Physics Letters B 531 (2002) 152-160

www. elsevier. com/locate/npe

Massless arbitrary spin fields in AdS5

R.R. Metsaev

Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, 119991, Moscow, Russia Received 5 February 2002; accepted 11 February 2002 Editor: P.V. Landshoff

Abstract

Free arbitrary spin massless and self-dual massive fields propagating in AdS$ are investigated. We study totally symmetric and mixed symmetry fields on an equal footing. Light-cone gauge action for such fields is constructed. As an example of application of light-cone formalism we discuss AdS/CFT correspondence for massless arbitrary spin AdS fields and corresponding boundary operators at the level of two point function. © 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

A study of higher spin gauge theory has two main motivations (see, e.g., [1,2]): firstly, to overcome the well-known barrier of N < 8 in d = 4 supergravity models and, secondly, to investigate if there is a most symmetric phase of superstring theory that leads to the usual string theory as a result of a certain spontaneous breakdown of higher spin symmetries. Another motivation came recently from conjectured duality of free large N conformal N = 4 SYM theory and a theory of massless higher spin fields in AdS5 [3,4]. Discussion of this theme in the context of various limits in AdS superstring may be found in [5,6]. In both the tensionless superstring theory and massless higher spin fields theory all types of massless fields appear in general. This implies that all types of massless fields should be studied. This is what we are doing in this Letter.

E-mail address: metsaev@lpi.ru (R.R. Metsaev).

Let us first formulate the main problem we solve in this Letter. Fields propagating in AdS5 space are associated with positive-energy unitary lowest weight representations of SO(4,2) group. A positive-energy lowest weight irreducible representation of the SO(4,2) group denoted as D(E0, h), is defined by E0, the lowest eigenvalue of the energy operator, and by h = (h1 ,h2), h1 > |h21, which is the highest weight of the unitary representation of the SO(4) group. The highest weights h1 ,h2 are integers and half-integers for bosonic and fermionic fields respectively. In this Letter we investigate the fields whose E0 and h are given by

E0 = h1 + 2, E0 >h1 + 1,

h1 > |h21, h1 = |h21,

|h21 > 1/2. (2)

The fields in (1) are massless fields while the ones in (2) are massive self-dual fields. The massless fields with h1 > 1, h2 = 0, ±1/2 are referred to as totally symmetric fields while both the massless and massive fields with |h21 > 1/2 we refer to as mixed symmetry fields. In manifestly Lorentz covariant formulation

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0370-2693(02)01344-8

the totally symmetric and mixed symmetric fields are the tensor fields whose SO(4, 1) space-time indices have the structure of the respective Young tableauxes with one and two rows.1 Actions for the massless totally symmetric integer and half-integer spin fields in AdSd space were found in [7,8]. The gauge invariant equations of motion for all types of massless fields (totally and mixed symmetric ones) in AdSd space for even d have been found in [9,10].2 Massless mixed symmetry AdS5 fields with arbitrary h and self-dual massive fields with arbitrary E0 have not been described at the field theoretical level so far.3 In this Letter we develop a light-cone gauge formulation for such fields at the action level. Using the light-cone gauge formalism in AdS space developed in [14], we describe both the totally and mixed symmetry fields on an equal footing. Since, by analogy with flat space, we expect that a quantization of the Green-Schwarz AdS superstring with Ramond-Ramond charge will be available only in the light-cone gauge [23,24] it seems that from the stringy perspective of AdS/CFT correspondence the light-cone approach is the fruitful direction to go.

2. Light-cone gauge action

'covariantform' [14].4

Sl.c. =

d x ié(x) I

□ = 29+d - + df + d.

where an A is some operator not depending on spacetime coordinates and their derivatives. This operator acts only on spin indices of fields collected in ket-vector . We call the operator A the AdS mass operator. In the remainder of this Letter we find it convenient to Fourier transform to momentum space for all coordinates except for the radial z and the light-cone time x+

|<£(x+,x ,x',z)) _ j d2pdfi i(ex-+pixi)UL +

(2n)3/2

where we use ¡3 to denote a momentum in light-cone direction: ¡3 = p+. Now we rewrite the action in the Hamiltonian form

Sl.c. = Jdx+ d2pdpdz{0{x+, -p,z)\

x 3(i9- + P-)\$(x+,p,z)), (5)

where P- is the (minus) Hamiltonian density

Let $(x) be a arbitrary spin field propagating in AdS5 space. If we collect spin degrees of freedom in a generating function l$(x)} then a light-cone gauge action for this field can be cast into the following

1 We note that the case h = (1,0) corresponds to spin one Maxwell field, the case h = (2, 0) is the graviton, and the cases h = (1, ±1) correspond to 2-form potentials.

