Two-twistor particle models and free massive higher spin fields

We present D=3 and D=4 models for massive particles moving in a new type of enlarged spacetime, with D-1 additional vector coordinates, which after quantization lead to the towers of massive higher spin (HS) free fields. Two classically equivalent formulations are presented: one with a hybrid spacetime/bispinor geometry and a second described by a free two-twistor dynamics with constraints. After quantization in the D=3 and D=4 cases, the wave functions are given as functions on the SL(2,R) and SL(2,C) group manifolds respectively, and describe arbitrary on-shell momenta and spin degrees of freedom. Finally, the D=6 case and possible supersymmetric extensions are mentioned.


Introduction
The development of higher spin (HS) theory was predominantly associated with massless (conformal) HS fields. One of the important methods for the description of HS fields consists of introducing master fields on an enlarged spacetime, which then lead to spacetime fields with all possible values of helicity (when the mass m=0) or spin (when m =0). In particular, a collection of D=4 massless HS fields with arbitrary helicities was described by quantizing particles propagating in tensorial spacetime x M = (x µ ∼ x αβ , y µν ∼ (y αβ ,ȳαβ)) extended by commuting Weyl spinor coordinates y α , yα, α,α = 1, 2 (see e.g. [1][2][3][4][5]). We shall mostly consider here non-supersymmetric theories, without Grassmann spinors (for the spinorial notation, see Appendix A).
The most general D = 4 model in D=4 tensorial spacetime describing free HS multiplets is provided by the following action S = dτ π απβẋ αβ + a π α π βẏ αβ +āπαπβẏαβ + b π αẏ α +bπαẏα , where a, b are complex parameters,πα ≡ (π α ) * , etc. We recall that the model (1.1) with b = 0 was considered in [2], and that the last two terms (b = 0) were first introduced in [3]. The advantage of having b = 0 is the much simpler structure of the constraints in phase space and the easier quantization procedure. It turns out that for a = 0 and/or b = 0 the action (1.1) can be rewritten (modulo boundary terms) as the one-twistor free particle model (see e.g. [6,7]) dτ ω απ α −παωα + h.c.
= − dτ ω απ α −παωα + boundary term , (1.2) and the D=4 twistor Z A , A = 1, . . . , 4 (conformal basic spinor) is described by a pair of Weyl spinors where the conformally invariant scalar product is obtained by the particular choice of the anti-hermitian antisymmetric U(2, 2) metric 1 . (1.5) The passage from the hybrid spacetime/spinor description (1.1) to the twistorial one (1.2) is achieved by a modified Penrose incidence relation. For the actions (1.1) and (1.2) a suitably chosen incidence relation is: ω α = x αβπβ + 2a y αβ π β + b y α , ωα = π β x βα + 2āȳαβπβ +bȳα . (1.6) After inserting eqs. (1.6) into (1.1), the free twistorial particle action (1.2) follows modulo boundary terms. Besides, since x αβ in the action (1.1) has to be hermitian for x µ to be real, inserting (1.6) in eq. (1.4) we see that Using the realization of the Poincaré algebra in terms of the twistor coordinates Z A ,Z A (see [6,7,11]), and using the canonical Poisson brackets (PB) following from (1.2), it follows that in the D = 4 massless case the helicity h is given by (1.8) When a = b = 0 we obtain the Shirafuji model [7] with twistor coordinates restricted, due to (1.7), by the zero helicity constraintZ A Z A = 0. In the twistor formulation of the Shirafuji model (1.2), this helicity constraint has to be added by a Lagrange multiplier. We add that the zero value of helicity can be shifted after quantization (h →ĥ) to a non-zero one by using various orderings for the quantized twistors in the helicity operatorĥ [12]. If a = 0 and/or b = 0 the value of h (see (1.7)) is not kinematically restricted in the twistor framework and the action describes an infinite massless multiplet with all helicities (see e.g. [2]). In this paper we describe D=3 and D=4 HS particle models which, after quantization, lead to free massive HS fields with arbitrary values of spin. The application of the ideas presented in [2][3][4][5] to the massive case requires the doubling of spinor indices in the hybrid (eq.(1.1) ) actions (see e.g. [13][14][15][16]) and the enlargement to the free two-twistor action (see e.g. [17][18][19][20][21][22][23]). In our study we provide the generalizations of the actions (1.1) and (1.2) by incorporating the mass-shell constraints and by introducing a suitable form of the incidence relations. In this way, we obtain HS particle models with the right number of physical phase space degrees of freedom, namely six in D = 3 (abelian spins) and twelve degrees of freedom in D=4 (SU(2)-spins). It will follow that describing massive HS fields by an extension of the 'hybrid' (eq. (1.1)) and purely twistorial (eq. (1.2)) actions produces equivalent models with the same number of degrees of freedom.
The plan of the paper is as follows. In Sec. 2 we study D=3 massive HS models. After some kinematic results about D=3 two-twistor space we describe our D=3 counterpart of the model (1.1). It is shown that the standard two-twistor Shirafuji model without additional coordinates only provides spinless massive D=3 particles (see also [16]). To modify this conclusion and obtain D=3 massive particles with arbitrary spin, we introduce a spinorial action with a pair of additional three-vector coordinates and suitable mass constraints. Further, we describe the model in phase space and show that after solving the first class constraints providing the unfolded equations [24], we obtain a wave function on the three-dimensional D=3 spinorial Lorentz group SL(2; R) ≈ SO(2, 1) manifold, with three independent coordinates, two related with the three-momentum on the mass-shell, and the third with arbitrary D=3 Abelian spin values. After introducing suitable incidence relations we obtain the two-twistor formulation with eight-dimensional phase space containing one first-class mass constraint. If we quantize the twistorial model we obtain as well the wave function defined on the SL(2; R) group manifold. After providing the realization of the D=3 spin operator we get that the power expansion of the wave function (see (2.45)) provides in momentum space a D=3 infinite-dimensional multiplet with all values of spin.
