Propagating modes of non-Abelian tensor gauge field of second rank
aa r X i v : . [ h e p - t h ] J un NRCPS-HE-57-07
June, 2007
Propagating ModesofNon-Abelian Tensor Gauge Field of Second Rank
Spyros Konitopoulos and
George SavvidyInstitute of Nuclear Physics,Demokritos National Research CenterAgia Paraskevi, GR-15310 Athens, Greece
Abstract
In the recently proposed extension of the YM theory, non-Abelian tensor gaugefield of the second rank is represented by a general tensor whose symmetric part de-scribes the propagation of charged gauge boson of helicity two and its antisymmetricpart - the helicity zero charged gauge boson. On the non-interacting level these po-larizations are similar to the polarizations of the graviton and of the Abelian antisym-metric B field, but the interaction of these gauge bosons carrying non-commutativeinternal charges cannot be directly identified with the interaction of gravitons or Bfield. Our intention here is to illustrate this result from different perspectives whichwould include Bianchi identity for the corresponding field strength tensor and theanalysis of the second-order partial differential equation which describes in this the-ory the propagation of non-Abelian tensor gauge field of the second rank. Analyzingthe interaction between two tensor currents caused by the exchange of these tensorgauge bosons we shall demonstrate that the residue at the pole is the sum of threeterms each of which describes positive norm polarizations of helicities two and zerobosons.
Introduction
An infinite tower of massive particles of high spin naturally appears in the spectrum ofdifferent string field theories [1, 2, 3, 5, 4]. From the point of view of quantum field theory,string field theories seem to contain an infinite number of nonrenormalizable interactionsbetween these fields, which are represented in the string action by nonlocal cubic interactionterms containing an exponential of a quadratic form in the momenta [6, 7].It is generally expected that in the tensionless limit or, what is equivalent, at highenergy scattering [8, 9, 10] the string spectrum becomes effectively massless and it is ofgreat importance to find out the corresponding action and its genuine symmetries [11, 12,13, 14, 15, 16, 17, 18]. On the quantum field theory language this should be a field theorywith infinite many massless fields.In quantum field theory the Lagrangian formulation of free massless Abelian tensorgauge fields has been constructed in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. The problemof introducing interactions appears to be much more complex and there has been importantprogress in defining self-interaction of higher-spin fields in the light-cone formalism and inthe covariant formulation of the theories [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].In this respect it is appealing to extend the gauge principle so that it will define theinteraction of gauge fields which carry not only non-commutative internal charges, but alsoarbitrary spins. For that purpose it seems reasonable to define extended non-Abelian gaugetransformations acting on charged tensor gauge fields and the corresponding field strengthtensors, which will enable the construction of a gauge invariant Lagrangian quadratic infield strength tensors [43], as it is the case in the Yang-Mills theory [42]. The resultinggauge invariant Lagrangian will contain kinetic terms for higher-spin fields and will uniquelydefine their cubic and quartic interaction terms [43, 44, 45].Here we shall follow the construction which is based on this direct extension of non-Abelian gauge transformations [43, 44, 45, 46, 47]. Recall that non-Abelian gauge fieldsare defined as rank-(s+1) tensor gauge fields A aµλ ...