POINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS

Transcription

1 POINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS arxv:hep-th/0055v 7 Oct 00 N.BELALOUI, AND H.BENNACER LPMPS, Département de Physque, Faculté des Scences, Unversté Mentour Constantne, Constantne, Algera. October 6, 00 Abstract In ths paper, we paraquantze the spnnng strng theory n the Neuveu-Shwarz model. Both the center of mass varables and the exctaton modes of the strng verfy paracommutaton relatons. Except the p,p ν commutator, the two other commutators of Poncaré algebra are satsfed. Wth the sole use of trlnear relatons we fnd exstence possbltes of spnnng strngs at space-tme dmensons other than D = 0. Ths work was supported by the Algeran Mnstry of Educaton and Research under contract No.D50//000. E-mal : n.belalou@wssal.dz

2 Introducton In the parabosonc strng case, a crtcal study of the Poncaré algebra was done and space-tme crtcal dmensons D as functons of the order of the paraquantzaton Q were obtaned. Ths work consst n dong an extenton of all these questons to the paraspnnng strng case n the Neuveu-Shwarz model. To set the notatons, we begn wth a bref summary of some famlar results n spnnng strng theory 3. wth and The acton s postulated as : S = π ψ = ρ 0 = ψ 0 ψ dσdτ a X a X īψ ρ a a ψ 0 ı ı 0 The solutons are : ; ψ = ψ T ρ 0 ; ρ = X σ, τ = x p τ n 0 0 ı ı 0 ; { ρ a, ρ b} = η ab n n 0exp ınτcos nσ where x and p are respectvely the center of mass coordnates and the total energy momentum of the strng. In the Neuveu-Shwartz model : ψ 0 σ, τ = b r exp ır τ σ r Z ψ σ, τ = 3 b r exp ır τ σ r Z The total angular momentum s gven by : M ν = M ν 0 x K ν where M ν 0 x s the bosonc part gven by : M ν 0 x = x p ν x ν p ı n n ν n ν n n and K ν = ı r= n= b r, b ν r bν r, b r 5 6

3 In the same way as n the parabosonc case, a frst study of the paraquantum Poncaré algebra was done by F.Ardalan and F.Mansour 5. These authors paraquantze the exctaton modes of the strng and mpose to the center of mass varables to satsfy the ordnary quantum commutaton relatons. Ths s done by the choce of a specfc drecton n the paraspace of the Green components 6, 7, 8, whch requres relatve paracommutaton relatons between the center of mass varables and the exctaton modes of the strng. In order that the theory s Poncaré nvarant, the space-tme dmenson D and the order of the paraquantzaton Q are related by the relaton D = 8. Here, we nvestgate the paraquantzaton of the spnnng strng theory n the Neuveu-shwarz model wthout the Ardalan and Q Mansour hypothess on the center of mass varables 5. We construct the paraspnnng strng formalsm and we dscuss the three commutators of the Poncaré algebra. We fnd that, except the p, p ν commutator, the other results are the same as n the Ardalan and Mansour work 5. In partcular, the relaton between the space-tme dmensons D and the order of the paraquantzaton Q s agan D = 8. Q Paraquantum formalsm of spnnng strng. Covarant gauge The paraquantzaton of the theory s carred out by renterpretng the classcal dynamcal varables n, p, x and b r as operators satsfyng the so called trlnear commutaton relatons : x, p ν, A x, p ν, p ρ n, ν m, ρ l n, ν m, B b r, b ν s, b ρ q b r, b ν s, C = ıg ν A 7-a = ı g ν p ρ g ρ p ν 7-b = g ν nδ nm,0 ρ l gρ nδ nl,0 ν m 7-c = ng ν δ nm,0 B 7-d = g ν δ rs,0 b ρ q g ρ δ rq,0 b ν s 7-e = g ν δ rs,0 C 7-f and all the other commutators are null. Here l, n Z and r, s, q C represent the followng operators : A= ρ n, x ρ or b ρ r B= p ρ, x ρ or b ρ r C= p ρ, x ρ or ρ n Z and A, B, and 3

4 . Transverse gauge In ths gauge, the paraquantum operators x, p, x, p, n and b r verfy the trlnear relatons : b r, b j s, b k q = δ j δ rs b k q δ k δ rq b j s 8-a n, j m, k l = δ j nδ nm,0 k l nδ k δ nl,0 j m 8-b x, p j, pk = δ j pk δ k p j 8-c n, j m, D = δ j nδ nm D 8-d b r, b j s, E = δ j δ rs E 8-e x, p j, F = ıδ j F 8-f x, p, G = ıg 8-g and all the others commutators are null. Here D, E, F, and G represent the followng operators : D = x, p, x k, p k or b k q. E = x, p, x k, p k or k n. F = x, p, x k, k n or b k r G = x, x k, p k k n or b k r 3 Paraquantum Poncaré Algebra In vew of the form of the relatons 7, the quantum form of the Poncaré algebra generators M ν,5,6 presents an order ambguty problem so that there must bee rewrten on the bass of an adequate symetrsaton whch takes the form where now M ν 0 x = l ν E ν wth: l ν = x, p ν xν, p 0 and and E ν = ı n= n n, ν n ν n, n

