Constrained BRST-BFV Lagrangian formulations for higher spin fields in Minkowski spaces

BRST-BFV method to construct constrained Lagrangian formulations for (ir)reducible half-integer higher-spin Poincare group representations in Minkowski space is suggested. The procedure is derived by two ways: first, from the unconstrained BRST-BFV method for mixed-symmetry higher-spin fermionic fields subject to an arbitrary Young tableaux with k rows (suggested in Nucl. Phys.B 869 (2013) 523, arXiv:1211.1273) by extracting the second-class constraints subsystem, Ôα = (Ôa, Ôa+), from a total super-algebra of constraints, second, in self-consistent way by means of finding BRST-extended initial off-shell algebraic constraints, Ôa. In both cases, the latter constraints supercommute on the constraint surface with constrained BRST operator QC and spin operators σCi . The closedness of the superalgebra {QC, Ôa, σCi} guarantees that the final gauge-invariant Lagrangian formulation is compatible with the off-shell algebraic constraints Ôa imposed on the field and gauge parameter vectors of the Hilbert space not depending from the ghosts and conversion auxiliary oscillators related to Ôa, in comparison with the vectors for unconstrained BRST-BFV Lagrangian formulation. The suggested constrained BRST-BFV approach is valid for both massive HS fields and integer HS fields in the second-order formulation. It is shown that the respective constrained and unconstrained Lagrangian formulations for (half)-integer HS fields with a given spin are equivalent. The constrained Lagrangians in ghost-independent and component (for initial spin-tensor field) are obtained and shown to coincide with the Fang-Fronsdal formulation for totally-symmetric HS field with respective off-shell gamma-traceless constraints. The triplet and unconstrained quartet Lagrangian formulations for the latter field are derived. The constrained BRST-BFV methods without off-shell constraints describe reducible half-integer HS Poincare group representations with multiple spins as a generalized triplet and provide a starting point for constructing unconstrained Lagrangian formulations by using the generalized quartet mechanism. A gauge-invariant Lagrangian constrained description for a massive spin-tensor field of spin n + 1/2 is obtained using a set of auxiliary Stueckelberg spin-tensors. A concept of BRST-invariant second-class constraints for dynamical systems with mixed-class constraints is suggested, leading to equivalent (w.r.t. the BRST-BFV prescription) results of quantization both at the operator level and in terms of the partition function.