λ s ∗ and that one can construct infiniteseries of forms L s ( s = 1 , , .. ) and L ′ s ( s = 2 , , .. ) which are invariant with respect to thegauge transformations. These forms are quadratic in the field strength tensors G aµν,λ ...λ s . ∗ Tensor gauge fields A aµλ ...λ s ( x ) , s = 0 , , , ... are totally symmetric with respect to the indices λ ...λ s . A priori the tensor fields have no symmetries with respect to the first index µ . In particular wehave A aµλ = A aλµ and A aµλρ = A aµρλ = A aλµρ . The adjoint group index a = 1 , ..., N − SU ( N )gauge group. G aµν,λ ...λ s transform homogeneously with respect to the gauge transformations. Therefore the gaugeinvariant Lagrangian describing dynamical tensor gauge bosons of all ranks has the form[43, 44, 45] L = L + g L + g ′ L ′ + .... (1.1)where L is the Yang-Mills Lagrangian. This Lagrangian contains kinetic terms of thetensor gauge fields A aµ , A aµλ , .. and nonlinear terms which describe their interactions, cubicand quartic interactions between lower- and higher-rank tensor gauge fields. For the lower-rank tensor gauge fields the Lagrangian has the following form [43, 44, 45]: L = − G aµν G aµν , L = − G aµν,λ G aµν,λ − G aµν G aµν,λλ , (1.2) L ′ = + 14 G aµν,λ G aµλ,ν + 14 G aµν,ν G aµλ,λ + 12 G aµν G aµλ,νλ , where generalized field strength tensors are: G aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν ,G aµν,λ = ∂ µ A aνλ − ∂ ν A aµλ + gf abc ( A bµ A cνλ + A bµλ A cν ) ,G aµν,λρ = ∂ µ A aνλρ − ∂ ν A aµλρ + gf abc ( A bµ A cνλρ + A bµλ A cνρ + A bµρ A cνλ + A bµλρ A cν ) . In the present paper we shall focus our attention on the lower-rank tensor gauge field A aµλ ,which in this theory is a general nonsymmetric tensor with 4x4=16 space-time components( A aµλ = A aλµ and it would have d × d = d components in the d-dimensional space-time) † .It has been found that if g ′ = g , then the quadratic part of the Lagrangian (1.2), whichdefines the kinetic energy of the field A aµλ , has the following form [43, 44, 45]: K = − F aµν,λ F aµν,λ + 14 F aµν,λ F aµλ,ν + 14 F aµν,ν F aµλ,λ , (1.3)where F aµν,λ = ∂ µ A aνλ − ∂ ν A aµλ , (1.4)and describes the propagation of helicity-two and helicity-zero λ = ± , charged gaugebosons . This means that within the 16 fields of nonsymmetric tensor gauge field A aµλ only † One should multiply these numbers by the dimension of the gauge group, N − SU ( N ). hree positive norm polarizations are propagating and that the rest of them are pure gaugefields. On the non-interacting level, when we consider only the kinetic term (1.3) of thefull Lagrangian (1.1), these polarizations are similar to the polarizations of the gravitonand of the Abelian antisymmetric B field [48, 49, 50, 51, 53], but the interaction of thesegauge bosons carrying non-commutative internal charges is uniquely defined by the fullLagrangian (1.1) and cannot be directly identified with the interactions of gravitons or Bfield [43, 44, 45].Our intention here is to illustrate this result from different perspectives which wouldinclude Bianchi identity for the field strength tensor F aµν,λ and the direct analysis of thesecond-order partial differential equation which describes in this theory the propagation ofthe free tensor gauge field A aµλ . This equation follows from the variation of the Lagrangian(1.3) with respect to A aνλ [43, 44, 45]. With this in mind we shall present a general methodfor counting the propagating modes in gauge field theories and apply it to the tensor gaugefield theory in order to illustrate the fact that the second-rank tensor gauge field A aµλ describes three polarizations λ = ± , λ = ± , Gauge Symmetries and Current Conservation