2 The particular case of massless field corresponding to h = (2, 1), for d ^ 6 was described at the action level in [11]. Gauge invariant actions for AdS4 massless fields have been established in [12,13]. Various gauge invariant descriptions of totally symmetric massless fields in terms of the potentials in AdSd may be found in [14-16]. Massive self-dual arbitrary spin fields in AdS3 were investigated in [17]. Discussion of massive totally symmetric fields in AdSd, d ^ 4, may be found in [18,19].

3 Group theoretical description of representation of so(4,2) algebra for arbitrary h and discrete values of Eo via oscillator method [20] may be found, e.g., in [21]. Lorentz covariant equations of motion for AdS5 self-dual massive fields with special values of

E0 were discussed in [22].

In Eq. (5) and below a momentum p as argument of wave function $(x+, p, z) designates the set {p',P}. All that remains to fix action explicitly is to find the AdS mass operator. The explicit form of this operator depends on what kind of realization of spin degrees of freedom we use. Let us first consider the bosonic fields.

4 We use parametrization of AdS5 space in which ds2 = (—dx0 + dxf + dxf + dz2)/z2. Light-cone coordinates in ± directions are defined as x^ = (x3 ± x°)/V2 and x+ is taken to be a light-cone time. Unless otherwise specified, we adopt the conventions: i,j = 1, 2. d' = d/dx', dz = d/dz, d± = d^ = d/dxT. We use indices I, J for so(3) tangent space vectors X1 = (X' ,Xz). Sometimes, instead of so(2) vector indices i, j = 1, 2 we use complex frame indices i, j = x,x. In this notation a so(2) vector X' is representable by Xx, Xx and for scalar product we have the decomposition X' Y' = XxYx + X-1 Yx.

3. Bosonic fields

To describe light-cone gauge formulation of the D(E0, h) massless or self-dual massive bosonic field we introduce a complex-valued totally symmetric traceless so(3) tensor field ^Il-Ihi. As usual to avoid cumbersome tensor expressions we introduce creation and annihilation oscillators a1 and a1

[a I,aJ] = 5

a110) = 0,

and the generating function |0) defined by

\$(x+,p, z)) = a11 ••• aIhi ^Il-Ihi (x+,p, z) |0). (8) The AdS mass operator is found then to be

1 ;; ;; 1

A =--mJmJ--,

¿J = Mij - i€ijc,

e12 = —e21 = 1, where the so(2) spin operator M'J takes the standard form

MJ = a'aJ — aJ a1,

and the number c is given by

Ih2, for massless fields (1), (Eo — 2) sign h2, (11)

for massive self-dual fields (2),

where signh = +1 (—1) for h > 0 (h < 0). Details of derivation may be found in Appendix B. A few comments are in order.

(i) The light-cone gauge action for the mixed symmetry D(E0, h) field, (|h2| > 0), is formulated in terms of the complex-valued rank h1 totally symmetric traceless so (3) tensor field. Number of physical degrees of freedom (DoF) of such field is equal to 2(2h1 + 1), i.e., for the field in (1), it depends only on h1 and does not on h2.

(ii) For the case of massless totally symmetric field (h2 = 0) we can impose reality condition on the complex-valued so(3) tensor field and this gives the standard light-cone description of the massless totally symmetric field [14]. Number of physical DoF of such field is equal to 2h1 + 1.

(iii) The h2 of AdS5 fields is similar to helicity of massless fields in flat four-dimensional space. As is well known number of physical DoF of the latter fields does not depend on helicity and is equal to two. If AdS5 field |0) has helicity

h2 then its complex-conjugated (01 has opposite helicity —h2, i.e., the light-cone gauge action (5) is formulated in terms of fields having opposite values of h2.