Conclusion 49
A On quantization of the dynamical system with mixed-class constraints 52 1 Introduction Higher-spin (HS) field theory, in its various reflections, has been under a long and intense study in order to re-analyse the problems of a unified description for the variety of elementary particles, thereby discovering approaches to a unification of the known and new possible fundamental interactions, and remains part of the LHC experimental program. HS field theory is in close relation to superstring theory due to its tensionless limit [1], which uses a BRST operator to handle an infinite set of HS fields with integer and halfinteger generalized spins in d-dimensional (d ≥ 4) space-time and incorporates HS field theory into superstring theory, providing the consideration of HS theory as an instrument for investigating the structure of superstring theory (for the current status of HS field theory and recent advances in HS theory, see the reviews [2][3][4][5][6][7]). A powerful systematic JHEP09(2018)104 tool used to reconstruct Lagrangian formulations for HS fields described by (spin-)tensor fields as elements of an irreducible representation of the Poincare ISO(1, d − 1) or (anti)-de-Sitter ((A)dS) groups in the respective d-dimensional Minkowski or (A)dS space-time by means of second-or first-order non-Lagrangian equations for free propagating HS integer and half-integer fields, respectively, is based on a BRST construction, developed initially for HS fields with lower spins [8][9][10][11] in R 1,d−1 , extended for higher integer spin fields in R 1,d−1 [12][13][14][15] and (A)dS d [16][17][18][19], and for the fields with higher half-integer spin [20][21][22] (for review, see [24]). This approach is based on the BRST-BFV method [25][26][27][28][29][30][31], originally developed to solve the problem of Hamiltonian quantization for dynamical systems with first-class constraints in Yang-Mills theories, suggested in [32], and is usually known as the BRST construction. The application of the BRST construction to free HS field theory consists of four steps. First, the conditions that determine representations with a given spin are regarded as a topological (i.e., without Hamiltonian) gauge system of first-and second-class constraints with a spin operator in an auxiliary Fock space. Second, the subsystem of the initial constraints, which contains only the second-class constraints (for massless case), together with the spin operator, is converted, while preserving the initial algebraic structure, into a system of first-class constraints alone, in an enlarged Fock space (for the conversion methods, see [33][34][35]). Third, with respect to the converted constraints, one constructs a BRST operator, which is a more involved problem for HS fields in (A)dS spaces due to quadratic constraint algebras [36][37][38][39]. Fourth, the Lagrangian and the reducible gauge transformations for an HS field are constructed in terms of the BRST operator in such a way that the corresponding equations of motion reproduce the initial constraints. We emphasize that the approach leads automatically to a gauge-invariant Lagrangian description with all the necessary auxiliary and Stuckelberg fields. Applying the BRST-BFV approach to the HS field theory, one usually works within a metric-like formulation due to Fronsdal's results in totally-symmetric HS fields with integer [40,41] and half-integer spins [42,43], whereas in the frame-like formulation [45] the results for the Lagrangian dynamics of HS fields [44] were obtained beyond this construction. In constructing Lagrangian formulations for free and interacting HS fields, one examines the cases of unconstrained or constrained dynamics, which means, respectively, the absence or presence of consistent usually off-shell holonomic (traceless, γ-traceless, mixedsymmetry) constraints. As a rule, most of the results in the metric- [46][47][48][49][50][51] and frame-like formulations [52][53][54][55][56][57] were obtained for off-shell constraints. There exists so-called Maxwelllike formulations for metric-like tensor fields (on flat and to some extent on (A)dS spaces) developed originally with off-shell differential constraints on the gauge parameters and with their resolution by means of a tower of reducible gauge transformations with higher derivatives [58,59]. We recall that irreducible Poincare or (A)dS group representations in constant curvature space-times is described by mixed-symmetric (MS) HS fields with an arbitrary Young tableaux of k rows, Y (s 1 , . . . , s k ) (symmetric basis), determined by more than one spin-like parameters s i [60][61][62][63][64][65], and, equivalently, by mixed-antisymmetric (spin-)tensor fields with an arbitrary Young tableaux of l columns, Y [ŝ 1 , . . . ,ŝ l ] (antisymmetric basis), the integers or half-integersŝ 1 ≥ŝ 2 ≥ . . . ≥ŝ l having a spin-like interpretation [66]. BRST-BFV Lagrangian formulations for arbitrary free MS HS fields with integer JHEP09(2018)104 and half-integer spins were constructed in an unconstrained form (complete with all the algebraic constraints that follow from the Lagrangian and the tower of the respective gauge transformations after a partial gauge fixing) in our papers [67,68]. In turn, the notion "unconstrained" has numerous interpretations, and was introduced originally as the geometric formulations of the field equations and Lagrangians for totally-symmetric HS fields both in non-local and local (minimal) representations [69,70] within the metric-like formulation, and in more symmetric form for Lagrangians with totally-symmetric HS fields in [71,72] as a "quartet unconstrained formulation" obtained from the triplet formulation with off-shell algebraic constraints [73]. Another usage of this term is due to Lagrangian formulations for MS HS fields in R 1,d−1 [74,75]. Constrained Lagrangians, together with minimal BV actions for totally and mixed-symmetric HS fields with integer spin, were studied within the so-called BRST-BV approach [76][77][78][79], which includes the constrained BRST-BFV Lagrangian approach itself. At the same time, no constrained BRST-BFV or BRST-BV constructions for half-integer HS fields in constant curvature spaces have been suggested so far, due to the yet inknown form of consistent (with constrained first-order Lagrangians) holonomic constraints. Moreover, explicit relations between the unconstrained and constrained BRST-BFV approaches to Lagrangian formulations for the same HS field with a given spin have not been established, despite the fact that the respective Lagrangians are to describe dynamics equivalent to the initial relations of an irreducible representation for the HS field. The same problem (which is left out of the paper's scope) arises as one examines unconstrained (so far undeveloped) and constrained BRST-BV constructions for both integer and half-integer HS fields.
The paper is devoted to solving the following problems: 1. Derivation of constrained BRST-BFV approaches to constrained Lagrangian formulations for MS HS fields in R 1,d−1 with given integer and half-integer spins from the respective unconstrained BRST-BFV approaches; 2. Derivation of constrained BRST-BFV approaches to constrained Lagrangian formulations for integer and half-integer MS HS fields in R 1,d−1 in a self-consistent way; 3. Study of equivalence between unconstrained and constrained BRST-BFV Lagrangian formulations for the same MS HS field; 4. Derivation of constrained gauge-invariant Lagrangian actions for totally symmetric half-integer massless and massive HS field in the metric formulation with a single spin-tensor, in the triplet and unconstrained quartet forms; 5. Study of a BRST invariant extension of second-class constraints for a general dynamical system with independent sets of first-and second-class constraints and its application to the quantization procedure within both the conventional path integral approach and generalized canonical quantization.
The organization of the paper is as follows. In the section 2 we remind the crucial points of BFV method application to the quantization of dynamical systems subject to the first and second-class constraints in operator and functional integral forms. In section 3, we briefly review the ingredients of the unconstrained BRST-BFV method for gauge-invariant

