Two-spinor description of massive particles and relativistic spin projection operators

On the basis of the Wigner unitary representations of the covering group ISL(2,C) of the Poincar\'{e} group, we obtain spin-tensor wave functions of free massive particles with arbitrary spin. The wave functions automatically satisfy the Dirac-Pauli-Fierz equations. In the framework of the two-spinor formalism we construct spin-vectors of polarizations and obtain conditions that fix the corresponding relativistic spin projection operators (Behrends-Fronsdal projection operators). With the help of these conditions we find explicit expressions for relativistic spin projection operators for integer spins (Behrends-Fronsdal projection operators) and then find relativistic spin projection operators for half integer spins. These projection operators determine the nominators in the propagators of fields of relativistic particles. We deduce generalizations of the Behrends-Fronsdal projection operators for arbitrary space-time dimensions D>2.


Introduction
In this paper, using the Wigner unitary representations [1] of the group ISL(2, C), which covers the Poincaré group, we construct spin-tensor wave functions of a special form. These spin-tensor wave functions form spaces of irreducible representations of the group ISL(2, C) and automatically satisfy the Dirac-Pauli-Fierz wave equations [2], [3], [4] for free massive particles of arbitrary spin. In our work, we use the approach set forth in the book [23]. The construction is carried out with the help of Wigner operators (a similar construction was developed in [5]; see also [6]), which translate unitary massive representation of the group ISL(2, C) (induced from the irreducible representation of the stability subgroup SU(2)) acting in the space of Wigner wave functions to a representation of the group ISL(2, C), acting in the space of special spin-tensor fields of massive particles. In our paper, following [23], a special parametrization of Wigner operators is proposed, with the help of which the momenta of particles on the mass shell and solutions of the Dirac-Pauli-Fierz wave equations are rewritten in terms of a pair of Weyl spinors (two-spinor formalism [12], [13]; see also [18], [30] and references therein). The expansion of a completely symmetric Wigner wave function over a specially chosen basis provides a natural recipe for describing polarizations of massive particles with arbitrary spins. As the application of this formalism, a generalization of the Behrends-Fronsdal projection operator is constructed, which determines the spintensor structures of the two-point Green function (propagator) of massive particles with any higher spins in the case of arbitrary space-time dimension D. We would also like to stress here that spin projection operators are employed for analysis of the high energy scattering amplitudes, differential cross sections, etc. ( [21]; see also [7], [8], [22] and references therein).
The work is organized as follows. In Section 2, we recall the definition of the group ISL(2, C), the universal covering of the Poincaré group, and build spin-tensor wave functions ψ (r) (β 1 ...βr) (α 1 ...αp) (k) (in the momentum representation) for free massive particles of arbitrary spin. Further in this section we prove that the constructed spin-tensor wave functions satisfy the Dirac-Pauli-Fierz equation system and show that these wave functions have a natural parametrization in terms of a pair of Weyl spinors (two-spinor formalism). At the end of Section 2, we prove that the wave functions ψ (r) (β 1 ...βr) (α 1 ...αp) (k) are eigenvectors for the Casimir operatorŴ nŴ n of the Poincaré group (Ŵ n are the components of the Pauli-Lubanski vector) with eigenvalues proportional to j(j + 1), where the parameter j = (p + r)/2 is called spin. In Section 3, as examples, we discuss in detail the construction of spin-tensor wave functions for spins j = 1/2, 1, 3/2 and 2. In particular, we show how to derive contributions from different polarizations and calculate the density matrices (the relativistic spin projectors) for particles with spins j = 1/2, 1, 3/2 and 2 as sums over the polarizations of quadratic combinations of polarization spin-tensors. In Section 4, we find the general form of the polarization tensors for arbitrary integer spin j, and we also establish the conditions which uniquely fix the form of density matrices (the relativistic spin projectors or the Behrends-Fronsdal projection operators) for integer spin j. In Section 5, in the case of integer spins and arbitrary space-time dimensions D > 2, an explicit formula of the density matrices is derived. This formula is a generalization of the Behrends-Fronsdal formula for the projection operator known (see [20], [21]) for D = 4. At the end of Section 5, an explicit expression for the density matrix of relativistic particles with arbitrary half-integer spin j is deduced. This expression will be obtained as the solution of the conditions to which the density matrix (the sum over the quadratic combinations of polarization spin-tensors) obeys.
2 Massive unitary representations of the group ISL(2, C) 2.1 Covering group ISL(2, C) of the Poincaré group.
To fix the notation, we recall the definition of the covering group ISL(2, C) of the Poincare group ISO ↑ (1, 3) and introduce its Lie algebra isℓ(2, C) (see, for example, [10]). The group ISL(2, C) is the set of all pairs (A, X), where A ∈ SL(2, C), and X is any Hermitian 2 × 2 matrix which can always be represented in the form With the use of (2.1) each Hermitian matrix X is uniquely associated with the four-vector x = (x 0 , x 1 , x 2 , x 3 ) in the Minkowski space R 1,3 . Sometimes below we use the notation (A, x) instead of (A, X).

