Non-Abelian Tensor Gauge Fields. Enhanced Symmetries

NRCPS-HE-06-35 Non-Abelian Tensor Gauge Fields arXiv:hep-th/0604118v1 17 Apr 2006 Enhanced Symmetries George Savvidy Demokritos National Research Center Institute of Nuclear Physics Plato Laboratory for Theoretical Physics Ag. Paraskevi, GR-15310 Athens,Greece E-mail: [email protected] Abstract We define a group of extended non-Abelian gauge transformations for tensor gauge fields. On this group one can define generalized field strength tensors, which are transforming homogeneously with respect to the extended gauge transformations. The generalized field strength tensors allow to construct two infinite series of gauge invariant quadratic forms. Each term of these infinite series is separately gauge invariant. The invariant Lagrangian is a linear sum of these forms and describes interaction of tensor gauge fields of arbitrarily large integer spins 1, 2, .... It does not contain higher derivatives of the tensor gauge fields, and all interactions take place through three- and four-particle exchanges with dimensionless coupling constant. The first term in this sum is the Yang-Mills Lagrangian. The invariance with respect to the extended gauge transformations does not fix the coefficients - the coupling constants - in front of these forms. There is a freedom to vary them without breaking the extended gauge symmetry. We demonstrate that by an appropriate tuning of these coupling constants one can achieve an enhancement of the extended gauge symmetry. This leads to highly symmetric equations. We present the explicit form of the free equations for the rank-2 and rank-3 gauge fields. Their relation to the Schwinger free equation for the rank-3 gauge fields is discussed. Contents 1 Introduction 2 2 Non-Abelian Tensor Fields 4 3 Enhanced Symmetry. Rank-2 Gauge Field 8 4 Rank-3 Tensor Gauge Field 10 5 Enhanced Symmetry. Rank-3 Gauge Field 12 6 Schwinger Equation for rank-3 Gauge Field 18 7 Appendix A 19 8 Appendix B 20 1 Introduction It is well understood, that the concept of local gauge invariance allows to define nonAbelian gauge fields [1], to derive their dynamical field equations and to develop a universal point of view on matter interactions as resulting from the exchange of gauge quanta of different forms. It is appealing to extend the gauge principle so that it will define the interaction of matter fields which carry not only non-commutative internal charges, but also arbitrary half-integer spins. This extension will induce the interaction of matter fields mediated by a charged gauge quanta carrying a spin larger than one [2, 3]. In our recent approach the gauge fields are defined as rank-(s + 1) tensors [2, 3, 4, 5, 6] Aaµλ1 ...λs and are totally symmetric with respect to the indices λ1 ...λs . A priory the tensor fields have no symmetries with respect to the first index µ. This is an essential departure from the previous considerations, in which the higher-rank tensors were totally symmetric [7, 8, 9, 10, 11, 12, 13, 14, 15, 17]. The index s runs from zero to infinity. The first member of this family of the tensor gauge bosons is the Yang-Mills vector boson Aaµ . The extended non-Abelian gauge transformation of the tensor gauge fields [2, 3] δξ Aaµλ1 λ2 ...λs is defined by the equation (4) and comprise a closed algebraic structure, because the commutator of two transformations can be expressed in the form [ δη , δξ ] Aµλ1 λ2 ...λs = − ig δζ Aµλ1 λ2 ...λs where the gauge parameters {ζ} are given by the matrix commutators (6). This allows to define generalized field strength tensors (7) [2, 3] Gaµν,λ1 ...λs 2 which are transforming homogeneously (8) with respect to the extended gauge transformations (4). The field strength tensors Gaµν,λ1 ....λs are used to construct two infinite series of gauge invariant quadratic forms [2, 3] Ls s = 1, 2, 3... and [3, 4, 5] ′ Ls s = 2, 3, ... Each term of these infinite series is separately gauge invariant with respect to the generalized gauge transformations (4). These forms contain quadratic kinetic terms and nonlinear terms describing nonlinear interaction of the Yang-Mills type. In order to make all tensor gauge fields dynamical one should add all these forms together. Thus the gauge invariant Lagrangian describing dynamical tensor gauge bosons of all ranks has the form L= ∞ X gs Ls + ∞ X ′ ′ gs Ls , (1) s=2 s=1 where L1 ≡ LY M is the Yang-Mills Lagrangian. It is important that: i) the Lagrangian does not contain higher derivatives of tensor gauge fields ii) all interactions take place through the three- and four-particle exchanges with dimensionless coupling constant g iii) the complete Lagrangian contains all higherrank tensor gauge fields and should not be truncated iv) the invariance with respect to the ′ extended gauge transformations does not fix the coupling constants gs and gs . ′ The coupling constants gs and gs remain arbitrary because every term of the sum is separately gauge invariant and the extended gauge symmetry alone does not fix them. This means that there is a freedom to vary these constants without breaking the initial gauge symmetry. The important question to which we should address ourselves here is the following: Can we achieve the enhancement of the initial gauge symmetry properly ′ tuning the coupling constants gs and gs ? ′ Let us consider a simple example: the sum of two Z2 invariant forms gx2 +g y 2 exhibits ′ the U(1) invariance if we choose g = g so that the initial symmetry is elevated to a one parameter family of continuous transformations. A less trivial example is a linear sum of Poincaré invariant forms comprising a SUSY invariant Lagrangian. One can find other examples of the same phenomena when a linear sum of invariant forms of the initial group G exhibits a symmetry with respect to a larger group G ⊃ G when the coefficients are properly tuned. A similar phenomena appears in our system. Indeed let us consider a linear sum of two gauge invariant forms in (1) ′ ′ g2 L2 + g2 L2 which describes the rank-2 tensor gauge field Aaµλ . As we have found in [3, 4, 5] one can ′ ′ ′ chose the coupling constants g2 and g2 so that the sum g2 L2 + g2 L2 exhibits invariance with respect to a bigger gauge group1 . This means that in addition to full extended gauge group (4), which we had initially, now we have bigger gauge group with double number of gauge parameters [3, 4, 5]. The explicit form of the free field equation for the rank-2 1 ′ c2 = g2 /g2 = 1. 