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Lagrangian formulations with free half-integer MS HS fields in Minkowski space, which is the starting point to obtain a constrained BRST-BFV operator, a spin operator and off-shell algebraic constraints in subsection 4.1 of section 4. A self-consistent way to construct the basic elements of the constrained BRST-BFV method is examined in subsection 4.2. Section 5 is devoted to the construction of constrained Lagrangian formulations for free massless and massive half-integer HS fields subject to Y (s 1 , . . . s k ). The case of constrained Lagrangian formulations for integer MS HS fields is examined in subsection 5.2. In section 6, we consider ghost-independent, component spin-tensor and triplet forms of constrained Lagrangians for totally-symmetric fermionic HS fields, in subsection 6.1, and quartet unconstrained Lagrangians for the same fields, together with massive fields, in subsection 6.2. In Conclusion, we present a review of our basic results. Finally, in appendix A, we suggest a new algorithm for quantizing dynamical systems with mixed-class constraints.

On BRST-BFV method for dynamical systems subject to constraints
Here, we briefly consider some specific points of the BRST-BFV construction (following in part to [25][26][27][28][29][30][31], see as well [80]) as applied to the solution of the direct problem of generalized canonical quantization of the dynamical systems subject to the first and second-class constraints in order to calculate average expectation values of the physical quantities on appropriate Hilbert space in gauge-invariant way and to the inverse problem of reconstruction of the Lagrangian formulation for initial non-Lagrangian equations on the (spin)tensor fields when applying to the HS field theory, firstly on the free and then on the interacting levels.
The constrained dynamical system is described, according to Dirac proposal [81,82] by the Hamiltonian, H 0 (Γ), and finite set of the first-class T A (Γ) = 0 and second-class constraints, Θ α (Γ) = 0, A = (A + , A − ) = 1, . . . , M ; M = (M + , M − ), α = (α + , α − ) = 1, . . . , m; m = (m + , m − ) with Grassman gradings ε(H 0 ; T A ; Θ α ) = (0; ε A ; ε α ) depending on the phase-space coordinates Γ p = (q i , p i ), i = 1, . . . , n, n = (n + , n − ) for (m + 2M ) ≤ 2n in the 2n-dimensional phase-space (M, ω) with Grassmann-even non-degenerate closed 2form ω, dω = 0, and fundamental Poisson superbrackets . The dynamics and gauge transformations are determined by the equations, with arbitrary functions ξ A (t) (functionally independent for linear independent set of T A (Γ)) meaning the necessity to introduce gauge conditions χ B (Γ) = 0 such that: and appropriate Cauchy problem: Γ p (t) |t=t 0 = Γ p 0 . There exists representation for the system of mixed constraints (T A , Θ α ) in the form (T c|A , Φ α )(Γ c ) given in the extended phase-space M c with coordinates Γ P c = (Γ p , ζ α ) for ε(ζ α ) = ε(Θ α ) being subject to new fundamental Poisson superbrackets, such that the set of (T c|A , Φ α )(Γ c ) appears by the first-class constraints system, known as the converted constraints in M c with deformed Hamiltonian H c|0 (Γ c ). The converted functions satisfy to the involution relations in M c for the system of the first-class constraints: with, in general, new structure functions, subject to analogous symmetry properties as for unconverted ones in (2.1)-(2.4) and with boundary conditions: The dynamics and gauge transformations for converted system are determined by the equations, with arbitrary functions ξ A (t), ̟ α (t), so that after introduction of the gauge conditions it leads to solution of appropriate Cauchy problem: Γ P c (t 0 ) = Γ P c|0 with admissible Γ P c|0 . The crucial moment in the conversion procedure that the dynamics of initial and converted dynamical systems are equivalent.

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The quantization problem for the dynamical system in terms of the functional integrals in both forms: initial and converted should be equivalent when calculating vacuum average expectation values for the quantities given on original phase-space M.
We recall that the total phase space M tot , (M ⊂ M tot ) underlying the BRST-BFV generalized canonical Hamiltonian quantization [25][26][27][28] is parameterized (for linearly independent constraints T A , Θ α ) by the canonical phase-space variables, Γ P T , ε(Γ P T ) = ε P , with canonical pairs of ghost, C A , P A , antighost, C A , P A and Lagrangian multipliers, π A , λ A for χ A and T A respectively with non-vanishing fundamental Poisson superbrackets: The generating functional of Green's functions for a dynamical system in question has the form with functional measure dµ introduced according to [84,85] and determines the partition function Z Ψ = Z Ψ (0) at the vanishing external sources I P (t) to Γ P T . In (2.17), integration over time is taken over the range t in ≤ t ≤ t out ; the functions of time Γ P T (t) ≡ Γ P T for t in ≤ t ≤ t out are trajectories,Γ P T (t) ≡ dΓ P T (t)/dt; the quantities ω T |PQ = (−1) (ε P +1)(ε Q +1) ω T |QP compose an even supermatrix inverse to that with the (constant) elements ω PQ ) is determined by three t-local functions: even-valued H(t) with gh(H) = 0, odd-valued functions Ω(t), with gh(Ω) = 1, and Ψ(t), with gh(Ψ) = −1, known as the BRST-BFV operator and gauge-fixing Fermion, given by the equations in terms of Dirac superbracket [81,82] constructed with help of the second-class constraints Θ α : with the boundary conditions

