TENSOR GAUGE FIELDS OF DEGREE THREE

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1 TENSOR GAUGE FIELDS OF DEGREE THREE E.M. CIOROIANU Department of Physcs, Unversty of Craova, A. I. Cuza 13, 2585, Craova, Romana, EU E-mal: Receved February 2, 213 Startng from a generc second-order Lagrangan that descrbes the dynamcs of an Abelan tensor gauge feld of degree three and depends on two arbtrary real constants) we perform the Drac analyss. The procedure allows the determnaton of the number of degrees of freedom and also a generatng set of gauge transformatons for the ntal theory. Key words: gauge felds, constraned systems, Drac algorthm. PACS: 11.1Ef. 1. MOTIVATION Ths paper offers a framework for the untary approach of the tensor gauge felds of degree three that transforms under rreducble representatons of the Lorentz group. It fnds ts roots n an old attempt to unfy gravty wth electromagnetsm proposed by Ensten and developed by hmself 1 and others 2,3. The aforementoned unfcaton scheme nvolves a tensor gauge feld of degree two wth no symmetry. In vew of ths, our startng pont s represented by a tensor feld of degree three A µν α antsymmetrc n the frst two Lorentz ndces A µν α = A νµ α ) that transforms under reducble representaton A µν α of the Lorentz group. We postulate for the consdered tensor feld the gauge transformatons δ ɛ A µν α = µ ɛ ν α. 1) In the above, the bosonc gauge parameters ɛ ν α have no symmetry. In vew of these, the most general second-order Lagrangan densty nvarant under the gauge Paper presented at The 8 th Workshop on Quantum Feld Theory and Hamltonan Systems, September 19-22, 212, Craova, Romana. RJP Rom. 58Nos. Journ. Phys., 5-6), Vol , Nos. 5-6, 213) P , c) Bucharest, 213

2 53 E.M. Coroanu 2 transformatons 1) reads as L = 1 6 F µνρ αf µνρ α + k 1 4 F µνρ λf µνλ ρ + k 2 4 F µν F µν, 2) where k 1 and k 2 are arbtrary real constants and F µνρ α s the feld-strength of the tensor gauge feld A µν α F µνρ α µa νρ α. 3) By F µν we denoted the trace of the feld-strength, F µν σ αβ F µνα β. Throughout the paper we work wth the flat metrc of mostly mnus sgnature σ µν = + ). The notaton µ...ν sgnfes full antsymmetry wth respect to the ndces between brackets wthout normalzaton factors.e. the ndependent terms appear only once and are not multpled by overall numercal factors). It has been shown 4 that f we decompose the gauge feld A µν α nto ts rreducble components A µν α t µν α + B µνα A µν α 1 ) ) 1 3 A µν α + 3 A µν α, 4) then the Lagrangan acton correspondng wth the densty 2) takes the form S L 4 tµν α,b µνρ = d D 7k1 k 2 x H µνρλ H µνρλ k 1 + k 2 12 µ B νρλ ) µ B νρλ k 1 12 F µνρ λf µνρ λ + k 2 4 F µνf µν k 1 + k 2 F 6 µνρ λ λ B. µνρ In the local functonal 5) H µνρλ stands for the feld-strength of the 3-form B µνρ H µνρλ µ B νρλ ), the tensor F µνρ λ s defned by F µνρ λ µ t νρ α and F µν s nothng but the trace of F µνρ λ F µν σ ρλ F µνρ λ ). A quck look on the Lagrangan acton 5) reveals two lmt stuatons namely 5) k 1 = 2, k 2 =, 6) and k 1 = 1, k 2 = 3 7) For the choce 6) the gauge feld t µν α wth the mxed symmetry 2,1) becomes a pure one t does not appear n the Lagrangan acton). On the other hand, f we set 7) nto the local functonal 5) then the 3-form B µνρ becomes a pure gauge feld. The above analyss suggests the tensor gauge felds of degree three that transform under rreducble representatons of the Lorentz group can be treated n un- RJP 58Nos. 5-6), ) c)

