LAGRANGEAN sp(3) BRST FORMALISM FOR MASSIVE VECTORIAL BOSONIC FIELDS

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1 Dedicted to Professor Oliviu Ghermn s 80 th Anniversry LAGRANGEAN sp(3) BRST FORMALISM FOR MASSIVE VECTORIAL BOSONIC FIELDS R. CONSTANTINESCU, C. IONESCU Dept. of Theoreticl Physics, University of Criov, 13 A. I. Cuz Str., Criov, RO , Romni Received August 23, 2010 In the 90s professor Ghermn promoted mny PhD theses in the field of the BRST quntiztion, very modern topic t tht time, nd contributed to the cretion of strong reserch group in this field t his university. This pper summrizes one of the spects developed by this group, the extended formlism for the Lgrngin BRST quntiztion. The procedure will be illustrted on generlized version of the mssive bosonic field with spin 1, model known s Proc field, nmed from nother fmous Romnin physicist. We shll strt from the Hmiltonin formlism nd we shll end with the Lgrngin quntum mster ction. This wy, from Hmilton to Lgrnge, hs double motivtion: the Proc model is not covrint nd the guge fixing procedure in the BRST Lgrngin context is simpler following this wy, s fr s the ghost spectrum. Key words: extended BRST symmetry, Proc model, guge fixing procedure. PACS: Ef 1. INTRODUCTION The BRST quntiztion method is one of the most powerful tools for describing the quntiztion of constrined dynmicl systems nd, in prticulr, the guge field theories. The BRST symmetry is expressed either s differentil opertor s, or in cnonicl form, by the ntibrcket (, ) in the Lgrngen (Btlin-Vilkovisky) cse [1] nd by the extended Poisson brcket [, ] in the Hmiltonin (Btlin-Frdkin- Vilkovisky) formultion [2]: s = [,Ω] = (,S). (1) The BRST chrge Ω nd the BRST genertor S re both defined in extended spces generted by the rel nd by the ghost-type vribles. In order to perform pthintegrl clcultions in this frme, it is necessry to remove the redundnt guge vribles, tht is to sy to guge-fix the ction by choosing suitble guge fermion Y. The elimintion of guge vribles ssures the BRST invrince of the ction [3] but not of the mesure. A BRST trnsformtion of the coordintes could generte Rom. Journ. Phys., Vol. 55, Nos. 9 10, P , Buchrest, 2010

2 962 R. Constntinescu, C. Ionescu 2 non-trivil terms in the ction. These terms cn be exponentited (Fdeev-Popov trick), leding to wht is clled the quntum ction. The guge fixing procedure in the stndrd BRST supposes the introduction of some supplementry vribles from non-miniml sector. As this extension is not lwys very simple nd cn generte difficulties, n extended sp(2) BRST symmetry hs been formulted, both in Hmiltonin [4] nd in Lgrngin formlisms [5]. Lter on, members of the Criov reserch group proposed even more extended BRST symmetries [6], [7]. Mny interesting models of guge theories hve been successfully investigted using these extended formlisms. The simplest exmple of guge theory is the belin guge field, or more precisely the electromgnetic field. In the four dimensionl spce-time it is described by the qudripotentil {A µ,µ = 0,1,2,3} nd by the tensor field F µν = µ A ν ν A µ (2) Not ll the components of the qudripotentil A µ (r,t) re independent. In the Hmiltonin description, using the Dirc terminology, the electromgnetic field cn be seen s constrined dynmicl system. In the erly 1930 s Proc considered model of mssive bosonic field, ssuming tht the photon hd some smll but non-zero mss [8]. The Lgrngin ction hd the form: S0Pr L = d 4 x( 1 4 F µν F µν m2 A µ A µ j µ A µ ) (3) It is known s Proc ction nd fr from the sources (j µ = 0) it genertes Euler- Lgrnge equtions of the form: A ν ν ( µ A µ ) + m 2 A ν = 0 (4) We note tht the mss term m 2 A µ A µ breks the guge invrince of the model nd this is why coupling with sclr field φ is usully considered. A more generl ction, coupling the Proc field with sclr field but lso with its higher order derivtives, hs been proposed in [9]. This lst model is described by the ction: S 0 [A µ, A µ,φ] = d 4 xl = d 4 x ( 1 4 F µνf µν k λ F αλ ρ F ρ α ( µφ ma µ )( µ φ ma µ )) When the sclr field φ nd the coupling constnt k vnish, we recover the Proc eqution (4). The sp(3) BRST formlism for the model (5) hd been proposed in [10]. In this pper we shll obtin guge fixed Lgrngin in the sp(3) BRST (5)