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The sign ′′ ≃ ′′ means weak equality, modulo arbitrary linear combination of the secondclass constraints Θ α [81]. From equations (2.20) and the Jacobi identities for the Dirac superbracket, it follows that The solutions for the generating equations (2.20) exist [34,80,83] in the form of series in powers of minimal ghost coordinates and momenta C A , P A with use of CP-ordering up to the second order in Γ gh : which encode in H and Ω min = Ω min (Γ, Γ gh|m ) the structure functions with the terms proportional to only T A from the algebra of the constraints (2.1)-(2.4). The BRST-BFV operator Ω min depends only on ghost coordinates and momenta from minimal sector Γ gh|m = C A , P A and satisfies to the equation (2.20). In turn, the simple form for the quadratic in powers of Γ P T gauge fermion to be sufficient for existence of Z Ψ (I) looks as with the gauge conditions χ A (Γ(t)) = 0 satisfying to (2.6). The integrand in (2.17) for I = 0 is invariant with respect to the infinitesimal BRST transformations, for odd-valued µ, µ 2 = 0 with nilpotent generator ← − s : ← − s 2 = 0 (by virtue of Jacobi identity for Dirac superbracket and equation (2.20)), realized on the phase-space trajectories Γ P T (t) (to be solutions of the first equations in (2.5), but with Hamiltonian H Ψ , with respective Poisson bracket and on the surface Θ α = 0) as The invariance of the integrand is due to additional (as compared for the Poisson superbracket) property for Dirac superbracket: {F, Θ α } D ≃ 0 for any function F given on M tot . For the converted dynamical system with first-class constraints (T c|A , Φ α )(Γ c ) given in M c the total phase space M c|tot , (M c ⊂ M tot and M tot ⊂ M c|tot ) is parameterized by the canonical phase-space variables, Γ P c|T , with account for the representation (2.14), (2.15) and properties (2.16) for Γ gh which correspond to only initial first-class constraints T A subsystem, with canonical pairs of ghost,

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C α , P α , antighost, C α , P α and Lagrangian multipliers, π α , λ α for Ξ α and Φ α respectively with the same non-vanishing fundamental Poisson superbrackets as in (2.16) determined now in M c|tot , with respect to the coordinates Γ P c|T so that {Γ gh , Γ 2|gh } T = 0. The generating functional of Green's functions for the dynamical system with converted constraints has the representation and determines the partition function Z Ψc = Z Ψc (0) at the vanishing external sources I T |P (t) to Γ P c|T with the same properties as for (2.17), whereas the quantities ω cT |PQ = (−1) (ε P +1)(ε Q +1) ω cT |QP compose an even supermatrix inverse to that with (constant) ele- , Ω c (t), with (ε, gh)(Ω c ) = (1, 1), and Ψ c (t), with (ε, gh)(Ψ c ) = (1, −1), defined for dynamical system with the converted constraints T c|A , Φ α : subject to the boundary conditions From the generating equations (2.33) it follows that total Hamiltonian commutes with BRST charge Ω c : {H Ψc , Ω c } T = 0. The solutions for the equations (2.33) exist in the form of series in powers of minimal ghost coordinates and momenta C A , P A , C α , P α and up to the second order in Γ gh looks as: which encode in H c and BRST-BFV operator Ω c| min ≡ Ω c| min (Γ c , Γ gh|m ) (depending on the ghost coordinates and moments in minimal sector Γ gh|m ≡ C A , P A ; C α , P α ) the structure functions from the algebra of the converted constraints (2.8)-(2.10). The quadratic in powers of Γ P c|T gauge fermion Ψ c to be sufficient for existence of Z Ψc (I T ) may be chosen as