3 tensor gauge field is given by equation (30). It was then demonstrated that it describes propagation of two polarizations of helicity-two massless charged tensor gauge boson and of the helicity-zero ”axion”. This result will be recapitulated in the third section. Our aim now is to extend this construction to the rank-3 tensor gauge field. We shall consider the linear sum ′ ′ g3 L3 + g3 L3 and shall demonstrate that for an appropriate choice of the coupling constants c3 = ′ g3 /g3 = 4/3 the system have an enhanced gauge symmetry. The explicit description of this symmetry together with the corresponding free field equation (51) for the rank3 tensor gauge field will be given in the forth and fifth sections. Its relation to the Schwinger equation for the symmetric rank-3 tensor gauge field is discussed in the last seventh section. First let us recapitulate the construction of the general Lagrangian L in (1). Non-Abelian Tensor Fields 2 The gauge fields are defined as rank-(s + 1) tensors [2, 3] Aaµλ1 ...λs (x), s = 0, 1, 2, ... and are totally symmetric with respect to the indices λ1 ...λs . A priory the tensor fields have no symmetries with respect to the first index µ. The index a numerates the generators La of the Lie algebra ğ of a compact2 Lie group G. One can think of these tensor fields as appearing in the expansion of the extended gauge field Aµ (x) over the unite tangent vector eλ [2, 3]: Aµ (x) = ∞ X Aaµλ1 ...λs (x) Laλ1 ...λs . (2) s=0 The gauge field Aaµλ1 ...λs carries indices a, λ1 , ..., λs labeling the generators of extended current algebra G associated with compact Lie group G. It has infinite many generators Laλ1 ...λs = La eλ1 ...eλs and the corresponding algebra is given by the commutator [4] 3 [Laλ1 ...λs , Lbρ1 ...ρk ] = if abc Lcλ1 ...λs ρ1 ...ρk . (3) The extended non-Abelian gauge transformations of the tensor gauge fields are defined by the following equations [2, 3]: δAaµ = (δ ab ∂µ + gf acb Acµ )ξ b , (4) δAaµν = (δ ab ∂µ + gf acb Acµ )ξνb + gf acb Acµν ξ b , b + gf acb (Acµν ξλb + Acµλ ξνb + Acµνλ ξ b ), δAaµνλ = (δ ab ∂µ + gf acb Acµ )ξνλ ......... . ............................ The algebra ğ possesses an orthogonal basis in which the structure constants f abc are totally antisymmetric. 3 See also the alternative expansions in [10, 11, 18, 19, 20, 30, 31] and the algebras based on diffeomorphisms group in [32, 33, 35, 34]. 2 4 where ξλa1 ...λs (x) are totally symmetric gauge parameters. These extended gauge transformations generate a closed algebraic structure. To see that, one should compute the commutator of two extended gauge transformations δη and δξ of parameters η and ξ. The commutator of two transformations can be expressed in the form [2, 3] [ δη , δξ ] Aµλ1 λ2 ...λs = − ig δζ Aµλ1 λ2 ...λs (5) and is again an extended gauge transformation with the gauge parameters {ζ} which are given by the matrix commutators ζ = ζλ1 = ζνλ = ...... . [η, ξ] [η, ξλ1 ] + [ηλ1 , ξ] [η, ξνλ] + [ην , ξλ] + [ηλ , ξν ] + [ηνλ , ξ], .......................... (6) The generalized field strengths are defined as [2, 3] Gaµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν , (7) Gaµν,λ = ∂µ Aaνλ − ∂ν Aaµλ + gf abc ( Abµ Acνλ + Abµλ Acν ), Gaµν,λρ = ∂µ Aaνλρ − ∂ν Aaµλρ + gf abc ( Abµ Acνλρ + Abµλ Acνρ + Abµρ Acνλ + Abµλρ Acν ), ...... . ............................................ and transform homogeneously with respect to the extended gauge transformations (4). The field strength tensors are antisymmetric in their first two indices and are totally symmetric with respect to the rest of the indices. The inhomogeneous extended gauge transformation (4) induces the homogeneous gauge transformation of the corresponding field strength (7) of the form [2, 3] δGaµν = gf abc Gbµν ξ c (8) δGaµν,λ = gf abc ( Gbµν,λ ξ c + Gbµν ξλc ), c δGaµν,λρ = gf abc ( Gbµν,λρ ξ c + Gbµν,λ ξρc + Gbµν,ρ ξλc + Gbµν ξλρ ) ...... . .........................., The field strength tensors are antisymmetric in their first two indices and are totally symmetric with respect to the rest of the indices. The symmetry properties of the field strength Gaµν,λ1 ...λs remain invariant in the course of this transformation. These tensor gauge fields and the corresponding field strength tensors allow to construct two series of gauge invariant quadratic forms. The first series is given by the formula [2, 3]: 1 a G Ga + ....... 4 µν,λ1 ...λs µν,λ1 ...λs 2s X 1X = − asi Gaµν,λ1 ...λi Gaµν,λi+1 ...λ2s ( η λi1 λi2 .......η λi2s−1 λi2s ) , 4 i=0 p′ s Ls+1 = − P (9) where the sum p runs over all nonequal permutations of i′ s, in total (2s − 1)!! terms and s! . the numerical coefficients are asi = i!(2s−i)! 5 The second series of gauge invariant quadratic forms is given by the formula [3, 4, 5]: ′ 1 a G Ga + ....... 4 µλ1 ,λ2 ...λs+1 µλ2 ,λ1 ...λs+1 ′ X X 1 2s+1 s a a a G G ( η λi1 λi2 .......η λi2s+1 λi2s+2 ) , (10) = 8 i=1 i−1 µλ1 ,λ2 ...λi µλi+1 ,λi+2 ...λ2s+2 p′s Ls+1 = P′ where the sum p runs over all nonequal permutations of i′ s, with exclusion of the terms which contain η λ1 ,λi+1 . In order to make all tensor gauge fields dynamical one should add the corresponding kinetic terms. Thus the invariant Lagrangian describing dynamical tensor gauge bosons of all ranks has the form ∞ ∞ L= X gs Ls + X ′ ′ gs Ls , (11) s=2 s=1 where L1 ≡ LY M . It is important that: i) the Lagrangian does not contain higher derivatives of tensor gauge fields ii) all interactions take place through the three- and four-particle exchanges with dimensionless coupling constant g iii) the complete Lagrangian contains all higherrank tensor gauge fields and should not be truncated iv) the invariance with respect to ′ the extended gauge transformations does not fix the coupling constants gs and gs . ′ The coupling constants gs and gs remain arbitrary because every term of the sum is separately gauge invariant and the extended gauge symmetry alone does not fix them. This means that we have a freedom to chose these constants without breaking the initial gauge symmetry. The important question which should be addressed here is the following: Can we achieve the enhancement of the gauge symmetry tuning the coupling constants gs ′ and gs ? Let us consider a linear sum of two gauge invariant forms in (1) ′ ′ g2 L2 + g2 L2 , which describes the rank-2 tensor gauge field Aaµλ . As we have found in [3, 4, 5] one can ′ ′ ′ choose the coupling constants g2 and g2 so that the sum g2 L2 + g2 L2 exhibits invariance with respect to a bigger gauge group (see the next section for details, where we will ′ demonstrate that c2 = g2 /g2 = 1). This means that in addition to full extended gauge group (4), which we had initially, now we have a bigger gauge group with double number of gauge parameters [3, 4, 5]. It was then demonstrated that the corresponding kinetic term describes propagation of two polarizations of helicity-two massless charged tensor gauge boson and the helicity-zero ”axion”. In summary the gauge invariant Lagrangian for the lower-rank tensor gauge fields has the form [3, 4, 5]: ′ 1 a a G G 4 µν µν 1 a G Ga − − 4 µν,λ µν,λ 1 a G Ga + + 4 µν,λ µλ,ν (12) L = L1 + L2 + L2 = − 6 1 a a G G 4 µν µν,λλ 1 a 1 Gµν,ν Gaµλ,λ + Gaµν Gaµλ,νλ . 