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On The Deser-Siegel-Townsend Notivarg ∗ M. Bakalarska, W. Tybor 8 9 Department of Theoretical Physics I 9 University of L ód´z 1 ul.Pomorska 149/153, 90-236 L ód´z , Poland n a J 5 Abstract 1 The interaction of the notivarg with an external Weyl current is dis- v cussed. The continuity equation for the Weyl current is obtained. The 6 canonicalanalysisofthetheoryofthenotivarginteractingwiththeexter- 1 nal Weyl current is performed. The covariant propagator of the notivarg 2 is found. 1 0 8 9 / h p - p e h : v i X r a ∗SupportedbyL o´d´zUniversityGrantNo505/486 1 1 Introduction The notion of the notivarg has been introduced by Deser, Siegel and Townsend [1] as a parallel to the Ogievetsky-Polubarinov notoph [2]. The notivarg is a scalar particle described by the gauge theory. The notivarg field is a twenty component tensor Kµναβ with symmetries of the Riemann tensor. The La- grangian density for the Deser-Siegel-Townsend theory of the free notivarg is [1] 1 1 = (∂ Kµναβ∂ K κ ∂ Kµνα ∂ Kσλ ). (1) 0 µ α νκβ µ ν α σλ L −2 − 3 There exists another description of the notivarg [3,4] given by the Lagrangian density = (∂ Kσναβ)2+(∂ Kσνα )2. (2) 0 σ σ ν L − The descriptions (1) and (2) are not connected by the point transformation [3,4]. The notivarg theory based on the Lagrangian (2) has been investigated with some details: (i) the interaction of the notivarg with the external Weyl current has been discussed in Ref. [5]; (ii) the canonical analysis of the free theory and the theory of the notivarg interacting with the external Weyl current has been performed in Ref. [6]; (iii) the covariant form of the notivarg propagator has been fixed in Ref. [7]. In the present paper the similar program of investigations is performed for the Deser-Siegel-Townsend notivarg. In Section 2 we obtain the conservation law for the external Weyl current. In Section 3 we carry out the canonical analysis of the theory. Its gauge invariance is discussed in Section 4. In Section 5 we obtain the physical Lagrangian demonstrating the pure spin-0 content of the theory. In Section 6 we fix the covariantform of the notivarg propagator. 2 Interaction with external Weyl current Letusdiscussthe notivargtheoryinthe Deser-Siegel-Townsenddescription.We take into accountthe interactionof the notivarg with an external Weyl current jµναβ. The action integral has the form I = d4x( + )= d4x , (3) 0 int L L L Z Z where the free Lagrangian density is given by Eq. (1) and the interaction 0 L term is 1 1 = j Kµναβ = j Cµναβ. (4) int µναβ µναβ L 4 4 The 20-component field Kµναβ has the symmetry of the Riemann tensor, i.e. Kµναβ = Kνµαβ =Kαβµν, ε Kµναβ =0; µναβ − 2 the 10-componentcurrent jµναβ has the symmetry of the Weyl tensor, i.e. it is the Riemann tensor with jµνα = 0 . Cµναβ is the Weyl part of Kµναβ (see ν Appendix I). The free part of the action I = d4x (5) 0 0 L Z is invariant under the following gauge transformations δKµναβ = [εµν ∂λ(∂αωηβ ∂βωηα)+εαβ ∂λ(∂µωην ∂νωηµ)]+ λη λη − − 1 εµναβ(2ωλ ∂ ∂ ωλη); (6) − 3 λ − λ η δKµναβ = gµα(∂νηβ +∂βην)+gνβ(∂µηα+∂αηµ) gµβ(∂νηα+∂αην)+ − gνα(∂µηβ +∂βηµ) 2(gµαgνβ gµβgνα)∂ ησ, (7) σ − − − where the gauge tensor