Pure kinetic term of the Lagrangian (1.2), (1.3), which describes the propagation of thetensor gauge field ( A aµλ = A aλµ ), has the following form [43, 44, 45]: K = − F aµν,λ F aµν,λ + 14 F aµν,λ F aµλ,ν + 14 F aµν,ν F aµλ,λ (2.5)and is invariant with respect to a pair of complementary gauge transformations δ and ˜ δ .When the coupling constant g is taken to vanish, these gauge transformations reduce tothe following form [45]: δA aµλ = ∂ µ ξ aλ (2.6)and ˜ δA aµλ = ∂ λ η aµ . (2.7)3he field strength tensor F aµν,λ (1.4) transforms with respect to these transformations asfollows: δ ξ F aµν,λ = 0 , ˜ δ η F aµν,λ = ∂ λ ( ∂ µ η aν − ∂ ν η aµ ) . (2.8)Therefore the kinetic term K is obviously invariant with respect to the first group of gaugetransformations δ and we have δ K = 0, but it is less trivial to see that it is also invariantwith respect to the complementary gauge transformations ˜ δ . The ˜ δ transformation of K is ˜ δ K = − F aµν,λ ∂ λ ( ∂ µ η aν − ∂ ν η aµ ) + 12 F aµν,λ ∂ ν ( ∂ µ η aλ − ∂ λ η aµ ) + 12 F aµν,ν ∂ λ ( ∂ µ η aλ − ∂ λ η aµ ) == 12 F aµν,λ ∂ λ ∂ ν η aµ + 12 F aµν,ν ∂ λ ( ∂ µ η aλ − ∂ λ η aµ ) , where we combined the first, the second and the forth terms and used the fact that the thirdterm is identically equal to zero. Just from the symmetry properties of the field strengthtensor it is not obvious to see why the rest of the terms are equal to zero. Therefore weshall use the explicit form of the field strength tensor F aµν,λ , which gives˜ δ K = 12 ( ∂ µ A aνλ − ∂ ν A aµλ ) ∂ λ ∂ ν η aµ + 12 ( ∂ µ A aνν − ∂ ν A aµν ) ∂ λ ( ∂ µ η aλ − ∂ λ η aµ ) . From the corresponding action S = R K dx after partial differentiation we shall get that theterm ∂ µ A aνν · ∂ λ ( ∂ µ η aλ − ∂ λ η aµ ) gives a zero contribution and the rest of the terms cancel eachother R ( ( ∂ µ A aνλ − ∂ ν A aµλ ) · ∂ λ ∂ ν η aµ − ∂ ν A aµν · ∂ λ ( ∂ µ η aλ − ∂ λ η aµ )) dx = 0. This demonstratesthe invariance of the action with respect to the δ and ˜ δ transformations defined by (2.6)and (2.7) when g = g ′ in (1.1) [43].Let us now consider the interaction of the tensor gauge field A aµλ with the current J aµν , asit is defined by the Lagrangian (1.1) ( J aµν = J aνµ ). In order to see what type of restrictionsare imposed on the current by the gauge symmetries of the action we shall consider theequation of motion which follows from the Lagrangian (1.1) ( see [43, 44, 45] for details): ∂ µ F aµν,λ −
12 ( ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ ) = J aνλ . (2.9)This equation contains two terms ∂ µ F aµν,λ and − ( ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ ),which arise from L and L ′ respectively. The derivatives over ∂ ν of both terms in the4quation are equal to zero separately. Indeed, due to the antisymmetric properties of thefield strength tensor F aµν,λ under exchange of µ and ν we have ∂ ν ∂ µ F aµν,λ = 0 , as well as − ∂ ν { ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ } = 0 . Thus it follows from (2.9) that ∂ ν J aνλ = 0 . (2.10)Hence, the current J aνλ must be divergenceless over its first index. Now let us take derivativeover ∂ λ of the left-hand side of the equation (2.9), that is, the derivative over the secondindex of the nonsymmetric current J aνλ . We see that ∂ λ ∂ µ F