4 2 The equations of motion which follow from this Lagrangian are: 1 ab b ∇ (G + Gbνλ,µλ + Gbλµ,νλ ) + gf acbAcµλ Gbµν,λ 2 µ µν,λλ 1 acb c b gf (Aµλ Gµλ,ν + Acµλ Gbλν,µ + Acλλ Gbµν,µ + Acµν Gbµλ,λ ) − 2 1 acb c + gf (Aµλλ Gbµν + Acµµλ Gbνλ + Acµνλ Gbλµ ) = 0 2 b ∇ab µ Gµν + (13) and for the second-rank tensor gauge field Aaνλ : 1 ab b ab b ab b ab b b ∇ab µ Gµν,λ − (∇µ Gµλ,ν + ∇µ Gλν,µ + ∇λ Gµν,µ + ηνλ ∇µ Gµρ,ρ ) 2 1 + gf acb Acµλ Gbµν − gf acb(Acµν Gbµλ + Acµµ Gbλν + Acλµ Gbµν + ηνλ Acµρ Gbµρ ) = 0. (14) 2 The variation of the action with respect to the third-rank gauge field Aaνλρ will give the equations 1 ab b 1 ab b ab b ab b b ηλρ ∇ab µ Gµν − (ηνρ ∇µ Gµλ + ηλν ∇µ Gµρ ) + (∇ρ Gνλ + ∇λ Gνρ ) = 0. 2 2 (15) Representing this system of equations in the form 1 a a a a + Fνλ,µλ + Fλµ,νλ ) = jνa ∂µ Fµν + ∂µ (Fµν,λλ 2 (16) 1 a a a a a a ∂µ Fµν,λ − (∂µ Fµλ,ν + ∂µ Fλν,µ + ∂λ Fµν,µ + ηνλ ∂µ Fµρ,ρ ) = jνλ 2 1 1 a a a a a a + ηνλ ∂µ Fµρ ) + (∂ρ Fνλ + ∂λ Fνρ ) = jνλρ , ηλρ ∂µ Fµν − (ηνρ ∂µ Fµλ 2 2 a a a where Fµν = ∂µ Aaν − ∂ν Aaµ , Fµν,λ = ∂µ Aaνλ − ∂ν Aaµλ , Fµν,λρ = ∂µ Aaνλρ − ∂ν Aaµλρ , we can find the corresponding conserved currents jνa = − gf abc Abµ Gcµν − gf abc ∂µ (Abµ Acν ) (17) 1 abc b c 1 a a a − gf Aµ (Gµν,λλ + Gcνλ,µλ + Gcλµ,νλ ) − ∂µ (Iµν,λλ + Iνλ,µλ + Iλµ,νλ ) 2 2 1 − gf abc Abµλ Gcµν,λ + gf abc (Abµλ Gcµλ,ν + Abµλ Gcλν,µ + Abλλ Gcµν,µ + Abµν Gcµλ,λ ) 2 1 abc b gf (Aµλλ Gcµν + Abλµλ Gcνµ + Abµλν Gcλµ ), − 2 a where Iµν,λρ = gf abc ( Abµ Acνλρ + Abµλ Acνρ + Abµρ Acνλ + Abµλρ Acν ) and 1 a jνλ = − gf abc Abµ Gcµν,λ + gf abc (Abµ Gcµλ,ν + Abµ Gcλν,µ + Abλ Gcµν,µ + ηνλ Abµ Gcµρ,ρ ) 2 1 − gf abc Abµλ Gcµν + gf abc (Abµν Gcµλ + Abλµ Gcµν + Abµµ Gcλν + ηνλ Abµρ Gcµρ ) 2 1 − gf abc ∂µ (Abµ Acνλ + Abµλ Acν ) + gf abc [∂µ (Abµ Acλν + Abµν Acλ ) + ∂µ (Abλ Acνµ + Abλµ Acν ) 2 + ∂λ (Abµ Abνµ + Abµµ Acν ) + ηνλ ∂µ (Abµ Abρρ + Abµρ Acρ )], (18) 7 1 a (19) jνλρ = − ηλρ gf abc Abµ Gcµν + gf abc (ηνρ Abµ Gcµλ + ηνλ Abµ Gcµρ − Abρ Gcνλ − Abλ Gcνρ ) 2 1 − ηλρ gf abc ∂µ (Abµ Acν ) + gf abc [∂µ (ηνλ Abµ Acρ + ηνρ Abµ Acλ ) − ∂λ (Abν Acρ ) − ∂ρ (Abν Acλ )]. 2 The conservation of the corresponding currents follows from the fact that we have enhancement of the gauge group and therefore the partial derivatives of the l.h.s. of the equations (16) are equal to zero ∂ν jνa = 0, a ∂ν jνλ = 0, a ∂ν jνλρ = 0, a ∂λ jνλ = 0, a ∂λ jνλρ = 0, a ∂ρ jνλρ = 0. (20) Our aim now is to extend this analysis to the case of rank-3 tensor gauge field. We shall consider the linear sum ′ ′ g3 L3 + g3 L3 and demonstrate that for an appropriate choice of the coupling constants ratio c3 = ′ g3 /g3 = 4/3 the system will have an enhanced symmetry. The explicit form of the Lagrangian for the third-rank tensor gauge field Aaµνλ can be obtained from our general formulas (9), (10) and (11) when we substitute s = 2. The Lagrangian is: 1 1 1 1 a Gµν,λρ Gaµν,λρ − Gaµν,λλ Gaµν,ρρ − Gaµν,λ Gaµν,λρρ − Gaµν Gaµν,λλρρ + 4 8 2 8 1 1 1 a (21) + c3 { Gµν,λρ Gaµλ,νρ + Gaµν,νλ Gaµρ,ρλ + Gaµν,νλ Gaµλ,ρρ + 4 4 4 1 1 1 1 a Gµν,λ Gaµλ,νρρ + Gaµν,λ Gaµρ,νλρ + Gaµν,ν Gaµλ,λρρ + Gaµν Gaµλ,νλρρ }, + 4 2 4 4 ′ L 3 + c3 L 3 = − ′ where c3 = g3 /g3 is a constant4 . Enhanced Symmetry. Rank-2 Gauge Field 3 As we have seen above there are two invariant forms for the rank-2 tensor gauge field L2 ′ ′ ′ and L2 and the general Lagrangian is a linear combination L2 + c2 L2 , where c2 = g2 /g2 is a constant coefficient. Let us review how this coefficient has been fixed by the requirement of an enhanced symmetry. For that let us consider the situation at the linearized level when the gauge coupling constant g is equal to zero. The free part of the L2 Lagrangian 1 1 Lf2 ree = Aaαά (ηαγ ηάγ́ ∂ 2 − ηάγ́ ∂α ∂γ )Aaγγ́ = Aaαά Hαάγγ́ Aaγγ́ , 2 2 where the quadratic form in the momentum representation has the form Hαάγγ́ (k) = (−k 2 ηαγ + kα kγ )ηάγ́ , is obviously invariant with respect to the gauge transformation δAaµλ = ∂µ ξλa , but it is not invariant with respect to the alternative gauge transformations δ̃Aaµλ = ∂λ ηµa . This can be seen, for example, from the following relations in momentum representation kα Hαάγγ́ (k) = 0, kά Hαάγγ́ (k) = −(k 2 ηαγ − kα kγ )kγ́ 6= 0. 4 (22) It is not difficult to present the explicit form of the Lagrangian for any higher-rank tensor field using expressions (9), (10) and (11). 8 Let us consider now the free part of the second Lagrangian ′ 1 L2f ree = Aaαά (−ηαγ́ ηάγ ∂ 2 − ηαά ηγγ́ ∂ 2 + ηαγ́ ∂ά ∂γ + ηάγ ∂α ∂γ́ + ηαά ∂γ ∂γ́ + 4 1 ′ +ηγγ́ ∂α ∂ά − 2ηαγ ∂ά ∂γ́ )Aaγγ́ = Aaαά Hαάγγ́ Aaγγ́ , 2 (23) where 1 1 ′ Hαάγγ́ (k) = (ηαγ́ ηάγ + ηαά ηγγ́ )k 2 − (ηαγ́ kά kγ + ηάγ kα kγ́ + ηαά kγ kγ́ + ηγγ́ kα kά − 2ηαγ kά kγ́ ). 2 2 It is again invariant with respect to the gauge transformation δAaµλ = ∂µ ξλa , but it is not invariant with respect to the gauge transformations δ̃Aaµλ = ∂λ ηµa , as one can see from analogous relations ′ ′ kά Hαάγγ́ (k) = (k 2 ηαγ − kα kγ )kγ́ 6= 0. kα Hαάγγ́ (k) = 0, (24) ′ As it is obvious from (22) and (24), the total Lagrangian Lf2 ree + L2f ree now poses new enhanced invariance with respect to the larger, eight-parameter, gauge transformations δAaµλ = ∂µ ξλa + ∂λ ηµa , (25) where ξλa and ηµa are eight arbitrary functions, because ′ ′ kα (Hαάγγ́ + Hαάγγ́ ) = 0, kά (Hαάγγ́ + Hαάγγ́ ) = 0. (26) Thus our free part of the Lagrangian is Ltot 2 f ree 1 ∂µ Aaνλ ∂µ Aaνλ + 2 1 + ∂µ Aaνλ ∂µ Aaλν − 4 1 ∂µ Aaνν ∂µ Aaλλ − + 4 = − 1 ∂µ Aaνλ ∂ν Aaµλ + 2 1 ∂µ Aaνλ ∂λ Aaµν − 4 1 ∂µ Aaνν ∂λ Aaµλ + 2 1 1 ∂ν Aaµλ ∂µ Aaλν + ∂ν Aaµλ ∂λ Aaµν 4 4 1 ∂ν Aaµν ∂λ Aaµλ (27) 4 or, in equivalent form, it is Ltot 2 f ree 1 1 1 = Aaαά {(ηαγ ηάγ́ − ηαγ́ ηάγ − ηαά ηγγ́ )∂ 2 − ηάγ́ ∂α ∂γ − ηαγ ∂ά ∂γ́ + 2 2 2 1 + (ηαγ́ ∂ά ∂γ + ηάγ ∂α ∂γ́ + ηαά ∂γ ∂γ́ + ηγγ́ ∂α ∂ά )}Aaγγ́ 2 (28) and is invariant with respect to the larger gauge transformations δAaµλ = ∂µ ξλa + ∂λ ηµa , where ξλa and ηµa are eight arbitrary functions. In momentum representation the quadratic form is 1 1 Hαtotάγγ́ (k) = (−ηαγ ηάγ́ + ηαγ́ ηάγ + ηαά ηγγ́ )k 2 + ηαγ kά kγ́ + ηάγ́ kα kγ 2 2 1 − (ηαγ́ kά kγ + ηάγ kα kγ́ + ηαά kγ kγ́ + ηγγ́ kα kά ). 