ωαβ is symmetric ωαβ = ωβα. Not all components of ωαβ act effecively because the transformation (6) is invariant under δωαβ =∂αλβ +∂βλα where λα is an arbitrary vector. We note that the transformation (6) varies some components ofthe Weyl partofKµναβ, andthe transformation(7)varies some components of the other parts of the field Kµναβ. The action integral describing the interaction with the external Weyl current I = d4x (8) int int L Z isinvariantunderthegaugetransformation(6)ifthesourceobeysthefollowing condition εαλµν∂ ∂ jσβ =0 (9) λ σ µν where the dual properties of the Weyl tensor are taken into account. Using the decomposition of the Weyl tensor (see Appendix II) jµναβ =(λij,σij) we can rewrite the conservation law (9) in the form ∂ ∂ σij =0, i j ∂0σi+εikp∂kλp =0, (10) [(∂0)2+∆]σij +∂iσj +∂jσi+∂0[εikp∂ λj +εjkp∂ λi]=0, k p k p where σi ∂ σji, λi ∂ λji. j j ≡ ≡ 3 In the helicity components (see Appendix IV) we get σ =0, L ∂0σTi +εikp∂kλTp =0, (11) [(∂0)2+∆]σij( 2)+∂0[εikp∂ λj( 2)+εjkp∂ λi( 2)]=0. ± k p ± k p ± The conservation law (9) can be obtained as well from the field equation following from the variational principle δI = 0. We do not write down this field equation. For futher aim we write the field equation in the covariant gauge Kµν =0, µν ∂ Kµνα =0, (12) µ ν 1 ∂ Kµναβ = (∂αKµν β ∂βKµν α) µ µ µ −2 − It has the following form 2Cµναβ =4jµναβ (13) We note that the gauge conditions (12) lead to εαλµν∂ ∂ Cσβ =0. (14) λ σ µν So, the current conservation law (9) follows from Eq. (13). 3 Canonical analysis Using the decomposition of the Weyl tensor (see Appendix II) jµναβ = λij,σij and the Riemann tensor (see Appendix(cid:0)III) (cid:1) Kµναβ = Tij,Rij,Sij,Ai,T,R wecanrewritetheaction(3)inth(cid:0)ecomponentform. Af(cid:1)tersomeintegrationsby partsweremovethevelocities∂0Ai,∂0Sij and∂0Rfromtheaction. Performing the Legendre transformationwe obtain I = d4x(Pij∂0T +Πij∂0R +P∂0T ), (15) ij ij c −H Z where the canonical momenta are ∂ 1 3 Pms L = ∂0Tms+ ∂0Rms+ (∂sAm+∂mAs)+ ≡ ∂∂0T − 2 2 ms (cid:26) 1 gms∂ Ak [εmnp∂ Ss+εsnp∂ Sm] , − k − 2 n p n p (cid:27) 4 ∂ 1 1 1 Πms L = ∂0Tms+ (∂sAm+∂mAs) gms∂ Ak+ ≡ ∂∂0R − 2 2 − 3 k ms (cid:26) 1 [εmnp∂ Ss+εsnp∂ Sm] , (16) −2 n p n p (cid:27) ∂ 1 4 P L = ∂0T ∂ Ak, ≡ ∂∂0T 3 − 3 k and the canonical Hamiltonian density is 3 1 1 1 = 2(Πij)2 2Π Pij + P2+ Ti∂ T + (∂iT)2+ (∂kRij)2+ c ij i H − 2 6 18 2 1 1 + ∂kRij∂ T (Ri)2 RiT Ri∂ T λ (Tij Rij)+ k ij i i ij 2 − − − 6 − − 1 1 1 +2(Pi+Πi 2∂iP)A ( ∆T ∂ Ri ∂ Ti)R+ i i i − − 9 − 12 − 4 +2[εpnm∂ (Ps Πs )+σps]S , (17) n m− m ps where the following abbreviations are introduced Ti ∂ Tji, Ri ∂ Rji,Pi ∂ Pji, Πi ∂ Πji. The momenta conjugated j j j j ≡ ≡ ≡ ≡ to Ai, Sij and R vanish because the Lagrangian density is independent of the corresponding velocities ∂ ∂ ∂ pi L =0, pij L =0, p L =0. A ≡ ∂∂0A S ≡ ∂∂0S R ≡ ∂∂0R i ij So, there are the following primary constraints Φi =pi , Φij =pij, Φ =p . (18) (1) A (2) S (3) R We introduce the total Hamiltonian [8] H = d3x( +λ Φi +λ Φij +λΦ ), (19) tot Hc i (1) ij (2) (3) Z where λ , λ and λ are Lagrange multipliers. The dynamics is expressed by i ij ∂0a= a,H , (20) { tot}Φ(1)=Φ(2)=Φ(3)=0 where ...,... isthePoissonbracketandaisafunctionofdynamicalvariables. { } The