aµν,λ = 0 , as well as − ∂ λ { ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ } = − ∂ λ { ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ } 6 = 0 . Thus, is not obvious to see the conservation of the nonsymmetric current J aνλ with respectto its second index λ alone from the antisymmetric properties of the field strength tensor.Therefore we have to use the explicit form of the field strength tensor F aµν,λ = ∂ µ A aνλ − ∂ ν A aµλ ,this gives ∂ λ ∂ µ F aµν,λ − ∂ λ { ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ } == ∂ λ ∂ µ F aµν,λ − ∂ λ ∂ µ F aλν,µ − ∂ F aµν,µ − ∂ ν ∂ µ F aµρ,ρ == 12 ∂ λ ∂ µ ( ∂ µ A aνλ − ∂ ν A aµλ ) − ∂ ( ∂ µ A aνµ − ∂ ν A aµµ ) − ∂ ν ∂ µ ( ∂ µ A aρρ − ∂ ρ A aµρ ) = 0 . Therefore the sum of the two nonzero expressions presented above are equal to zero, thus[43, 44, 45] ∂ λ J aνλ = 0 . (2.11)The natural question which arises here is connected with the fact that in order to seethese cancellations one should use the explicit form of the field strength tensor F aµν,λ , andit remains a mystery, why this takes place only when the relative coefficient between the5nvariant forms L and L ′ is equal to one ( g = g ′ in (1.1)) [43]. Our main concern thereforeis to understand the general reason for these cancellations without referring to the explicitform of the field strength tensor. As we shall see, the deep reason for this cancellations liesis the Bianchi identity (5.52), (5.53) for the field strength tensor ∂ µ F νλ,ρ + ∂ ν F λµ,ρ + ∂ λ F µν,ρ = 0 , (2.12)which we shall derive in the Appendix A. Indeed, we can evaluate the derivative of thel.h.s of the equation (2.9) to the following form: ∂ λ { ∂ µ F µν,λ −
12 ( ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F aµν,µ + η νλ ∂ µ F aµρ,ρ ) } = (2.13)= − { ∂ F µν,µ + ∂ µ ∂ ν F µρ,ρ + ∂ µ ∂ λ F νλ,µ } , where we have used only the antisymmetric property of F µν,λ to cancel the second termand to combine the first one with the third one of the l.h.s of the above equation. Now, weshall take advantage of the Bianchi identity. Taking the derivative of the Bianchi identity(2.12) over ∂ µ and setting ν = ρ we get ∂ F µν,µ + ∂ µ ∂ ν F µρ,ρ + ∂ µ ∂ λ F νλ,µ ≡ ∂ λ J aνλ = 0 . In other words, if one repeats these calculations for arbitrary coefficients g and g ′ inthe Lagrangian (1.1) g L + g ′ L ′ , then the last expression in parenthesis (2.13) will takethe form ∂ F µν,µ + ∂ µ ∂ ν F µρ,ρ + (2 g g ′ − ∂ µ ∂ λ F νλ,µ . Comparing it with Bianchi identity (2.14) one can see that it is equal to zero only if g = g ′ and therefore only in that case (2.11) holds.It seems that this situation is similar to that in gravity, where both tensors R µν and g µν R have correct transformation properties and therefore can be present in the equationof motion [52] R µν − c g µν R = T µν (2.15)with unknown coefficient c , but the Bianci identity R µν ; µ − (1 / R ; ν = 0 tells that thecoefficient c should be taken equal to 1 / J aµν = J aνµ ) current (2.10) and (2.11) ∂ ν J aνλ = 0 , ∂ λ J aνλ = 0is adequate to cast the theory unitary at free level. In the next section we shall present ageneral method for counting the propagating modes in gauge field theories and shall seehow the gauge invariance guarantees the elimination of all negative-norm states. Counting Propagating Modes
As we have seen above, the equation of motion (2.9) which describes the propagation ofthe second-rank tensor gauge field ( A aµλ = A aλµ ) has the following form ‡ [43, 44, 45]: ∂ µ F aµν,λ −