2 9 (29) Free equations of motion which follow from the Lagrangian (28) will take the form 1 1 1 1 ∂ 2 (Aaνλ − Aaλν ) − ∂ν ∂µ (Aaµλ − Aaλµ ) − ∂λ ∂µ (Aaνµ − Aaµν ) + ∂ν ∂λ (Aaµµ − Aaµµ ) 2 2 2 2 1 + ηνλ (∂µ ∂ρ Aaµρ − ∂ 2 Aaµµ ) = 0 (30) 2 and describe the propagation of massless particles of spin 2 and spin 0. It is also easy to see that for the symmetric tensor gauge fields Aaνλ = Aaλν our equation reduces to the Einstein-Fierz-Pauli-Schwinger-Chang-Singh-Hagen-Fronsdal equation ∂ 2 Aνλ − ∂ν ∂µ Aµλ − ∂λ ∂µ Aµν + ∂ν ∂λ Aµµ + ηνλ (∂µ ∂ρ Aµρ − ∂ 2 Aµµ ) = 0, which describes the propagation of massless boson with two physical polarizations, the s = ±2 helicity states. For the antisymmetric fields it reduces to the equation ∂ 2 Aνλ − ∂ν ∂µ Aµλ + ∂λ ∂µ Aµν = 0, and describes the propagation of one physical polarization s = 0, the zero helicity state. The above consideration brings the final form of the gauge invariant Lagrangian for the lower-rank tensor gauge fields to the form (12). 4 Rank-3 Tensor Gauge Field ′ The Lagrangian L1 + g2 (L2 + L2 ) contains the third-rank gauge fields Aaµνλ , but without corresponding kinetic term. In order to make the fields Aaµνλ dynamical we have to add ′ ′ the corresponding Lagrangian g3 L3 + g3 L3 presented in (21), so that at this level the total Lagrangian is the sum [3, 4, 5] ′ ′ L = L1 + g2 (L2 + L2 ) + g3 (L3 + c3 L3 ) + ..., ′ where c3 = g3 /g3. The Lagrangian L3 has the form (21): 1 1 1 1 L3 = − Gaµν,λρ Gaµν,λρ − Gaµν,λλ Gaµν,ρρ − Gaµν,λ Gaµν,λρρ − Gaµν Gaµν,λλρρ , 4 8 2 8 where the field strength tensors (7) are (31) Gaµν,λρσ = ∂µ Aaνλρσ − ∂ν Aaµλρσ + gf abc { Abµ Acνλρσ + Abµλ Acνρσ + Abµρ Acνλσ + Abµσ Acνλρ + +Abµλρ Acνσ + Abµλσ Acνρ + Abµρσ Acνλ + Abµλρσ Acν } and Gaµν,λρσδ = ∂µ Aaνλρσδ − ∂ν Aaµλρσδ + gf abc { Abµ Acνλρσδ + + X Abµλ Acνρσδ + λ↔ρ,σ,δ X Abµλρ λ,ρ↔σ,δ Acνσδ + X Abµλρσ Acνδ + Abµλρσδ Acν }. λ,ρ,σ↔δ The terms in parentheses are symmetric over λρσ and λρσδ respectively. The Lagrangian L3 is invariant with respect to the extended gauge transformations (4) of the low-rank gauge fields Aµ , Aµν , Aµνλ and of the fourth-rank gauge field (4) δξ Aµνλρ = ∂µ ξνλρ − ig[Aµ , ξνλρ] − ig[Aµν , ξλρ] − ig[Aµλ , ξνρ] − ig[Aµρ , ξνλ ] − −ig[Aµνλ , ξρ] − ig[Aµνρ , ξλ ] − ig[Aµλρ , ξν ] − ig[Aµνλρ , ξ] 10 and of the fifth-rank tensor gauge field (4) δξ Aµνλρσ = ∂µ ξνλρσ − ig[Aµ , ξνλρσ ] − ig −ig X X [Aµν , ξλρσ ] − ν↔λρσ [Aµνλ , ξρσ ] − ig νλ↔ρσ X [Aµνλρ , ξσ ] − ig[Aµνλρ , ξ], νλρ↔σ where the gauge parameters ξνλρ and ξνλρσ are totally symmetric rank-3 and rank-4 tensors. The extended gauge transformation of the higher-rank tensor gauge fields induces the gauge transformation of the fields strengths of the form (8) δGaµν,λρσ = gf abc ( Gbµν,λρσ ξ c + Gbµν,λρ ξσc + Gbµν,λσ ξρc + Gbµν,ρσ ξλc + +Gbµν,λ and δGaµν,λρσδ = gf abc ( Gbµν,λρσδ ξ c + X c ξρσ + Gbµν,ρ X + Gbµν,σ (32) c ξλρ Gbµν,λρσ ξδc + λρ,σ↔δ + c ξλσ c Gbµν,λρ ξσδ + λρ↔σ,δ + Gbµν c ξλρσ ) (33) X c c Gbµν,λ ξρσδ + Gbµν ξλρσδ ). λ↔ρ,σ,δ Using the above homogeneous transformations for the field strengths tensors one can demonstrate the invariance of the Lagrangian L3 with respect to the extended gauge transformations (see reference [2] for details). The second invariant Lagrangian can also be constructed explicitly in terms of the above field strength tensors. The following seven Lorentz invariant quadratic forms can be constructed by the corresponding field strength tensors [3, 4, 5] Gaµν,λρ Gaµλ,νρ , Gaµν,νλ Gaµρ,ρλ , Gaµν,νλ Gaµλ,ρρ , Gaµν,λ Gaµλ,νρρ , Gaµν,λ Gaµρ,νλρ , Gaµν,ν Gaµλ,λρρ , Gaµν Gaµλ,νλρρ . (34) Calculating the variation of each of these terms with respect to the gauge transformation (8), (32) and (33) one can get convinced that the particular linear combination 1 1 1 a Gµν,λρ Gaµλ,νρ + Gaµν,νλ Gaµρ,ρλ + Gaµν,νλ Gaµλ,ρρ 4 4 4 1 a 1 a 1 a 1 a a + Gµν,λ Gµλ,νρρ + Gµν,λ Gµρ,νλρ + Gµν,ν Gaµλ,λρρ + Gaµν Gaµλ,νλρρ . (35) 4 2 4 4 forms an invariant Lagrangian (see Appendix A). In summary we have the following Lagrangian for the third-rank gauge field Aaµνλ ′ L3 = 1 1 1 1 a Gµν,λρ Gaµν,λρ − Gaµν,λλ Gaµν,ρρ − Gaµν,λ Gaµν,λρρ − Gaµν Gaµν,λλρρ + 4 8 2 8 1 a 1 1 + c3 { Gµν,λρ Gaµλ,νρ + Gaµν,νλ Gaµρ,ρλ + Gaµν,νλ Gaµλ,ρρ + (36) 4 4 4 1 1 1 1 a Gµν,λ Gaµλ,νρρ + Gaµν,λ Gaµρ,νλρ + Gaµν,ν Gaµλ,λρρ + Gaµν Gaµλ,νλρρ }, + 4 2 4 4 where c3 is an arbitrary constant. Our intention is to investigate the dependence of ′ symmetries of the system L3 + c3 L3 as a function of constant c3 . The system is always invariant with respect to the initial, extended gauge group of transformations (8), (32) and (33) for any value of the constant c3 . We wish to know if there exists a special value of the constant c3 at which the system will have even higher symmetry, as it happens in the case of the rank-2 gauge field. We shall see that this indeed takes place. ′ L 3 + c3 L 3 = − 11 Enhanced Symmetry. Rank-3 Gauge Field 5 The free part of the Lagrangian L3 comes from the terms 1 1 − Gaµν,λρ Gaµν,λρ − Gaµν,λλ Gaµν,ρρ 4 8 (37) and has the form 1 1 1 1 ∂µ Aaνλρ ∂µ Aaνλρ + ∂µ Aaνλρ ∂ν Aaµλρ − ∂µ Aaνλλ ∂µ Aaνρρ + ∂µ Aaνλλ ∂ν Aaµρρ 2 2 4 4 1 1 1 1 a A ′ ′′ (ηαγ ∂ 2 − ∂α ∂γ )( ηα′ γ ′ ηα′′ γ ′′ + ηα′ γ ′ ηα′′ γ ′′ + ηα′ α′′ ηγ ′ γ ′′ )Aaγγ ′ γ ′′ = 2 αα α 2 2 2 1 a (38) A ′ ′′ H ′ ′′ ′ ′′ Aa ′ ′′ , = 2 αα α αα α γγ γ γγ γ Lf3 ree = − where the quadratic form in the momentum representation is 1 Hαα′ α′′ γγ ′ γ ′′ (k) = − Hαγ (ηα′ γ ′ ηα′′ γ ′′ + ηα′ γ ′′ ηα′′ γ ′ + ηα′ α′′ ηγ ′ γ ′′ ), 2 where Hαγ = k 2 ηαγ − kα kγ and by construction is invariant with respect to the gauge transformation a δAaµνλ = ∂µ ξνλ because we have kα Hαα′ α′′ γγ ′ γ ′′ (k) = 0. But it is not invariant with respect to the alternative gauge transformations a a δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , a where the gauge parameter ζµλ is a totaly symmetric tensor. This can be seen from the following relation in momentum representation 1 kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = − Hαγ (kγ ′ ηα′′ γ ′′ + kγ ′′ ηα′′ γ ′ + kα′′ ηγ ′ γ ′′ ) 6= 0. 