3 3 Tensor Gauge Felds of Degree Three 531 fed manner through the gauge feld A µν α whose dynamcs s governed by the Lagrangan densty 2). Ths remark justfes the Drac analyss of the gauge model 2). 2. RESULTS In ths secton we perform the canoncal analyss of the model subject to the Lagrangan densty 2). In vew of ths, f we denote by π µν α the canoncal momenta assocated wth the felds A µν α, the defntons of the formers read as π µν α 1 L 2 A L µν α A νµ α = F µν α + k 1 2 F αµν + k 2 2 σ αµf ν, ) 8) where by overdot we denoted the dervatve n respect wth the temporal coordnate x. From the defntons n the above, we nfer the prmary constrants and also the relatons G 1) π, G 1) j π j,j = 1,D 1 9) π j = 2 + k 1 + k 2 F 2 j + k 2 2 F jk k,,j = 1,D 1, 1a) π j k = F j k + k 1 2 F k j + k 2 2 σ k F l jl,,j,k = 1,D 1. 1b) Startng wth the equatons 1b), only by algebrac computatons, we derve π j j = 2 + k 1 + D 2)k 2 F 2 j j, = 1,D 1, 11a) π j k = 1 k 1 )F j k k 1F jk,,j,k = 1,D 1. 11b) The frst step n the canoncal analyss s completed by solvng the system 1) n respect wth some of the generalzed veloctes. In vew of ths, the results 11) lead to seven dstnct stuatons dctated by the coeffcents 2 + k 1 + k 2, 2 + k 1 + D 2)k 2 RJP 58Nos. 5-6), ) c)

4 532 E.M. Coroanu 4 and 1 k 1 of the temporal components of the feld-strength n 1a) and 11), namely k 1 = 2, k 2 = ; 12) k 1 = 1, k 2 = 3; 13) k 1 = 1, k 2 = 3 D 2 ; 14) k 1 = 1, k 2 k R\ { 3, 3 D 2 } ; 15) k 1 k R\{ 2,1}, k 2 = 2 + k D 2 ; 16) k 1 k R\{ 2,1}, k 2 = 2 + k ) ; 17) k 1 + k 2 2 k 1 + D 2)k 2, k 1 R\{1}. 18) In the remanng part of ths secton we wll complete the canoncal analyss of the model n each of the seven stuatons delmted n the above CASE I In the stuaton 12) the defntons of the canoncal momenta 8) lead to the ndependent prmary constrants 9) and γ 1) j π j,,j = 1,D 1, 19a) γ 1) j k π j k 1 3 π j k,,j,k = 1,D 1, 19b) γ 1) π j j, = 1,D 1. 19c) Expressng from the equatons 8) correspondng to the choce 12)) some of the generalzed veloctes, we get the canoncal Hamltonan well defned only on the prmary constrant surface) H I) x ) = 1 3 d D 1 x 1 ) 6 πj k π j k + 6F jk 1 2 F jk lf jk l 2A j k π j k ). By drect computaton t smply verfes that the constrants 9) and 19) are Abelan so that the consstency of the prmary constrants reduces only to the computaton of the Posson brackets between them and the canoncal Hamltonan. Based on these arguments, consstency of the prmary constrants produces secondary constrants G 2) j 1 3 k π j k, = 1,D 1, 21) RJP 58Nos. 5-6), ) c) )

5 5 Tensor Gauge Felds of Degree Three 533 that are second-order and off-shell reducble, wth the reducblty functons Z j ) k = δ j k, Zk = k. 22) The consstency of the secondary constrants 21) does not reveal any new constrant as the constrants 9), 19), 21) are Abelan and G 2) j,hi) =. 23) The arguments n the above allow us to conclude that the canoncal Hamltonan s exactly the frst-class Hamltonan of the system. The rreducble character of the frst-class constrants 9) and 19a), together wth the frst-order reducbltes among the constrants 19b) 19c) 1 2 δ l σjk γ 1) j k δ l γ1) = and the second-order ones correspondng to the constrants 21) allow us to compute the number of degrees of freedom for the model under study N I) DOF D 2)D 3)D 4) =. 24) 6 Fnally, on behalf of the Drac s conjecture accordng to whch any frst-class constrant generates gauge transformatons), f we pass agan to the Lagrangan formulaton va extended acton), we derve for the Lagrangan acton correspondng to the choce 12) the generatng set of gauge transformatons δ I) ɛ,ξ A µν α = µ ɛ να + ξ µν α, 25) where the gauge parameters ξ µν α have the mxed symmetry 2,1) ξ µν α = ξ νµ α, ξ µν α =. 26) To conclude wth, ths stuaton s nothng but the lmt case where the tensor gauge feld s part wth the mxed symmetry 2,1) becomes a pure gauge one CASE II In ths part, we complete the canoncal analyss of the model 2) n whch the real parameters k 1 and k 2 are taken as n 13). Wth ths settng, the defntons of the canoncal momenta 8) produce the ndependent prmary constrants 9) and γ 1) j π j F jk k,,j = 1,D 1, 27a) γ 1) jk π j k 3 2 F jk,,j,k = 1,D 1. 27b) RJP 58Nos. 5-6), ) c)

