EXTENDED BRST SYMMETRIES IN THE GAUGE FIELD THEORY

Transcription

1 Romnin Reports in hysics olume 53 Nos EXTENDED BRST SYETRIES IN TE GUGE FIELD TEORY UREL BBLEN RDU CONSTNTINESCU CREN IONESCU Deprtment of Theoreticl hysics University of Criov 3.I. Cuz Criov Romni E-mil: rconst@centrl.ucv.ro Received September bstrct: The pper presents the structure of the extended phse spce suitble for the BRST cnonicl quntiztion of -reducible guge theory in the frme of BRST symmetry of order three. The corresponding BRST chrges nd the extended hmiltonin re lso constructed. Key words: reducible guge theory BRST quntiztion.. INTRODUCTION The BRST procedure offers one of the most powerful methods for the quntum description of the guge field theories. s hs been lredy stted the unphysicl degrees of freedom tht pper in this cse cn be esily cnceled by the introduction of the ghost type vribles. In the hmiltonin formlism the structure of the ghosts tht must be used minly depends on two fctors: the type of the theory tht is the reltions tht exist mong the constrints of the theory; if ll the constrints re linerly independent the theory is clled irreducible nd if there re some reducibility reltions mong them the theory is reducible 3; the extension of the symmetry tht we should lie to implement. The stndrd BRST symmetry 4 hs been extended nd sp-symmetry 56 or even spn-symmetry 7 hs been proposed. In this pper the exct ghost structure tht could be used in order to describe -reducible guge theory in the frme of the sp3 BRST symmetry will be presented. The pper is orgnized s follows: in Section the extended phse spce is built strting from the structure of the sme spce for the cse of sp3 BRST symmetry in n irreducible theory. The explicit form of the BRST chrges ttched to the three BRST differentils re obtined in Section 3 nd in Section 4 the extended hmiltonin is lso presented. Some concluding remrs concerning the guge fixing procedure end the pper.

2 . Bblen et l.. TE EXTENDED SE-SCE Let us consider guge theory chrcterized by the cnonicl hmiltonin i q pi i... n nd by the first clss constrints G q p ;... m. One supposes tht the following reducibility reltions could be estblished mong the constrints: where Z re G ;... m. Z C -functions on the phse spce. If these functions re independent the set { G Z } defines -reducible hmiltonin theory. If we consider tht the constrints. re linerly independent the extended i phse q ; spce llowing the implementtion of sp3 BRST symmetry will hve the structure 8: i { p q ; 3}.3 i The moment { } pi define differentil complex endowed by the Koszul differentils { δ 3 }. Cnonicl conjugtion reltions cn be estblished mong these moment nd the ghost coordintes { q i } :.4 ; b δ δ b b δ For two rbitrry functionls F q p nd G q p the oisson brcet is defined s: δf δg F G δg δf ε F ε G.5 δ δ δ δ Becuse of reltions. supplementry non-trivil cycles pper nd they s for new genertors bc ;... m ; b c 3} { : Z G b b δ δ Z.6 The cycles δ Z b b could be eliminted if we define the vribles: μ ε Z.7 bc bc bcd d

3 3 Extended BRST symmetries in the guge field theory so tht: bc c Z b δ μ δ.8 The cyclicity of δ is ccomplished if we lso introduce the vribles nd bd b tht stisfy the requirements: δ ε μ.9 bd cd bc ; δ b δ d bd Conclusions: The Koszul prt of the extended phse spce hs for -reducible theory endowed with sp3 BRST symmetry the structure: K { p ; b 3}. b The corresponding longitudinl complex is generted by the ghosts: b b b i { o L q ; b 3}. In order to obtin grduted differentil complex the ghost-type moment { } nd the ghost genertors b b b b { } will be ssigned with the degrees: - ghost number gh gh g - resolution degree res gh ; res - level number lev lev l Remr: The condition of first clss constrints imposes to G ss for the vlidity of the reltions: i G G f l q p G G q p G ;... m. 3. TE BRST CRGES Let us pss now to the problem of finding the expressions of the three BRST chrges 3. These chrges represent the cnonicl genertor of the BRST differentils s 3 : s 3.

4 . Bblen et l. 4 The requirement of nticommuttivity of { s 3} imposes similr property on the chrges: b ; b 3 3. The determintion of could be done using the homologicl perturbtion theory procedure tht requires the decomposition of the chrges s sum of terms with different resolution degrees: ; 3; res 3.3 The equtions 3. will be ccompnied by dequte boundry conditions expressing the fct tht the differentils s strt with δ nd lso contin the longitudinl derivtives d cting on the longitudinl complex: s δ d "more" 3.4 By combining these reltions with those of the previous section we hve: 3 G b b b b ε bc... δ ; ε mb bc mc b Z bc δ c δ...; b c δ bc bc To these boundry terms will be joined other higher order interction terms. The concrete form of these terms depend on the type of the theory under considertion. For -reducible first rn theory tht is theory with constnt structure functions the explicit form of the BRST chrges re presented in TE EXTENDED ILTONIN We will obtin now n extension of the cnonicl hmiltonin for first rn bosonic theory. It will be the solution of n eqution of the form 3. with : The boundry condition will hve the form: s ; 3 4. G q p We will use gin the homologicl perturbtion theory nd we will write: 4.

5 5 Extended BRST symmetries in the guge field theory 3 ; ; gh res 4.3 ; res B B By projecting 4. on the resolution degree we will obtin: s is given by3.5 nd s p q the previous eqution gets: 4.4 For l the eqution 4. hs the form: This eqution gives the term:... b c bc m m p q ε 4.5 where the coefficients must be determined. The following terms in the decomposition 4.3 will be obtined by projecting 4. on the vlues nd 3. They will be the lst terms occurring in nd they will hve the forms:......; 4 3 b b mb mc bc b b p q ε The complete form for these pieces of the extended hmiltonin depends of the type of the theory. 5. CONCLUSIONS The min conclusion of our pper consists in the possibility of finding n extended phse spce suitble for the sp3 quntiztion of -reducible theory. oreover one could construct the three BRST chrges nd the extended

6 4. Bblen et l. 6 hmiltonin for such theory. They contin ll the terms of the irreducible guge theory the supplementry ones being due to the reducibility functions. It is simple to prove using the sme method s for n irreducible guge theory 8 tht the extended hmiltonin dmits guge fixing procedure tht llows to void the unphysicl degrees of freedom in its expression. The problem will be detiled elsewhere 9. REFERENCES C. Becchi. Rouet R. Stor hys. Lett. 5 B I.. Tyutin Lebedev reprint 39/975 unpublished. 3. enneux C. Teitelboim untiztion of Guge Systems rinceton U I.. Btlin G.. ilovisy hys. Lett.B I.. Btlin.. Lvrov I.. Tyutin J. th. hys I.. Btlin.. Lvrov I.. Tyutin J. th. hys Bblen R. Constntinescu C. Ionescu hysics.u.c R. Constntinescu L. Ttru hys. Lett. B R. Constntinescu in preprtion.