2 (39) ′ Let us consider now the free part of the Lagrangian L3 which comes from the terms 1 a 1 1 Gµν,λρ Gaµλ,νρ + Gaµν,νλ Gaµρ,ρλ + Gaµν,νλ Gaµλ,ρρ , 4 4 4 thus ′ 1 1 ∂µ Aaνλρ ∂µ Aaλνρ − ∂µ Aaνλρ ∂λ Aaµνρ − 4 4 1 1 ∂µ Aaννλ ∂µ Aaρρλ − ∂µ Aaννλ ∂ρ Aaµρλ − + 4 4 1 1 + ∂µ Aaννλ ∂µ Aaλρρ − ∂µ Aaννλ ∂λ Aaµρρ − 4 4 1 a ′ A ′ ′′ H ′ ′′ ′ ′′ Aa ′ ′′ , = 2 αα α αα α γγ γ γγ γ L3 f ree = + 12 1 ∂ν Aaµλρ ∂µ Aaλνρ + 4 1 ∂ν Aaµνλ ∂µ Aaρρλ + 4 1 ∂ν Aaµνλ ∂µ Aaλρρ + 4 1 ∂ν Aaµλρ ∂λ Aaµνρ 4 1 ∂ν Aaµνλ ∂ρ Aaµρλ 4 1 ∂ν Aaµνλ ∂λ Aaµρρ 4 (40) ′ ′ ′′ ′ ′′ where one should symmetrize the Hαα′ α′′ γγ ′ γ ′′ over the α ↔ α , γ ↔ γ and the exchange ′ ′′ ′ ′′ of two sets of indices αα α ↔ γγ γ , so that the second quadratic form in the momentum representation is (see also Appendix B for derivation) 1 ′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + (k 2 ηαα′ − kα kα′ )(ηα′′ γ ηγ ′ γ ′′ + ηα′′ γ ′ ηγγ ′′ + ηα′′ γ ′′ ηγγ ′ ) 8 + (k 2 ηαα′′ − kα kα′′ )(ηα′ γ ηγ ′ γ ′′ + ηα′ γ ′ ηγγ ′′ + ηα′ γ ′′ ηγγ ′ ) + (k 2 ηαγ ′ − kα kγ ′ )(ηα′ γ ηα′′ γ ′′ + ηα′ γ ′′ ηα′′ γ + ηα′ α′′ ηγγ ′′ ) + (k 2 ηαγ ′′ − kα kγ ′′ )(ηα′ γ ηα′′ γ ′ + ηα′ γ ′ ηα′′ γ + ηα′ α′′ ηγγ ′ ) } 1 − { + 8 + + + 1 + { + 4 + kγ kα′ (ηαγ ′ ηα′′ γ ′′ + ηαγ ′′ ηα′′ γ ′ + ηαα′′ ηγ ′ γ ′′ ) kγ kα′′ (ηαγ ′ ηα′ γ ′′ + ηαγ ′′ ηα′ γ ′ + ηαα′ ηγ ′ γ ′′ ) kγ kγ ′ (ηαα′ ηα′′ γ ′′ + ηαα′′ ηα′ γ ′′ + ηαγ ′′ ηα′ α′′ ) kγ kγ ′′ (ηαα′ ηα′′ γ ′ + ηαα′′ ηα′ γ ′ + ηαγ ′ ηα′ α′′ ) } ηαγ (kα′ kγ ′ ηα′′ γ ′′ + kα′ kγ ′′ ηα′′ γ ′ + kα′′ kγ ′ ηα′ γ ′′ kα′′ kγ ′′ ηα′ γ ′ + kα′ kα′′ ηγ ′ γ ′′ + kγ ′ kγ ′′ ηα′ α′′ ) }. (41) a It is again invariant with respect to the transformation δAaµνλ = ∂µ ξνλ because we have ′ kα Hαα′ α′′ γγ ′ γ ′′ (k) = 0, a a but it is not invariant with respect to the transformation δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , as one can see from the analogous relation (see also Appendix B for derivation) 1 ′ kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + 8 + + 1 − { + 4 1 + { + 4 Hαα′′ (kγ ′ ηγγ ′′ + kγ ′′ ηγγ ′ ) Hαγ ′ (kγ ′′ ηα′′ γ + kα′′ ηγγ ′′ ) Hαγ ′′ (kγ ′ ηα′′ γ + kα′′ ηγγ ′ ) } kγ kα′′ (kγ ′′ ηαγ ′ + kγ ′ ηαγ ′′ ) + kγ kγ ′ kγ ′′ ηαα′′ − 3ηαγ kα′′ kγ ′ kγ ′′ } 6 0. Hαγ (kγ ′ ηα′′ γ ′′ + kγ ′′ ηα′′ γ ′ + kα′′ ηγ ′ γ ′′ ) } = (42) We have to see now whether the sum ′ ′ kα′ ( Hαα′ α′′ γγ ′ γ ′′ + c3 Hαα′ α′′ γγ ′ γ ′′ ) can be made equal to zero by an appropriate choice of the coefficient c3 . For that let us compare the expressions (39) and (42) for divergences. As one can see, only the last term in (42) Hαγ (kγ ′ ηα′′ γ ′′ + kγ ′′ ηα′′ γ ′ + kα′′ ηγ ′ γ ′′ ) and the whole term (39) can cancel each other if we choose c3 = 2, but this will leave the rest of the terms in (42) untouched, thus ′ ′ kα′ ( Hαα′ α′′ γγ ′ γ ′′ + c3 Hαα′ α′′ γγ ′ γ ′′ ) 6= 0 13 for all values of c3 . This situation differs from the case of the rank-2 gauge field Aaµν . In the last case we were able to choose the coefficient c2 = 1 so that the divergences (22) and (24) cancel each other and we got (26) ′ kά (Hαάγγ́ + Hαάγγ́ ) = 0. ′ Therefore the Lagrangian Lf2 ree + Lf2 ree has enhanced invariance with respect to a large gauge group of transformations δAaµλ = ∂µ ξλa + ∂λ ηµa . In order to understand the reason why in the case of the rank-3 gauge field it is impossible fully cancel divergences we have to analyze the corresponding field equations. We shall compare the resulting equation with the equation derived by Schwinger [12] for the totally symmetric Abelian rank-3 tensor field in order to get better insight into the problem. It has been proved by Schwinger [12] that it is impossible to derive field equation for the totally symmetric rank-3 tensor which is invariant with respect to the a full gauge group of transformations δAµνλ = ∂µ ξνλ + ∂ν ξµλ + ∂λ ξµν without imposing some restriction on the gauge parameters ξµν . As he demonstrated, the gauge parameter should be traceless ξµµ = 0. We shall see that similar phenomena take place also in our case, that a is, the second gauge parameter ζµλ should fulfill constraint which we shall derive below (see equation (44)). What we would like to prove is that the equation has enhanced invariance with respect to the gauge group of transformations a a δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , (43) a but in our case the gauge parameters ζµλ should fulfill the following constraint: a a ∂ρ ζρλ − ∂λ ζρρ = 0. (44) This takes place when we choose the coefficient c3 = 4/3 . Indeed, let us consider the equation of motion. From the first form Lf3 ree we have the following contribution to the field equation: 1 Hαα′ α′′ γγ ′ γ ′′ Aaγγ ′ γ ′′ = ∂ 2 Aaαα′ α′′ − ∂α ∂ρ Aaρα′ α′′ + ηα′ α′′ (∂ 2 Aaαρρ − ∂α ∂ρ Aaρλλ ), 2 (45) ′ and from the second one L3 f ree we have 1 ′ Hαα′ α′′ γγ ′ γ ′′ Aaγγ ′ γ ′′ = − {∂ 2 (Aaα′ αα′′ + Aaα′ α′′ α + Aaα′′ αα′ + Aaα′′ α′ α ) − 8 −∂α ∂ρ (Aaα′ ρα′′ + Aaα′ α′′ ρ + Aaα′′ ρα′ + Aaα′′ α′ ρ ) − −∂α′ ∂ρ (Aaραα′′ + Aaρα′′ α − Aaαρα′′ − Aaαα′′ ρ − Aaαα′′ ρ − Aaαρα′′ ) − −∂α′′ ∂ρ (Aaραα′ + Aaρα′ α − Aaαρα′ − Aaαα′ ρ − Aaαα′ ρ − Aaαρα′ ) − −∂α ∂α′ (Aaα′′ ρρ + Aaρα′′ ρ + Aaρρα′′ ) − ∂α ∂α′′ (Aaα′ ρρ + Aaρα′ ρ + Aaρρα′ ) + 2∂α′ ∂α′′ Aaαρρ } − 1 − {ηαα′ [∂ 2 (Aaα′′ ρρ + Aaρα′′ ρ + Aaρρα′′ ) − ∂α′′ ∂ρ Aaρλλ − ∂λ ∂ρ (Aaρα′′ λ + Aaρλα′′ )] + 8 +ηαα′′ [∂ 2 (Aaα′ ρρ + Aaρα′ ρ + Aaρρα′ ) − ∂α′ ∂ρ Aaρλλ − ∂λ ∂ρ (Aaρα′ λ + Aaρλα′ )] + +ηα′ α′′ [∂ 2 (Aaραρ + Aaρρα ) − ∂α ∂ρ (Aaλρλ + Aaλλ ρ) − ∂λ ∂ρ (Aaραλ + Aaλρα − 2Aaαλρ )]}. (46) 14 Summing these two pieces together we shall get the following free field equation of motion for the rank-3 tensor gauge field: c3 c3 ′ (Hαα′ α′′ γγ ′ γ ′′ + c3 Hαα′ α′′ γγ ′ γ ′′ )Aaγγ ′ γ ′′ = ∂ 2 (Aaαα′ α′′ − Aaα′ α′′ α − Aaα′′ αα′ ) − 4 4 c3 a c3 a c3 a a a −∂α ∂ρ (Aρα′ α′′ − Aα′ α′′ ρ − Aα′′ ρα′ ) − ∂α′ ∂ρ (Aαρα′′ + Aαα′′ ρ − Aaραα′′ ) − 4 4 4 c3 c3 a a a − ∂α′′ ∂ρ (Aαρα′ + Aαα′ ρ − Aραα′ ) + ∂α ∂α′ (Aaα′′ ρρ + Aaρα′′ ρ + Aaρρα′′ ) + 4 8 c3 c3 a + ∂α ∂α′′ (Aα′ ρρ + Aaρα′ ρ + Aaρρα′ ) − ∂α′ ∂α′′ Aaαρρ − 8 4 c3 − ηαα′ (∂ 2 Aaα′′ ρρ − ∂α′′ ∂ρ Aaρλλ + 2∂ 2 Aaρρα′′ − 2∂λ ∂ρ Aaρλα′′ ) − 8 c3 − ηαα′′ (∂ 2 Aaα′ ρρ − ∂α′ ∂ρ Aaρλλ + 2∂ 2 Aaρρα′ − 2∂λ ∂ρ Aaρλα′ ) − 8 c c3 c3 c3 1 3 + ηα′ α′′ (∂ 2 Aaαρρ − ∂α ∂ρ Aaρλλ − ∂ 2 Aaρρα + ∂α ∂ρ Aaλλρ − ∂λ ∂ρ Aaαλρ + ∂λ ∂ρ Aaλρα ) = 0. 2 2 2 2 2 We shall prove that this equation is invariant with respect to the gauge transformation a a δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν if we choose the coefficient c3 = 4/3. Performing the above gauge transformation of the field one can see that the terms which originate from differential operators ∂ 2 , ∂α ∂ρ , ∂α′ ∂ρ and ∂α′′ ∂ρ in the above equation cancel each other if we choose c3 = 4 . 