In Sec. 3, the D=4 case is considered. First, we provide variables useful in the relativistic kinematics of massive particles with spin (four-momenta, Pauli-Lubański four-vector, orthonormal bases in four-momentum space called also Lorentz harmonics) in terms of twotwistor geometry. Secondly, we consider the extension of the D=4 hybrid action (1.1) to two-twistor space. In the most general case, the auxiliary coordinates present in (1.1) can be enlarged by the replacements (1.9) The standard Shirafuji model with spacetime coordinates x αβ and a pair of spinors (π α → π i α , πα →πα i ) leads, after using the standard incidence relation (see e.g. [6]), to a two-twistorial D=4 free particle model with four first class constraints. If the two spinorial mass constraints are further added, where π α i = ǫ αβ ǫ ij π j β and M is a complex mass parameter 2 , one obtains a model with four first class constraints and two second class ones describing D= 4 spinless massive particle. To relax the constraints that require the spin to be zero, we introduce three additional auxiliary four-vector coordinates y r αβ (r=1, 2, 3) (see (1.9)). Arranging correctly the generalized incidence relations we obtain the two-twistorial free model with one first class and two second class constraints, which reduce the 16 twistor real coordinates (eq. (3.1)) to 12 physical degrees of freedom. These new versions of the hybrid model can be quantized and solved by using the 'spinorial roots' (π i α ,πα i ) of the four-momenta as independent variables, which provides the reduced D=4 wave function ψ(π i α ,πα i ). If we take into consideration the mass constraints (1.10) we obtain that the manifold of the spinorial coordinates is described by the group manifold of SL(2; C), the cover of the D= 4 Lorentz group, with its six real parameters being half of the twelve physical phase space degrees of freedom that are left in the bitwistorial formulation. We show that such a wave function can be identified with a D= 4 master field describing an infinite-dimensional multiplet of massive HS fields with arbitrary D= 4 spin spectrum (for an analogy see [25]).
Finally, in Sec. 4 we present some comments going beyond D = 3, 4, on possible D=6 and supersymmetric extensions. The paper is supplemented with two appendices. Appendix A specifies in detail our conventions; Appendix B presents an interpretation of our N=2 D=3 spinorial model in Sec. 2.2 as described by an N=1 D=4 vectorial model for the nonstandard O(2, 2) Lorentz group. 2 It is related with the mass parameter m of the particle through 2|M | 2 = m 2 (see also (3.6)). 2 D = 3 bispinorial particle models and HS massive fields from their quantization 2.1 Summary of D=3 two-twistor kinematics D=3 twistors are real four-dimensional Sp(4; R) = SO(3, 2) spinors. We introduce a pair of D=3 real twistors where the contravariant spinor is constructed using the Sp(4; R)-invariant antisymmetric metric (see also footnote 1 ) If we only employ the spinors λ i α we can construct the following D=3 bilinears describing composite three-vectors in internal N=2 (i, j=1,2) space where the 2 × 2 matrices (γ a ) ij are internal space SO(2, 1) Dirac matrices (eq. (A.10)) and form the basis in space of symmetric 2×2 matrices (see Appendix A). Further, according to Penrose twistor theory (see e.g. [6]) we take u 0 αβ = p αβ (three-momentum). We shall further impose the following spinorial mass constraint which implies that the three-vectors (2.5) describe, after suitable normalization e a αβ = 1 m u a αβ , the D=3 vectorial harmonics (see e.g. [26,27]) 4 describing the D=3 Lorentz orthonormal vector frame e a αβ e b αβ = η ab , It is easy to check that the set of three-vectors u a αβ has three independent degrees of freedom equal to the number of spinorial degress of freedom constrained by the relation (2.6). In particular, if a=b=0 we obtain from (2.7) the mass-shell condition for the D=3 momenta where In order to describe the realizations of Lorentz group and the Abelian scalar D=3 spin S we should use all twistor components (see (2.1)). The Lorentz algebra generators M µν = x µ p ν − x ν p µ , M µ = 1 2 ǫ µνλ M νλ are given in spinorial notation by (2.10) and the scalar spin S for the massive particle with mass m is described by the D=3 counterpart of the Pauli-Lubański operator given by (µ, ν, ̺ = 0, 1, 2) where we used the bitwistor representation of momenta (2.9). We see that D=3 spin is described by the unique nonvanishing conformal-invariant twistor norm provided by formula (2.2). We shall further consider the field equations that determine the mass and spin eigenvalues of the D=3 Casimirs (2.8) and (2.11). Such field equations were also considered in quantum theory as describing anyons, with arbitrary fractional value of s (see e.g. [28][29][30]). In the next section we obtain these equations with fixed m and half-integer values of s as a result of the quantization of the new particle action. We will not consider here the anyonic fractional spin values that come from representations of the universal cover R of the D=3 Abelian spin group U(1).