Spin-tensor representations of group ISL(2, C) and Dirac-Pauli-Fierz equations
Further in this paper we will consider only the massive case when m > 0. In this case the unitary irreducible representations of the group ISL(2, C) are characterized by spin j = 0, 1 2 , 1, 3 2 , . . . and act in the spaces of Wigner wave functions φ (α 1 ...α 2j ) (k), which are components of a completely symmetric SU(2)-tensor of rank 2j. Here the brackets (.) in the notation of multi-index (α 1 . . . α 2j ) indicate the full symmetry in permutations of indices α ℓ , and k = (k 0 , k 1 , k 2 , k 3 ) denotes the four-momentum of a particle with mass m: Let us fix some test momentum q = (q 0 , q 1 , q 2 , q 3 ) such that (q) 2 = m 2 , q 0 > 0. For each momentum k belonging to the orbit (k) 2 = m 2 , k 0 > 0 of the Lorentz transformations (2.4) which transfer the test momentum q to the momentum k, we choose representative A (k) ∈ SL(2, C): where (kσ) = k n σ n , (qσ) = q n σ n . The relation between the matrices A (k) and Λ k ≡ Λ(A (k) ) is standard (see (2.6)). One can rewrite the Lorentz transformation (2.15) in an equivalent form where (kσ) = (k nσ n ) and (qσ) = (q nσ n ). This form of the transformation will be needed later.
Define a stability subgroup (little group) G q ⊂ SL(2, C) of the momentum q as the set of matrices A ∈ SL(2, C) satisfying the condition which, by means of the identity (qσ)(qσ) = m 2 , is equivalently rewritten as In the massive case (q) 2 = m 2 , m > 0, one can prove that the stability subgroup G q is isomorphic to SU(2) regardless of the choice of test momenta q. Now we note that the matrix A (k) ∈ SL(2, C), which transfers the test momentum q to momentum k is not determined by (2.15) uniquely. Indeed, A (k) can be multiplied by any element U of the stability subgroup G q = SU(2) from the right since we have For each k we fix a unique matrix A (k) satisfying (2.15). The fixed matrices A (k) numerate left cosets in SL(2, C) with respect to the subgroup G q = SU(2), i.e. they numerate points in the coset space SL(2, C)/SU (2). Let T (j) be a finite-dimensional irreducible SU(2) representation with spin j, acting in the space of symmetric spin-tensors of the rank 2j with the components φ (α 1 ...α 2j ) . The Wigner unitary irreducible representations U of the group ISL(2, C) with spin j are defined in [1] (see also [6], [9], [23] and references therein) by the following action of the element (A, a) ∈ ISL(2, C) in the space of wave functions φ (α 1 ,...,α 2j ) (k): Here we use the concise notation the indicesᾱ,ᾱ ′ must be understood as multi-indices (α 1 . . . α 2j ), (α ′ 1 . . . α ′ 2j ), the matrix Λ ∈ SO ↑ (1, 3) is related to A ∈ SL(2, C) by eq. (2.6), and the element belongs to the stability subgroup SU(2) ⊂ SL(2, C). In formula (2.19) the element h A,Λ −1 ·k of the stability subgroup is taken in the representation T (j) as matrix ||T (2.22) Here we split the tensor product of 2j = (p + r) factors h A,Λ −1 ·k into two groups. The first group consists of the p factors h A,Λ −1 ·k , and in the second group we use the identity (2.18) and write r multipliers Further, we substitute (2.21) into (2.22) and split the result as follows: (2.23) where we introduced the concise notation In the operator form the matrix (2.23) is represented as where instead of the Wigner wave functions φ (δ 1 ...δ p+r ) (k) we introduced spin-tensor wave functions of ( p 2 , r 2 )-type (with r dotted and p undotted indices): The upper index (r) of the spin-tensors ψ (r) distinguishes these spin-tensors with respect to the number of dotted indices. The operators A ⊗p (k) ⊗ A †−1 (k) (qσ) ⊗r , used in (2.27) to translate the Wigner wave functions into spin-tensor functions of ( p 2 , r 2 )-type, are called the Wigner operators.
In the massive case, the test momentum can be conveniently chosen in the form q = (m, 0, 0, 0). Then, relations (2.15) and (2.16) for the elements A (k) , A †−1 (k) ∈ SL(2, C) are written as: Since σ 0 andσ 0 are the unit matrices, eqs. (2.28) have a concise form: but we should stress here that the balance of dotted and undotted indices is violated in (2.29).
Proof. The proof of the first equation in (2.30) is given by the chain of relations: where we applied the second formula of (2.28) and used the symmetry of the Wigner wave functions φ (δ ′ 1 ...δ ′ p+r ) (k) with respect to any permutations of indices δ ′ k . The second equation in (2.30) is proved analogously (we need to use the first relation in (2.28)). The consistence conditions for the system (2.30) follow from the chain of relations where we used the identities (2.8) and equations (2.30). Comparing the left and right parts in (2.31), we obtain the mass-shell condition (k n k n − m 2 )ψ (r) (k) = 0.