3 3 Lgrngen sp(3) BRST formlism for mssive vectoril bosonic fields 963 formlism nd its quntum version for the mssive vectoril bosonic field, seen s the cse k = 0 of the model described by (5). In order to void the lost of the guge invrince, we shll keep the sclr field φ 0 in (5) nd t the end only the limit φ = 0 could be considered. The pper hs the following structure: in the next section we shll mke the cnonicl nlysis of the model, then, in section 3, we shll demonstrte how the guge fixed Lgrngin cn be generted through the Hmiltonin formlism. As conclusions of the pper, we shll see how the quntum mster equtions cn be written for the mssive belin fields in our sp(3) BRST context. 2. THE CANONICAL ANALYSIS Let us consider the Lgrngin ction (5) in the cse k = 0. The cnonicl nlysis of this ction leds to the following irreducible first clss constrints G (1) (x) F 00 (x) = p 0 (x) 0 (6) G (2) (x) = i p i (x) + mp φ 0 (7) where we considered the conjugted moment of the mssive vectoril fields: p i L A i (8) The conjugted moment ttched to the sclr field φ hs the form p φ L φ = φ ma 0 (9) It do not generte new constrint, llowing to express the velocity : φ = p φ + ma 0 (10) The first clss Hmiltonin will be: ( 1 H 0 (A,φ,p,p φ ) = d 3 x 4 F ij F ij 1 2 p ip i + A 0 ( i p i + mp φ ) p φp φ 1 ) (11) 2 ( iφ)( i φ) ma i ( i φ) + m2 2 A ia i The guge lgebr of the constrints nd of the Hmiltonin is expressed by the reltions: [G (1) (x),g (1) (x )] x 0 =x 0 = 0,[G(1) (x),g (2) (x )] x 0 =x 0 = 0, (12)

4 964 R. Constntinescu, C. Ionescu 4 [G (2) (x),g (2) (x )] x 0 =x 0 = 0 [H 0,G (1) (x)] x 0 =x 0 = G(2) (x), [H 0,G (2) (x)] x 0 =x 0 = 0. (13) As we see, it is n belin lgebr nd, moreover, the limit φ 0 do not ffect this lgebr. For simplicity reson, we shll introduce the condensed nottion {G ( ), = 1,2} where the superscript = 1 designtes the primry constrints nd = 2 the secondry ones. 3. THE sp(3) BRST SYMMETRY 3.1. THE sp(3) HAMILTONIAN FORMALISM To implement sp(3) BRST symmetry supposes to find not one, but three nticommuting differentils so tht: s T = s 1 + s 2 + s 3 ;s s b + s b s = 0,,b = 1,2,3 (14) The reltion (1) is generlized now s: s = [,Ω ] = (,S) ; = 1,2,3 (15) As we mentioned, our im is to obtin for our model Lgrngin sp(3) symmetry nd, in order to recover the Lgrngin guge fixed ction, we shll strt from the Hmiltonin ction nd we shll use the equivlence between the two formlisms. There is direct modlity of obtining the sp(3) BRST Lgrngen formlism, but it ssumes the use of very lrge spectrum of ghost genertors. To void this unuseful extension, it is simpler t the clssicl level to construct the Lgrngen formlism following its equivlence with the Hmiltonin one [11]. The sp(3) Hmiltonin formlism is esier to be built, without the need of introducing nonminiml sector, nd, on the other hnd, the Lgrngin guge fixed ction is useful in the quntum description of the model. So, we shll strt by developing the sp(3) BRST Hmiltonin formlism [11] for our theory. The ction (5) cn be written in the following cnonicl form: S cn [A,p,u] = where the Hmiltonin H (0) (q,p,u) hs the form dt( A i p i H (0) (A,p,u)), i = 1,,n (16) H (0) (A,p,u) = H 0 (A,p) + u ( ) G ( ). (17) The Lgrnge multipliers {u ( ), = 1,2} ply key role in estblishing n equivlence between the Hmiltonin nd the Lgrngin formlisms. The condition (14)