D=3 bispinorial generalization of the Shirafuji model
We propose the following action for our D=3 model (i, j = 1, 2; r = 1, 2) where λ α i = ǫ αβ ǫ ij λ j β etc. and ℓ is a Lagrange multiplier imposing the constraint Λ in eq. (2.6). The parameters c, f may be set equal to one by rescaling the coordinates, but we shall keep them arbitrary in order to consider various variants of the model (actually, the most interesting values are 0 and 1). In particular, if we set c=1 the first two terms in (2.12) collapse into λ i α (γ a ) ij λ j βẏ αβ a where y αβ a = (ẋ αβ , y αβ r ) with a = (0, r) = (0, 1, 2). If c = f = 0, after using the standard incidence relation and inserting (2.13) into (2.11), we get S = 0, i.e. we obtain the model describing a spinless particle. In the general case the incidence relation (2.13) has to be generalized as follows 5 µ αi = 2x αβ λ i β + 2c (γ r ) ij y αβ r λ j β + f y αi . (2.14) After using relations (2.5) in (2.11) we obtain Setting c = f = 1, the constraints defining the momenta follow from (2.12) with the result Eqs. (2.17) and (2.18) determine pairs of second class constraints. After introducing for them Dirac brackets we obtain that the variables (y αβ a , y α i ), a = 0, 1, 2, are canonically conjugate to (p a αβ , λ α i ) so that the non-vanishing PBs are given by The model (2.12) has ten first class constraints expressed by the formula (2.16) and the mass-shell constraint (2.6). After quantization the above PB relations can be realized in terms ofŷ αβ a = y αβ a ,λ i α = λ i α and the following differential operatorŝ where, by definition, ∂ ∂y αβ β . As a result, the quantized constraints (2.16) after using equations (2.5) deterimine the following three unfolded equations for the wave function Φ ≡ Φ(y αβ a , λ α i ), with the following solution expressing explicitly the dependence on y αβ a , Using, instead of (2.21), the dual differential realization in spinorial sector one obtains from (2.6) a single field equation for the reduced wave function In the 'spinorial momentum' picture described by the spinors λ i α the reduced wave function φ(λ i α ) depends on the spinorial momenta restricted by the algebraic equation (2.6). We see that the wave function describing the quantum mechanical solution of the model (2.12) depends on three degrees of freedom, two describing the on-shell three-momenta and a third one being the (arbitrary) value of the D=3 spin. In order to express the spin operator (2.11) as a differential operator in spinorial momentum space one has to consider the quantum version of the twistorial description of model (2.12).
Let us now compare the models (2.12) with f = 0 and f = 0 (for simplicity we set c=1). From expression (2.15) it follows that in both models S is a composite dynamical variable that describes arbitrary D=3 spin; however, the limit f → 0 changes the structure of the constraints. Indeed, if f = 0, those in (2.17) are not present; only the constraints (2.16), (2.18) and the mass-shell constraint (2.6) appear. The alternative constraint structure is well illustrated if the nine relations (2.16) are replaced by the equivalent set of nine Abelian constraints Similarly, the four constraints (2.18) can be replaced by four equivalent ones as follows where the D=3 gamma matrices (γ a ) i j satisfy the so(1, 2) commutation relations it is seen that the thirteen new constraints (T b a , F a , F ) have the following non-vanishing PBs: We see from the second and fourth equations of (2.30) that four out of the ten first class constraints T b a and Λ (eq. (2.6)) present when f =0 become second class due to the appearance of four constraints (2.28) in the limit f =0. These four constraints (F a , F ) are second class and describe the gauge fixing of four gauge transformations present if f =0. We can conclude that putting f =0 in (2.12) leads to the partial gauge fixing of four out of the ten gauge degrees of freedom generated when f =0 by the ten first class constraints T a αβ (or T a b ) and Λ. If f =0 the ten first class constraints remove 2×10 = 20 real degrees of freedom; for f =0 the six first class constraints plus the eight second class remove the same number of d.o.f., 2×6 + 8 = 20. Thus, both models have the same physical (i.e. without gauge degrees of freedom) content. This proves the equivalence of the classical models considered for f =0 and f =0.
Finally, we point out that for c=1 our model (2.12) describes a vectorial SO(2, 2)-particle model, as discussed in Appendix B.

D=3 bitwistorial description
In order to introduce the twistor coordinates (2.1), we insert in (2.12) the generalized incidence relation (2.14). Modulo boundary terms, we obtain for c =0 and/or f =0 the following twistorial free action with Sp(4, R) D=3 twistorial metric (1.5) is obtained: The action (2.31) describes an infinite tower of D=3 free massive particles with any spin (see e.g. [16]). Let us prove it. The action (2.31) describes a system with canonical variables µ αi and λ i α , {µ αi , λ j β } = δ ij δ α β , and the constraint (2.6) which generates the gauge transformations in bitwistor space. Let us fix this gauge freedom by the constraint Introducing Dirac brackets incorporating the constraints Λ ≈ 0 and G ≈ 0 we obtain that they become strong and we get the following Dirac brackets for the twistor variables

(2.33)
A quantum realization of the algebra (2.33) withλμ ordering is the followinĝ We point out that the second class constraints (2.6) and (2.32) are fulfilled in strong sense, i.e.Ĝ =λ i αμ αi ≡ 0. If we use the formulae (2.34), the spin operator (2.11) is realized as Our aim will be to decompose the Fourier transform (2.26) of the reduced wave functioñ φ(y α i ) satisfying eq. (2.25) into a superposition of momentum-dependent eigenfunctions of the operator (2.35) (see eqs. (2.53), (2.54) below).