We now return back to the discussion of the matrices A (k) ∈ SL(2, C) which, according to (2.15), transfer the test momentum q to the momentum k. The matrices A (k) numerate points of the coset space SL(2, C)/SU (2). The left action of the group SL(2, C) on the coset space SL(2, C)/SU(2) is given by the formula where the matrices A ∈ SL(2, C) and Λ ∈ SO ↑ (1, 3) are related by condition (2.6) and the element U A,k ∈ SU(2) depends on the matrix A and momentum k. Under this action the point A (k) ∈ SL(2, C)/SU(2) is transformed to the point A (Λ·k) ∈ SL(2, C)/SU (2). We note that formula (2.32) is equivalent to the definition (2.21) of the element h A,k of the stability subgroup in Wigner's representation (2.19).
The left action (2.32) of the element A ∈ SL(2, C) transforms two columns of the matrix A (k) as Weyl spinors. Therefore, it is convenient to represent the matrix A (k) by using two Weyl spinors µ and λ with components µ α , λ α (the matrix A † (k) will be correspondingly expressed in terms of the conjugate spinors µ and λ) in the following way [23]: In eqs. (2.33) we fix the normalization of matrices . From formulas (2.29) it follows that the momentum k is expressed in terms of the spinors µ α , λ β , µβ, µβ as follows: where |µ ρ λ ρ | = z z * . Thus, in view of (2.27) and (2.35) the wave functions of massive relativistic particles, which are functions of the four-momentum k, can be considered as functions of two Weyl spinors λ and µ. The two-spinor expression (2.35) for the four-vector k (k 2 = m 2 and k 0 > 0) is a generalization of the well-known twistor representation for momentum k of a massless particle [11]. We will see below that the two-spinor description of massive particles (about two-spinor formalism see also papers [12] - [19], [30]) based on the representation (2.35) proves to be extremely convenient in describing polarization properties of massive particles with arbitrary spin j. In the next Section, we apply this two-spinor formalism to describe relativistic particles with spins j = 1/2, 1, 3/2 and j = 2.
At the end of this Subsection, we demonstrate why, in the case of p + r = 2j, the system of spin-tensor wave functions (2.27), which obey the Dirac-Pauli-Fierz equations (2.30), does describe relativistic particles with spin j.
Take the representation (2.26) of the group ISL(2, C), acting in the space of spin-tensor wave functions ψ (r) (β 1 ...βr) (α 1 ...αp) (k), and consider this representation in the special case when the element (A, a) ∈ ISL(2, C) is close to the unit element (I 2 , 0), i.e., we fix the vector a = 0 and take the matrices A, A †−1 ∈ SL(2, C) such that where ω nm = −ω mn are small real parameters, and the matrices σ nm andσ nm are the spinor representations (2.12) of the generators M nm ∈ so(1, 3). As a result, we obtain for (2.26) the expansion where we used the notationᾱ andβ for multi-indices (α 1 ...α p ) and (β 1 ...β r ). In eq. (2.36) the operators M mn = (k m ∂ ∂k n − k n ∂ ∂k m ) +Σ mn are the generators of the algebra so(1, 3) in the representation U and the matriceŝ describe the spin contribution to the components of the angular momentum M mn . The operators (σ mn ) a and (σ mn ) b in (2.37) are defined as follows: .
, which obey the Dirac-Pauli-Fierz equations (2.30), automatically satisfy the equations ,Ŵ m are the components of the Pauli-Lubanski vector (2.14) andŴ mŴ m is the casimir operator for the group ISL(2, C).
Proof. The components (2.14) of the Pauli Lubanski vector are equal to where u α andūβ are auxiliary Weyl spinors. Then the action of the spin operators (2.37) on on the generating functions (2.40). In eq. (2.41) we have used the notation ∂ β = ∂ u β and ∂β = ∂ūβ . We set W m (u,ū) = 1 2 ε mnij P nΣij (u,ū) and use the identitŷ which is obtained from (2.39) by direct computation. The terms Σ nmP n Σ kmP k and Σ nmΣ nm in the right-hand side of (2.42) are reduced to the forms: where (P σ) = (P n σ n ) and (Pσ) = (P nσ n ). To obtain (2.43) and (2.44), it is necessary to apply identities of the following type: The substitution of (2.43) and (2.44) into the right-hand side of (2.42) gives Finally, the result of the action of the operator (2.46) on the generating function (2.40) of the spin-tensor fields of ( p 2 , r 2 )-type can be calculated directly and we havê that is equivalent to (2.38) for j = p+r 2 . To obtain (2.47), we used the equalityP n ψ (r) (k, u,ū) = k n ψ (r) (k, u,ū), the Dirac-Pauli-Fierz equations (2.30), and the relations: following from the calculation of the degree of homogeneity of polynomials (2.40).