6 534 E.M. Coroanu 6 Expressng from the equatons 8) some of the generalzed veloctes, we derve the canoncal Hamltonan well defned only on the prmary constrant surface) x ) = d D 1 x 2A j ) ) π j 2A j k π k j H II) 1 4 F jk lf jl k F jk k F jl l π j kπ j k 1 6 F jk µf jk µ 1 6 π j kf jk 2 3D 3) π j j πk k Smple computatons lead to the Abelan character of the prmary constrants 9) and 27) so that ther consstency reduces only to the calculatons of the Posson brackets between canoncal Hamltonan and them. By drect computatons one obtans G 1),H II) G 1) j,hii) γ 1) j,hii) γ 1) j,hii). 28) = j π j G 2), 29a) = k π k j G 2) j, The results 29) dsplay the secondary constrants 29b) = 5 6 k γ 1) jk G2) j, 29c) =. 29d) G 2), G 2) j, 3) that together wth 9) and 27) consttute an Abelan set of constrants. The secondary constrants dsplayed n the above are off-shell frst-order reducble wth the reducblty functons Z, Z j) k δ j k. The consstency of the secondary constrants 3) does not produce any new constrant so the Drac algorthm stops at ths level. At ths stage, we can conclude that the canoncal Hamltonan 28) s exactly the frst-class Hamltonan of the system. The rreducble character of the frst-class constrants 9) and 27) together wth the frst-order reducbltes of the secondary constrants 3) allow us to count the degrees of freedom for the model under study N II) D D 2)D 4) DOF =. 31) 2 Fnally, f we pass agan to the Lagrangan formulaton va extended acton), we derve for the correspondng functonal assocated wth the local functon 2) the generatng set of gauge transformatons δ II) ɛ,ε A µν α = µ ɛ να + µ ε να α ε µν + ε µνα, 32) RJP 58Nos. 5-6), ) c)

7 7 Tensor Gauge Felds of Degree Three 535 where the gauge parameters ɛ µν are symmetrc whle ε µν and ε µνρ are completely antsymmetrc. To conclude wth, ths stuaton s nothng but the lmt case where the antsymmetrc part of the tensor gauge feld becomes a pure gauge one CASE III Wth the settngs 14) of the real parameters k 1 and k 2, the defntons 8) of the canoncal momenta furnsh the prmary constrants 9) and γ 1) π j j, = 1,D 1, 33a) γ 1) jk π j k 3 2 F jk,,j,k = 1,D 1. 33b) Performng, on the prmary constrants surface, the Legendre transformaton of the Lagrangan densty 2) n respect wth some of the generalzed veloctes those that can be expressed from the defntons of the canoncal momenta) we get the canoncal Hamltonan H III) x ) = d D 1 x 2A j ) ) π j 2A j k π k j 1 4 F jk lf jl k 3 + 4D 3) F jk k F jl l π j kπ j k 1 6 π j kf jk 2 3D 3) π j j πk k 1 6 F jk µf jk µ + D 2 3D 3) π j π j + 1 D 3 πj F jk k As n the other two cases, the prmary constrants 9) and 33) are Abelan so ther consstency reduces to the computaton of the Posson brackets between them and the canoncal Hamltonan 34). By drect calculatons we nfer G 1),H III) G 1) j,hiii) γ 1),H III) γ 1) jk,hiii) 34) = j π j G 2), 35a) = k π k j G 2) j, 35b) = G 2), 35c) = π jk 32 ) m F jk m γ 2) jk. 35d) The results 35) put nto evdence the secondary constrants G 2), G 2) j, γ2) jk,,j,k = 1,D 1. 36) RJP 58Nos. 5-6), ) c)

