 G Nm c Thus, ccording to Ng nd vn Dm the tot error is 1/2 δ tot = δ QM + δ C = 2 + G N m m c c 2. 17

3 472 Brziin Journ of Physics, vo. 35, no. 2B, June, 2005 This error cn be minimized gin by choosing suitbe vue for the mss of the cock, 1/2 tot m m = 0 m min = c 18 G 2 N nd, inserting m min in Eq. 17, we then hve δ tot min = 3 2 p 1/3. 19 The gob uncertinty on contins therefore term proportion to 1/ Hoogrphic properties We now see immeditey the beuty of the Ng nd vn Dm GUP: it obeys the hoogrphic scing. In fct in cube of size the number of degrees of freedom is given by 3 3 nv = = = 2 δ tot min 2 p 1/3 2, 20 p s required by the hoogrphic principe. 4 Modes with n extr dimensions We sh now generize the procedure outined in previous section to spce-time with 4 + n dimensions, where n is the number of spce-ike extr dimensions [4]. The first probem we shoud dess is how to rete the grvittion constnt G N in four dimensions with the one in 4 + n, henceforth denoted by G 4+n. This of course depends on the mode of spce-time with extr dimensions tht we consider. Modes recenty ppered in the iterture mosty beong to two scenrios: the Arkni-Hmed Dimopouos Dvi ADD mode [1], where the extr dimensions re compct nd of size ; the Rnd Sunum RS mode [2], where the extr dimensions hve n infinite extension but re wrped by non-vnishing cosmoogic constnt. A feture shred by the origin formutions of both scenrios is tht ony grvity propgtes ong the n extr dimensions, whie Stndrd Mode fieds re confined on fourdimension sub-mnifod usuy referred to s the brneword. In the ADD cse the ink between G N nd G 4+n cn be fixed by compring the grvittion ction in four dimensions with the one in 4+n dimensions. The spce-time topoogy in such modes is M = M 4 R n, where M 4 is the usu four-dimension spce-time nd R n represents the extr dimensions of finite size. The spce-time brne hs no tension nd therefore the ction S 4+n cn be written s c 3 S 4+n = d 4+n x g R 16 π G 4+n M 4 R n c 3 d 4 x g n R, π G 4+n M 4 where R, g re the projections on M 4 of R nd g. Here n is the voume of the extr dimensions nd we omitted unimportnt numeric fctors. On compring the bove expression with the purey four-dimension ction we obtin c 3 S 4 = d 4 x g 16 π G R, 22 N M 4 G 4+n G N n. 23 The RS modes re more compicted. It cn be shown [2] tht for n = 1 extr dimension we hve G 4+n = σ 1 G N, where σ is the brne tension with dimensions of ength 1 in suitbe units. The grvittion force between two point-ike msses m nd M on the brne is now given by F = G N m M r e σr σ 2 r 2, 24 where the correction to Newton w comes from summing over the extr dimension grviton modes in the grviton propgtor [2]. However, since Eq. 24 is obtined by perturbtive ccutions, not immeditey ppicbe to nonperturbtive structure such s bck hoe, we sh consider ony the ADD scenrio in this pper. To be more precise, from tbe-top tests of the grvittion force one finds tht n 2 in ADD [1, 8]. On the other hnd, bck hoes with mss M σ 1 re ikey to behve s pure five-dimension in RS [9], therefore our resuts for n = 1 shoud ppy to such cse. 5 Ng nd Vn Dm GUP in 4 + n dimensions Ng nd vn Dm s derivtion cn be generized to the cse with n extr dimensions. The Wigner retion 13 for the quntum mechnic error is not modified by the presence of extr dimensions nd we just need to estimte the error δ C due to curvture. We ought not to consider micro bck hoes creted by the fuctutions E in energy, s in Section 2, but we hve rther to de with more or ess mcroscopic cocks nd distnces. This impies tht we hve to distinguish four different cses: 1. 0 < < r g < < ; 2. 0 < r 4+n < < < ; 3. 0 < r 4+n < < < ; 4. 0 < r 4+n < < < ;