5 5 Lgrngen sp(3) BRST formlism for mssive vectoril bosonic fields 965 requires t the first step the introduction for ech constrint G ( ) of the nticommuting ghosts {Q ( ), = 1,2,3} nd of their conjugted moment {P ( ), = 1,2,3}. As the originl fields {A,p} re bosonic, the new vribles must hve the Grssmnn prities ε(q ( ) ) = ε(p ( ) ) = 1. The next steps sks for the introduction of ghosts of ghosts λ ( ) with the conjugted moment π ( ) nd of ghosts of ghosts of ghosts η ( ) with their conjugted moment π ( ). We shll use the following condensed nottion for ll these genertors of the extended phse spce: - for moment (rel moment nd ghost ones): P A {p i,p A } {p i,p ( ) 1,P ( ) 2,P ( ) 3,π ( ) 1,π ( ) 2,π ( ) 3,π ( ) }. (18) - for fields (rel nd ghost type): Q A {A i,q A } {A i,q ( )1,Q ( )2,Q ( )3,λ ( )1,λ ( )2,λ ( )3,η ( ) } (19) These genertors hve to obey to the reltions (in the de Witt nottion): [ pi,a j] = δ j i,[p ( ) b,q ( ) ] = δ δb [, π ( ) b,λ ( )] = δ δb,[π( ),η ( ) ] = δ. (20) The BRST chrges Ω, = 1,2,3 re defined s solutions of the mster equtions [ Ω,Ω b] = 0,,b = 1,2,3 (21) with boundry conditions Ω = δb G( ), Q=λ=0 Q ( )b Ω λ ( )b = ε bc P ( ) αb, Ω = π ( ). (22) Q=λ=0 Q=λ=0 η ( ) The solution of the problem (21)-(22) cn be obtined using the homologicl perturbtions theory [3] nd, by doing tht in our cse, we shll obtin the following concrete expressions for the BRST chrges: Ω = d 3 x(p 0 Q (1)b δ b +( i p i +mp φ )Q (2)b δ b +ε bc P c ( ) λ ( )b +π ( ) η ( ) ). (23) The second problem, consisting in finding the BRST Hmiltonin, hs the solution given by [H,Ω ] = 0, = 1,2,3 with the boundry condition H P =Q=π=λ=π=η=0 = H 0

6 966 R. Constntinescu, C. Ionescu 6 It leds to the sp(3) BRST invrint Hmiltonin H = H 0 + d 3 x (Q (1) P (2) + λ (1) π (2) + η (1) π (2) ). (24) By choosing for the guge-fixing fermion function the form Y = d 3 x ( i A i )π (1) (25) we obtin the following guge fixing term K = 1 3! εbc [Ω,[Ω b,[ω c,y ]]] = d 3 x ( p 0 ( i A i ) ( i P (1) )( i Q (2) )+ + ( i π (1) )( i λ (2) ) ( i π (1) )( i η (2) )). The functions Ȳ nd Y which stisfy to the reltions s T Ȳ = s Y hve the concrete forms Ȳ = 1 3 d 3 x( [P (1) ( i A i ) ( i π (1) )( i λ (2) ) 3 =1 (27) 3 b=1 ( i π (1) )( i Q (2)b )] 3 b=1 ε bcd ( i π (1) d )( i Q (2)c ) respectively Y = d 3 x(p (1) ( i A i ) ε bc ( i π (1) b )( i Q (2)c ) ( i π (1) )( i λ (2)b )δ b. (28) By eliminting the uxiliry fields P (2), Q (1), π (2), λ (1), π (2) nd η (1) on the bsis of their equtions of motion, we shll obtin the following covrint guge fixed ction S K = d 4 x( 1 4 F µνf µν m2 A µ A µ + p 0 ( µ A µ )+ (29) + ( µ P (1) )( µ Q (2) ) ( µ π (1) )( µ λ (2) ) + ( µ π (1) )( µ η (2) )). The ction (29) is invrint to the BRST trnsformtions (26) s A µ = [A µ,ω ] = µ Q (2)b δ b,s φ = [φ,ω ] = mq (2)b δ b, s Q (1)c = [Q (1)c,Ω ] = ε bc λ (1)b,s Q (2)c = [Q (2)c,Ω ] = ε bc λ (2)b, s λ (1)c = [λ (1)c,Ω ] = δη c (1),s λ (2)c = [λ (2)c,Ω ] = δη c (2), s η (1) = [η (1),Ω ] = 0,s η (2) = [η (2),Ω ] = 0. (30)