8 536 E.M. Coroanu 8 that together wth the prmary constrants 9) and 33) consttute an Abelan set of constrants. Ths output combned wth the property that the Posson brackets between the canoncal Hamltonan and secondary constrants 36) vansh on-shell allow us to conclude that the Drac algorthm stops at ths level. Moreover, the canoncal Hamltonan s exactly the frst-class Hamltonan. The concrete expressons of the frst-class constrants 9), 33) and 36) evdence frstly, the constrants γ 2) jk are off-shell reducble of order D 4) wth the reducblty functons Z 1 ) k j 1 j k+1 = j1 δ 1 j2 δ k jk+1, k = 3,D 2 37) secondly, the constrants γ 1), G 2) and G 2) j are of-shell frst order reducble ) γ 1) + σ j) G 2) j ) =, 38a) δ j k G 2) j =, 38b) ) G 2) = 38c) and thrdly 9) and 33b) are rreducble. The aforementoned reducbltes of the frst-class constrants mply that the system possesses the same number of physcal degrees of freedom as n the prevous stuaton 31). If we return agan to the Lagrangan formulaton va extended acton), we derve for the functonal correspondng to 2) the generatng set of gauge transformatons δ III) ɛ,ε A µν α = σ αµ ɛ ν + µ ɛ ν α + ρ ε µναρ, 39) where the gauge parameters ε µναρ are completely antsymmetrc. In addton, the aforementoned generatng set of gauge transformatons s Abelan and off-shell reducble of order D 4). A short look on the gauge transformatons 39) reveals new conformal 8 and topologcal BF-lke 9 behavours of the massless tensor gauge feld of degree three A µν α. Precsely, the frst term n the rght-hand sde of 39) s nothng but a knd of flat-conformal gauge transformaton for the analysed tensor gauge feld whle the last term mmc the gauge transformatons of the 3-form n a topologcal BF-model CASE IV From the dynamcal pont of vew, ths stuaton s qute smlar to the prevous one. Wth the choce 15), the defntons 8) of the canoncal momenta put nto evdence the prmary constrants 9) and 33b) that are Abelan) and also lead to the RJP 58Nos. 5-6), ) c)

9 9 Tensor Gauge Felds of Degree Three 537 canoncal Hamltonan well defned only on the prmary constrant surface) H IV ) x ) = d D 1 x 2A j ) ) π j 2A j k π k j 1 4 F jk lf jl k 3k 4k + 3) F jk k F jl l π j kπ j k 1 6 π j kf jk 2k 33 + k D 3) π j j πk k 1 6 F jk µf jk µ + 1 k + 3 π j π j k k + 3 πj F jk k The consstency of the prmary constrants dsplays the same secondary constrants as n the prevous stuaton 36) because the followng Posson brackets hold G 1),H IV ) G 2), G 1) j,hiv ) G 2) j, γ 1) jk,hiv ) γ 2) jk. 41) Concernng the consstency of the secondary constrants 36) ths does not mply new constrants because frstly, the constrants 9), 33b) and 36) are Abelan and secondly, the Posson brackets between the canoncal Hamltonan 4) and secondary constrants 36) weakly vansh. As n the thrd stuaton, the canoncal Hamltonan 4) s exactly the frst-class Hamltonan. Also, the frst-class constrants 9), 33b) and 36) possess the same reducbltes as n the prevous case, namely 9) and 33b) are rreducble, G 2) and G 2) j are of-shell frst-order reducble the reducblty relatons 38b) 38c) hold) and the constrants γ 2) jk are off-shell reducble of order D 4) wth the reducblty functons 37). Based on these results we compute the number of degrees of freedom for the system under study N IV ) DOF 4) D 1)D 2)D 3) =. 42) 3 Fnally, on behalf of the Drac s conjecture, f we pass agan to the Lagrangan formulaton va extended acton), we derve the off-shell D 4)-reducble generatng set of gauge transformatons δ IV ) ɛ,ε A µν α = µ ɛ ν α + ρ ε µναρ, 43) where the gauge parameters ε µναρ are completely antsymmetrc. As n the prevous stuaton, the gauge transformatons 43) reveals new topologcal BF-lke 9 behavour of the massless tensor gauge feld of degree three A µν α. The aforementoned behavour s due to the last term n the rght-hand sde of 43) that mmc the gauge transformatons of the 3-form n a topologcal BF-model. RJP 58Nos. 5-6), ) c)