12 ( ∂ µ F aµλ,ν + ∂ µ F aλν,µ + ∂ λ F a µµν, + η νλ ∂ µ F a ρµρ, ) = J aνλ , (3.16)where F aµν,λ = ∂ µ A aνλ − ∂ ν A aµλ and is invariant with respect to a pair of complementarygauge transformations δ (2.6) and ˜ δ (2.7) A aµλ → A aµλ + ∂ µ ξ aλ + ∂ λ η aµ (3.17)with arbitrary gauge parameters ξ aλ and η aµ . The equivalent form of the equations of motion(2.9) in terms of the gauge field is [43, 44, 45] ∂ ( A aνλ − A aλν ) − ∂ ν ∂ µ ( A aµλ − A aλµ ) − ∂ λ ∂ µ ( A aνµ − A aµν ) + ∂ ν ∂ λ ( A a µµ − A a µµ ) ++ 12 η νλ ( ∂ µ ∂ ρ A aµρ − ∂ A a µµ ) = J aνλ . (3.18)In momentum space this type of second-order partial differential equations can always berepresented as matrix equation of the following general form H γ ´ γα ´ α ( k ) A aγ ´ γ = J aα ´ α , (3.19)where H γ ´ γα ´ α ( k ) is a matrix operator quadratic in momentum k µ . In our case it has thefollowing form [43, 44, 45]: H α ´ αγ ´ γ ( k ) = ( − η αγ η ´ α ´ γ + 12 η α ´ γ η ´ αγ + 12 η α ´ α η γ ´ γ ) k + η αγ k ´ α k ´ γ + η ´ α ´ γ k α k γ −
12 ( η α ´ γ k ´ α k γ + η ´ αγ k α k ´ γ + η α ´ α k γ k ´ γ + η γ ´ γ k α k ´ α ) , (3.20) ‡ The Lorentz indexes of the tensor fields are raised and lowered with flat metric η µν = ( − , , , H α ´ αγ ´ γ = H γ ´ γα ´ α . First of all we shall solve the equation in the casewhen there are no interactions, J aα ´ α = 0: H γ ´ γα ´ α ( k ) A aγ ´ γ = 0 . (3.21)The vector space of independent solutions A γ ´ γ of this system of equations crucially dependson the rank of the matrix H γ ´ γα ´ α ( k ). If the matrix operator H has dimension d × d and itsrank is rankH = r , then the vector space has the dimension N = d − r. Because the matrix operator H γ ´ γα ´ α ( k ) explicitly depends on the momentum k µ its rankH = r also depends on momenta and therefore the number of independent solutions N dependson momenta N ( k ) = d − r ( k ) . (3.22)Analyzing the rankH of the matrix operator H one can observe that it depends on thevalue of momentum square k µ . When k µ = 0 - off mass-shell momenta - the vector spaceconsists of pure gauge fields . When k µ = 0 - on mass-shell momenta - the vector spaceconsists of pure gauge fields and propagating modes . Therefore the number of propagatingmodes can be calculated from the following relation: ♯ of propagating modes = N ( k ) | k =0 − N ( k ) | k =0 = rankH | k =0 − rankH | k =0 . (3.23)Before considering the equation of motion for the tensor gauge field (3.21), let us considerfor illustration some important examples. Vector Gauge Field
The kinetic term of Lagrangian which describes the propagation of free vector gauge fieldis K = − F µν F µν (3.24)and the corresponding equation of motion in momentum space is H γα e γ = ( − k δ γα + k α k γ ) e γ = 0 , (3.25)8here A µ = e µ exp ( ikx ). We can always choose the momentum vector in the third direction k µ = ( ω, , , k ) and the matrix operator H takes the form H γα = − k − kω ω − k ω − k kω ω . If ω − k = 0, the rank of the 4-dimensional matrix H γα is rankH | ω − k =0 = 3 and thenumber of independent solutions is 4-3=1. As one can see from the relation H γα ( k ) k γ = 0this solution is proportional to the momentum e µ = k µ = ( − ω, , , k ) and is a pure-gaugefield. This is a consequence of the gauge invariance of the theory e µ → e µ + ak µ . If ω − k = 0, then the rank of the matrix drops, rankH | ω − k =0 = 1, and the number ofindependent solutions increases: 4-1=3. These three solutions of equations (3.25) are e ( gauge ) γ = 1 √ − , e (1) γ = , e (2) γ = , from which the first one is a pure-gauge field ( ∼ k γ ), while the remaining two are thephysical modes, perpendicular to the direction of the wave propagation. The generalsolution at ω − k = 0 will be a linear combination of these three eigenvectors: e γ = ak γ + c e (1) γ + c e (2) γ , where a, c , c are arbitrary constants. We see that the number of propagating modes is rankH | ω − k =0 − rankH | ω − k =0 = 3 − , as it should be. Symmetric Tensor Gauge Field