Formula (2.38) shows that the spin-tensor wave functions ψ (r) (k) are eigenvectors for the Casimir operator (Ŵ mŴ m ) of the algebra iso(1, 3). According to Remark 3 of Section 2.1, these wave functions generate the space of unitary representation of the Lie algebra iso(1, 3) (of the group ISL(2, C)) with spin j.
In Section 3, we consider in detail four special examples j = 1/2, j = 1, j = 3/2 and j = 2 of the general construction presented above. These examples are important from the point of view of applications in physics.
According to the general construction developed in Subsection 2.2, the unitary Wigner representation (2.19) of the group SL(2, C) with spin j = 1/2 is realized in the space of SU(2) spinors φ: where the components φ α (k) are functions of the 4-momentum k. Further it is convenient to write these components as follows: Then, taking into account (2.27) and (2.33), we obtain that for j = 1/2 the spin-tensor representation (2.26) is realized in the spaces of Weyl spinors: whereσ 0 is the unit matrix I 2 (see (2.7)) and we used the identities that follow from (2.33): (3.5) As it was shown in Proposition 1, the Weyl spinors ψ (0) α (k), ψ (1)α (k) satisfy the system of equations (a particular case of equations (2.30)): It is well known that this system is equivalent to the Dirac equation: where γ n are the Dirac (4 × 4) matrices and Ψ is the Dirac bispinor: .
since here we only used the properties of the commuting Weyl spinors: λ α λ α = 0, µβµβ = 0, etc. Taking into account (3.3) and (3.4), the spinor Ψ given in (3.8) can be written in the form where we introduced two Dirac spinors: We denote the components of these spinors as e

(+)
A and e In view of (2.33) it is easy to verify that spinors (3.11) are normalized as follows: (3.13) We note that the coefficients φ 1 (k) and φ 2 (k) of the Wigner wave function (3.2) (in the representation space of the stability subgroup SU(2) with spin j = 1/2) correspond to the projections +1/2 and −1/2 of the operator of the third spin component S 3 = 1 2 σ 3 . Therefore, comparing the expansions (3.2) and (3.10), it is natural to interpret the spinors e (+) and e (−) in the expansion of the Dirac spinor Ψ(k) as "vectors" of polarization of a particle with spin j = 1/2.

Spin j = 1.
As it was shown in Section 2.2, the unitary representation of the group ISL(2, C) with spin j = 1 acts in the space of spin-tensor wave functions ψ (2)αβ (k), ψ (1)β α (k) and ψ (0) αβ (k). These functions satisfy the system of Dirac-Pauli-Fierz equations (2.30): where the vector field components A m (k) are defined by the spin-tensor wave function ψ (1) (k): Proof. The spin-tensor wave functions where the matrices σ mn ,σ mn were introduced in (2.12) and we use them in the forms which are symmetric under permutations α ↔ β andα ↔β. Note that equation (3.22) is equivalent to the definition (3.20). We also note that in view of the properties (2.13) of the matrices σ mn andσ mn the antisymmetric vector-tensors F (+) mn (k) and F (−) mn (k) are self-dual and anti-self-dual, respectively: then multiply both sides of (3.25) by ε δα (σ p ) δβ and contract the indices α andβ. As a result, we have the relation which in view of the identities Tr(σ r σ p ) = 2δ r p , Tr(σ b σ mn σ p ) = (η np η mb − η nb η mp − i ε mnpb ) and (3.24) is written as the equation In the same way, the remaining equations of the system (3.15)-(3.18) can be transformed into the following equations: where we introduced the notation F ℓp (k) = m (F ℓp (k) into (3.30)), we deduce the Proca equation (3.19) which describes the dynamics of free spin-1 particles with mass m.