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where the upper sign "T " denotes the matrix transposition and the gauge conditions χ A c (Γ c (t)) = 0, Ξ α (Γ c (t)) = 0 should satisfy to (2.13). The integrand in (2.31) for I T = 0 is invariant with respect to the infinitesimal BRST transformations, with nilpotent generator ← − s c : ← − s 2 c = 0, realized on phase-space trajectories Γ P c|T (t) (to be solutions of the first equations in (2.5), but with Hamiltonian H Ψc and respective Poisson bracket) analogously to the rule (2.28) but with Poisson superbracket determined on M c|tot instead of Dirac one.
The equivalence of both quantizations for the initial with (T A , Θ α , H 0 )(Γ) in M and converted with (T c|A , Φ α , H c|0 )(Γ c ) in M c dynamical systems means that the vacuum average expectation values for any quantity A(Γ) determined on the initial phase-space in M coincide when calculating with respect to path integrals Z Ψ (2.17) and Z Ψc (2.31): On the operator level for the quantization problem let us consider, first, a constrained dynamical system with only second-class constraints Θ α satisfying to the relations (2.2) and (2.4) for T A ≡ 0. In case of existing the splitting of Θ α (at least, locally) on two subsystems Θ α (Γ) → Θ ′ α (Γ) = Λ β α (Γ)Θ β (Γ)= θᾱ, θ α with non-degenerate (on the surface Θ α = 0) supermatrix of rotation of the constraints: sdet Λ β α = 0, for the index α division: α = (ᾱ, α) forᾱ = 1, . . . , 1 2 m and α = 1 2 m + 1, . . . , m such that each subsystems θᾱ, θ α appear by the first-class constraints ones: with use of the Leibnitz rule for the Poisson brackets and for ε(Λ β β ) = ε β + ε β : In the corresponding Hilbert space H Γ [for the correspondence Γ p →Γ p : [Γ p ,Γ q } = ı ω pq , ω pq = const, with representation respecting the division for Θ α , but without exception of the degrees of freedom related to the second-class constraints and with choice of some qp-ordering for Θ α (Γ)] according to Dirac approach [81,86]  The physical states should satisfy to the Schrodinger equation with Hamiltonian not depending on the rest constraints θ α , playing the role of the gauge conditions for θᾱ: In turn, for the classically equivalent dynamical system of converted operator first-class constraints (Φ α )(Γ c ) [for the correspondence Γ P c →Γ P c = (Γ p ,ζ α ): with choice of some ordering for the products in powers of ζ α additional to qp-ones for Φ α (Γ c ) and without T c|A ] in M c with operator of Hamiltonian H c|0 (Γ c ) (2.9), (2.10) it is valid the Statement 1. The second-class constraints system Θ α (Γ) converted into first-class constraints one Φ α (Γ,ζ) with additional toΓ p operatorsζ α (2.7), whose number coincides with one of Θ α , satisfying to the superalgebra: [ζ α ,ζ β } = ı ω αβ with constant ω αβ : selects the same set of the physical states in H Γ as the converted constraints Φ α (Γ,ζ) select from H Γ ⊗ H ζ : Note, first, the operator functions F γ αβ (Γ,ζ) in (2.48) obey to the properties analogous to ones for f γ αβ (Γ) (2.3) and there are no anomalies in the right-hand side of (2.48) which in opposite case should be proportional to 2 D αβ (Γ,ζ). Second, for special (but interesting) cases the additional phase-space operatorsζ α may be chosen, as respecting the division of the second-class constraints: Θ ′ α = (θᾱ, θ α ) as followsζ α = (q α , pᾱ): [q α , pᾱ} = ı δᾱ α , in particular, as for the additional operators (case of additive conversion) Φ α (Γ c ) = Θ α (Γ) + ϑ α (ζ) for [Θ α , ϑ β } = 0, for Fock space Hζ with oscillators satisfying to B a , B b+ = δ ab . Third, the presentation (2.49) permits the boundary conditions for any |χ ∈ H Γ ⊗ H ζ : |χ ζ=0 = |ψ ∈ H Γ . Fourth, the quantum evolution of the converted constrained system is described by the Schrodinger equation analogous to (2.47) Fifth, the Statement 1 guarantees a preservation of explicit Poincare covariance for fieldtheoretic models, like QED, gravity, models with HS fields when working with converted constraints.

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For proper gauge dynamical system with some operator first-class constraints system T A (Γ) only (without reference to any second-class constraints) it is valid the following Statement 2. Nilpotent Grassman-odd BRST-BFV operator Q =Ĉ A T A (Γ) + "more ′′ , Q ≡ Ω min (Γ,Γ gh|m ) from (2.25) with gh(Q) = 1 constructed with respect to the system of T A (Γ) in the Hilbert space H Q = H Γ ⊗ H gh|m admitting Z-grading: H Q = k H k Q , gh(|χ k ) = −k, for any |χ k ∈ H k Q ) with Hilbert space H gh|m generated by the operatorŝ C A ,P A from the minimal sector subject to (2.15) and [Ĉ A ,P B } = ı δ A B permits to find the physical Hilbert subspace H phys 1 as follows: for the quotient of the subspace of Q-closed vectors (ker Q ∈ H Q ) with respect to subspace of Q-exact ones (Im Q ∈ H Q ). The evolution of the system is described by the Corollary 1. The physical states in H Γ for the dynamical system with second-class constraints, permitting the division on two sets with only first-class constraints, Θ α (Γ) → Θ ′ α (Γ) = θᾱ, θ α , maybe equivalently presented by nilpotent BRST-BFV operator Q c|2 = C α Φ α (Γ c ) + "more ′′ , Q c ≡ Ω c| min (Γ c ,Γ gh|m ) (2.37) for T A ≡ 0, constructed with respect to the system of converted second-class constraints Φ α (Γ,ζ) in the Hilbert space H(Q c|2 ) = H Γ ⊗ H ζ ⊗ H 2gh|m in the form: for k = 0, . . . , M , (ε, gh)(|χ k ) = (k mod 2, −k).