The corresponding SU(1, 1) matrix is obtained by the complex similarity transformation In terms of the variables (2.39) the spin operator (2.35) takes the form (2.40) The matrix g α i in (2.38) describes SU(1, 1) spinorial harmonics, where first column g α ) describes a SU(1, 1) spinor with spin eigenvalue s = − 1 2 (s = 1 2 ). One can introduce the natural parametrization of the SU(1, 1) matrices (2.38) [31] In terms of the angle ψ, the operator (2.40) takes the simple form i.e., it describes the D=3 U(1) spin. After the transformation (2.36), the twistorial wave function Ψ(g) is defined on SU(1, 1). The SU(1, 1) regular representation is given by its action of on the (wave) functions Ψ(g) defined on the SU(1, 1) manifold. To obtain the Hilbert space of the quantized model (2.31) we may use the theory of special functions on matrix group manifolds (see e.g. [31]) and require that the wave function Ψ(g) = Ψ(ϕ, r, ψ) is square-integrable, |Ψ(g)| 2 dg < +∞, dg = sinh r dr dϕ dψ. Due to eq. (2.41), the wave function satisfies the periodicity conditions Ψ(ϕ, r, ψ) = Ψ(ϕ + 4π, r, ψ) = Ψ(ϕ, r, ψ + 4π) = Ψ(ϕ + 2π, r, ψ + 2π) , (2.44) which eliminate the anyonic quantum states with arbitrary fractional spin. One can use the double Fourier expansion The summation is over all pairs (k, n) such that the numbers k and n are both integer or half-integer. The eigenvalues of the operator S defined by (2.43) coincide with parameter n in the expansion (2.45). As a result, the spin in our model takes quantized integer and half-integer values. The functions F n (r, ϕ) describe states with definite D=3 spin equal to n. The r-dependent fields in (2.45) are expressed by and the Plancherel formula gives Square integrable functions f kn (r) have an (integral) expansion on the matrix elements of the SU(1, 1) infinite-dimensional unitary representations (see [31,32] for details). Using (2.9) and (2.36), (2.41) we obtain that p 0 = m aā + bb = m cosh r , We see that the on-shell momentum components (2.48) do not depend on the angle ψ and thus define the coset manifold SU(1, 1)/U(1), the hyperboloid which is the base manifold of the (trivial) U(1)-fibration of SU(1, 1). The wave function (2.44) with the Fourier expansion (2.45) in the U(1) ψ-variable describes an infinite-dimensional tower of D=3 higher spin fields.
The coefficient fields in the expansion in (2.45) are defined on the coset SU(1, 1)/U(1) as functions of the on-shell three-momenta p µ , Let us analyze the expansion (2.45) in a Lorentz covariant form. We recall that the transformation (2.37) describes the isomorphism between SL(2; R) and SU(1, 1) matrix group (see, for example, [33]). Using eq. (2.37) one can transform D=3 spinors and γ-matrices from Majorana (real) representation to a complex representation. We get in such a way the D=3 framework which uses the SU(1, 1) spinor coordinates 6 In the variables (2.51) the D=3 spin operator (2.40) takes the form We find easily that in terms of the SU(1, 1) spinors (2.51) the three-momentum (2.48) is given by is the D=3 counterpart of the standard Penrose formula for the four-momenta in the D=4 case, in which the D=4 SL(2; C) Weyl spinors have been replaced by D=3 SU(1, 1) spinors.
We note that the SU(1, 1) spinorial formalizm is more convenient for the description of spin states than the SL(2; R) framework because it diagonalizes the spin eigenvalues. Formally the wave function (2.54) (or the reduced wave function φ(λ i α ) in (2.26)), after using (2.6), can be written as follows However, the monomials λ i 1 α 1 . . . λ i N α N are not eigenvectors of the spin operator (2.35). We point out that the expansions (2.54) include both states with positive (n > 0) and negative (n < 0) spin values and that it is infinitely degenerate because a spin n is generated by all monomials One can remove the degeneracy in N, K for a given n by projecting on the spaces with definite sign of spin if we consider anti-holomorphic wave functions satisfying the condition A solution of (2.56) is provided by the power serie which depends only onξ and contains only positive spins. Alternatively, we may impose the condition which can also be interpreted as another SU(1, 1) harmonic expansion condition. The spacetime dependent fields are obtained in the standard way by means of a generalized Fourier transform with exponent e ipµx µ = e −i(ξγµξ) x µ and measure µ 3 (ξ) = d 4 ξ δ(ξσ 3 ξ − m) (see eq. (2.51)). We get in such a way the Fourier-twistor transform for D=3 massive fields. The corresponding spacetime fields are then given by The fields (2.59) are symmetric with respect to their spinorial indices and satisfy the D=3 Bargmann-Wigner equations where the γ-matrices are taken in the complex SU(1, 1) representation (A.7). The negative (n < 0) spin (helicity) states are described by the holomorphic twistor wave function which is a solution of equation (2.58). The twistor transform can be obtained by the complex conjugation of (2.59) and defines spacetime fields with symmetric spinorial indices that satisfy the Bargmann-Wigner equations (2.60) with m → −m.