10 538 E.M. Coroanu CASE V We shall see that, as n the prevous stuaton, the present case manfests smlartes wth the thrd stuaton. Here, the constants that parametrze 2) take the values 16). In ths realm, the defntons of the canoncal momenta 8) furnsh the prmary constrants 9) and 33a) that are Abelan) and also produce the canoncal Hamltonan H V ) x ) = d D 1 x 2A j ) ) π j 2A j k π k j k 4 F jk lf jl k + k + 2 4D 3) F k jk F jl l + 1 k + 2 π j kπ j k 1 6 F jk µf jk µ + + k k )F jk F jk + πj D 3 k 2 k + 2 ) 1 k)π j k π j k k + 2 ) Fjk ) ) D 2 k + 2 π j + Fjk k. The requrement of preservaton n tme for the prmary constrants 9) and 33a) leads to the secondary constrants G 2) and G 2) j respectvely defned n 35a) and 35b). By drect computatons we nfer the Abelan character of the constrants 9), 33a), G 2) and G 2) j. Moreover, the consstency of the secondary constrants G 2) and G 2) j no longer produces tertary constrants G 2),H V ) = = G 2) j,hv ). 45) The rreducble character of the frst-class constrants 9), together wth the frst-order reducbltes 38) of the constrants 33a), G 2) and G 2) j allow us to compute the number of physcal degrees of freedom 44) N V ) D D 1)D 4) P hys = ) 2 Usng the same method as n the prevous subsectons, one can deduce for the functonal Lagrangan acton the off-shell frst-order reducble generatng set of gauge transformatons δ V ) ɛ A µν α = σ αµ ɛ ν + µ ɛ ν α. 47) The frst term n the rght-hand sde of the gauge transformatons 47) puts nto evdence a conformal-lke 8 behavour of the massless tensor gauge feld of degree three n the present stuaton. RJP 58Nos. 5-6), ) c)

11 11 Tensor Gauge Felds of Degree Three CASE VI Here the real constants that label the local functon 2) reads as n 17). Replacng the choce 17) n the defntons 8), the lasts lead to the set of prmary constrants consttuted by 9) and γ 1) j π j k 2 F jk k,,j = 1,D 1. 48) Expressng from the equatons 8) wrtten for the choce 17)) some of the generalzed veloctes, we get the canoncal Hamltonan well defned only on the prmary constrant surface) V I) H x ) = d D 1 x 2A j ) ) π j 2A j k π k j 1 6 F jk µf jk µ k 4 F jk lf jl k + k F jk k F jl l + 1 k + 2 π j kπ j k k ) + ) 2 k k )π j k π j k k + 2 )F jk k k )F jk F jk 2 ) π k + 2 D 3) j j πk k. 49) It smply verfes that the constrants 9) and 48) are Abelan so that the consstency of the prmary constrants reduces only to the computaton of the Posson brackets between them and the canoncal Hamltonan. By drect computaton one obtans G 1),H V I) = G 2), G 1) I) j,hv = G 2) j, γ 1) j I),HV = γ 2) j, 5) where the functons n the rght-hand sdes are gven n formulas 35a), 35b) and γ 2) j k π j k k k F jk. 51) These results derved n the above allow to dsplay the secondary constrants possessed by the model under study G 2), G 2) j, γ2) j. 52) It can be checked that the constrants 9), 48) and 52) are Abelan. Usng ths remark, the requrement of preservaton n tme for the secondary constrants reduces to the computaton of the Posson brackets between them and the canoncal RJP 58Nos. 5-6), ) c)