5 K ν = ı r= b r, b ν r b ν r, b r Wth a drect applcaton of the trlnear relatons 7, let us perform the second and the thrd commutators of the algebra : p, M νρ = p, M νρ 0 p, K νρ By the use of 7-a,b, t s easy to see that and p, K νρ = 0 p, M νρ 0 = ı g ρ p ν g ν p ρ 5 so that p, M νρ = ı g ρ p ν g ν p ρ Now, for the thrd commutator M ν, M β = M ν 0 x, M β 0 x M ν 0 x, K β K ν, M β 0 x K ν, K β 7 the frst term s gven by : M ν 0, Mρσ 0 = ıgνρ M σ 0 ıg σ M νρ 0 ıg νσ M ρ 0 ıg ρ M νσ 0 8 It s agan clear from 7 that : M ν 0 x, K β = K ν, M β 0 x = 0 9 Let us now consder the commutator : K ν, K β = 6 r,s { b r, b ν r, b s, b β s, ν, β ν,, β ν, β} 5

6 The frst term gves : A = b r, b ν r, b s, b β s r,s = { b r b ν r, b s, s bβ b r, b s, s bβ b ν r b ν r b r, b s, s bβ b ν r, b s, } bβ s b r r,s Wth the use of 7-e, becomes : A = r= g ν b r, br β g β b r, b ν r g νβ b r, b r g b β r, b ν r In the same way, one can perform the other terms of 0 and obtan : K ν, K β = ı r= { ı {g ν b β r, b r b r, br β g β b r, r bν b ν r, b r g νβ b r, b r b b r, b r }} g ν r, b β r b β r, b ν r then K ν, K β = ı g ν K β g β K ν g νβ K g K νβ When combned wth 8 and 9, 7 gves : M ν, M β = ı g ν M β g β M ν g νβ M g M νβ 5 Now, for the frst commutator p, p ν of the algebra, one can only wrte p, p ν, p σ = 0 and not p, p ν = 0! Space-tme crtcal dmensons As n the ordnary case, one can obtan the space-tme crtcal dmenson by performng, n the transverse gauge, the commutator M, M j. Let us ntroduce, n the transverse gauge, the generators M n the form : where M = M 0 x K 8 M 0 x = l E 9 6

7 wth and l = E = ı n = x, n= l= p n 0 n, p n l, l x, p 30 n n r= n, p 3 r n b n r, b a r δ n,0 3 K = ı r= b r, G p r G r b r, p 33 wth G r = n= n, b r 3 Let us now perform ths commutator : M, M j = M0, Mj 0 M 0, K j K, M j 0 K, K j 35 Projectng the equaton M, M j = 0 on the physcal states k m 0 and b k s 0, one can wrte : 0 l m M, M j k m 0 0 b l s M, M j b k s 0 = 0 36 We begn by computng the frst commutator of 35. Frst, notce that one can wrte : 0 b l s M 0, M j 0 b k s For the other mean value on the physcal states k m 0, except the term whch s done by 9 : C = 0 l n m n, p n,n = p δl δ jk n n j n n, Q D m m ma 8 p k m 0 all the other terms are analog to parabosonc case. Then, we obtan : M 0, M j 0 = p n= n, j n j n, n n Q D 8 n a n n

8 For the second and the thrd terms of 35, one can wrte: M 0, K j K, M j 0 = l E, K j j Before computng the mean value of the commutator l, K j, we frst transform t as follows : we notce that by the use of the relatons 8-a,b,d,e, one can verfy that : n, br = so that : r n n, G r = r n 0, b j r, b nr 0 G rn G p r = r b j r, G p r p, bj r rg r = 0 On the other hand, and agan by the use of 8-f,g, one can verfy that : x, G r x, p, H = ıb r 3 = ı p H where the operator H = x, x k, p k k n or b k r. Then we obtan : l, K j = r= p, bj r b r, p p, b r p, bj r 0 p r= b j r, p G r G r b j r, p 5 It s easy to see that : 0 l m l, K j k m 0 = 0 6 Now, for the other mean value, one can prove that : 0 b l s l, K j b k s 0 = p Notce that one can agan wrte : 0 b l s r b j r, br b r, br j b k s 0 r= 0 b l s l, K j K, l j b k s 0 = 0 b l s p δlj p p k p δjk p p l 7 l, K j b k s 0 8 8