3 D= 4 bispinorial models and HS massive fields

Summary of D= 4 two-twistor kinematics
The standard D= 4 Penrose twistors are complex four-dimensional SU(2, 2) = SO(4, 2) spinors Z Ai ,Z Ai that can be expressed by two pairs of two-component Weyl spinors (π i α ,ωα i ) One can introduce four conformal-invariant scalar products (a = 0, 1, 2, 3) where the hermitian 2×2 matrices above σ a are defined in Appendix A and act in the internal bidimensional space. Using the complex Weyl spinors π i α ,πα i we can define the following set of real composite four-vectors u a αβ = π i α (σ a ) i jπβ j , a = 0, 1, 2, 3 , which for a=0 give the Penrose formula for the composite four-momentum [6] u 0 We shall impose (see (1.10)) two complex spinorial mass constraints by means of the complex mass parameter M = M 1 + iM 2 . From (3.4) and (1.10) it follows easily that
The Pauli-Lubański four-vector W µ describing the D = 4 relativistic spin, can be written after using expressions (3.9) and (3.2) as an expression in twistorial coordinates as follows W αβ = S r u αβ r , r = 1, 2, 3 , Further, using the relations (1.10), (3.5) and (3.6) it follows that After quantization, as it is shown in Sec. 3, we obtain the well known relativistic spin square spectrum with S 2 replaced by s(s + 1) (s = 0, 1 2 , 1, . . . ). We observe that the covariant generators S r , which (see (3.11) and (3.8)) can be expressed as and describe the su(2) spin algebra in a Lorentz frame-independent way.

D= 4 bispinorial generalization of Shirafuji model
Following the choice made in the D=3 case (see (2.12)), we shall generalize the standard D= 4 bispinor Shirafuji action by adding three additional terms depending on the supplementary four-vectors y µ r (r = 1, 2, 3) and on the spinorial kinetic terms, plus the pair of spinorial mass shell constraints M,M in eq. (1.10): When c = f = 0, S (4) describes the standard bispinorial Shirafuji model, with the pair of standard incidence relations The reality of the spacetime coordinates x µ implies, after multiplying the first equation above on the right side by A i jπα j and the second one on the left side by π j α A ji , the constraint which depends on the arbitrary hermitian 2 × 2 matrix A i j , i.e. (A i j ) † = A j i . Using the σ a basis of 2×2 hermitian matrices (Appendix A), eq. (3.17) gives the following four linearly independent constraints (a = (0; r) = (0; 1, 2, 3)) which can also be expressed by the four conformal scalar products of the twistors If relation (3.19) is valid, we see that the twistors generated by the incidence relation (3.16) are null twistors located on the null plane. The four constraints (3.19) and two spinorial mass constraints (1.10) provide four first class constraints and two of second class (see also [16]), i.e. if c = f = 0 we obtain 16 − 2×4 − 2 = 6 physical degrees of freedom describing the physical phase space of massive spinless particle.
In the general case when c = 0 and f = 0 the proper generalization of the incidence relations is the followingωα (3.20) Repeating the derivation of the constraints (3.17), we obtain in place of the formulae (3.18) the following relations (i, j = 1, 2 ; r = 1, 2, 3): where u r αβ is given by formula (3.3). The independence of the first expression in (3.21) on parameter c follows from the reality of the four-vector coordinates y αβ a ∼ (x µ , y µ r ). To describe the phase space structure of the model (3.14) we calculate the momenta p a αβ , p (π) α i , p (π)α i , p (y) i α , p (y)αi conjugate to y αβ a , π i α ,πα i , y α i ,ȳα i . This leads to the constraints (we set c = f = 1 for simplicity) T a αβ = p a αβ − u a αβ ≈ 0 , (3.22) The remaining two (mass) constraints are given by (1.10).
where the reduced wave functions ψ(π,π) depend on complex D=4 spinorial momenta satisfying the mass constraints in (1.10). For the general model (3.15) (c =0, f =0) it follows from (3.21) that all four variables S a are dynamical and that the reduced wave function ψ(π,π) does not satisfy any further constraints besides (1.10).
We add that for a D= 4 particle of mass m and fixed spin s the physical phase space has eight degrees of freedom, with the spin degrees represented e.g. by the coordinates on the sphere S 2 [38,25]. In such a theory the relation (3.13) that determines the fixed spin value s is first class constraint. If this constraint is removed, the resulting theory with arbitrary spin s has then ten degrees of freedom. It will be shown in Sec. 3.4 that the wave function solving the model (3.15) describes twelve degrees of freedom due to the multiplicity that is associated with each value of the different spins. We shall reduce the twelve degrees of freedom to ten, as required by a HS theory with nondegenerate spin spectrum, by imposing an harmonicity constraint (see (3.81) below) on the wave function.