The free gravitational field is described in terms of a symmetric second-rank tensor field h µν and is governed by the Einstein and Pauli-Fierz equation: ∂ h νλ − ∂ ν ∂ µ h µλ − ∂ λ ∂ µ h µν + ∂ ν ∂ λ h µµ + η νλ ( ∂ µ ∂ ρ h µρ − ∂ h µµ ) = 0 , (3.26)9hich is invariant with respect to the gauge transformations δh µλ = ∂ µ ξ λ + ∂ λ ξ µ , (3.27)which preserve the symmetry properties of A µν . The corresponding matrix operator is: H α ´ αγ ´ γ ( k ) = { η α ´ α η γ ´ γ −
12 ( η αγ η ´ α ´ γ + η α ´ γ η ´ αγ ) } k − η α ´ α k γ k ´ γ − η γ ´ γ k α k ´ α ++ 12 ( η α ´ γ k γ k ´ α + η ´ α ´ γ k α k γ + η αγ k ´ α k ´ γ + η ´ αγ k α k ´ γ ) (3.28)and is a 10 ×
10 matrix in four-dimensional space-time with the property H α ´ αγ ´ γ = H ´ ααγ ´ γ = H α ´ α ´ γγ = H γ ´ γα ´ α and is presented in Appendix B.If ω − k = 0, the rank of the 10-dimensional matrix H γ ´ γα ´ α ( k ) is equal to rankH | ω − k =0 =6 and the number of independent solutions is 10 − k γ = ( ω, , , k ), then one can find the following four linearly independent solutions: e γ ´ γ = − ω k , − ω − ω k k , − ω
00 0 0 0 − ω k k , − ω k k , (3.29)pure-gauge field solutions of the form (3.27) e γ ´ γ = k γ ξ ´ γ + k ´ γ ξ γ as one can see from therelation H γ ´ γα ´ α ( k )( k γ ξ ´ γ + k ´ γ ξ γ ) = 0 . (3.30)When ω − k = 0, then the rank of the matrix H α ´ αγ ´ γ ( k ) drops and is equal to rankH | ω − k =0 =4. This leaves us with 10 − e (1) γ ´ γ = − , e (2) γ ´ γ = . (3.31)10hus the general solution of the equation on mass-shell is: e γ ´ γ = ξ γ k ´ γ + ξ ´ γ k γ + c e (1) γ ´ γ + c e (2) γ ´ γ , where c , c are arbitrary constants. We see that the number of propagating modes is rankH | ω − k =0 − rankH | ω − k =0 = 6 − , as it should be. Antisymmetric Tensor Gauge Field
The antisymmetric second-rank tensor field B µν is governed by the equation [48, 49, 50, 51]: ∂ B νλ − ∂ ν ∂ µ B µλ + ∂ λ ∂ µ B µν = 0 , (3.32)which is invariant with respect to the gauge transformations δB µλ = ∂ µ η λ − ∂ λ η µ , (3.33)which preserve the symmetry properties of B µν . The corresponding matrix operator is: H α ´ αγ ´ γ ( k ) = −
12 ( η αγ η ´ α ´ γ − η α ´ γ η ´ αγ ) k −−
12 ( η α ´ γ k γ k ´ α − η ´ α ´ γ k α k γ + η ´ αγ k α k ´ γ − η αγ k ´ α k ´ γ ) (3.34)and is 6 × H α ´ αγ ´ γ = − H ´ ααγ ´ γ = − H α ´ α ´ γγ = H γ ´ γα ´ α and is presented in Appendix B.If ω − k = 0, the rank of the 6-dimensional matrix H γ ´ γα ´ α ( k ) is equal to rankH | ω − k =0 =3 and the number of independent solutions is 6 − k γ = ( ω, , , k ) one can findthe following three solutions: e γ ´ γ = ω
00 0 0 0 − ω k − k , ω − ω k − k , − , (3.35)11ure-gauge fields of the form (3.33) e γ ´ γ = k γ η ´ γ − k ´ γ η γ , as one can see from the relation H γ ´ γα ´ α ( k )( k γ η ´ γ − k ´ γ η γ ) = 0 . (3.36)When ω − k = 0, then the rank of the matrix H α ´ αγ ´ γ ( k ) drops and is equal to rankH | ω − k =0 =2. This leaves us with 6 − e ( A ) γ ´ γ = − . (3.37)Thus on mass-shell the general solution of the equation is: e γ ´ γ = k γ η ´ γ − k ´ γ η γ + c e ( A ) γ ´ γ , where c is arbitrary constant. We see that the number of propagating modes is rankH | ω − k =0 − rankH | ω − k =0 = 3 − . After this parenthetic discussion we shall turn to the tensor gauge theory.