4. Fermionic fields

In this case h1 and h2 are half-integers and in what follows we use notation h[ = h1 — 1/2. To discuss light-cone gauge formulation of D(E0, h) fermionic field we introduce complex-valued traceless so(3) h...lh'

spin-tensor field 0Y 1, y = 1, 2, which is symmetric in I1,..., Ih'. In addition to the bosonic creation and annihilation1 oscillators (7) we introduce fermionic oscillators nY, ny

{na,nY} = 5aY, nY |0) =0. (12)

For this case the Fock vector |0) defined by

I1...Ih'

|0(x+,p, ^ = a11

• a h1 nY0Y

1 (x+,p,z) |0)

is assumed to satisfy the constraint

(a ioi —az)o 3n|0) = 0,

where a1, a3 are Pauli matrices. This constraint

h...ih'

amounts to requiring the spin-tensor field 0a 1 transforms in irreducible spin h1 representation of so(3) algebra. The operator A and the number c take then the form given by formulas (9), (11) in which the spin operator MiJ (10) should be replaced by

M,] =a'd]-a]d'+ -e,Jr]a3i]. (15)

As a side of remark we note that these results can be presented in an equivalent form which is somewhat convenient for evaluation. Instead of tensor fields (8), (13) we can introduce 2h1 + 1 complex-valued fields which with the help of a complex-valued variable Z we collect in the |0) as

|0(x+,p,z)) = Zhl—mMx+,P,z),

m=—h\

where for bosons m = 0, ±1,...,±h1, while for fermions m = ±1/2, ±3/2,..., ±h1. The Z, which

we refer to as projective variable, is used to formulate results entirely in terms of the generating function |<). The scalar product for such generating functions is defined to be

(1+ l^)2A1+2

t(Z)<p(Z)-

The AdS mass operator for this realization takes the form

a = c — ht

where the number c is given by (11). In the complex notation the spin operator Mlj is representableby Mxx which takes the form

Mxx = —zdZ + hi, dZ = d/dz.

We note that the representation for the operators A (18) and Mxx (19) is valid for both the integer and half-integer fields, i.e., while using the projective variable Z it is not necessary to introduce anticommuting oscillators (12) to discuss the half-integer fields.

5. Global symmetries of light-cone gauge action

We turn now to discussion of global so(4,2) symmetries of the light-cone gauge action. The choice of the light-cone gauge spoils the manifest global symmetries, and in order to demonstrate that these global invariances are still present one needs to find the Noether charges which generate them.5 The field theoretical representation for Noether charges in AdS light-cone gauge formalism takes the standard form [14]

G = J dzd2pdpp[<(x+, -p,z)\G\<(x+,p,z)),

where G are differential operators acting on physical field |<). Noether charges (generators) canbe split into two groups: kinematical and dynamical generators (see Appendix A). Representation of the kinematical

5 These charges play a crucial role in formulating interaction vertices in field theory. Application of Noether charges in formulating superstring field theories may be found in [25,26].

generators is given by

P' = p', p + = p,

— dp'p' + zdz +

D = ix+P— — J+— = ix+P — + J+' = ix+p' + dp'P,

J'j = p'dpj — pjdp' + M'j,

K+ = \ {2ix+dp - d2pl + z2)p + ix+D,

K' = -(2ix+dfi-d2p}+z2)p' -dplD

— M'Jdpj — Mz'z + iM'—x

(21) (22)

where <//;i; = <)/■■, = ^7. The dynamical generator P- is given by (6) while the remaining ones are

J= —dpp' + d'P — + M,

K~ = -(2ix+3p - d2pl +z2)P~ - dpD

dp' r t 1

2zj3 J P

where 'nonlinear' spin operator M 1 = —Ml is given by

M~l = J ^Mijpj + Mzi dz + Y [Mzi, A]^. (30)

The operators B, Mzi are acting only on spin degrees of freedom and are given by

B = -MZ1MZ1 + -ce'J M'J, 2

Mzl = azâ' + a'az + -rja'fj.

For the case of bosonic fields we should simply ignore the fermionic term in expression for Mz' (32). We note that the spin operators M'j, Mz' satisfy commutation relations of so(3) algebra

[Mz',Mzj] = M'j, [M'j,Mzk] = SjkMz' — S'kMzj,

and a realization of the spin operator Mz' in terms of projective variable Z takes the form

Mzx = dt;, Mzx = (34)

Since the light-cone gauge action (5) is invariant with respect to the global symmetries generated by so(4, 2) algebra, 5g|0) = G0), the formalism we discuss is sometimes referred to as an off shell light-cone formulation [25].