4 Fbio Scrdigi nd Roberto Csdio 473 where r 4+n is the Schwrzschid rdius of the detector in 4 + n dimensions, nd of course r g = r 4. The curvture error wi be estimted s before by computing the optic pth from r 0 to r. Of course, we wi use metric which depends on the retive size of with respect to nd, tht is the usu four-dimension Schwrzschid metric in the region r >, nd the 4 + n dimension Schwrzschid soution in the region r < where the extr dimensions py n ctu roe. In cses 1. nd 2. the ength of the optic pth from to cn be obtined using just the four-dimension Schwrzschid soution nd the resut is given by Eq. 19. In cses 3. nd 4. we insted hve to use the Schwrzschid soution in 4 + n dimensions [11], ds 2 = + 1 C r n+1 c 2 dt 2 1 C 1 r n r 2 dω 2, 25 t est for prt of the optic pth. In the bove, C = 16 π G 4+n m n + 2 A c 2, 26 nd A is the re of the unit n + 2-sphere, tht is Besides, we note tht, for n = 0, A = 2 π n+3 2 Γ n C = 2 G N m c 2 = r g, 28 tht is, C coincides in four dimensions with the Schwrzschid rdius of the detector. The Schwrzschid horizon is octed where 1 C/r n+1 = 0, tht is t r = C 1/n+1 r 4+n, or where [ 2 G4+n m r 4+n = αn c 2 [ 8 π αn = n + 2 A ] 1 n+1 ] 1 n is n unimportnt numeric fctor. Since mesurements cn be performed ony on the brne, to the uncertinty x in position we cn sti ssocite n energy given by Eq 1. The corresponding Schwrzschid rdius is now given by Eq. 29 with m = E/c 2 nd the critic ength such tht x = r 4+n is the Pnck ength in 4 + n dimensions, x = [αn] 1+n 2+n = [αn] 1+n 2+n G4+n c n 2 p n For ske of simpicity becuse α0 = 1 nd in ny cse αn 1 we define the Pnck ength in 4 + n dimensions s 4+n 2 p n The energy ssocited with 4+n is nogousy the Pnck energy in 4 + n dimensions, ɛ 4+n = n c n 1 G N n = 1 2 [ 4 ɛ 2 p n ] 1 33 where ɛ p is the Pnck energy in four dimensions given in Eq. 4. In cse 3. we obtin the ength of the optic pth from to by dding the optic pth from to nd tht from to. We must use the soution in 4 + n dimensions for the first prt, nd the four-dimension soution for the second prt of the pth, c t = C 1 + r n+1 + C = + + C + r g 1 + r g r r g r n+1 C r r g. 34 It is not difficut to show tht from r 4+n < which hods in cses 3. nd 4. we cn infer r g < r 4+n <. 35 Now, suppose n+1 C = r n+1 4+n, tht is r 4+n, so tht we re not doing mesures inside bck hoe. Then r g < r 4+n < < nd c t + C r n+1 + r g r = + C 1 n n 1 n + r g og = n n 1 16 π G4+n n n + 2A c 2 m 2 GN + c 2 og m. 36 The error cused by the curvture when < < is therefore iner in m, [ 1 1 δ C = n n 1 16 π GN n n n + 2 A c G N c 2 og ] m K m. 37 We rec tht the curvture error in four dimensions does not contin the size of the cock. On the contrry, this error in 4 + n dimensions depends expicity on the size of the

5 474 Brziin Journ of Physics, vo. 35, no. 2B, June, 2005 cock nd on the size of the extr dimensions. Hence the tot error is given by δ tot = δ QM + δ C = 2 1/2 + K m m c = J m 1/2 + K m, 38 where J = 2 /c 1/2 nd K is defined bove. This error cn be minimized with respect to m, Finy, 2/3 δ tot J m = 0 m min = K δ tot min = 3 2 2/3 K J 2 1/3 [ 1 1 = 3 2 1/3 n n 1 n 8 π 2+n 4+n n + 2 A + 2 p og ] 1/3 40 where we used the definition of J nd K. In cse 4., the optic pth from to cn be obtined by using simpy the Schwrzschid soution in 4+n dimensions. We get c t = 1 + = + C C r n+1 C r n+1 C. 41 Suppose now, s before, tht n+1 C = r n+1 4+n, tht is r 4+n i.e. our cock is not bck hoe. We then hve c t + C r n+1 = + C 1 n n 1 n. 42 If the distnce we re mesuring is, t est, of the size of the cock 2, we cn write c t + C 2 n 1 n 2 n n. 43 The error cused by the curvture is therefore when < < δ C = C 2 n 1 n 2 n n. 44 Here we gin note tht the curvture error in 4 + n dimensions expicity contins the size of the cock. The gob error cn be computed s before δ tot = δ QM + δ C = 2 1/2 + C m c n 2 n 1 2 n n, 45 where C is iner in m. Minimizing δ tot with respect to m cn be done in perfect nogy with the previous ccution. The resut is 2 δ tot min = 3 2 2/3 n 1/ π 2 n n n + 2 A 4+n 1/3 n. 46 We note tht the expression 40 coincides in the imit with Eq. 19 tking 2, whie, in the imit, we recover from Eq. 40 the expression 46 of course, supposing so tht Hoogrphic properties We finy exmine the hoogrphic properties of Eq. 46 for the GUP of Ng nd vn Dm type in 4 + n dimensions. We just consider the expression in Eq. 46 becuse it so represents the imit of Eq. 40 for nd 2. Moreover, for n = 0, Eq. 46 yieds the four-dimension error given in Eq. 19. Since we re just interested in the dependence of nv on nd the bsic constnts, we cn write 4+n δ tot min 1/3 n. 47 We then hve tht the number of degrees of freedom in the voume of size is 3+n 2 n nv = = δ tot min 2 p n 1+ n 3, 48 nd the hoogrphic counting hods in four-dimensions n = 0 but is ost when n > 0. In fct we do not get something s nv = 4+n 2+n, 49 s we woud expect in 4 + n dimensions. Even if we tke the ide cse 4+n we get nv = 4+n 2 1+ n 3, 50 nd the hoogrphic principe does not hod for n > 0. 6 Concuding remrks In the previous Sections, we hve shown tht the hoogrphic principe seems to be stisfied ony by uncertinty retions in the version of Ng nd vn Dm nd for n = 0. Tht is, ony in four dimensions we re be to formute uncertinty principes which predict the sme number of degrees of freedom per spti voume s the hoogrphic counting. This