3 (47) The rest of the terms have the following form: 1 a ′′ + ∂α ∂α′ ∂ρ ζρα 3 4 ′ (Hαα′ α′′ γγ ′ γ ′′ + Hαα′ α′′ γγ ′ γ ′′ )δ̃Aaγγ ′ γ ′′ = 3 2 1 4 a a a + ∂α ∂α′ ∂α′′ ζρρ + ∂α ∂α′′ ∂ρ ζρα ′ − ∂α′ ∂α′′ ∂ρ ζρα 3 3 3 1 2 a a a ′′ − 4∂ ′′ ∂λ ∂ρ ζ − ηαα′ (2∂ρ ∂ 2 ζρα λρ + 2∂α′′ ∂ ζρρ ) α 6 1 2 a a a ′ − 4∂ ′ ∂λ ∂ρ ζ − ηαα′′ (2∂ρ ∂ 2 ζρα λρ + 2∂α′ ∂ ζρρ ) α 6 1 a a − ∂α ∂λ ∂ρ ζλρ ) + ηα′ α′′ (∂ρ ∂ 2 ζρα 3 (48) and can be rewritten in the form which makes the desired invariance explicit: 1 4 ′ a a ′′ − ∂ ′′ ζ (Hαα′ α′′ γγ ′ γ ′′ + Hαα′ α′′ γγ ′ γ ′′ )δAaγγ ′ γ ′′ = + ∂α ∂α′ (∂ρ ζρα α ρρ ) 3 3 1 4 a a a a + ∂α ∂α′′ (∂ρ ζρα ′ − ∂ ′ζ α ρρ ) − ∂α′ ∂α′′ (∂ρ ζρα − ∂α ζρρ ) 3 3 1 a a a a ′′ − ∂ ′′ ζ − ηαα′ [∂ 2 (∂ρ ζρα α ρρ ) + 2∂α′′ ∂λ (∂λ ζρρ − ∂ρ ζρλ )] 3 1 a a a a − ηαα′′ [∂ 2 (∂ρ ζρα ′ − ∂ ′ ζρρ ) + 2∂ ′ ∂λ (∂λ ζρρ − ∂ρ ζρλ )] α α 3 1 a a a a − ∂α ζρρ ) + ∂α ∂λ (∂λ ζρρ − ∂ρ ζρλ )]. + ηα′ α′′ [∂ 2 (∂ρ ζρα 3 15 (49) From that we see that if the gauge parameter satisfies the conditions (44) a a ∂ρ ζρλ − ∂λ ζρρ =0 the equation is indeed invariant with respect to a larger group of gauge transformations a a δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , because 4 ′ (Hαα′ α′′ γγ ′ γ ′′ + Hαα′ α′′ γγ ′ γ ′′ )δ̃Aaγγ ′ γ ′′ = 0. 3 (50) The final form of the equation is 1 1 1 1 ∂ 2 (Aaαα′ α′′ − Aaα′ α′′ α − Aaα′′ αα′ ) − ∂α ∂ρ (Aaρα′ α′′ − Aaα′ α′′ ρ − Aaα′′ ρα′ ) − (51) 3 3 3 3 1 1 a a a a a − ∂α′ ∂ρ (Aαρα′′ + Aαα′′ ρ − Aραα′′ ) − ∂α′′ ∂ρ (Aαρα′ + Aαα′ ρ − Aaραα′ ) + 3 3 1 1 1 a a a a + ∂α ∂α′ (Aα′′ ρρ + Aρα′′ ρ + Aρρα′′ ) + ∂α ∂α′′ (Aα′ ρρ + Aaρα′ ρ + Aaρρα′ ) − ∂α′ ∂α′′ Aaαρρ − 6 6 3 1 a 2 a 2 a − ηαα′ (∂ Aα′′ ρρ − ∂α′′ ∂ρ Aρλλ + 2∂ Aρρα′′ − 2∂λ ∂ρ Aaρλα′′ ) − 6 1 − ηαα′′ (∂ 2 Aaα′ ρρ − ∂α′ ∂ρ Aaρλλ + 2∂ 2 Aaρρα′ − 2∂λ ∂ρ Aaρλα′ ) − 6 2 2 2 2 1 + ηα′ α′′ (∂ 2 Aaαρρ − ∂α ∂ρ Aaρλλ − ∂ 2 Aaρρα + ∂α ∂ρ Aaλλρ − ∂λ ∂ρ Aaαλρ + ∂λ ∂ρ Aaλρα ) = 0 2 3 3 3 3 and it is invariant with respect to the gauge group of transformations a δAaµνλ = ∂µ ξνλ , a a δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , a a ∂ρ ζρλ − ∂λ ζρρ = 0. (52) a One should stress that there are no restrictions on the gauge parameters ξµν . The above invariance of the equation now can be checked directly without referring to the previous analysis. In summary, we have the following Lagrangian for the third-rank gauge field Aaµνλ : 4 ′ 1 a L3 + L3 = − G Ga − 3 4 µν,λρ µν,λρ 1 a G Ga + + 3 µν,λρ µλ,νρ 1 a G Ga + + 3 µν,λ µλ,νρρ 1 a 1 1 Gµν,λλ Gaµν,ρρ − Gaµν,λ Gaµν,λρρ − Gaµν Gaµν,λλρρ + 8 2 8 1 a 1 G Ga + Ga Ga + (53) 3 µν,νλ µρ,ρλ 3 µν,νλ µλ,ρρ 2 a 1 1 Gµν,λ Gaµρ,νλρ + Gaµν,ν Gaµλ,λρρ + Gaµν Gaµλ,νλρρ . 3 3 3 We shall present the free equation of motion (51) also in terms of field strength tensors. The kinetic term of the above Lagrangian is 4 ′ 1 a 1 a a a L3 + L3 |f ree = − Fµν,λρ Fµν,λρ − Fµν,λλ Fµν,ρρ 3 4 8 1 a 1 a 1 a a a a Fµν,λρ Fµλ,νρ + Fµν,νλ Fµρ,ρλ + Fµν,νλ Fµλ,ρρ , + 3 3 3 where a Fµν,λρ = ∂µ Aaνλρ − ∂ν Aaµλρ . 16 (54) The variation of the above Lagrangian over the field Aaνλρ gives the free equation written a in terms of field strength tensor Fµν,λρ and it is identical to the equation (51) 1 1 1 1 a a a a a ∂µ Fµν,λρ − ∂µ Fµλ,νρ − ∂µ Fµρ,νλ + ∂µ Fνλ,µρ + ∂µ Fνρ,µλ + 3 3 3 3 1 1 1 1 a a a a + ∂ρ Fνµ,µλ + ∂λ Fνρ,µµ + ∂ρ Fνλ,µµ − + ∂λ Fνµ,µρ 3 3 6 6 1 1 1 1 a a a a −ηλν ( ∂µ Fµσ,σρ + ∂µ Fµρ,σσ ) − ηνρ ( ∂µ Fµσ,σλ + ∂µ Fµλ,σσ )+ 3 6 3 6 1 1 1 a a a a − ∂µ Fµσ,σν + ∂µ Fνσ,σµ ) = jνλρ . +ηλρ ( ∂µ Fµν,σσ 2 3 3 (55) As we demonstrated, this equation is invariant with respect to the following gauge transformations: a a a δAaµνλ = ∂µ ξνλ , δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν , where the gauge parameters are totally symmetric tensors satisfying the condition (44). a The initial invariance of the equation δAaµνλ = ∂µ ξνλ imposes restriction on the current a jνλρ , in particular, on its conservation over the first index: a ∂ν jνλρ = 0. (56) There are also additional constraints on the current which follow from the enhanced a a invariance δ̃Aaµνλ = ∂ν ζµλ + ∂λ ζµν . In Fourier components the constraints on the group parameters are ωζ03 + κζ33 + κ(ζ00 − ζ11 − ζ22 − ζ33 ) = 0, ωζ01 + κζ31 = 0, ωζ02 + κζ32 = 0, ωζ00 + κζ30 − ω(ζ00 − ζ11 − ζ22 − ζ33 ) = 0, where k µ = (ω, 0, 0, κ), therefore ω ζ03 , κ ω ζ31 = − ζ01 , κ ω ζ32 = − ζ02 , κ κ = −ζ11 − ζ22 − ζ03 , ω ζ00 = ζ11 + ζ22 − ζ33 (57) and we have six independent gauge parameters ζ01 , ζ02 , ζ03 , ζ11 , ζ22 , ζ12 . From this it follows that the current components fulfill the following six relations: ω j1λ3 + κ ω kλ (j2λ0 + j0λ2 + j2λ3 + κ kλ (j1λ0 + j0λ1 + 17 ω j3λ1 ) = 0, κ ω j3λ2 ) = 0, κ (58) ω κ kλ ( j0λ0 + j3λ3 ) = 0, κ ω kλ (j0λ0 − j3λ3 + j1λ1 ) = 0, kλ (j0λ0 − j3λ3 + j2λ2 ) = 0, kλ (j1λ2 + j2λ1 ) = 0, One can also use a different set of independent parameters, in particular: ζ11 , ζ12 , ζ13 , ζ22 , ζ23 , ζ33 . 6 Schwinger Equation for rank-3 Gauge Field The Schwinger equation for symmetric massless rank-3 tensor gauge field has the form [12] + ∂ 2 Aαα′ α′′ − ∂α ∂ρ Aρα′ α′′ − ∂α′ ∂ρ Aαρα′′ − ∂α′′ ∂ρ Aαα′ ρ + + ∂α ∂α′ Aα′′ ρρ + ∂α ∂α′′ Aα′ ρρ + ∂α′ ∂α′′ Aαρρ − 3∂α ∂α′ ∂α′′ A − 1 − ηαα′ (∂ 2 Aρρα′′ − ∂λ ∂ρ Aρλα′′ + ∂α′′ ∂ρ Aρλλ ) − 2 1 2 − ηαα′′ (∂ Aρρα′ − ∂λ ∂ρ Aρλα′ + ∂α′ ∂ρ Aρλλ ) − 2 1 2 (59) − ηα′ α′′ (∂ Aρρα − ∂λ ∂ρ Aρλα + ∂α ∂ρ Aλλρ ) = jαα′ α′′ 2 and contains the scalar field A which should satisfy the high-order differential equation 2 (60) ∂ 2 ∂ 2 A − ∂ 2 ∂λ Aλρρ + ∂α ∂λ ∂ρ Aαλρ = 0. 3 Taking derivatives of the l.h.s. of the above equation one can get convinced that we have conservation of the totally symmetric current jαα′ α′′ ∂α jαα′ α′′ = 0 (61) and the invariance of the equation with respect to the full gauge transformation δAµνλ = ∂µ ξνλ + ∂ν ξµλ + ∂λ ξµν (62) without any restrictions on the symmetric gauge parameter ξνλ . The great advantage of this formulation is that we have conservation of current and full gauge symmetry (62). The disadvantage of this formulation is the appearance of the scalar field A and its highorder differential equation. The illuminating remark of Schwinger was to make a change of field variable of the form [12] Aαα′ α′′ → Aαα′ α′′ − 3∂ −2 ∂α ∂α′ ∂α′′ A, which allows to eliminate the scalar field A from the field equation without changing its actual form! The equation will take a unique form [12]: + ∂ 2 Aαα′ α′′ − ∂α ∂ρ Aρα′ α′′ − ∂α′ ∂ρ Aαρα′′ − ∂α′′ ∂ρ Aαα′ ρ + + ∂α ∂α′ Aα′′ ρρ + ∂α ∂α′′ Aα′ ρρ + ∂α′ ∂α′′ Aαρρ − 1 − ηαα′ (∂ 2 Aρρα′′ − ∂λ ∂ρ Aρλα′′ + ∂α′′ ∂ρ Aρλλ ) − 2 1 − ηαα′′ (∂ 2 Aρρα′ − ∂λ ∂ρ Aρλα′ + ∂α′ ∂ρ Aρλλ ) − 2 1 − ηα′ α′′ (∂ 2 Aρρα − ∂λ ∂ρ Aρλα + ∂α ∂ρ Aλλρ ) = jαα′ α′′ , 2 18 (63) but it is not invariant any more with respect to the unrestricted gauge transformations (62). The gauge parameter should be traceless: ξµµ = 0. (64) This leads to the modification of the current conservation law: 1 ∂α jαα′ α′′ − ηα′ α′′ jαρρ = 0. (65) 4 The conservation law for the current became more sophisticated because of the traceless restriction on the gauge parameters (64). One can see that the same phenomenon also happens in our case where the restriction on the gauge parameters ζνλ has the form (52) and the conservation law takes the form (58). Recent discussion of the Schwinger equation can be found in [44, 45]. I would like to thank Luis Alvarez-Gaume, Ignatios Antoniadis, Ioannis Bakas, Lars Brink, Ludwig Faddeev, Sergio Ferrara, Peter Minkowski and Raymond Stora for discussions and CERN Theory Division, where part of this work was completed, for hospitality. This work was partially supported by the EEC Grant no. MRTN-CT-2004-005616. 