6. Ground state of AdS5 fields

In this section we would like to demonstrate that the massless and self-dual massive fields discussed above are indeed carriers of the positive-energy unitary lowest weight representations of SO(4, 2) group labeled by D(E0, h) where E0 is given by Eqs. (1), (2). To do this we evaluate ground state of AdS fields. All that requires is to demonstrate that E0 for ground states of massless and self-dual fields satisfy the basic relations (1), (2). To this end we begin with definition of ground state | v) (vacuum). The most convenient way to do this is to use so(4,2) algebra taken to be in energy basis (see Appendix A). In this basis one has energy operator Jww, deboost operators Jwy, Jwy, Jwl, boost operators, Jwy, Jwy, Jwl and six operators, Jiy, J'y, Jyy, JlJ, which are generators of the so(4) algebra.6 The vacuum forms a linear space which is invariant under the action of the energy operator Jww and generators of the so(4) algebra. In other words the | v) is (i) the eigenvalue vector of Jww; (ii) a weight h representation of the so (4) algebra. Now the vacuum | v) is defined by the relations

Jww |v) = E0|v),

Jwy |v)=0, Jwy |v)=0,

Jwi |v)=0.

Eq. (35) reflects the fact that the | v) is the eigenvector of the energy operator Jww, while Eqs. (36), (37) tell us that E0 is the lowest value of the energy: since the deboost operators decrease the energy they should evidently annihilate the |v). Just as in the

6 In complex notation jij is representable by jxx while so(2) vectors like jw' are representable by jwx and jwx.

ordinary Poincare quantum field theory one can build the representation Fock space D(E0, h) by acting with boost operators Jwy, Jwy, Jwi on the vacuum states

Without loss of generality and to simplify our presentation we (i) evaluate representative of ground state denoted by |vE°,h) which is a highest weight vector of representation of the so (4) algebra; (ii) evaluate ground state for x+ = 0 and j > 0; (iii) drop the |vE°,h) in equations and instead of equations like X|vE0 ,h) = Y |vE0 ,h) we write simply X « Y .In addition to (35)-(37) the |vE0,h) satisfies, by definition, the following constraints

Jyy « h1, Jyi « 0.

Next step is to rewrite Eqs. (35)-(39) in terms of generators in light-cone basis (21)-(29). Making use of (37), (39) and formulas (A.12), (A.14) we get the equations

Pi + J+i « 0,

Ki — J ~

Using Eqs. (36) and formulas (A.8), (A.9) we can express P — and K — in terms of P+, K+, D and J+—. Exploiting this representation for P—, K— in (A.6), (A.7) we get from (35) and the 1st equation in (38) the following equations

P+ + \{~D + J+-K+ + ±(D + J+-)

(E0 + h1),

(E0 — h1).

Now making use of concrete representation for generators in terms of differential operators (21)-(29) we can find ground state explicitly. Using Eqs. (40), (42), (43) we find respective dependence of the ground state | vE°,h) on momenta pl, j and radial coordinate z:

I v—xp, p, z)) = exp( -^p'p1 ~ \pz2 - p

.,E0,h(

E0— hi —3/20E0 — 1

where | v0) depends only on Z .To find a dependence of | v0) on Z we use the 2nd equation in (38) which with the help of (19) gives

|v0) = Z

_ Zh1—h2

Formulas (44), (45) give explicit representation for the ground state which we use to find desired constraint on E0 (1), (2). To this end we analyze Eqs. (41) whichfor i = x and i = x lead to the equations

(Eo - hi - 1 + a + Zdz)Mzx K> = 0, (E0 - h1 - 1 - a - Zdz)MzX K} = 0.

From now on we analyze the cases h1 > |h2| and h1 = |h2| in turn.

(i) If h1 > |h2| then Eqs. (46), (47) give

Eo - 2 + a - h2 = 0, Eo - 2h1 - 2 - a + h2 = 0.

It is easy to see that solution to these equations is E0 = h1 + 2, a = h2 — h1. These E0 and a correspond to massless fields (see (1), (11), (18)).

(ii) If h1 = h2, h2 > 0 then Eq. (46) is satisfied automatically while Eq. (47) gives a = E0 — 2 — h1.

(iii) If h1 = —h2, h2 < 0 then Eq. (47) is satisfied automatically while Eq. (46) gives a = 2 — E0 — h1.

Thus the cases (ii) and (iii) lead to values of a corresponding to the self-dual massive fields (see (18), (11)). As to restriction on E0 (2) it follows simply from the requirement that the ground state (44) be square-integrable with respect to z: / dz||vEo,b)|2 < to. A requirement of square-integrability with respect to remaining variables does not lead to new constraints.