7 7 Lgrngen sp(3) BRST formlism for mssive vectoril bosonic fields THE LAGRANGIAN FORMALISM VIA THE HAMILTONIAN ONE If in (16) we shll look t the moment {p i,i = 1,,n} s uxiliry vribles, we cn eliminte them on the bsis of their eqution of motion. We obtin the ction S 0 [A,φ,u] = dt L 0 (A i, i A j,φ,u (1),u (2) ) (31) where the Lgrnge multipliers {u ( ), = 1,2} re considered now s rel fields. Strting from (31) we will develop the sp(3) BRST Lgrngen formlism [7]. The complete spectrum of the ntifields in our cse is given by Q A {Q ( ) Q A {,u ( ) Q ( ),ū ( ) } = {A µ,φ,u ( ) } = {Āµ, φ,ū ( ),Q ( ) b, Q( ) b,λ ( ) b,η ( ), λ ( ) b,,b = 1,2,3}, (32), η( ) = 1,2,3}. (33) Q A { Q ( ),ū ( ) } = {Āµ, φ,ū ( ), Q( ) ( ), λ, η ( ), = 1,2,3}. (34) It is well known tht the Lgrngin dynmics is generted in nticnonicl structure. The genertor of the Lgrngin BRST symmetry is S = S 0 [A µ,φ] + Becuse S 0 is unique, we shll consider tht S is unique too, nd we will introduce three ntibrcket structures which hve the sme properties s in the stndrd theory: (F,G) = δr F δq A δ l G δq A δr F δ l G δq A δq A The functionls F nd G re dependent on Q A nd Q A. On the bsis of grdution properties nd of Grssmnn prities [7] we define the pirs cnoniclly conjugted in respect with these ntibrckets (Q Ab,QB ) = δ B Aδ b (35) where Q A re s in (32) nd QA represents the fields of the theory (rel A i,u ( ) nd ghosts Q A ) Q A {A i,φ,u (1),u (2) A 0,Q ( ),λ ( ),η ( ), = 1,2,3}. (36) For the Lgrnge multipliers we shll hve ε(u ( ) ) = 0,gh(u ( ) ) = 0. We note tht some ntifields of the theory hve cnonicl conjugte in the ntibrcket structure (35) while other ntifields do not hve cnonicl pirs (33). So, the BRST differentils {s, = 1,2,3} will present the following decomposition s = (s ) cn +V = (,S) +V, = 1,2,3 (37)

8 968 R. Constntinescu, C. Ionescu 8 where the non-cnonicl opertors V hve the form [7] δ r V ( ) ε(qa) ε bc Q Ac δ Q +( ) ε(qa )+1 δ b QAb Ab δ Q. (38) A The nilpotency condition for s (37) leds to the mster equtions 1 2 (S,S) + V S = 0, = 1,2,3. (39) For our irreducible theory, the proper solution of the mster eqs. (39), till terms liner in the ntifields, is ( S = S 0 + dt u ( ) G ( ) + A i i Q (2) + u (1) Q (1) + u (2) ( Q (2) Q (1) ) + ε bc λ ( )c Q ( ) b + δ b λ ( ) b η ( ) + Āi i λ (2) + ū (1) λ (1) + ū (2) ( λ (2) λ (1) ) +Āi i η (2) + ū (1) η (1) + ū (2) ( η (2) η (1) (2) ) + Q λ (2) + Q (2) η (2)). On the bsis of grdution rules nd Grssmnn prities we cn identify the following vribles P ( ) δ r (40) u ( ),π ( ) ū ( ),π ( ) ū ( ). (41) These identifictions will be very useful in the guge fixing procedure. For exmple, on the bsis of the identifictions (41), we cn consider the following form of the guge fixing functionl: Y = ( µ A µ )ū (1). (42) The following reltions re vlid [12]: A µ = δr δa µ (1 2 ε bcv b V c Y ) = δr ( α A α ) δa µ u (1), (43) Ā µ = δr (V Y ) δa µ = δr ( α A α ) δa µ ū (1) (44) Ā µ = δr Y δa µ = δr ( α A α ) δa µ ū (1) (45) u (1) = δl δu (1) ( 1 2 ε bcv b V c Y ) = µ A µ (46) the remining ntifields vnishing becuse of the choice (42) for Y.