According to formula (2.27), the spin-tensor wave functions ψ (1)β α , which are related to the vector fields A m (see equations (3.20) and (3.22)), are determined by the corresponding Wigner wave function φ: In the space of SU(2) symmetric tensors of the second rank, we introduce the normalized basis vectors where spinors ǫ + and ǫ − were defined in (3.1), and expand the symmetric Wigner wave function φ(k) with the components φ (α 1 α 2 ) (k) over these basis vectors are eigenvectors with eigenvalues +1, 0, −1 of the operator S 3 of the third spin component in the representation of the group SU(2) for spin j = 1. Now we substitute the decomposition (3.32) into (3.31) and fix the test momentum as q = (m, 0, 0, 0). As a result, expansion (3.31) is written as where, according to (2.33), the components of the spin-tensors e (k) and (−) e (k) have the following form: In the space of spin-tensors ψ (1)β α we define the Hermitian scalar product One can deduce relations (3.38) by using the normalization (2.33) of the spinors λ and µ. Let us find the expansion of the vector potential A m (k) over the components φ (αβ) of the Wigner wave function φ. To do this, we substitute the expansion (3.33) into formula (3.20) and obtain where the vectors e (+) can be naturally interpreted (bearing in mind that the components φ (11) , φ (12) , φ (22) of the Wigner wave function (3.32) correspond to the projections +1, 0, −1 of the spin generator S 3 of the stability subgroup SU (2)) as polarization vectors of a massive vector particle. The polarization vectors e (+) (k), e (0) (k), e (−) (k) with components (3.40) are orthonormal and transverse to the momentum k. The normalization and orthogonality of these vectors follow from the conditions (3.38) for the corresponding spin-tensors. Indeed, for any two complex four-vectors e m and e ′ m , which are related to spin-tensors eβ γ and e ′β γ by means of equations (3.40), we have the following identity between two scalar products: where we used formulas (2.35), (3.34) and the properties of the commuting Weyl spinors: µ α µ α = ε αβ µ α µ β = 0 , λβλβ = εβαλβλα = 0, which follow from the definition (2.9) of the metrics ||ε αβ || and ||εαβ||. The transversality of the vectors e (0) m and e (−) m is proved analogously. Thus, e (+) (k), e (0) (k) and e (−) (k) are indeed interpreted as polarization vectors, and expression (3.39) gives the expansion of the vector potential with the components A m (k) over these polarization vectors.
At the end of this Subsection devoted to a detailed discussion of unitary representations of the group ISL(2, C) with spin j = 1 , we calculate the sum over polarizations of the product of vectors (3.40): Here the common minus sign is chosen in accordance with the normalization (3.42) of the vectors e (+) (k), e (0) (k) and e (−) (k) so that the matrix (Θ (1) ) r m = η rn Θ (1) nm (k) satisfies the projection property (Θ (1) ) n r (k) (Θ (1) ) r m (k) = (Θ (1) ) n m (k) . (3.45) The matrix ||Θ (1) nm (k)|| is called the density matrix of massive vector particles and plays an important role in the relativistic theory. To calculate the sum (3.44), we use the two-spinor formalism. We substitute formulas (3.40) and (3.34)-(3.36) into (3.44) and obtain: (3.46) Then we expand the brackets in the right-hand side of (3.46) and group terms in a different way: (3.47) whereupon we use the two-spinor representations (2.35) for the momentum k and the identities: to deduce the final expression for Θ (1) nm (k): (3.49) Here we take into account the identities Tr(σ n σ m ) = 2η nm , Tr(σ nσl σ mσs ) = 2(η nl η ms − η nm η ls + η ns η ml − iε nlms ) , (3.50) and the fact that the momentum k belongs to the mass shell: (k) 2 = m 2 .

Spin j = 3/2.
According to Proposition 2, the unitary representation ISL(2, C) with spin j = 3/2 is realized in the space of spin-tensor wave functions: given in (2.27). In view of the Dirac-Pauli-Fierz equations (2.30) only two functions ψ (1)β1 α 1 α 2 (k) and ψ (2)β1β2 α 1 (k) can be regarded as independent. We transform these two wave functions into spin-vectors (ψ − n ) α 2 (k), (ψ + n )β 2 (k) by converting in the standard way one dotted and one undotted indices into a vector index n: From this set of Weyl spinors we form a bispinor wave function which simultaneously is a 4-vector. The wave function (Ψ n ) A (k) possesses not only the spinor index A = 1, 2, 3, 4 but in addition it also has the vector index n = 0, 1, 2, 3. We define the Dirac conjugate wave function Ψ B n (k) in the standard way: where we used the notation:ψ + n = (ψ + n ) * ,ψ − n = (ψ − n ) * . Proposition 4 The spin-vector wave function Ψ n (k), defined in (3.52), satisfies the Rarita-Schwinger equation describing the field of a free massive relativistic particle with spin j = 3/2. In formula (3.54) we used the notation γ [mrs] and γ [mn] for antisymmetrized products of γ-matrices (k) and ψ (3)β1β2β3 (k). The proof of these facts is straightforward and employs the same methods which we have already used in proving Proposition 3. Now by using the two-spinor formalism, we construct the expansion of the spin-vector wave function Ψ n (k) over polarizations. We will make it in the same way as we constructed such expansions for spins j = 1/2 and j = 1 in Sections 3.1 and 3.2. The spin-tensor wave functions ψ (1)β1 α 1 α 2 (k), ψ (2)β1β2 α 1 (k), which define the spin-vector Ψ n (k) in (3.52), are related to the Wigner wave functions φ (γ 1 γ 2 γ 3 ) by formula (2.27): In the space of symmetric third-rank SU(2)-tensors we introduce normalized basis vectors 58) where the spinors ǫ + and ǫ − were defined in (3.1), and we expand the symmetric Wigner wave function φ(k) with the components φ (α 1 α 2 α 3 ) (k) over these basis vectors We substitute expansion (3.59) into formulas (3.56), (3.57), use (3.5) and then the result is substituted into (3.52). Finally, we obtain n (k) were defined in (3.40), while bispinors e (+) , e (−) were defined in (3.11). As before, it is natural to assume that the bispinor functions e 2 ) n (k) are polarization spin-vectors (below we simply call them polarizations). The spin-vector wave function Ψ n (k) which describes particles with spin j = 3/2 is decomposed into a linear combination of polarization spin-vectors (3.61).