9 In the same way, for the commutator E, K j, we compute the mean value 0 l m E, K j k m 0 0 bl s E, K j b k s 0 9 To do ths, we need to perform the followng non null expressons, whch gve the results H = r= n= 0 l m p, n n p, bj r G r k m = 3m3 p δ l δ jk H = 0 b l n s n, p n, G r r,n = δ jk δ l s p s H 3 = = r,n 0 b l s n p δlj δ k n n, p s b j r, b k p s 0 p δjk p l p 5 b j n, G p r b k s 0 It follows that : 0 l m E, K j k m 0 0 bl s E, K j b k s 0 = { 3 p m3 s δ l δ jk δ lj δ k δjk δ s l } s δ jk p l p 5 Then we can wrte the result : 0 l m E, K j K, E j k m 0 0 bl s E, K j K, E j b k s { 0 = 3 p m3 s 8 s δ l δ jk δ lj δ k } δ jk p l p δ k p l p j 55 Lastly, one can perform the mean value of the last commutator of 35 and obtan Appendx A : 0 b l s K, K j b k s 0 0 l m K, K j k m 0 = δ l p δ jk δ lj δ k s s Q D s 8a m 3 δ l p p j p k δ lj p p k δ jk p l p δ k p l p j

10 Now by regroupng all these results, n terms of operators, one obtan : M, M j = p b p r, bj r r= n= n, j n j n, n Q D 8 b j r, b r Q D r 8 n an n n a We are thus led to conclude that, n order to have M, M j = 0, one must have : 57 { D = 8 Q a = 58 5 Concluson Ths work consst n paraquantzng the spnnng strng by mposng paracommutaton relatons to the classcal varables X σ, τ, P σ, τ and ψ A σ, τ. Unlke n Ardalan and Mansour work 5, ths requres that both the center of mass varables and the exctaton modes of the strng verfy paracommutaton relatons.to satsfy ths, one must have X σ, τ,p ν σ, τ = ıg ν δσ σ, But the Ardalan and Mansour hypothess, characterzed by the anzatz x β = x δ β and p β = p δ β, leads to the result X σ, τ, P ν σ, τ = ıg ν δσ σ δ, whch s dfferent from the latter. Wth the only use of the trlnear relatons, one can prove that : p, M νρ = ıg ν p ρ ıg ρ p ν M ν, M ρσ = ıg νρ M σ ıg σ M νρ ıg νσ M ρ ıg ρ M νσ In the transverse gauge, n order to have M, M j = 0, the space-tme crtcal dmenson D s calculated wth the only use of the trlnear relatons. Lke n Ardalan and Mansour work 5, D s agan gven as functon of the paraquantzaton order through the relaton D = 8. Thus, one can have paraspnnng strngs wth crtcal dmensons Q D = 0, 6,, 3 respectvely n orders Q =,,, 8. Ths concde wth the dmensons n whch fractonal superstrngs can be formulated 0,. Some questons arses; n order to satsfy p, p ν = 0, one adopt the Ardalan and Mansour anzatz, then the queston s can one speak about the paraquantum formalsm on the classcal varables X σ, τ, P σ, τ and ψ A σ, τ? On the other hand, f we paraquantze these classcal varables by mposng them to satsfy paracommutaton relatons, what we may only wrte s p, p ν, p σ = 0. Then, what about the space-tme propertes? 0

11 Acknowledgments We are pleased to thank professor Keth R. Denes from the Department of Physcs of the Unversty of Arzona for pontng out some works on fractonal superstrngs, and for hs advses. Appendx Let us set 0 l n K K j k m 0 0 bl s K K j b k s 0 = = D A- computng D = 0 b l s r,r = p δl δ jk b r, p s G p r b k s 0 s p δl p j p k A- G r b j r, D = r,r 0 b l s = p δl δ jk D 3 = b r, Q r,r 0 l m G r = m3 p δlj δ k p D G r G r b j r, p s a b k s 0 b r, b j p r, G p r k m 0 A-3 A- D = 0 b l s G r p, b r r,r = p δjk δ l s G r b j r, b k p s 0 s p δjk p l p A-5 t s then straghtforward to obtan the relaton 56

12 References N. Belalou, H. Bennacer, Poncaré algebra and space-tme crtcal dmensons for parabosonc strngs, arxv:hep-th/009. P. Ramond, Phys. Rev. D3, A. Neuveu, J.H. Schwarz, Nucl. Phys. B3, A. Neuveu, J.H. Schwarz and C.B. Thorn, Nucl.Phys. B3, F. Ardalan and F. Mansour, Phys. Rev. D9, E.P. Wgner, Phys. Rev. 77, H.S. Green, Phys. Rev. 90, Y. Ohnuk, S. Kamefuch, Quantum Feld Theory and parastatstcs, Sprnger-Verlag, See expresson 66 of reference. 0 Keth R. Denes, Nucl. Phys. B3 : 03-6, 99. Phlp C. Argyres, Keth R. Denes, Phys. Rev. Lett. 7 : 89-8, 993.