9 9 Lgrngen sp(3) BRST formlism for mssive vectoril bosonic fields CONCLUSIONS Following the pth from Hmilton to Lgrnge, we obtined sp(3) Lgrngin guge fixed ction depending on the originl fields nd on some ntifields. The identifiction (41), s well s the choice (42) of the guge fixing function, s it ws suggested by the Hmiltonin formlism, drsticlly reduced the spectrum of the supplementry ghost-type fields required by pure sp(3) Lgrngin pproch. Prcticlly, the guge fixed ction will tke the form: S 1Y = S 1 [A µ,φ,q A,u (1),ū (1),ū (1),A µ,āµ,āµ] (47) where u (1),A µ, Ā µ,āµ comply the reltions (43)-(46). It leds to n effective ction which is s -invrint nd tht cn be further used in the pth integrl. This pth integrl cn be written s ZY L = DA µ DφDQ (2) Dλ (2) Dη (2) Du (1) Dū (1) Dū (1) exp(i S 1Y ). (48) If we introduce the condensed nottions φ A {A µ,φ,q (2),λ (2),η (2),u (1),ū (1),ū (1) } (49) we cn consider the following BRST trnsformtions φ A φ A = φ A (s φ A )µ ( 1) ε A. (50) where µ re smll fermionic constnt prmeters. The Jcobin of these trnsformtions cn be pproximted through the supertrce (becuse it involves both fermionic nd bosonic fields, the Jcobin is superdeterminnt) nd the integrting mesure will be [ Dφ A 1 (( 1) ε r A = r S φ A φ A ) ((1 S)µ Dφ A. ] + V S )µ Dφ A = In the previous reltion, the opertors, = 1,2,3 hve the form (51) ( 1) ε r r S A φ A + V = φ A = ( 1) ε A r φ A r S φ A r r + ε bc φ Ac φ + ( 1) ε A δ b φab Ab φ A (52)

10 970 R. Constntinescu, C. Ionescu 10 nd they re nilpotent b + b = 0,,b = 1,2,3 (53) Such opertors determine the sp(3) quntum mster equtions in the sp(3) symmetric formultion of the guge theories: where e i W = (W,W ) = i W, = 1,2,3 W = S 1Y + W W represents the quntum ction, the order corrections being generted by the integrting mesure. Acknowledgements. The uthors re grteful to professor Oliviu Ghermn for the support offered long the erly period of their scientific ctivity. REFERENCES 1. I.A. Btlin, G. A. Vilkovisky, Phys. Lett. B 69, 309 (1977). 2. E.S. Frdkin, G. A. Vilkovisky, Phys. Lett. B 55, 224 (1975). 3. M. Henneux, C. Teitelboim, Quntiztion of Guge Systems (Princeton Univ. Press, 1992). 4. I.A. Btlin, P. M. Lvrov, I.V. Tyutin, J. Mth. Phys.31, 1487 (1990). 5. Ph. Gregoire, M. Henneux, Phys. Lett. B 277, 459 (1992). 6. R. Constntinescu, C. Ionescu, Int. J. Mod. Phys. A 21(32), 6629 (2006). 7. C. Bizdde, S. O. Sliu, Mod. Phys. Lett. A 17, 269 (2002). 8. F. Zmni nd A. Mostfzdeh, J. Mth. Phys. 50, (35 pp) (2009). 9. Zi-ping Li, Rui-jie Li, Int. J. of Theor. Phys., Vol. 45, No. 2, 395 (2006). 10. C. Ionescu, Phys. AUC 16(prt I), 97 (2006). 11. R. Constntinescu, C. Ionescu, Int.J. Mod.Phys. A 21, 1567 (2006). 12. R. Constntinescu, C. Ionescu, J. Phys. A: Mth. Theor. 42, (9pp) (2009).