theory is consistent if constraints hold for all times. This leads to the secondary constraints: Φi = Pi+Πi 2∂iP, (4) − Φij = εinp∂ Pj Πj +εjnp∂ Pi Πi +2σij, (5) n p − p n p− p 3 9 Φ = (cid:2)∆T (cid:0)∂ Ri ∂(cid:1) Ti, (cid:0) (cid:1)(cid:3) (21) (6) − 4 i − 4 i Φij = εinm∂ ∆ Tj +Rj +∂j(T +R ) + (7) n m m m m +εjnm(cid:2)∂n (cid:0)∆ Tmi +R(cid:1)mi +∂i(Tm+R(cid:3)m) + +4 ∂0σij(cid:2)+ε(cid:0)inm∂nλjm+(cid:1) εjnm∂nλim , (cid:3) (cid:2) (cid:3) 5 where the conservation law (9) is taken into account. We note that Φij = Φij ( 2)+Φij ( 1), (5) (5) ± (5) ± Φij = Φij ( 2), (22) (7) (7) ± because ∂ ∂ Φij =0 and ∂ Φij =0. The dynamics of the constraints is i j (5) j (7) ∂0Φi = 2Φi , (1) − (4) ∂0Φij = Φij , (2) − (5) 1 ∂0Φ = Φ , (3) 9 (6) 2 ∂0Φi = ∂iΦ , (4) 9 (6) 3 ∂0Φ = ∂ Φi , (23) (6) 2 i (4) 1 ∂0Φij = Φij , (5) 2 (7) ∂0Φij = 2(∆Φij +∂i∂ Φkj +∂j∂ Φki ). (7) − (5) k (5) k (5) Becausetheconstraintscanbe addedtoHamiltonian[9],weconstructthenew Hamiltonian density new = +V(4)Φi +V(5)Φij +V(6)Φ +V(7)Φij , (24) H H0 i (4) ij (5) (6) ij (7) where V(4), V(5), V(6) and V(7) are the Lagrange multipliers, and has the i ij ij H0 following form = 2(Πij)2 2Π Pij + 3P2+ 1Ti∂ T + 0 ij i H − 2 6 + 1 (∂iT)2+ 1(∂kRij)2+ 1∂kRij∂ T + (25) k ij 18 2 2 (Ri)2 RiT 1Ri∂ T λ (Tij Rij). i i ij − − − 6 − − We observe that the variables Ai, Sij and R disappear in the new description. Let us note that according to (22) we have (see Appendix IV) V(5) = V(5)( 2)+V(5)( 1), ij ij ± ij ± V(7) = V(7)( 2). ij (ij) ± TheHamiltoniandensity new isderivable[10]fromthephase-spaceLagrangian H density new =P ∂0Tij +Π ∂0Rij +P∂0T new, (26) ij ij L −H 6 where the Lagrangian multipliers V(4), V(5), V(6) and V(7) are treated as dy- i ij (ij) namical variables. So, passing to the canonical formalism, we find the primary constraints πi =0, πij =0, π =0, πij =0, (27) (4) (5) (6) (7) where πi , πij , π and πij are the canonical momenta conjugated to V(4), (4) (5) (6) (7) i V(5), V(6) and V(7) respectively. Thus the new total Hamiltonian is ij ij Hnew = d3x( new +λ(4)πi )+λ(5)πij +λ(6)π +λ(7)πij ), (28) tot H i (4) ij (5) (6) ij (7) Z where λs are the Lagrange multipliers. The dynamics is expressed by ′ ∂0a= a,Hnew . (29) { tot }π=0 In particular we have ∂0πi = Φi , ∂0πij = Φij , (4) − (4) (5) − (5) ∂0π = Φ , ∂0πij = Φij . (30) (6) − (6) (7) − (7) The time derivatives of Φ , Φ , Φ and Φ are given by Eqs. (23). (4) (5) (6) (7) 4 Gauge transformations Let us discuss the gauge transformations of the free notivarg theory described by the action integral I = d4x new, (31) free Lfree Z where new is obtained from new putting λij = σij = 0. In this limit the Lfree L constraints are Φi = Pi+Πi 2∂iP, (4) − Φij = εinp∂ (Pj Πj)+εjnp∂ (Pi Πi), (5) n p − p n p− p 3 9 Φ = ∆T ∂ Ri ∂ Ti, (32) (6) − 4 i − 4 i Φij = εinm∂ [∆(Tj +Rj )+∂j(T +R )]+ (7) n m m m m +εjnm∂ [∆(Ti +Ri )+∂i(T +R )], n m m m m and they obey the relations Φ ,Φ =0, a,b=4,5,6,7. (a) (b) (cid:8) (cid:9) 7 So, we have the theory with the first class constraints. The generator of the gauge transformations is G= d3x α(4)πi +α(5)πij +α(6)π +α(7)πij (33) i (4) ij (5) (6) ij (7) Z (cid:16) +η(4)Φi +η(5)Φij +η(6)Φ +η(7)Φij , i (4) ij (5) (6) ij (7) (cid:17) where αs and η s are gauge functions. They have the helicity structure as the ′ ′ corresponding constraints. The generator (33) obeys the consistency condition d G =0. (34) dt (cid:12)π=0 (cid:12) Using Eq. (29) we obtain (cid:12) (cid:12) 3 α(4) = ∂0η(4) ∂ η(6), i i − 2 i 2 α(6) = ∂0η(6) ∂iη(4), (35) − 9 i α(5)( 2) = ∂0η(5)( 2) 2∆η(7)( 2), ij ± ij ± − ij ± α(5)( 1) = ∂0η(5)( 1), ij ± ij ± 1 α(7) = ∂0η(7)+ η(5)( 2). ij ij 2 ij ± The gauge transformations are δV(4) V(4),G =α(4), i ≡ i i δV(5) = nα(5), o ij ij δV(6) = α(6), δV(7) = α(7), ij ij δT = 2∂iη(4), i δP = ∆η(6), − 1 1 δTij = ∂iη(4)j +∂jη(4)i + gij∂kη(4)+ −2 3 k +ε(cid:16)inm∂ η(5)j +εjnm(cid:17)∂ η(5)i, (36) n m n m 1 1 δRij = ∂iη(4)j +∂jη(4)i + gij∂kη(4)+ −2 3 k (cid:16) (cid:17) εinm∂ η(5)j +εjnm∂ η(5)i , − n m n m 9 h 1 i δPij = ∂i∂j + gij∆ η(6) ∆ εinm∂ η(7)j +εjnm∂ η(7)i , 4 3 − n m n m (cid:18) (cid:19) h i 3 1 δΠij = ∂i∂j + gij∆ η(6) ∆ εinm∂ η(7)j +εjnm∂ η(7)i . 4 3 − n m n m (cid:18) (cid:19) h i 8 In Appendix V we impose the noncovariant conditions to remove completely the gauge freedom. Using the conservation law of the current (9) we can verify that the action I = d4x new (37) L Z is invariant under the gauge transformations (36). 5 Physical Lagrangian Solving the constraints (21) we obtain from Φ =0: (4) P +Π +2P =0, L L Pi +Πi =0, T T from Φ =0: (5) 2 Pi Πi = εikp∂ σ , T − T ∆ k Tp 1 Pij( 2) Πij( 2) = εikp∂ σj( 2)+εjkp∂ σi( 2) , (38) ± − ± 2∆ k p ± k p ± (cid:2) (cid:3) from Φ =0: (6) 9 3 T T R =0, L L − 4 − 4 from Φ =0: (7) 1 4 Tij( 2)+Rij( 2)= ∂0[εikp∂ σj( 2)+εjkp∂ σi( 2)] λij( 2). ± ± ∆2 k p ± k p ± − ∆ ± Inserting these solutions to the action (37) we get I = d4x , (39) phys L Z where =p∂0ϕ (40) phys free int L −H −H and √3 √3 (p,ϕ)= (P 3Π ), (T R ) , (41) L L L L 2 − 4 − ! or other 8 pairs that can be obtained from (41) using the constraints: 9 3 P +Π +2P =0, and T T R =0. L L L L − 4 − 4 9 The free Hamiltonian density is = 1p2 1(∂iϕ)2 (42) free H 2 − 2 and the interacting one is 1 = 2√3λ ϕ+4λ ( 2) λij( 2)+ (43) int L ij H − ± ∆ ± 1 1 1 +(∂0σij( 2)) (∂0σ ( 2)) 3σij( 2) σ ( 2) 8σi σ . ± ∆2 ij ± − ± ∆ ij ± − T∆2 Ti In the momentum space we get int = 2√3λL( k)ϕ(k)+k2(k0)−2 ~k −2σij( 2, k)σij( 2,k)+ H − − | | ± − ± +8 ~k −4σi ( k)σ (k). (44) | | T − Ti 6 Notivarg propagator Letusconsidertheexchangeofthenotivargbetweentwoexternalcurrents. The genaralstructure ofthe amplitude describing the processin the secondorderof the perturbation theory is [7] a b = jµναβ( k)j (k)+ k jµναβ( k)kσj + A − k2 − µναβ k4 µ − σναβ (cid:18) c + k k jµναβ( k)kσkκj (k) , (45) k6 µ α − σνκβ (cid:17) where a, b, c are number factors. Due to the conservation law (9) we have 1 k k jµναβkσkκj = k2k jµναβkσj . µ α σνκβ µ σναβ 2 Using the following identity for the Weyl tensor 1 jµναβj = δµjκναβj σναβ 4 σ κναβ weobserveonlythefirstterminEq.(45)isindependent. Assumingthefollowing form of the notivarg propagator 1 1 D (k) = (g g g g +g g g g + µναβ,σλγδ −8k2 µσ νλ αγ βδ µλ νσ αδ βσ +g g g g +g g g g g g g g + µγ νδ ασ βλ µδ νγ αλ βσ µλ νσ αγ βδ − g g g g g g g g g g g g ) (46) µσ νλ αδ βγ µγ νδ αλ βσ µδ νγ ασ βλ − − − we obtain the amplitude = jµναβ( k)D (k)jσλγδ(k) (47) µναβ,σλγδ A − − 10