REGULARIZATION IN QUANTUM GAUGE THEORY OF GRAVITATION WITH DE SITTER INNER SYMMETRY

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1 THEORETICAL PHYSICS REGULARIZATION IN QUANTUM GAUGE THEORY OF GRAVITATION WITH DE SITTER INNER SYMMETRY V. CHIRIÞOIU 1, G. ZET 1 Poltehn Unversty Tmºor, Tehnl Physs Deprtment, Romn E-ml: vorel.hrtou@et.upt.ro Gh. Ash Tehnl Unversty, Deprtment of Physs, Iº, Romn E-ml: gzet@phys.tus.ro Reeved Otoer 10, 008 We study the regulrzton of the guge theory of grvtton usng the de Stter group s symmetry of the model. The method of generlzed zet-funton s used to relze the regulrzton nd the guge group s onsdered s n nternl symmetry. An effetve ntegrl of ton s otned nd omprson wth other results s gven. 1. INTRODUCTION Most of the exstng guge theores of grvtton dopt geometrl desrpton of grvty. Nmely, the Ponré group s onsdered prtly s spe-tme prtly s n nternl symmetry group. The lol extenson of ts spe-tme prt eomes then the dffeomorphsm group nd the guge theory s nvrnt under generl oordnte trnsformtons nd lol Lorentz frme rottons. Therefore, ths lol symmetry group s onneted wth the geometry of the spe-tme. It s possle lso to onsder spe-tme symmetres (for exmple Ponré or de Stter n ths pper) s purely nner symmetres [1, ]. Ths leds to desrpton of the guge theory of grvtton whh s n omplete nlogy wth the desrpton of nner symmetres s groups of generlzed rottons n feld spe. In ths pper we onsder the group de Stter (DS) s purely nner symmetry nd develop guge theory of grvtton. We otn n effetve ntegrl of ton whh utomtlly nludes the osmologl onstnt. The method of generlzed zet-funton s used to study the regulrzton of the theory. Pper presented t the Ntonl Conferene of Physs, Septemer 10 13, 008, Buhrest Mãgurele, Romn. Rom. Journ. Phys., Vol. 5, Nos. 9 10, P , Buhrest, 009

2 790 V. Chrþou, G. Zet In Seton we ntrodue the DS guge group nd gve n n expltly form ts equton of strutures. The guge ovrnt dervtve s ntrodued s usully, onsderng the DS group s n nternl symmetry nd ntrodung the orrespondng guge felds. The strength feld s defned s the ommuttor of two guge ovrnt dervtves. The regulrzton of the theory s studed n Seton 3, usng the method of generlzed zet funton. The hnge of the prtton funton wth respet to sle trnsform s lulted for the se of spnor Dr feld ntertng wth the grvttonl feld desred y the guge potentls. Then, mnml feld guge ton, omptle wth regulrzton requrements nd nludng the osmologl onstnt, s determned. Fnlly, some onludng remrks re gven n the Seton. It s emphszed tht n our model there s no ny dret nterrelton etween grvty nd the struture of spe-tme. At quntum level t my oneptully e eser to del wth feld theoretl desrpton of grvtton free of ny geometrl spets.. DE SITTER GAUGE THEORY We onsder guge theory of grvtton hvng de Stter (DS) group s lol symmetry. Let X A, A = 1,,, 10, e ss of DS Le lger wth the orrespondng equtons of struture gven y [1] where C A B AB C X X f X (1) f C AB re the onstnts of struture whose expressons wll e gven elow [see Eq. (3)]. In order to wrte the onstnt of strutures the followng nottons for the ndex A: f C AB n ompt form, we use A () Ths mens tht A n stnd for sngle ndex lke s well s for pr of ndes lke [01], [1], et. The nfntesml genertors X A re nterpreted s: X A = P (energy-momentum opertors) nd X M (ngulr momentum opertors) wth the property M M. The onstnts of struture f C AB hve then the followng expressons:

3 3 Grvtton wth de Stter nner symmetry 791 f f f 0 de de fd d d (3) 1 fd f d d d ef 1 e f e f e f e f f d d d d d e f where s rel prmeter, nd dg s the Mnkowsk metr of the spe-tme. In ft, here we hve deformton of the de-stter Le lger hvng s prmeter. Consderng the ontrton 0 we otn the Ponré Le lger,.e., the group DS ontrts to the Ponré group. Now we ntrodue the lol DS guge trnsformton nd the orrespondng guge ovrnt dervtve, onsderng DS s n nternl group of symmetry. As usully n ny guge theory, we hve B () together wth the followng deomposton of B wth respet to the nfntesml genertors P nd M B B P B M (5) The orrespondng genertors of the DS group n the feld spe hve the expressons: 1 P K M x x (6) where K re the trnslton de Stter genertors nd the spn ngulr momentum opertors. The lst one stsfy ommutton reltons of the sme form s M nd K hve the expresson []: K x x x x (7) We n lso deompose B wth respet to nd s follows: B d B B dx x d B (8) Introdung (8) nto Eq. () nd denotng d d d e B x x B x (9)

4 79 V. Chrþou, G. Zet we otn e B (10) Beuse n our model the oordnte nd DS guge trnsformtons re strtly seprted, we emphsze tht the ntroduton of B, B nd guge felds hs no mpltons on the struture of the underlyng spe-tme, whh s ssumed to e (M, ) endowed wth the Mnkowsk metr. Arevtng e d e B B (11) where must e onsdered nto the Lorentz group representton t ts on, we n wrte the guge ovrnt dervtve (10) under the smple form: d B (1) The dervtve d n e just onsdered s trnslton guge ovrnt dervtve [3]. In order to otn the tensor (feld strength opertor) F of the guge felds, we ntrodue the non-ovrnt deomposton The quntty d d H d (13) H s expressed n terms of e s: d m d m m d d H e e e e e (1) n n n where e m s the mtrx nverse of e,.e. em e m. Usng the defnton of the feld strength opertor n guge theory, we hve: F H d B B d d B d B B B (15) If we ntrodue the tensor then we n rewrte F s where R d hs the expresson T B B H (16) F d T R d (17) d d d de de e d e e e R d B d B B B B B H B (18)

5 5 Grvtton wth de Stter nner symmetry 793 In wht follows we wll use the shorthnd notton R (19) R d As F n (17) hs deomposton wth respet to nd d t ts n generl not only s mtrx ut lso s frst order dfferentl opertor n feld spe. But, f we suppose tht d H B B (0) tht s we tke T 0, then we n wrte Eq. (15) under the form: F R R (1) d d We n verfy tht T nd R d trnsform homogeneous under nfntesml lol DS guge trnsformtons. Then, s onsequene, the hoe T 0 s ndeed guge ovrnt sttement s mpltly ssumed ove. 3. REGULARIZATION In order to nlyze the regulrzton of our DS guge theory, we wll onsder frst the glolly DS nvrnt ton for Dr spnor feld (mtter feld): S D d x m () Then, f we wnt to otn guge (lol) nvrnt ton, we hve to hnge the usul dervtve n () y the guge ovrnt dervtve defned n Eq. (1): S 1 D d xe m (3) nd to use the new volume element 1, prtlly ntegrtng the Dr ton: where we used the hoe T 0. d xe where e e 1 det. Then, n the seond term of (3), we otn the usul form of D 1 () S d xe m

6 79 V. Chrþou, G. Zet 6 The ssumpton tht the nterton of the DS guge felds wth the mtter felds (n our se wth the Dr feld) n e regulrzed, mposes strong ondtons on the lssl guge feld dynms. Nmely, we know tht the hnge of the prtton funton of the whole system under reslng n e sored n ts lssl ton yeldng t most nontrvl sle dependene of the dfferent ouplngs, msses nd wve funton regulrztons. As onsequene, the hnge of one-loop mtter prtton under reslng wll llow us to onstrn the lssl guge feld dynms. The ontruton of the Dr feld to the prtton funton s gven y the followng funtonl ntegrl []: where Z eb DDe (5) S D eb Then, we my perform forml Grssmnn ntegrl n (5) nd otn: Here, M e B 1 ln det M e B Z e B e (6) M ebdd R m (7) s nmed hyperol flututon opertor nd ts expresson n (7) s otn s usully [] y squrng the Dr opertor ntrodued n Eq. (). For the se T 0 we onsder here, the opertor D n Eq. (7) s gven y the formul: D B (8) The guge feld (Le lger vlued) shll only t on the spnor ndes nd the ovrnt dervtve only on vetor ndes. The spnor ontruton to the prtton funton regulrzed t sle s gven then y [5]: 1 0 M e B Z e B e (9) where s M e B to the hyperol flututon opertor M e B nd 0 M e B s the generlzed zet funton of prmeter s ssoted s the dervtve of the generlzed zet funton wth respet to s tken for s = 0. We onsder now new sle nd determne the orrespondng hnge of Z e B. To end ths, we use the very well known property [5]