D= 4 bitwistorial description of HS massive multiplets
Following the procedure in Sec. 2 for D=3, we now express the action (3.15) just in terms of a pair of D= 4 twistor coordinates (eq. (3.1)) by postulating the incidence relations (3.20). With f =0 (c may be arbitrary) this leads to the following two-twistorial action with two complex-conjugated Lagrange multipliers µ,μ The model (3.42) contains only two complex-conjugated spinorial mass constraints (1.10). When f = 0 and c = 0, as it follows from formulae (3.21), one still has to to impose one additional constraint via a Lagrange multiplier In order to find the first and second class constraints we use the canonical PB that follow from theS (4) The PBs in eq, (3.46) show that the generators S 0 , φ ′ of the generators of the translation sector of E(2) may be considered as producing spontaneously broken symmetries. Indeed, after quantization of PB (3.46) one can consider that the action of the E(2) generators (Ŝ 0 ,φ ′ 1 ,φ′ 2 ) annihilates the vacuum | 0 . Then, the quantized relations (3.46) are consistent only ifŜ 0 | 0 = 0, φ ′ 1,2 | 0 = 0 ⇒φ 1,2 | 0 = M 1,2 | 0 = 0. This means that if we look atφ 1 ,φ 2 as generating the two translational symmetries of E(2) these have to be spontaneously broken 8 . Similarly, if we introduce another choice of generators 8 We recall that the symmetry associated with a Lie algebra generatorX is spontaneously broken if X| 0 = 0 [39]. The phenomenon above described is that ifφ 1 ,φ 2 are considered as translation generators, then we cannot longer ignore that the true algebra is larger and that, in it, the constants determine a central subalgebra. Taking a basis that it is not a subalgebra led to the symmetry breaking above. the PB (3.46) will be rewritten as representing E(2) algebra broken spontaneously only in one translational direction generated by φ 1 We see from (3.48) that the constraint φ 1 is of first class, and φ 2 , S 0 form a pair of second class constraints.
It turns out nevertheless that the number of physical phase space degrees of freedom is the same and equal to twelve, irrespectively of the value of the parameter f . In fact, 1. if f = 0 we have two first class constraints (1.10), i.e. in 16-dimensional two-twistor phase space the number of degrees of freedom is 16 − 2 × 2 = 12.
2. if f = 0 and c = 0 we get three constraints satisfying the PBs (3.48), one first class and two second class. The count of degrees of freedom is the same: 16 − 1 × 2 − 2 × 1 = 12.
In the fist two cases we obtain the twelve dimensions of physical phase space by doubling the number of independent coordinates that parametrize the six-dimensional manifold SL(2; C); in accordance with (3.34), the reduced wave function is defined on this manifold.
To relate more closely our description with the spin degrees of freedom, let us recall the Lorentz-invariant spin variables S r defined by eq. (3.19). Using the PB relations in (3.44), one can show that the bilinears S r satisfy the so(3) ≃ su(2) PB algebra (q, p, r = 1, 2, 3) {S q , S p } = ǫ qpr S r .
(3. 49) In particular, if S r ≈ 0 ⇒ W αβ ≈ 0 (see (3.11)), i.e. the spin is equal to zero. In our twistorial model S r = 0 (see (3.21)) and after quantization (S r →Ŝ r )) we obtain from (3.49) the so(3) algebra of Lorentz-invariant spin generatorsŜ r The mass shell constraints, after using the bitwistor formula (3.4) for the four-momentum, provide the generalized Dirac equation with complex mass M and four-components complex Dirac spinors Further, in our two-twistor framework we obtain as well the generalization of eqs. (3.51) for the set of three auxiliary fourmomenta (r = 1, 2, 3), where e 2iϕ = M/M . The D= 4 mass constraints (1.10) are expressed in terms of π ′ i α ,π ′α i by (cf. eq. (2.6) for D=3) For the Weyl spinors π ′ i α ,π ′α i we get the equations (3.51), (3.52) with M replaced by m. The transformations (3.53) do not affect the SL(2; C) part of the variables π i α (see next section) because they change only the determinant of 2×2 matrix π i α , which is parametrized by the coset GL(2; C)/SL(2; C) ≃ GL(1; C), parametrized by an arbitrary complex mass parameter.

D= 4 bitwistor wave function of HS massive multiplet
Our D= 4 dynamical bitwistorial system is described by twistorial coordinates see (3.1)) in terms of the variables π j α ,πα k , ω α k ,ωα i endowed with the canonical PBs The constraints M,M can be equivalently described by One can check easily that the constraints V and F 2 are second class. For the local gauge transformations generated by the constraint F 1 we introduce the gauge fixing condition The PB of the constraints V , G, F 1 and F 2 are  This gives for the twistor components the DBs Below we will consider the (π,π)-realization of quantized version of the DB algebra (3.63)-(3.67). In such a realization, after using the ordering with π's at the left and ω's at the right, we obtainπ k α = π k α ,πα k =πα k and one checks that in the presence of D= 4 mass constraints (1.10) the constraints (3.56), (3.59) are satisfied in the strong sense:π k αω α k ≡ 0,πα kωα k ≡ 0. Taking into account the expressions (3.68) we obtain the quantum counterparts of the quantities (3.12) as the spin operatorŝ Using (3.13), the square of the Pauli-Lubański vector becomesŴ µŴ µ = −m 2ŜrŜr , which will be used later to define spin states.
Thus, the twistorial wave function is defined on the space parametrized by π i α ,πα i which satisfy the constraints M,M (eq. (1.10)), and the matrix defines the SL(2, C) group manifold. Thus, the twistorial wave function is defined on SL(2, C) parametrized by π i α , so that Ψ = Ψ(π i α ,πα i ). One can use the well known decomposition of SL(2, C) elements in terms of the product of an hermitian matrix h = h † with unit determinant and an SU(2) matrix v, v † v = 1 (in the above formulae, the v k i play the role of ψ in (2.41) for D=3). The three parameters of the matrix h describe four-momenta on the mass shell, and the three parameters of the matrix v correspond to the spin algebra (3.50). The matrix h parametrizes the coset SL(2, C)/SU(2) which defines the three-dimensional mass hyperboloid for timelike four-momenta which does not depend on the v k i variables (as in D=3 eqs. (2.48) do not depend on ψ). So, the definition (3.4) can be rewritten as follows wherehα i = (h α i ) * and α=1,2 and i=1,2.