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Finally, for initial dynamical system with mixed-class constraints T A , Θ α and Hamiltonian H 0 , satisfying to the operator analog of the superalgebra (2.1)-(2.4), we get to Corollary 2. The physical states in H Γ for the dynamical system with first -T A (Γ) and second-class Θ α constraints, permitting for the latter the division on two sets with only first-class constraints, Θ α (Γ) → Θ ′ α (Γ) = θᾱ, θ α subject to operator analog of the relations (2.43), maybe equivalently presented by nilpotent BRST-BFV operator Q c = C α Φ α (Γ c ) +Ĉ A T c|A (Γ c ) + "more ′′ , Q c ≡ Ω c| min (Γ c ,Γ gh|m ) (2.37), constructed with respect to the system of converted first T c|A (Γ c ) and second-class constraints Φ α (Γ c ) in the Hilbert space H(Q c ) = H Γ ⊗ H ζ ⊗ H gh|m ⊗ H 2|gh|m in the form: for k = 0, . . . , N . The above Statements and Corollaries are the crucial results in the application of the BRST-BFV method to a canonical quantization of any dynamical system with finite degrees of freedom subject to first-and second-class constraints. In fact, Statement 2 was proved in the excellent textbook [87] (see chapter 14 and Theorem 14.7 therein). The correctness of Statement 1 is based, first of all, on the previously established (see [34,35]) classical equivalence of a dynamical system with second-class constraints to the same dynamical system with converted first-class constraints. Secondly, the Statement is based on the representation given by the first line of (2.49) for the physical space H phys Γ of a dynamical system with second-class constraints known from Dirac's work [86]. Therefore, under the absence of anomalies in (2.48) a classically equivalent dynamic al system with converted first-class constraints leads to the same physical space H phys Γ given by the second line of (2.49), according to the same Dirac quantization concept for a first-class constraints system, also presented in [87] (see chapter 13 and section 13.3 therein).
In case of the first-class constraints subsystem {T A (Γ)} to be closed with respect to Hermitian conjugation: (T A ) + ∈ {T A (Γ)}, the results of the Statement 2 and Corollary 2 can be refined. Namely, the property above means that the presentation instead of the whole set of T A (Γ) in (2.52) and then in (2.56). The argumentation to prove the last condition is analogous to one considered for the case of conditions (from the Virasoro algebra) which have been used to select physical states from the total Hilbert space for bosonic string or for the superstring models [88,89], but now for the finite set of constraints.

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We stress that when analyzing the presentation the structure of the physical states we did not considered the BRST operator, Ω(Γ T ), Ω c (Γ c|T ) and unitarizing Hamiltonian, H Ψ (Γ T ), H Ψc (Γ c|T ) for the dynamical system in question, which act on more wider respective Hilbert spaces (than H(Q), H(Q c )) extended by additional operatorsĈ,P,π,λ from non-minimal sectors. Therefore, it is not necessary to operate with the gauge conditions χ A (Γ), Ξ β (Γ c ) to select physical states from H(Q), H(Q c ). There exists the question does it possible to select any physical vectors for the dynamical system with mixed-class constraints by means of intermediate procedure, with BRST-BFV operator Q for only firstclass constraints T A (Γ) imposed on such state, Q|ψ = 0, with |ψ ∈ H 0 (Q) together with special extension of the maximal subsystem of the first-class constraints θᾱ(Γ) from Θ α (Γ): θᾱ(Γ)|ψ = 0? We consider the solution for this problem within application of BRST-BFV approach for the construction of the unconstrained and constrained Lagrangian formulations for HS fields given on R 1,d−1 .