Non-Abelian Tensor Gauge Field
Now we are ready to consider the equation (3.21) for the tensor gauge field A µλ with thematrix operator (3.20), which in four-dimensional space-time is a 16 ×
16 matrix. In thereference frame, where k γ = ( ω, , , k ), it has the form presented in the Appendix B.If ω − k = 0, the rank of the 16-dimensional matrix H γ ´ γα ´ α ( k ) is equal to rankH | ω − k =0 =9 and the number of linearly independent solutions is 16 − e γ ´ γ = − ω k , ω k , ω k , ω
00 0 0 00 0 0 00 0 k , ω k , ω k , ω k (3.38)pure-gauge tensor potentials of the form (3.17) e γ ´ γ = k γ ξ ´ γ + k ´ γ η γ , (3.39)as one can get convinced from the relation H γ ´ γα ´ α ( k )( k γ ξ ´ γ + k ´ γ η γ ) = 0 , (3.40)which follows from the gauge invariance of the action and can be checked also explicitly.When ω − k = 0, then the rank of the matrix H α ´ αγ ´ γ ( k ) drops and is equal to rankH | ω − k =0 = 6. This leaves us with 16 − e (1) γ ´ γ = − , e (2) γ ´ γ = , e Aγ ´ γ = − (3.41)Thus the general solution of the equation on mass-shell is: e γ ´ γ = ξ ´ γ k γ + η γ k ´ γ + c e (1) γ ´ γ + c e (2) γ ´ γ + c e ( A ) γ ´ γ , (3.42)where c , c , c are arbitrary constants. We see that the number of propagating modes isthree rankH | ω − k =0 − rankH | ω − k =0 = 9 − . These are propagating modes of helicity-two and helicity-zero λ = ± , charged gaugebosons [43, 44, 45]. Indeed, if we make a rotation around the z-axisΛ αβ = θ − sin θ
00 sin θ cos θ
00 0 0 1 ,
13e shall get e (1) ′ = Λ e (1) Λ T = − cos 2 θ − sin 2 θ − sin 2 θ cos 2 θ
00 0 0 0 , e (2) ′ = Λ e (2) Λ T = − sin 2 θ cos 2 θ
00 cos 2 θ sin 2 θ
00 0 0 0 . Therefore the first two solutions describe helicity λ = ± A aµλ , as it was done in[43, 44, 45]. Indeed, one can observe that for the symmetric part of the tensor gauge fields A aνλ the equation reduces to the previously studied free equation in gravity (3.26), whichdescribes the propagation of massless tensor gauge bosons with two physical polarizations:the λ = ± A aνλ theequation reduces to the equation (3.33) which describes the propagation of massless bosonwith one physical polarization: the λ = 0 helicity state. Interaction of Currents
The interaction amplitude between two tensor currents caused by the exchange of thesetensor gauge bosons can be found from (3.16), (3.18) and (3.19) and has the following form[45] J ′ µλ ∆ µλνρ J νρ (4.43)where the propagator ∆ abµλνρ is∆ abµλνρ = δ ab η µν η λρ − η µλ η νρ ω − k , (4.44)therefore J ′ µλ ∆ µλνρ J νρ = J ′ µλ ω − k J µλ − J ′ µµ ω − k J λλ . (4.45)We shall evaluate the first term in the interaction amplitude. This gives J ′ µλ ω − k J µλ = 1 ω − k { J ′ J − J ′ J − J ′ J − J ′ J − J ′ J − J ′ J − J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J } . k µ = ( ω, , , k ) and using the conservation of the current (2.10) expressed in themomentum space k µ J µλ = 0 , ωJ λ = − k J λ we shall get1 ω − k [(1 − ω k ) J ′ J − (1 − ω k ) J ′ J − (1 − ω k ) J ′ J − (1 − ω k ) J ′ J − J ′ J − J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J + J ′ J ] . Now using the second conservation law (2.11) in momentum space k λ J µλ = 0 , ωJ µ = − k J µ we arrive at 1 ω − k [(1 − ω k ) J ′ J − (1 − ω k ) J ′ J − (1 − ω k ) J ′ J −− (1 − ω k ) J ′ J − (1 − ω k ) J ′ J − (1 − ω k )( ω k ) J ′ J ] ++ 1 ω − k [ J ′ J + J ′ J + J ′ J + J ′ J ]and after simple algebra at − k [(1 − ω k ) J ′ J − J ′ J − J ′ J − J ′ J − J ′ J ] ++ 1 ω − k [ J ′ J + J ′ J + J ′ J + J ′ J ] . Evaluating the second term in the interaction amplitude (4.44) in the same manner asabove, we shall finally get for the total amplitude: − k [(1 − ω k ) J ′ J − J ′ J − J ′ J − J ′ J − J ′ J ] ++ 1 ω − k [ 12 ( J ′ − J ′ )( J − J ) + J ′ J + J ′ J ] . (4.46)For the instantaneous term we get − k [(1 − ω k ) J ′ J − J ′ J − J ′ J − J ′ J − J ′ J ] (4.47)and for the retarded term ( J = J )1 ω − k [ 12 ( J ′ − J ′ )( J − J ) + J ′ J + J ′ J ] . (4.48)15he retarded term represents a sum of three independent products+ 14 [ J ′ − J ′ + i ( J ′ + J ′ )][ J − J − i ( J + J )] ++ 14 [ J ′ − J ′ − i ( J ′ + J ′ )][ J − J + i ( J + J )] ++ 12 ( J ′ − J ′ )( J ′ − J ′ ) (4.49)or polarizations corresponding to the helicities λ = ± ,
0. Thus all negative-norm statesare excluded from the spectrum of the second-rank tensor gauge field A µλ , due to the gaugeinvariance of the theory and we come to the conclusion that the theory does indeed respectunitarity at the free level. Appendix A. Bianchi identity
The non-Abelian tensor fields A aµλ ...λ s can be seen as appearing in the expansion of theextended gauge field A µ ( x, e ) over the unit tangent vector e λ [43, 44, 45]: A µ ( x, e ) = ∞ X s =0 s ! A aµλ ...λ s ( x ) L a e λ ...e λ s . and the extended field strength tensor can be defined in terms of the extended gauge field A µ ( x, e ) as follows: G µν ( x, e ) = ∂ µ A ν ( x, e ) − ∂ ν A µ ( x, e ) − ig [ A µ ( x, e ) , A ν ( x, e )] . Defining the extended covariant derivative: D µ = ∂ µ − ig A µ , one can get [45]:[ D µ , D ν ] = [ ∂ µ − ig A µ , ∂ ν − ig A ν ] = − ig G µν . (5.50)The operators D µ , D ν , D λ obey Jacobi identity:[ D µ , [ D ν , D λ ]] + [ D ν , [ D λ , D µ ]] + [ D λ , [ D µ , D ν ]] = 0 , which with the aid of (5.50) is transformed into the generalized Bianchi identity[ D µ , G νλ ] + [ D ν , G λµ ] + [ D λ , G µν ] = 0 . (5.51)Let us now expand equation (5.51) over e ρ up to linear terms. We have,[ ∂ µ − igA µ − igA µρ e ρ , G νλ + G νλ,ρ e ρ ] + cyc.perm. + O ( e ) = 016n zero order the above equation gives the standard Bianchi identity in YM theory:[ D µ , G νλ ] + [ D ν , G λµ ] + [ D λ , G µν ] = 0 , where D µ = ∂ µ − igA µ . The linear term in e ρ gives:[ D µ , G νλ,ρ ] − ig [ A µρ , G νλ ] + [ D ν , G λµ,ρ ] − ig [ A νρ , G λµ ] + [ D λ , G µν,ρ ] − ig [ A λρ , G µν ] = 0 (5.52)Using explicit form of the operators D µ , G µν and G µν,λ one can independently check thelast identity and get convinced that it holds. Now, if we expand the above equation overg, the zeroth order gives the Bianchi identity for the field strength tensor F νλ,ρ : ∂ µ F νλ,ρ + ∂ ν F λµ,ρ + ∂ λ F µν,ρ = 0 (5.53)These equations impose tight restrictions on the source currents and hence on the natureof interactions. Appendix B
The matrix operator in gravity (3.28) is of dimension 10 ×
10 and has the following form0 0 0 0 − k − k − k − kw − k − kw
00 0 0 0 kw kw − k − kw k − w − w ( − k + w ) 0 0 0 00 kw w − k − kw k − w − w kw w
00 0 0 0 − w − w × − k kw − k kw ( − k + w ) 0 0 − kw w − kw w . (6.55)The matrix operator for non-Abelian tensor gauge theory (3.20) is of dimension 16 × − k
22 0 0 0 0 − k
22 0 0 0 0 00 − k k
22 0 0 kω − kω − k k
22 0 0 kω − kω
00 0 0 0 0 kω kω k
22 0 0 − k − kω kω − k
22 0 0 − kω k − ω
2) 0 − kω − ω
220 0 0 0 0 0 − k ω k − ω
2) 0 0 0 0 0 00 − kω kω ω − ω
22 0 00 0 k
22 0 0 0 0 0 − k − kω kω k − ω
2) 0 0 − k ω − k
22 0 0 − kω k − ω
2) 0 0 0 0 0 0 − kω − ω
220 0 − kω kω ω − ω
22 00 0 0 0 0 kω kω kω − kω − ω
22 0 0 0 0 0 ω kω − kω − ω
22 0 0 ω − ω
22 0 0 0 0 − ω
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