The unitary matrix v paramerizes S 3 ∼ SU(2) and is linked with the spin degrees of a massive particle. In particular, the operators (3.69) expressed by the variables (3.71) take the formŜ We can consider the variables v i k as the harmonic variables that were introduced early to describe N=2 superfield formulations (see, for example, [41]). In particular, it is useful to introduce the notation Then, the operators (3.73) take the form (3.75) and the square of the Pauli-Lubański vector is given by the formulâ Since the variables v ± i parametrize a compact space, the general wave function on SL(2, C) has the following harmonic expansion (we use the SU(2)-covariant expansion from [41]) which follows from the second expression in the definition of harmonic variables (3.74). These coefficient fields depend on the on-shell four-momenta due to (3.72 . Such functions defined on the mass hyperboloid can be expanded into SL(2; C) irreducible representations belonging to the principal series of the first kind [40]. Each monomial of the variables v ± i in the expansion (3.77) is an eigenvector of the Casimir operator (3.76): where s = N +K 2 . So, the expression (3.77) is in fact the general expansion into arbitrary spin states. By means of the nonsingular transformation is degenerate. This degeneracy can be however removed by the harmonic condition on the wave function (see also [41]) form the basis, as a solution of (3.81), we obtain the following wave functioñ This twistor wave function rewritten in Lorentz covariant way takes the form Note that twistor wave function (3.83) also depends on π − α andπ ± α through p µ in the argument of the component fields.
Spin s=L/2 massive particles are described by the fields ψ α 1 ...α L (p µ ). The corresponding spacetime fields are obtained by an integral Fourier-twistor transform which combines the Fourier and twistor transformations. More explicitly, by means of these integral transformations we can obtain the following multispinor fields, all with a total of L undotted plus dotted indices, where p µ is defined by (3.4) as a bilinear product of twistors. In the integrals (3.84) for a given L, only the term π + α 1 . . . π + α L ψ α 1 ...α L (p µ ) in the twistorial wave function (3.83), with U(1) harmonic charge q = L (see [41]), gives a non-zero contribution. Note that as it follows by looking at the SU(2) representation contents of the finite irreducible representations of SL(2, C) (see e.g. [42]). The nonmaximal (s < L 2 ) spins are eliminated subjecting φ (N,M ) to the generalized Lorenz conditions which follow as well from the formulae (3.84), plus all the tracelessness conditions which are also consequences of (3.84). Dirac-Fierz-Pauli equations for spin s can be written in Weyl spinor notation as equations relating the φ (L,0) and φ (L−1,1) multispinor fields (3.87) alternatively, we can choose eq. (3.85) for the multispinors φ (0,L) and φ (1.L−1) . The second equation in (3.87) can be considered (for m = 0) as defining the fields φ (L−1,1) , so the whole set of fields (3.84) can be obtained from the fields φ (L,0) , which satisfy the massive Klein-Gordon equation and describe spins s = L/2 [42,43]. Indeed, using relations (3.85) subsequently for the fields φ (L−1,1) , ..., φ (L−M,M ) it may be shown that all these fields can be expressed in terms of φ (L,0) by . In particular if we choose L=1 in (3.87), we obtain the standard Dirac equation for a Dirac field in the Weyl realization as the sum of an undotted spinor and a dotted one, φ (1,0) ⊕ φ (0,1) in the our notation. If L=2 we obtain the Proca equations expressed in terms of φ (1,1) , φ (2,0) and φ (0,2) . Consider first (3.85) for N=0, M = 2, Eliminating φβ 1β2 , we see that the vector field φ α 1β2 satisfies the massive Klein-Gordon equation. Further, using the symmetry in the β 1 β 2 indices, it follows that the two equations above imply the Lorenz condition (see (3.86)) which eliminates the spin zero part of φ αβ . Thus, by virtue of eqs. (3.89), φ αβ is the spin one Proca field φ µ satisfying ( + m 2 )φ µ = 0 , ∂ µ φ µ = 0 (see eq. (A.15)). Similarly, if we now consider the case N=1=M in eqs. (3.85), we obtain We note that to obtain the Proca equations as a massive extension of the Maxwell equations it is sufficient to describe the free field dynamics in terms of the field strength φ µν = ∂ µ φ ν − ∂ ν φ µ . The tensor φ µν may be expressed in terms of its dual and antiselfdual parts, φ µν ∼ (φ (2,0) , φ (0,2) ). Using these two bispinor fields one obtains the Proca equations, ∂ µ φ µν + m 2 φ ν = 0.

Outlook
We have presented in this paper new massive particle models in D=3 and D=4 spacetimes enlarged in D=3 by two (y αβ r , r = 1, 2) or in D=4 by three (y αβ r , r = 1, 2, 3) additional copies of Minkowskian four-vector variables and their momenta. After quantization, the wave functions are defined on SL(2; K) manifolds (K = R for D=3, K = C for D=4) and describe towers of free massive HS fields. A natural extension of these models is the D=6 case, in which the wave functions would be defined on the SL(2; H) manifold, with 12 real parameters. In such a case, the complex D=4 twistors in Sec. 3 should be replaced by quaternionic D=6 twistors (see e.g. [44]), defined as fundamental spinorial realization of the D=6 conformal SO(6, 2) group with spinorial quaternionic covering U α (4; H) ≃ O * (8; C) group (see e.g. [10,45]).