7. AdS/CFT correspondence

In spite of its Lorentz noncovariance, a light-cone formalism offers conceptual and technical simplification of approaches to various problems of modern quantum field theory. One of problems which allows to demonstrate efficiency of a light-cone approach is AdS/CFT correspondence [27-29]. In the Lorentzian signature AdS/CFT correspondence at the level of state/operator matching between the bulk massless arbitrary spin totally symmetric fields and the corresponding boundary operators was demonstrated in [14]. Here we would like to demonstrate this correspondence in the Euclidean signature at the level of

two point function.7 In light-cone gauge the massless arbitrary spin field in AdSd is described by an totally symmetric traceless so(d — 2) tensorfield $l1..Js, I = 1,...,d — 2. This tensor can be decomposed into traceless totally symmetric tensors of so(d — 3) algebra 0l1...Is = ES'=0$'s1 ...'s', i = 1,...,d — 3. The Euclidean light-cone gauge action8 takes then the following simple form [14,32]

Sl.c. =

AA4>A'

where an eigenvalue of the AdS mass operator for 0'1...'s' is given by

= s' +

Attractive feature of this action is that there are no contractions of tensor indices of fields with the ones of space derivatives, i.e., the action looks like a sum of actions for 'scalar' fields with different mass terms. This allows us to extend the analysis of Ref. [29] in a rather straightforward way. Using Green's function method a solution to equations of motion

"9x ~dZ+^2

^...V (x,z) = 0,

x = (x1,..., is found to be

«^...''s' (x,z)

(z2 +|x - x'|2)V+(d-1)/2

' (x').

As was expected [14] this solution behaves for z ^ 0 like z-vs'+1/2O(x). Plugging this solution into the action (49) and evaluating a surface integral gives

. IV H--

7 Discussion of AdS/CFT correspondence for spin one Maxwell field, s = 1, and graviton, s = 2, may be found in [29] and [30,31], respectively.

8 Note that only in this section we use the Euclidean signature.

For d = 4 the AdS mass operator is equal to zero for all massless

fields. Here we restrict our attention to the dimensions d > 5.

As' = vt -

x / dxdx--———I—, (53)

J |x - x'\2vs'+d-1

This is so(d — 3) decomposition of two point function of spin s conserved current.

For the case of AdS5 this analysis can be generalized to arbitrary D(E0,h1,h2) fields (1), (2) in a rather straightforward way. The most convenient way to do this is to use representation for AdS mass operator given in (18). All that is required is to replace in above formulas the vs' by v = \Zdz + a\, and the fields <s' by \<) (16). Repetition of the procedure outlined above gives

SF = - f dxdx'l&(x)\ V + l{2 A |(9(x'))- (54) Lo" 2J 1 '|x-x'|2v+41 1 K '

As in (16) the ket-notation \O) is used to indicate dependence of conformal operator on the complex-valued variable Z. A scalar product (O\O') is defined as in (17).

Acknowledgements

This work was supported by the INTAS project 991590, by the RFBR Grant No. 02-02-17067, and RFBR Grant for Leading Scientific Schools, Grant No. 01-02-30024.

Appendix A

Generic commutation relations of so(4,2) algebra generators JCF, C, F = 0', 0, 1..., 4, we use are

[JABJCF] = nBCJAF + 3 terms,

nCF = (—, —, +,...,+). (A.1)

In the conformal algebra notation the JCF split into translations Pa, conformal boosts Ka, dilatation D, and so(3, 1) rotations Jab, a,b = 0, 1,2, 3. Commutation relations in this basis are

[D,Pa ] = —Pa, [D,Ka ] = Ka,

[Pa,Kb] = nabD — Jab, (A.2)

[Pa Jbc J = f^ab Pc — nacPb

[Ka,Jbc] = nabKc — nacKb, (A.3)

JJab,Jce] = nbcJae + 3 terms,

nab = (—, +, +, +). (A.4)

In the light-cone basis these generators split into two groups:

P+, Pl, K+, Kl, J+l, J+—, Jlj, D, (A.5)

which we refer to as kinematical generators, and P—, K—, J—l which we refer to as dynamical generators. The kinematical generators have positive or zero J+— charges, while dynamical generators have negative J+— charges. Commutation relations in light-cone basis are obtainable from (56)-(58) with the light-cone metric having the following non-vanishing elements n+ = n + = 1, nl] = .