Below in this paper, to simplify formulas, we do not often write the dependence of polarizations on the momentum k. The bispinors (3.61) are normalized as follows: here we used the normalization conditions (3.13) and (3.42). The sum over the polarizations (the density matrix for particles of spin 3/2) is defined by the expression: and in view of (3.62) satisfies the projector property (Θ Finally, we substitute (3.61) into formula (3.63) and group terms so that the density matrix (3.63) takes the form: (3.64) We will need this form of the spin j = 3/2 density matrix below in Section 5.

Spin j = 2.
According to Proposition 2, the unitary representation (with spin j = 2) of the group ISL(2, C) is realized in the space of spin-tensor wave functions: These wave functions were defined in (2.27) and satisfy the system of Dirac-Pauli-Fierz equations (2.30). The most important for us spin-tensor wave function is the function ψ (2) (k), which corresponds to a symmetric second-rank tensor h n 1 n 2 (k) in the Minkowski space-time.
The relation between ψ (2) (k) and h n 1 n 2 (k) is given by the standard formula

Proposition 5 The system of Dirac-Pauli-Fierz equations (2.30) for spin-tensor wave functions (3.65) is equivalent to the massive Pauli-Fierz equation [4]
for the wave functions h mn (k) of massive graviton: where h(k) = η rs h rs (k).

The polarization vector for the fields of arbitrary integer spin
The unitary irreducible representation (2.19) of the group ISL(2, C) with spin j, according to (2.20) ,acts in the space of symmetrized Wigner's wave functions φ (α 1 ···α 2j ) (k). It is convenient to write these symmetrized wave functions as a generating function: where v α are the components of the auxiliary Weyl spinor v = (v 1 , v 2 ). Introduce homogeneous monomials T j m (v) in the variables v 1 and v 2 : which can be considered as (2j + 1) basis elements in the space of polynomials (4.1) since any polynomial (4.1) can be expanded in terms of T j m (v): The relation between the coefficients φ m (k) and φ (α 1 ···α 2j ) (k) is given by the formula: In the space of polynomials (4.1) and (4.3) an irreducible representation of the algebra sℓ(2, C) is realized with generators: The monomials T j m (v) are the eigenvectors of the operator S 3 (the third component of the spin vector) given in (4.5). In fact, we have where p + r = 2j and ∂ (v) ρ i = ∂ ∂v ρ i . We fix as usual the test momentum q = (m, 0, 0, 0) and substitute expression (4.3) for the Wigner wave function to the formula (4.7). After that, expanding the left-and right-hand sides of (4.7) over u and u, we obtain It is clear that the tensor ǫ where the normalization factor 1 √   Proof. The proof is based on the use of the definition (4.11) and is carried out similarly to the proof of Proposition 1.
In this Section, we will mainly consider spin-tensor wave functions of type ( j 2 , j 2 ): ψ (j) (β 1 ···β j ) (α 1 ...α j ) (k) for which the number of dotted and undotted indices is the same. It gives us the possibility to suppress sometimes the index (j) in the notation of the spin-tensor ψ (j) (we have to restore this index in the proof of Proposition 7). The spin-tensor functions of ( j 2 , j 2 )-type are related to the vector-tensors in the Minkowski space by the following formula (cf. (3.20) and (3.66)): By vector-tensors we call the tensors with the components f n 1 ···n j (k) having only vector indices {n 1 , · · · , n j }. We note that in view of the symmetry of the components ψ (β 1 ···β j ) (α 1 ...α j ) (k) under all permutations of the spinor indices (β 1 · · ·β j ) and (α 1 · · · α j ), the vector-tensor f n 1 ···n j (k), given in (4.14), is completely symmetric with respect to the permutations of the vector indices.
As in (3.37), we define the Hermitian scalar product of the spin-tensor functions ψ(k) and ξ(k) of type ( j 2 , j 2 ) in the following way: Recall that under complex conjugation * the dotted indices of the spin-tensors are converted to undotted indices and vice versa. Now one can check that the spin-tensors (m) e (k) defined in (4.11) are orthonormal with respect to the scalar product (4.15). This follows from the chain of equations: where we used the identities: ρ 1 ···ρ 2j , which were discussed after eq. (4.10). We stress that in equations (4.16)-(4.18) it is not needed to put dots over the indices γ i since these indices correspond to the representations of the group SU(2). Now we convert the spin-tensor wave functions ψ (β 1 ···β j ) (γ 1 ...γ j ) (k) to the vector-tensor functions f n 1 ···n j (k) by means of relation (4.14) and substitute expression (4.12) for ψ (β 1 ···β j ) (γ 1 ...γ j ) (k) in terms of Wigner's coefficients φ m (k). As a result, we obtain the expansion where we used the notation: The vector-tensors e (m) n 1 ···n j (k) will be called polarization tensors for particles with spin j. This terminology is natural since the vector-tensors e (m) n 1 ···n j (k) form the basis in the expansion of the fields f n 1 ···n j (k) over Wigner's coefficients φ m (k), which in view of (4.3) and (4.6) are propotional to contributions of projection m of the spin component S 3 .