6 Fbio Scrdigi nd Roberto Csdio 475 coud be evidence for questioning the existence of extr dimensions. Moreover, such n rgument bsed on hoogrphy coud so be used to support the compctifiction of string theory down to four dimensions, given tht there seems to be no firm rgument which forces the ow energy imit of string theory to be four-dimension except for the obvious observtion of our word. In this respect, we shoud so sy tht the cses 3. nd 4. of Section 5 do not hve good probbiity to be reized in nture since, if there re extr spti dimensions, their size must be shorter thn 10 1 mm [8]. Therefore, cses 1. nd 2. of Section 5 re more ikey to survive the test of future experiments. A number of gener remrks re however in order. First of, we cnnot cim tht our ist of possibe GUP s is compete nd other retions might be derived in different contexts which ccommodte for both the hoogrphy nd extr dimensions. Further, one might find hrd to ccept tht quntum mechnics nd gener retivity enter the construction of GUP s on the sme footing, since the former is supposed to be fundment frmework for theories whie the tter cn be just regrded s effective theory of the grvittion interction. We might gree on the point of view tht GUP s must be considered s effective phenomenoogic bounds vid t ow energy beow the Pnck sce rther thn fundment retions. This woud in fct reconcie our resut tht four dimensions re preferred with the fct tht string theory s consistent theory of quntum grvity requires more dimensions through the compctifiction which must occur t ow energy, s we mentioned bove. et us so note tht gener retivity contrry to usu fied theories determines the spce-time incuding the cusity structure, nd the tter is n essenti ingredient in ctu mesurements. It is therefore t est equy hrd to conceive uncertinty retions which negect gener retivity t. This concusion woud become even stronger in the presence of extr dimensions, since the fundment energy sce of grvity is then owered [1, 2] possiby within the scope of present or ner-future experiments nd the grvittion rdius of mtter sources is correspondingy enrged [10]. A fin remrk regrds cses with ess thn four dimensions. Since Einstein grvity does not propgte in such spce-times nd no direct nogue of the Schwrzschid soution exists, one expects quittive difference with respect to the cses tht we hve considered here. For instnce, point-ike source in three dimensions woud generte ft spce-time with conic singurity nd no horizon [12]. Consequenty, one does expect tht the usu Heisenberg uncertinty retions hod with no corrections for grvity. References [1] N. Arkni-Hmed, S. Dimopouos nd G. Dvi, Phys. ett. B 429, ; Phys. Rev. D 59, ; I. Antonidis, N. Arkni-Hmed, S. Dimopouos nd G. Dvi, Phys. ett. B 436, [2]. Rnd nd R. Sunum, Phys. Rev. ett. 83, ; Phys. Rev. ett. 83, [3] Y.J. Ng nd H. vn Dm, Mod. Phys. ett. A 9, ; Mod. Phys. ett. A 10, ; Phys. ett. B 477, ; Y.J. Ng, Phys. Rev. ett. 86, [4] F. Scrdigi nd R. Csdio, Css. Qu. Grv. 20, [5] F. Scrdigi, Phys. ett. B 452, [6] E.P. Wigner, Rev. Mod. Phys. 29, ; H. Secker nd E.P. Wigner, Phys. Rev. 109, [7].D. ndu nd E.M. ifshitz, The cssic theory of fieds Pergmon Press, Oxford, [8] C.D. Hoye et., Phys. Rev. ett. 86, [9] S.B. Giddings, E. Ktz, nd. Rnd, JHEP 0003, ; J. Grrig nd T. Tnk, Phys. Rev. ett. 84, [10] P.C. Argyres, S. Dimopouos, nd J. Mrch-Russe, Phys. ett. B 441, [11] R.C. Myers nd M.J. Perry, Ann. Phys. 172, [12] In three dimensions with negtive cosmoogic constnt one so hs the BTZ bck hoe which forms when two point-ike prtices coide provided certin initi conditions re stisfied. For recent review, see Ref. [13]. [13] D. Birminghm, I. Schs, nd S. Sen, Int. J. Mod. Phys. D 10,