12 54 E.M. Coroanu 12 Hamltonan 49) G 2),H V I) =, G 2) I) j,hv = = γ 2) I) j,hv. 53) The outputs 53) allow us to conclude that the model under study possesses no tertary constrants and, n addton, the canoncal Hamltonan 52) concdes wth the frstclass Hamltonan. In vew of the countng the degrees of freedom, one observes that: ) the frstclass constrants 9) and 48) are rreducble and ) the secondary constrants 52) are off-shell second-order reducble wth the reducblty relatons ) G 2) =, 54a) ) δ j k G 2) j =, 54b) δ k j) ) G 2) 1 j + 2 δ k j γ 2) j = 54c) and respectvely k) ) δ j k + k) δ k j) =, 55a) k) ) 1 2 δ k j =. 55b) Puttng together the prevous results we determne the number of degrees of freedom V I) N DOF D 1)D 2)D 4) =. 56) 2 Employng the same procedure as n the prevous subsectons, we derve for the Lagrangan acton the off-shell second-order reducble generatng set of gauge transformatons V I) δ ε,ɛ A µν α = µ ε νρ + µ ɛ ν α, 57) where the bosonc gauge parameters ε µν are completely antsymmetrc CASE VII In the last stuaton, the real parameters k 1 and k 2 have the ranges defned by 18). Here, the defntons 8) of the canoncal momenta lead to the Abelan prmary RJP 58Nos. 5-6), ) c)

13 13 Tensor Gauge Felds of Degree Three 541 constrants 9) and produce the canoncal Hamltonan V II) H x ) = d D 1 x 2A j ) ) π j 2A j k π k j k 1 4 F jk lf jl k + k 2 k 1 + 2) 4k 1 + k 2 + 2) F jk k F jl l + 1 k π j kπ j k k 1 + ) 2k 1 + 2)1 k 1 ) πj k π j k k 1 + 2)F jk 58) + k k 1 ) F jk F jk 1 6 F jk µf jk µ + πj k 1 + k k 2 k 1 + 2)2 + k 1 + k 2 D 2) π j ) π j k 2 Fjk k. j πk k The consstency of the prmary constrants 9) dsplays the secondary constrants G 2) and G 2) j whose concrete expressons are respectvely wrtten n 35a) and 35b)) as the Posson brackets hold G 1) V II),H = G 2), G 1) II) j,hv = G 2) j. 59) The Abelan character of the constrants 9), 35a) and 35b) supplemented wth the Posson brackets V II),H = = 6) G 2) G 2) II) j,hv allow us to fnalze the Drac algorthm at ths stage. At ths stage, we can conclude that the canoncal Hamltonan 58) s exactly the frst-class Hamltonan of the system. The rreducble character of the frst-class constrants 9) supplemented wth the frst-order reducbltes 54a) 54b) of the secondary frst-class constrants G 2) and G 2) j allow us to count the degrees of freedom for the model under study V II) D D 2)D 3) N DOF =. 61) 2 Fnally, f we return to the Lagrangan formulaton va extended acton), we conclude that n ths stuaton 1) represents a generatng set of gauge transformatons for 2). 3. CONCLUSIONS In ths paper we have analysed the tensor gauge felds of degree three defned on an arbtrary Mnkowsk space-tme. Startng wth the most general second-order Lagrangan densty for the tensor gauge felds of degree three, we have performed the Drac analyss. The procedure has revealed seven dstnct stuatons that can RJP 58Nos. 5-6), ) c)

14 542 E.M. Coroanu 14 occur, cases dctated by the values of the constants that label the startng secondorder Lagrangan densty. It s remarkable that three of the seven cases manfest nterestng conformal 8 and/or topologcal BF-lke 9 behavours of the massless tensor gauge feld of degree three A µν α. REFERENCES 1. A. Ensten, E.G. Straus, Ann. Math. 47, ). 2. E. Schrödnger, Space-Tme Structure Cambrdge Unversty Press, 195). 3. J.W. Moffat, Phys. Rev. D 19, ). 4. Yu.M. Znovev, Frst Order Formalsm for Mxed Symmetry Tensor Felds, arxv:hep-th/ ). 5. P.A.M. Drac, Can. J. Math. 2, ). 6. P.A.M. Drac, Lectures on Quantum Mechancs Academc Press, 1967). 7. M. Henneaux, C. Tetelbom, Quantzaton of Gauge Systems Prnceton Unversty Press, 1992). 8. E.S. Fradkn, A.A. Tseytln, Phys. Rept. 119, ). 9. D. Brmngham, M. Blau, M. Rakowsk, G. Thompson, Phys. Rept. 29, ). RJP 58Nos. 5-6), ) c)

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