Proposition 7
The operator Θ(k), defined in (4.23), satisfies the following properties: 1) projective property and reality: 2) symmetry: Θ The second property follows from the symmetry of the vector-tensors e (m) n 1 ···n j (k) with respect to any permutation of vector indices {n 1 , · · · , n j }, as it follows from formula (4.20).
The third property is equivalent to the transversality of the vector-tensors e (m) (k), i.e., is equivalent to the condition k r 1 e (m) r 1 ···r j (k) = 0. This condition follows from the chain of relations: (here we need to restore the label (j) in the notation of the spin-tensors  The fourth property is equivalent to the statement that the vector-tensors e (m) (k) are traceless, i.e., η r 1 r 2 e (m) r 1 ···r j (k) = 0. This statement can be proven as follows: where we apply formula (2.45).
5 Spin projection operators for integer and half-integer spins.
In this Section, we find an explicit expression for the projection operator Θ n 1 ···n j r 1 ···r j (k) (see (4.23)) for any integer spin j > 1, in terms of the operator Θ n r (k). The operator Θ n r (k) is the projection operator for spin j = 1 and it was calculated in (3.49) . For the four-dimensional D = 4 space-time, the Behrends-Fronsdal projection operator Θ(k) was explicitly constructed in [20], [21]. Here we find a generalization of the Behrends-Fronsdal operator to the case of an arbitrary number of dimensions D > 2. Also in this Section we prove an important formula which connects the projection operators for half-integer spins j with the projection operators for integer spins j + 1/2. The construction will be based on the properties of this operator, which are listed in Proposition 7.
where [ j 2 ] -integer part of j/2, a (j) The generating function (5.2) satisfies the differential equation Proof. We recall (see (3.45), (3.49)) that the matrix Θ n r which is defined in (5.4) is a projection operator onto the subspace orthogonal to the D-dimensional vector with the components k r : Taking into account this fact, the most general covariant operator Θ n 1 ...n j r 1 ...r j (k), satisfying properties 2) and 3) from Proposition 7, is written as follows: where σ, µ ∈ S j are permutations of the indices {1, 2, . . . , j}, and the components B ℓ 1 ...ℓ j m 1 ...m j (k) are any covariant combinations of the metric η rm and coordinates k r of the D-vector of momentum. Since the matrices Θ m r used in the right-hand side of (5.7) are transverse to the D-momentum k (see (5.6)), the external indices of the tensor B ℓ 1 ...ℓ j m 1 ...m j (k) can be associated only with the indices of the metric η and, therefore, this tensor is represented in the form where the functions a (j) A (k) depend on the invariants (k) 2 = k r k r . Substitution (5.8) into (5.7) gives Finally, using expression (5.9) in (5.1), we obtain formula (5.2) for the generating function of the projection operator. The coefficients a (j) A in (5.2), as we will see below, do not depend on (k) 2 and their explicit form is fixed by properties 1) and 4) from Proposition 7.
Property 4) (traceless) in Proposition 7 for the tensor Θ is equivalent to the harmonic equation for the generating function (5.2): where (∂) 2 = ∂ x r ∂ x r and ∂ x r = ∂/∂x r . Let us substitute expression (5.2) into the equation (5.10) to find the conditions which fix the coefficients a (j) A . It is convenient to rewrite the series (5.2) in the form As a result of the substitution (5.2) into equation (5.10), we have where in the second equality we used the relations Thus, to fulfill the identity (5.10), according to (5.12), it is necessary to require the recurrence relation for the coefficientsã A : The solution of equation (5.17) has the form a (j) i.e., the condition (5.10) determines the coefficients a (j) A up to a single arbitrary factor a (j) 0 . Note that, firstly, the product of factors in square brackets in the denominator (5.18) must be considered equal to unity for A = 0 and, secondly, when substituting the coefficients (5.18) into the sum (5.2), it is clear that this sum is automatically terminated for A > j/2 since in this case an infinite factor (j − 2A)! = ∞ appears in the denominator.
Let us now verify condition 1) from Proposition 7. First of all, the property Θ(x, y) = Θ(y, x) for the function (5.2) and reality condition Θ * = Θ are equivalent to Θ † = Θ for the matrices (5.9). The projector condition Θ 2 = Θ (see property 1 in Proposition 7) for the matrix (5.9) can be checked directly: where in the second and third equalities we used the identities that follow from conditions 3) and 4) of Proposition 7. Thus, to fulfill the projector condition, we must fix the initial coefficient in the expansion (5.2) as a Finally, we prove the identity (5.5). For this we calculate where we used the equalities Taking into account the explicit formula for the coefficients (5.3), we obtain the relation a (j) substituting this into (5.19), we immediately derive the identity (5.5).