7 Appendix A ′ The invariance of the form L3 can be demonstrated by explicit variation of each term in the sum (35). Indeed, the variation of the first term is c δξ Gaµν,λρ Gaµλ,νρ = 2gf abc Gaµν,λρ Gbµλ,ν ξρc + 2gf abc Gaµν,λρ Gbµλ,ρ ξνc + 2gf abc Gaµν,λρ Gbµλ ξνρ , of the second term is c δξ Gaµν,νλ Gaµρ,ρλ = 2gf abc Gaµν,νλ Gbµρ,ρ ξλc + 2gf abc Gaµν,νλ Gbµρ,λ ξρc + 2gf abc Gaµν,νλ Gbµρ ξρλ , of the third term is c δξ Gaµν,νλ Gaµλ,ρρ = 2gf abc Gaµν,νλ Gbµλ,ρ ξρc + gf abc Gaµν,νλ Gbµλ ξρρ + gf abc Gaµλ,ρρ Gbµν,ν ξλc + c +gf abc Gaµλ,ρρ Gbµν,λ ξνc + gf abc Gaµλ,ρρ Gbµν ξνλ , of the forth term is δξ Gaµν,λ Gaµλ,νρρ = gf abc Gaµλ,νρρ Gbµν ξλc + 2gf abc Gaµλ,νρ Gbµν,λ ξρc + gf abc Gaµλ,ρρ Gbµν,λ ξνc + c c c , + gf abc Gaµν,λ Gbµλ ξνρρ + 2gf abc Gaµν,λ Gbµλ,ρ ξνρ +gf abc Gaµν,λ Gbµλ,ν ξρρ of the fifth term is δξ Gaµν,λ Gaµρ,νλρ = gf abc Gaµρ,νλρ Gbµν ξλc + gf abc Gbµρ,νλ Gaµν,λ ξρc + gf abc Gbµρ,νρ Gaµν,λ ξλc + gf abc Gbµρ,λρ Gaµν,λ ξνc + c c c c +gf abc Gaµν,λ Gbµρ,ν ξλρ + gf abc Gaµν,λ Gbµρ,λ ξνρ + gf abc Gaµν,λ Gbµρ,ρ ξνλ + gf abc Gaµν,λ Gbµρ ξνλρ , of the sixth term is δξ Gaµν,ν Gaµλ,λρρ = c + gf abc Gaµλ,λρρ Gbµν ξνc + 2gf abc Gbµλ,λρ Gaµν,ν ξρc + gf abc Gbµλ,ρρ Gaµν,ν ξλc + gf abc Gbµλ,λ Gaµν,ν ξρρ c c +2gf abc Gbµλ,ρ Gaµν,ν ξλρ + gf abc Gaµν,ν Gbµλ ξλρρ 19 and finally of the seventh term is δξ Gaµ,ν Gaµλ,νλρρ = c 2gf abc Gaµν Gbµλ,νλρ ξρc + gf abc Gaµν Gbµλ,νρρ ξλc + gf abc Gaµν Gbµλ,λρρ ξνc + gf abc Gaµν Gbµλ,νλ ξρρ + c c c 2gf abc Gaµν Gbµλ,νρ ξλρ + 2gf abc Gaµν Gbµλ,λρ ξνρ + gf abc Gaµν Gbµλ,ρρ ξνλ + c c c c . + gf abc Gaµν Gbµλ ξνλρρ + 2gf abc Gaµν Gbµλ,ρ ξνλρ + gf abc Gaµν Gbµλ,λ ξνρρ gf abc Gaµν Gbµλ,ν ξλρρ c c Some of the terms here are equal to zero, like: gf abc Gaµν,λ Gbµρ,λ ξνρ , gf abc Gaµλ,λ Gbµν,ν ξρρ and c abc a b gf Gµν Gµλ ξνλρρ . Amazingly, all nonzero terms cancel each other. 8 Appendix B ′ The quadratic form Hαα′ α′′ γγ ′ γ ′′ can be extracted from (40) and should be symmetrized ′ ′′ ′ ′′ ′ ′′ ′ ′′ over the α ↔ α , γ ↔ γ and over the exchange of two sets of indices αα α ↔ γγ γ , so that in the momentum representation it has the form ′ k2 { + 8 + + + 1 − { + 8 + + + + + + + 1 + { + 4 + Hαα′ α′′ γγ ′ γ ′′ (k) = ηαα′ (ηα′′ γ ηγ ′ γ ′′ + ηα′′ γ ′ ηγγ ′′ + ηα′′ γ ′′ ηγγ ′ ) ηαα′′ (ηα′ γ ηγ ′ γ ′′ + ηα′ γ ′ ηγγ ′′ + ηα′ γ ′′ ηγγ ′ ) ηαγ ′ (ηα′ γ ηα′′ γ ′′ + ηα′ γ ′′ ηα′′ γ + ηα′ α′′ ηγγ ′′ ) ηαγ ′′ (ηα′ γ ηα′′ γ ′ + ηα′ γ ′ ηα′′ γ + ηα′ α′′ ηγγ ′ ) } kα kα′ (ηα′′ γ ηγ ′ γ ′′ + ηα′′ γ ′ ηγγ ′′ + ηα′′ γ ′′ ηγγ ′ ) kα kα′′ (ηα′ γ ηγ ′ γ ′′ kα kγ ′ (ηα′ γ ηα′′ γ ′′ kα kγ ′′ (ηα′ γ ηα′′ γ ′ kγ kα′ (ηαγ ′ ηα′′ γ ′′ kγ kα′′ (ηαγ ′ ηα′ γ ′′ kγ kγ ′ (ηαα′ ηα′′ γ ′′ kγ kγ ′′ (ηαα′ ηα′′ γ ′ + ηα′ γ ′ ηγγ ′′ + ηα′ γ ′′ ηγγ ′ ) + ηα′ γ ′′ ηα′′ γ + ηα′ α′′ ηγγ ′′ ) + ηα′ γ ′ ηα′′ γ + ηα′ α′′ ηγγ ′ ) + ηαγ ′′ ηα′′ γ ′ + ηαα′′ ηγ ′ γ ′′ ) + ηαγ ′′ ηα′ γ ′ + ηαα′ ηγ ′ γ ′′ ) + ηαα′′ ηα′ γ ′′ + ηαγ ′′ ηα′ α′′ ) + ηαα′′ ηα′ γ ′ + ηαγ ′ ηα′ α′′ ) } ηαγ (kα′ kγ ′ ηα′′ γ ′′ + kα′ kγ ′′ ηα′′ γ ′ + kα′′ kγ ′ ηα′ γ ′′ kα′′ kγ ′′ ηα′ γ ′ + kα′ kα′′ ηγ ′ γ ′′ + kγ ′ kγ ′′ ηα′ α′′ ) }. or combining some of the terms together we shall get an equivalent form 1 ′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + (k 2 ηαα′ − kα kα′ )(ηα′′ γ ηγ ′ γ ′′ + ηα′′ γ ′ ηγγ ′′ + ηα′′ γ ′′ ηγγ ′ ) 8 + (k 2 ηαα′′ − kα kα′′ )(ηα′ γ ηγ ′ γ ′′ + ηα′ γ ′ ηγγ ′′ + ηα′ γ ′′ ηγγ ′ ) + (k 2 ηαγ ′ − kα kγ ′ )(ηα′ γ ηα′′ γ ′′ + ηα′ γ ′′ ηα′′ γ + ηα′ α′′ ηγγ ′′ ) + (k 2 ηαγ ′′ − kα kγ ′′ )(ηα′ γ ηα′′ γ ′ + ηα′ γ ′ ηα′′ γ + ηα′ α′′ ηγγ ′ ) } 1 − { + kγ kα′ (ηαγ ′ ηα′′ γ ′′ + ηαγ ′′ ηα′′ γ ′ + ηαα′′ ηγ ′ γ ′′ ) 8 + kγ kα′′ (ηαγ ′ ηα′ γ ′′ + ηαγ ′′ ηα′ γ ′ + ηαα′ ηγ ′ γ ′′ ) 20 (66) + kγ kγ ′ (ηαα′ ηα′′ γ ′′ + ηαα′′ ηα′ γ ′′ + ηαγ ′′ ηα′ α′′ ) + kγ kγ ′′ (ηαα′ ηα′′ γ ′ + ηαα′′ ηα′ γ ′ + ηαγ ′ ηα′ α′′ ) } 1 + { + ηαγ (kα′ kγ ′ ηα′′ γ ′′ + kα′ kγ ′′ ηα′′ γ ′ + kα′′ kγ ′ ηα′ γ ′′ 4 + kα′′ kγ ′′ ηα′ γ ′ + kα′ kα′′ ηγ ′ γ ′′ + kγ ′ kγ ′′ ηα′ α′′ ) }. (67) This expression can be used to calculate divergences. Indeed, 1 ′ kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + (k 2 ηαα′′ − kα kα′′ )(kγ ηγ ′ γ ′′ + kγ ′ ηγγ ′′ + kγ ′′ ηγγ ′ ) 8 + (k 2 ηαγ ′ − kα kγ ′ )(kγ ηα′′ γ ′′ + kγ ′′ ηα′′ γ + kα′′ ηγγ ′′ ) + (k 2 ηαγ ′′ − kα kγ ′′ )(kγ ηα′′ γ ′ + kγ ′ ηα′′ γ + kα′′ ηγγ ′ ) } 1 − { + 8 + + 1 + { + 4 k 2 kγ (ηαγ ′ ηα′′ γ ′′ + ηαγ ′′ ηα′′ γ ′ + ηαα′′ ηγ ′ γ ′′ ) kγ kα′′ (2kγ ′′ ηαγ ′ + 2kγ ′ ηαγ ′′ + kα ηγ ′ γ ′′ ) kγ kγ ′ (kα ηα′′ γ ′′ + 2kγ ′′ ηαα′′ ) + kγ kγ ′′ kα ηα′′ γ ′ } ηαγ (k 2 kγ ′ ηα′′ γ ′′ + k 2 kγ ′′ ηα′′ γ ′ + k 2 kα′′ ηγ ′ γ ′′ + 3kα′′ kγ ′ kγ ′′ } or using the operator Hαγ = k 2 ηαγ − kα kγ one can get 1 ′ kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + 8 + + 1 − { + 8 + + 1 + { + 4 Hαα′′ (kγ ηγ ′ γ ′′ + kγ ′ ηγγ ′′ + kγ ′′ ηγγ ′ ) Hαγ ′ (kγ ηα′′ γ ′′ + kγ ′′ ηα′′ γ + kα′′ ηγγ ′′ ) Hαγ ′′ (kγ ηα′′ γ ′ + kγ ′ ηα′′ γ + kα′′ ηγγ ′ ) } Hαα′′ kγ ηγ ′ γ ′′ + Hαγ ′ kγ ηα′′ γ ′′ + Hαγ ′′ kγ ηα′′ γ ′ kγ kα′′ (2kγ ′′ ηαγ ′ + 2kγ ′ ηαγ ′′ + 2kα ηγ ′ γ ′′ ) kγ kγ ′ (2kα ηα′′ γ ′′ + 2kγ ′′ ηαα′′ ) + 2kγ kγ ′′ kα ηα′′ γ ′ } ηαγ (k 2 kγ ′ ηα′′ γ ′′ + k 2 kγ ′′ ηα′′ γ ′ + k 2 kα′′ ηγ ′ γ ′′ + 3kα′′ kγ ′ kγ ′′ } and canceling the identical terms we shall get 1 ′ kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + 8 + + 1 − { + 4 1 + { + 4 + Hαα′′ (kγ ′ ηγγ ′′ + kγ ′′ ηγγ ′ ) Hαγ ′ (kγ ′′ ηα′′ γ + kα′′ ηγγ ′′ ) Hαγ ′′ (kγ ′ ηα′′ γ + kα′′ ηγγ ′ ) } kγ kα′′ (kγ ′′ ηαγ ′ + kγ ′ ηαγ ′′ ) + kγ kγ ′ kγ ′′ ηαα′′ } Hαγ kγ ′ ηα′′ γ ′′ + Hαγ kγ ′′ ηα′′ γ ′ + Hαγ kα′′ ηγ ′ γ ′′ 3ηαγ kα′′ kγ ′ kγ ′′ }. Again collecting terms we shall get the final expression: 1 ′ kα′ Hαα′ α′′ γγ ′ γ ′′ (k) = { + Hαα′′ (kγ ′ ηγγ ′′ + kγ ′′ ηγγ ′ ) 8 21 + Hαγ ′ (kγ ′′ ηα′′ γ + kα′′ ηγγ ′′ ) + Hαγ ′′ (kγ ′ ηα′′ γ + kα′′ ηγγ ′ ) } (68) 1 − { + kγ kα′′ (kγ ′′ ηαγ ′ + kγ ′ ηαγ ′′ ) + kγ kγ ′ kγ ′′ ηαα′′ − 3ηαγ kα′′ kγ ′ kγ ′′ } 4 1 + { + Hαγ (kγ ′ ηα′′ γ ′′ + kγ ′′ ηα′′ γ ′ + kα′′ ηγ ′ γ ′′ ) }, 4 which has been used in the main text. References [1] C.N.Yang and R.L.Mills. ”Conservation of Isotopic Spin and Isotopic Gauge Invariance”. Phys. Rev. 96 (1954) 191 [2] G. Savvidy, Generalization of Yang-Mills theory: Non-Abelian tensor gauge fields and higher-spin extension of standard model, arXiv:hep-th/0505033. [3] G. Savvidy, Non-Abelian tensor gauge fields: Generalization of Yang-Mills theory,” Phys. Lett. B 625 (2005) 341 [arXiv:hep-th/0509049]. [4] G. Savvidy, Non-Abelian tensor gauge fields and extended current algebra: Generalization of Yang-Mills theory, arXiv:hep-th/0510258. [5] G. Savvidy, Non-Abelian tensor gauge fields and higher-spin extension of standard model, Fortschr. Phys. 54 (2006) 472 [arXiv:hep-th/0512012]. [6] G. Savvidy and T. Tsukioka, Gauge invariant Lagrangian for non-Abelian tensor gauge fields of fourth rank, arXiv:hep-th/0512344. [7] M. Fierz. Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin, Helv. Phys. Acta. 12 (1939) 3. [8] M. Fierz and W. Pauli. On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field, Proc. Roy. Soc. A173 (1939) 211. [9] P. Minkowski. Versuch einer konsistenten Theorie eines Spin-2 Mesons, Helv. Phys. Acta. 32 (1966) 477 [10] H.Yukawa, Quantum Theory of Non-Local Fields. Part I. Free Fields, Phys. Rev. 77 (1950) 219 ; M. Fierz, Non-Local Fields, Phys. Rev. 78 (1950) 184; H.Yukawa, On the radius of the Elementary Particle, Phys. Rev. 76 (1949) 300; H.Yukawa, On the Theory of Elementary Particles.I., Prog. Theor. Phys. 2 (1947) 209; H.Yukawa, Reciprocity in Generalized Field Theory, 3 (1948) 205 H.Yukawa, Possible Types of Nonlocalizable Fields,3 (1948) 452 [11] E. Wigner, Invariant Quantum Mechanical Equations of Motion, in Theoretical Physics ed. A.Salam (International Atomic Energy, Vienna, 1963) p 59; E. Wigner, Relativistische Wellengleichungen, Zeitschrift für Physik 124 (1948) 665 [12] J.Schwinger, Particles, Sourses, and Fields (Addison-Wesley, Reading, MA, 1970) 22 [13] S. Weinberg, Feynman Rules For Any Spin, Phys. Rev. 133 (1964) B1318. [14] S. J. Chang, Lagrange Formulation for Systems with Higher Spin, Phys.Rev. 161 (1967) 1308 [15] L. P. S. Singh and C. R. Hagen, Lagrangian formulation for arbitrary spin. I. The boson case, Phys. Rev. D9 (1974) 898 [16] L. P. S. Singh and C. R. Hagen, Lagrangian formulation for arbitrary spin. II. The fermion case, Phys. Rev. D9 (1974) 898, 910 [17] C.Fronsdal, Massless fields with integer spin, Phys.Rev. D18 (1978) 3624 [18] G. K. Savvidy, Conformal Invariant Tensionless Strings, Phys. Lett. B 552 (2003) 72. [19] G. K. Savvidy, Tensionless strings: Physical Fock space and higher spin fields, Int. J. Mod. Phys. A 19, (2004) 3171-3194. [20] G. Savvidy, Tensionless strings, correspondence with SO(D,D) sigma model, Phys. Lett. B 615 (2005) 285. [21] W. Siegel and B. Zwiebach, “Gauge String Fields,” Nucl. Phys. B 263 (1986) 105. [22] J.Fang and C.Fronsdal. ”Deformation of gauge groups. Gravitation”. J. Math. Phys. 20 (1979) 2264 [23] F. A. Berends, G. J. H Burgers and H. Van Dam, “On the Theoretical problems in Constructing Interactions Involving Higher-Spin Massless Particles,” Nucl. Phys. B 260 (1985) 295. [24] B.deWit, F.A. Berends, J.W.van Halten and P.Nieuwenhuizen, “On Spin-5/2 gauge fields,” J. Phys. A: Math. Gen. 16 (1983) 543. [25] E. Sezgin and P. Sundell, “Holography in 4D (Super) Higher Spin Theories and a Test via Cubic Scalar Couplings,” hep-th/0305040 [26] A. Sagnotti, E. Sezgin and P. Sundell, “On higher spins with a strong Sp(2,R) condition,” arXiv:hep-th/0501156. [27] T. Curtright, “Are There Any Superstrings In Eleven-Dimensions?,” Phys. Rev. Lett. 60 (1988) 393 [28] A. K. Bengtsson, I. Bengtsson and L. Brink, “Cubic Interaction Terms For Arbitrary Spin,” Nucl. Phys. B 227 (1983) 31. [29] A. K. Bengtsson, I. Bengtsson and L. Brink, “Cubic Interaction Terms For Arbitrarily Extended Supermultiplets,” Nucl. Phys. B 227 (1983) 41. [30] A. K. H. Bengtsson, “An abstract interface to higher spin gauge field theory,” J. Math. Phys. 46 (2005) 042312 [arXiv:hep-th/0403267]. 23 [31] L. Edgren, R. Marnelius and P. Salomonson, “Infinite spin particles,” JHEP 0505 (2005) 002 [arXiv:hep-th/0503136]. [32] I. Bakas, B. Khesin and E. Kiritsis, “The Logarithm of the derivative operator and higher spin algebras of W-infinity type,” Commun. Math. Phys. 151 (1993) 233. [33] I. Bakas and E. Kiritsis, “Grassmannian Coset Models And Unitary Representations Of W(Infinity),” Mod. Phys. Lett. A 5 (1990) 2039. [34] E. A. Ivanov, “Yang-Mills Theory In Sigma Model Representation. (In Russian),” Pisma Zh. Eksp. Teor. Fiz. 30 (1979) 452. [35] X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, “Nonlinear higher spin theories in various dimensions,” arXiv:hep-th/0503128. [36] N. Turok, M. Perry and P. J. Steinhardt, “M theory model of a big crunch / big bang transition,” Phys. Rev. D 70 (2004) 106004 [arXiv:hep-th/0408083]. [37] J. Engquist and P. Sundell, “Brane partons and singleton strings,” arXiv:hepth/0508124. [38] L. Brink, “Particle physics as representations of the Poincare algebra,” arXiv:hepth/0503035. [39] J. Gamboa, M. Loewe and F. Mendez, “Quantum theory of tensionless noncommutative p-branes,” Phys. Rev. D 70 (2004) 106006. [40] G. Bonelli, “On the boundary gauge dual of closed tensionless free strings in AdS,” JHEP 0411 (2004) 059 [arXiv:hep-th/0407144]. [41] I. Bakas and C. Sourdis, “On the tensionless limit of gauged WZW models,” JHEP 0406 (2004) 049 [arXiv:hep-th/0403165]. [42] A. Bredthauer, U. Lindstrom, J. Persson and L. Wulff, ‘Type IIB tensionless superstrings in a pp-wave background,” JHEP 0402 (2004) 051 [arXiv:hep-th/0401159]. [43] M. Bianchi, J. F. Morales and H. Samtleben, “On stringy AdS(5)x S 5 and higher spin holography,” JHEP 0307 (2003) 062 [arXiv:hep-th/0305052]. [44] D. Francia and A. Sagnotti, “Minimal local Lagrangians for higher-spin geometry,” Phys. Lett. B 624 (2005) 93 [arXiv:hep-th/0507144]. [45] D. Francia and A. Sagnotti, “On the geometry of higher-spin gauge fields,” Class. Quant. Grav. 20 (2003) S473 [46] N. Boulanger, S. Cnockaert and M. Henneaux, “A note on spin-s duality,” JHEP 0306 (2003) 060 [47] S. Deser and A. Waldron, “Gauge invariances and phases of massive higher spins in (A)dS,” Phys. Rev. Lett. 87 (2001) 031601 [arXiv:hep-th/0102166]. [48] B. de Wit and D. Z. Freedman, “Systematics Of Higher Spin Gauge Fields,” Phys. Rev. D 21 (1980) 358. 24