In energy basis the generators JCF split into energy operator Jww, deboost operators Jwy, Jwy, Jwl, boost operators Jwy, Jwy, Jwl ,and so (4) algebra generators Jly, Jly, Jyy, Jlj. Commutation relations in this basis are obtainable from (A. 1) with the metric having the following non-vanishing elements nxx = nyy = —nww = 1 (in the so(2) notation nlj = ). Interrelation of the generators in light-cone and energy bases is given by

JWW = ^(P+ - p- + K+ - K~), (A.6)

Jyy = ^(P+ + p- - K+ - K~), (A.7)

Jwy = \{p~ + K+ + D + J+-), (A.8)

Jwy = ^(P+ + K~ -D + J+-), (A.9)

JWy = \{~P+ -K--D + /+"), (A.10)

Jwy = ^(-P~ -K+ + D + J+-), (A.11)

Jw' = ^(Pl + Kl + J+i - J~l), (A.12)

Jw' = ^(-Pi -Kl + J+i (A.13)

Jy' -- = -K[ + J+i + J~i), (A.14)

Jy' -- = + Kl + J+i + J-1). (A.15)

Appendix B

In this appendix we explain a procedure of derivation of the AdS mass operator and the operator B which enter in the definition of the light-cone gauge action and representation of the generators of so(4,2) algebra on physical fields. As was demonstrated in [14] the operators A and B should satisfy the basic defining equations

[.A,Mij ] = 0, (B.1)

2{Mzi,A} — [[Mzi,A],A] = 0, (B.2)

— [Mzi, [Mzj,A]] + {Mil,Mlj} + {Mzi,Mzj}

= —2SijB. (B.3)

To find solution of these equations we use representation for spin operators Mij, Mzi in terms of projective complex-valued variable Z given in (19), (34). In general the A is an operator depending on Z and dZ. Eq. (B.1) tells that the A depends only on N, where N = ZdZ .We now are looking for A = A(N) satisfying the remaining Eqs. (B.2), (B.3). Making use of the obvious relations

A(N)dZ = dZA(N — 1), A(N)Z = ZA(N + 1),

we get from Eqs. (B.2) for i = x and i = x the respective functional equations

2(A(N) + A(N — 1)) — (A(N) — A(N — 1))2 = 0,

2(A(N) + A(N + 1)) — (A(N) — A(N + 1))2 = 0.

It is easy to see that solution of these equations does not involve higher than second-order terms in NN . Making use of general anzats A = q n2 + r N + t we find that q = 1, r = 2a, t = a2 — 1/4, where a is arbitrary constant. Thus we get A = (N + a)2 — 1/4. To find the operator B we use Eq. (B.3) for i = x, j = x. Solution to this equation is found to be

B = —{Mzx,Mzx} — (a + h1)Mxx. (B.7)

The remaining unknown number a can be fixed in various ways. One way to fix a is to consider the ground state (see Eq. (34)). Here we would to outline procedure of finding a by using technique of Casimir

operators. Let Q be the second order Casimir operator. Making use of expressions for generators (21)-(29) the equation (Q — (Q))|0) = 0, where (Q) is the eigenvalue of the Q in D(E0, h)

— Q) = E0E0 — 4) + h1(h1 + 2) + h2,

can be cast into the form (for details see Section 4 in Ref. [14])

—A + 2B + —Mfj + —— (2) = 0-

Plugging solution for A and B into (B.9) we obtain the equation

(E0 — 2)2 — h2 + h2 — (a + h1)2 = 0.

(B.10)

In the same way an evaluation of the third order Casimir operator gives the equation h1(h1 + a) = (E0 — 2)h2. For massive fields (2) this equation and (B.10) lead to a given in (18), (11). For the case of h1 > |h2| we get E0± = 2 ± h1 and the physical relevant value E0+ (1) can be fixed by unitarity condition (see, e.g., [9,11]). Making use of normalization for Levi-Civita tensor in complex frame £xx = —i it is easy to see that the operators A (18) and B (B.7) can be cast into so(2) covariant form (9), (31). The most convenient way to relate the spin operators in so(2) covariant form (15), (32) and the ones in (19), (34) is to pass from the oscillators (7) to complex-valued variables a', az subject to the constraint a'a' — azaz = 0. For simplicity we consider the bosonic case. In terms of ai, az the so(2) covariant form of the spin operators is

Mij = aidaj — ajdai, Mzi = azdai + aidaz.

(B.11)

Introducing the Z = az/ax one can make sure that a representation of the operators (B.11) on a monomial having h1th power in ai, az takes the same form as in (19), (34).

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