Remark 1. Identity (5.5) for the generating functions (5.1) gives the equality that connects the projection operators (5.9) for the spins j and (j − 1): In other words the trace of the matrix Θ (j) over the pair of indices is proportional to the matrix Θ (j−1) . Using formula ( This trace is equal to the dimension of the subspace, which is cut out from the space of vectortensor wave functions f n 1 ...n j (k) by the projector Θ (j) . In other words, the trace (5.21) is equal to the number N of independent components of symmetric vector-tensor wave functions f (n 1 ...n j ) (k) that satisfy the conditions k n 1 f (n 1 ...n j ) (k) = 0 , η n 1 n 2 f (n 1 n 2 ...n j ) (k) = 0 .
On the space of these functions an irreducible massive representation of the D-dimensional rotation group is realized. For example, for D = 3 we have N = 2 (∀j), and for D = 4 formula (5.21) gives the well-known result N = (2j + 1), which coincides with the number of polarizations of the massive particles with spin j (it coincides with the dimension of the irreducible representation with spin j of a small subgroup SU(2) ⊂ SL(2, C)). Remark 2. From relation (5.2) the useful identity [21] immediately follows: A=0 a (j) where the coefficients a (j) A are defined in (5.5). The sum of the coefficients a (j) A in the right-hand side of (5.22) can be calculated explicitly by using relations (5.5) and (5.22). Indeed, we put x r = y r in (5.5), take into account (5.10) and apply the operator ∂ x r ∂ xr to both sides of equality (5.22). After that, comparing the results obtained in both sides of (5.22), we deduce the recurrence equation: A . Solving this equation with the initial condition S , j > 1 .
Now we will construct the spin projection operator Θ (j) (k) in the case of half-integer spins j. Recall that for spin j = 1/2 the operator Θ (1/2) (k) was explicitly calculated in (3.14). For j = 3/2 we found formula (3.64) for the operator Θ (3/2) (k) in terms of bispinors (3.11) and polarization vectors (3.40). To obtain the general formula for any half-integer spin j (in the case of 4-dimensional space-time), one can use the definition of the spin projection operator (Θ (j) ) Formula (5.25) is a generalization of (3.61) and is obtained from equations (4.11) and (4.12). After substitution of (5.25) into (5.24) one can calculate the sum over polarizations m in (5.24) and deduce explicit formula for the operator Θ (j) . For a special case j = 1/2 it was done in (3.14). However, it is rather a long way to obtain an explicit expression for the operator Θ (j) . Here we will use another method which is based on the ideas of the paper [21]. Moreover, this method gives us a possibility to find the spin projection operator Θ (j) (for half-integer spins j) for the general case of arbitrary space-time dimension D.

Conclusion
In this paper, on the basis of unitary representations of the covering group ISL(2, C) of the Poincaré group, we have constructed explicit solutions of the wave equations for free massive particles of arbitrary spin j (the Dirac-Pauli-Fierz equations). Then we proposed the method for decomposing of these solutions into a sum over independent components corresponding to different polarizations. The sum over the polarizations for the density matrix of particles with arbitrary integer spin is calculated explicitly. This density matrix (spin projection operator) coincides with the Behrends-Fronsdal projection operator for space-time dimension D = 4. The generalization of the Behrends-Fronsdal projection operator for any number of spacetime dimensions D > 2 was found. We also found the generalization of the explicit formula for the density matrix (spin projection operator) of particles with half-integer spins. The most interesting examples corresponding to spins j = 1/2, 1, 3/2 and j = 2 were discussed in detail.
We have to stress that the massless case can also be considered in a similar manner. Certain steps in this direction were made in [23]. Just as in the massive case, the spintensor wave functions of free massless particles with arbitrary helicity are constructed from the vectors of spaces of the unitary massless Wigner representations for the covering group ISL(2, C) of the Poincare group.
Moreover, formula (2.27) is carried over to the massless case practically unchanged (we need to remove the normalizing factor m −r and choose the test momentum as q = (E, 0, 0, E)). The corresponding spin-tensor wave functions satisfy the Penrose equations (these equations for fields of massless particles were formulated by Penrose in the coordinate representation; see [24] and [25]) instead of the Dirac-Pauli-Fierz equations. It is remarkable that instead of the two-spinor approach, which is suitable for the massive case and used in this paper, we arrive in the massless case at the twistor formalism [24].
We hope that the two-spinor formalism considered in this paper for describing massive particles of arbitrary spin will be useful in the construction of scattering amplitudes of massive particles in a similar way to the construction of spinor-helicity scattering amplitudes for massless particles [26], [27], [28] (see also [29] and references therein). Some steps in this direction have already been done in papers [30], [31], [32] where the analogous two-spinor formalism and its special generalization were used.