7 7 Grvtton wth de Stter nner symmetry M e B 0 M e B ln 0 M e B (30) equton Then, we otn: ln 0M eb Z e B Z e B e (31) In order to otn zet funton 0 M e B s x we strt wth het kernel M K sxy (3) together wth n symptotlly s-expnson for the het kernel Ksx y of the form r x y k exp d k s s (33) k0 K sxy s xy In eq. (3) the dfferentl opertor M ts on het kernel K nd the ndex x denotes the dervtve of het kernel wth respet to x. We rememer the ft tht zet funton n four dmensons s gven y 0M eb d xdete tr x nd the oeffent funton form 1 (3) x for the Dr feld n the se T = 0 hs the 1 1 d tr D R 30 R 7 Rd 7 d 1 d Rd R R d R 1 m R 3 m Usng equtons (3) nd (35) we otn the vlue of 0 M e B tr Next, ntrodung the vlue of 0 M e B n ntegrl over. D (35) s n the prtton funton (31) llow us to otn the nomlous terms n the spnorl se nd then to determne mnml feld guge ton omptle wth regulrzton requrements. Regulrzton of ny theory, nludng dynml guge felds, requres tht these ontrutons to the prtton funton lke (31) e expressed s lol DS guge nvrnt polynomls n the felds e nd

8 796 V. Chrþou, G. Zet 8 B. In our se, under the onstrnt T 0, we otn s mnml lssl ton for the guge felds: S guge eb 1 d d xe R R R R R Rd 1 16 G d d (36) Here, G s the grvttonl onstnt nd,, re the ouplng onstnts. We n see tht the DS guge group utomtlly enfores osmologl onstnt whh n our model s equl to 1, where s the deformton prmeter of the de Stter Le lger. We emphsze tht S guge n (36) s n ton for guge felds defned on the Mnkowsk spe-tme (M, ) nd s nvrnt on one hnd under lol DS guge trnsformtons, on the other hnd under glol Ponré symmetry refletng the symmetry of the underlyng spe-tme.. CONCLUSION Bsed on the hypothess tht DS s purely nner symmetry we hve developed guge theory of grvtton wth the onstnt osmologl utomtlly nluded. When the deformton prmeter 0, we otn the Ponré guge theory on the Mnkowsk spe-tme whh do not nlude the osmologl onstnt. The grvttonl nterton s medted y guge felds defned on fxed Mnkowsk spe-tme. Ther dynms hs een determned mposng onssteny requrements wth regulrzton propertes of mtter felds n the grvttonl kgrounds. In our model there s no ny dret nterrelton etween grvty nd the struture of spetme. At quntum level t my oneptully e eser to del wth feld theoretl desrpton of grvtton free of ny geometrl spets. Aknowledgments. The uthors knowledge the support of CNCSIS UEFISCSU Grnt ID-60 of the Mnstry of Eduton nd Reserh of Romn. REFERENCES 1. G. Zet, C. D. Oprºn, S. Beþ, Int. J. Mod. Phys. 15 C, (00).. R. Aldrovnd, R. Beltrn Almed, J. P. Pererr, Clss. Qunt. Grv., (006). 3. C. Wesendnger, Clss. Qunt. Grv. 13, (1996).. D. Bln, A. Love, Introduton to guge feld theory, IOP Pulshng, Brstol, E. Frdkn, Introduton to quntum feld theory, Unversty of Illnos, 005.