We would like to point out that it is possible to relate the D=3 and D=4 massive models with D=4 and D=5 massless ones by observing that massless fields in D + 1 spacetimes become massive in one less dimension D after dimensional reduction and interpreting the (D+1)-th momentum component as the mass in D-dimensional spacetime. There is a link between the description of helicity in massless theories and spin in massive case; i.e. In particular, the Abelian helicity operator in D = 4 (see (1.8)) corresponds to the spin operator (2.11) in D = 3 and further the SU(2) spin algebra in D = 4 could be used analogously to describe the generalized helicity states in D = 5. We add that recently it has been pointed out that the symplectic two-form describing the spin contribution to the D = 3 free massive spinning particle dynamics [16] can be identified with the symplectic two-form describing the helicity part of particle dynamics for massless D = 4 particles with nonvanishing helicity [46]. One can extend our considerations to the supersymmetric case. In it, as in the first example of a spinorial particle model in tensorial spacetime [2] extended to superspace, the additional variables are associated with the so-called tensorial central charges of the supersymmetry algebras (i.e., central but for the Lorentz subalgebra). These charges play an important role in the theory of supersymmetric extended objects [47]; the corresponding central tensorial generators of the superlagebras act as differential operators on the additional coordinates of the associated extended superspaces 9 . From this perspective, the two-twistor models introduced here can be related with the tensorial central charges of N = 2 supersymmetry and the variables of the suitably extended superspaces. The most general D = 3 N = 2 superalgebra extended by tensorial central charges is as follows The real vectorial 'central' charges Z (1) αβ , Z αβ may be considered as the momenta p r αβ generating the translations of our additional coordinates y αβ r (r = 1, 2; see (2.12) and (2.19)). In this view, the first formula (2.21) takes the form (4.3) In the D=4, N = 2 supersymmetry algebra with tensorial central charges, the generators associated with the coordinates listed in (1.9) (see also Sec. 3) appear as part of those of the extended superalgebra {Q i α , Q j β } = δ ijZ αβ + (σ 1 ) ijZ (1) αβ + (σ 3 ) ijZ (2) αβ + ǫ ij ǫ αβZ (4.5) (similarly for ({Q iα ,Q jβ }), where the 16 generators P a αβ are real and the 10 generatorsZ αβ , Z (1) αβ ,Z (2) αβ ,Z are complex (i.e. there are 36 bosonic real generators). In our D = 4 model we have only used the sixteen coordinates y a αβ (see eq. (2.12)) associated with P a αβ , a=0,1,2,3 and the remaining 10 complex tensorial charges were put equal to zero. Let us observe that for N = 1 only first term on r.h.s. of the relation (4.5) survives and describes the tensorial central charges used in [1][2][3][4]. If N = 2 we also note that the generatorZ in (4.2) and (4.5) that we did not include in our considerations is a truly central one (it is a Lorentz scalar). This generator, associated with a scalar central coordinate, has eigenvalues characterizing the mass; its role in N = 2 massive superparticle model was considered long ago [49].
The models discussed in this paper give the same mass for all HS fields, which of course is very restrictive. In a physical HS case, when considering e.g. spin excitations in string theory, the masses are spin-dependent. They lie on a Regge trajectory, which in the general case can be described by replacing the constant mass by an spin-dependent function m = m(s) (usually linear). In this case, the constant m in the mass-shell condition should be replaced by a spin-dependent operator (see (3.13), (3.14), (3.50)) i.e., m 2 → m 2 ( S 2 ) . (4.6) In the twistor formulation, the spinorial mass shell conditions (eq. (1.10) in D = 4) may be considered as 'complex roots' of the standard mass shell condition. It is an interesting problem to see how to introduce, in the complex mass parameter M appearing in (1.10), a dependence on the twistor variables that could lead to HS multiplets with masses on a Regge trajectory. Other problem which can be studied is related with the description of the interacting HS theories. For that purpose it should be useful to introduce in our formalism with additional coordinates the nonvanishing AdS radius, i.e. generalize the set of coordinates (x µ , x r µ ) (see (1.9)) to the case where x µ is endowed with constant spacetime curvature.
Finally, we note that the use of additional vector variables beyond the spacetime vector is also an important ingredient in the BRST approach to the Lagrangian formulation of HS fields 10 developed in [50][51][52].
Our model (2.12) defines therefore the extension of the N = 2 D = 3 Shirafuji model to the vectorial model in SO(2, 2) tensorial space y αβ a . Such a model cannot be however extended to a corresponding O(3, 3) twistorial model, because SO(3, 3) twistors are described by the pair of primary SO(2, 2) spinors ( 1 2 , 0)⊕(0, 1 2 ) which we denote as (λ α , λ i ) and (ω α , ω i ). The second pair of spinors should be defined in terms of SO(2, 2) spacetime coordinates x i α by the SO(2, 2) incidence relations However, in this paper we did not use neither of the simple spinors λ α , λ i nor the incidence relations (B.2) i.e., if we pass to an SO(2, 2) interpretation of our model (2.12), we loose the corresponding SO(2, 2) twistorial formulation.