BRST operator for HS symmetry superalgebra for mixed-symmetric fermionic fields
In this section, we shortly repeating a basic results of our research [68] consider a halfinteger spin irreducible representation of Poincare group in a Minkowski space R 1,d−1 within metric-like formalism which is to be described by a spin-tensor field: , with the Dirac index A (later suppressed) of rank k j=1 n j and the generalized spin s = (n 1 + 1/2, , which corresponds to a Young tableaux Y (s 1 , s 2 , . . . , s k ) with k rows of length n 1 , n 2 , . . . , n k , respectively, The field is symmetric with respect to permutations of each type of Lorentz indices µ i and obeys, respectively, to the Dirac, gamma-tracelessness and mixed-symmetry equations [for and, in addition for any neighbour rows with equal length: n i = n i+1 = . . . = n i+m , m = 1, . . . , k − i the field should be identical with respect to permutation of any two groups

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from the total group of the respective indices (µ i ) n i , (µ i+1 ) n i+1 , . . . , (µ i+m ) n i+m . The underlined figure bracket in (3.4) denotes that the indices included in it do not take part in symmetrization, which thus concerns only the indices ( Equivalently, in terms of the general state (being by a Dirac-like spinor) of the Fock space, H, generated by the k bosonic pairs of creation and annihilation operators a providing the symmetry property of Ψ (µ 1 )n 1 ,(µ 2 )n 2 ,...,(µ k )n k the set of the relations (3.2)-(3.4) are equivalent for any spin s to set of ( 1 2 k(k + 1) + 1) operator equations: Adding to the relations (3.7), the generalized spin constraints imposed on |Ψ in terms of number particle operator g i 0 : the irreducible massless of spin s = n + 1 2 Poincare group representation is equivalently to the eqs. (3.2)-(3.4) given by (3.7)-(3.9).
The bosonic character of the primary constraint operatorst 0 ,t i , ε(t 0 ) = ε(t i ) = 0 does not permit to obtain the second-order operator l 0 = ∂ µ ∂ µ in terms of a commutator constructed fromt 0 , l 0 , due to ε(γ µ ) = ε(a µ i ) = 0 (see footnote 5 in [68]). These operators are transformed into fermionic ones (originally proposed in [23], eqs. (16)- (19) for d = 2N , and partially following ref. [22]) by means of certain d + 1 Grassmann-odd gamma-matrix-like objectsγ µ ,γ, ε(γ µ ) = .ε(γ) = 1, whose explicit realization differs in even, d = 2N , and odd, d = 2N + 1, N ∈ N dimensions. In the former case, we have the following definition [22]: with a non-trivial realization forγ, where the completely antisymmetric Levi-Civita tensor ǫ µ 1 ...µ d is normalized by ǫ 01...d−1 = 1, and the odd unit matrix Π: Π 2 = 1, 1 changes the Grassmann parity of columns (or JHEP09(2018)104 In odd space-time dimensions the matrix γ d+1 is trivial, but the definition ofγ in (3.11) remains valid under the following modification of the commutation relations (3.10): In both cases, we have a realization of the Clifford algebra for Grassmann-odd gammamatrix-like objectsγ µ in R 1,d−1 . 2 The primary Grassman-odd constraints result in The set of primary constraints {t 0 , t i , t ij , g i 0 } is closed with respect to the [ , ]-multiplication if we add to them divergentless, l i , traceless, l ij , i ≤ j and D'alamber operators, l 0 : but for the reality of the Lagrangian we need, in addition, a closedness with respect to the appropriate hermitian conjugation defined by means of odd scalar product in H: that means the set: composes the first-class constraints subsystem. As a result, the superalgebra A f (Y (k), R 1,d−1 ), known as the half-integer HS symmetry algebra in Minkowski space with a Young tableaux having k rows [68] and containing the extra operators  Table 1. Even-even, odd-even and odd-odd parts of HS symmetry superalgebra these indices are raised and lowered using the Euclidian metric tensors δ ij , δ ij , δ i j . The quantity given explicitly by the relations of table 1 (for details, see [67,68]). Representing the basis elements of A f (Y (k), R 1,d−1 ), with allowance for the definitions (3.9), (3.13), (3.14), (3.16) and in agreement with (3.17), as follows:

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the table of basis [ , }-products can be presented as a Lie superalgebra: From the Hamiltonian analysis of dynamical systems it follows the operators o a , o + a are respective operator-valued 2k 2 bosonic and 2k fermionic secondclass, as well as o A are (2k + 1) bosonic and 1 fermionic first-class constraint subsystems among {o I } for a topological gauge system (one with vanishing Hamiltonian H 0 ≡ 0) . The symmetry properties of the real constants f c ab , f C AB , f C aB (being another than general ones in (2.1) , (2.2), (2.3) of the section 2) are described in [68]. We only remind the quantities ∆ ab (g i 0 ) form a non-degenerate (k × k; k 2 × k 2 ) supermatrix, ∆ ab , in the Fock space H on the surface Σ ⊂ H: ∆ ab |Σ = 0, which is determined by the equations (o a , t 0 , l 0 , l i )|Ψ = 0. The only operators g i 0 are not the constraints in H, due to eqs. (3.9), but they select a vector from |Ψ with fixed value of spin. The second-class constraints o a , o + a have already splitted on 2 first-class constraints subsystems due to (3.22).
In [68] it is shown the subsuperalgebra of {o a , o + a , g i 0 } is isomorphic to orthosymplectic superalgebra osp(1|2k), thus realizing the generalization of Howe duality [91,92] among whole the set of unitary half-integer HS representations of Lorentz subalgebra so(1, d − 1) and osp(1|2k). The rest elements isometries, which has the form of direct sum: [68] with account of (3.20) according to general formula [25][26][27][28] (with theĈP-ordering 3 ): (3.25) for ghost coordinates, C I , and momenta, P I from the minimal sector, subject to [C I , P J } = δ I J as for Q = Ω c| min (Γ,Γ gh|m ) in (2.37), but for ζ α ≡ 0 and with g i 0 considered as the JHEP09 (2018)104 constraints. Explicitly, it has the form, Here, the set of ghost operators with the Grassmann parity and ghost number given according to (3.25) respectively for the elements and their conjugated ghost momenta P I (composing Wick pairs of ghost operators for the {o I } \ {t 0 , l 0 , g i 0 } constraints) with the same properties as those for C I in (3.27) with the only nonvanishing commutation relations for bosonic and the anticommutation ones for fermionic ghosts By construction the property gh(Q ′ ) = 1 holds, whereas the Hermitian conjugation of zero-mode pairs: provides for Q ′ to be hermitian (Q ′ ) + = Q ′ .

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Decomposing Q ′ in powers of zero-mode ghosts η i g , P i g corresponding to the operators g i 0 not entered into the first-class {o A } and second-class {o a , o + a } subsystems of constraints where the operator σ(g) = (σ 1 , σ 2 , . . . , σ k )(g) playing the role of "ancient" of generalized spin operator σ [68] is Hermitian, (σ i ) + = σ i , as well as Q: Q + = Q. From the nilpotency condition for Q ′ it follows the operator equations in powers of η i g , P i g : where the both the operator nilpotency of the operator Q and its weak nilpotency on some subspacesH tot|N of a total Hilbert space H tot = H ⊗ H gh|m can not be realized, except for the case: σ i (g)H

Unconstrained Lagrangian formulation
To construct Lagrangian formulation it is impossible to use BRST operator Q ′ for the initial HS superalgebra A f (Y (k), R 1,d−1 ) of the elements {o I }. Instead, the additively con-

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verted first-class constraints, O I : as well as, so that after oscillator realization over the Heisenberg-Weyl superalgebra, parameter- = δ qr δ ts for even. Important for further construction additional number particle operators, g ′i 0 , contain real-valued quantities h i , i = 1, . . . , k being by the arbitrary dimensionless constants, introduced in the process of generalized Verma module construction [68], which permits to make the generalized spin equations for proper eigen-vectors in H c|tot : with spectra of proper eigen-values for h i . On the Hilbert subspace H σ(G) c|tot|N from H c|tot consisting from the proper eigen-vectors satisfying to the equations (3.38) the BRST-BFV operator Q (3.33), but for converted first-class constraints, O I , without converted number particle operators G i 0 = g i 0 + g ′i 0 is nilpotent. Namely, the latter operator generates the right BRST complex, whose cohomology in Hilbert subspace H σ(G)k c|tot|N with respective ghost number values spectrum, starting from, k = 0 generates correct Lagrangian dynamic for initial spin-tensor with appropriate set of auxiliary spin-tensor fields.
The unconstrained gauge-invariant Lagrangian formulation for HS field Ψ (µ 1 )n 1 ,(µ 2 )n 2 ,...,(µ k )n k is derived from the sequence of the equations, following from BRST-like equations for (3.31) and vector |χ with choice the standard representation in H c|tot : the following spectral problem is derived (3.40) imply, for instance, for (n 0 ) c and (n) ij , the sets of indices (n 0 1 , . . . , n 0 k ) and (n 11 , . . . , n 1k , . . . , n k1 , . . . , n kk ). The above sum is taken over n b0 , n ae , n bg , h l , n ij , p rs , running from 0 to infinity, and over the remaining n's from 0 to 1. Thus, the physical state |χ 0 for the vanishing of all auxiliary operators B a+ and ghost variables The middle set of equations (3.41)-(3.44), has the general solution for the set of proper eigenvectors |χ 0 (n) k , |χ 1 (n) k , . . ., |χ s (n) k , with (n) k = (s) k − 1 2 , and the set of the corresponding eigenvalues for values of the parameters h i , so that the vector |χ 0 (n) k contains the physical field (3.5) and all of its auxiliary fields. Explicitly, the spin and ghost number gh distributions are given by the respective relations being valid for a general case of HS fields subject to Y (s 1 , . . . , s k ) and for the subset of "ghost" numbers in (3.40) and (3.5) for fixed values of n i for |χ l (n) k , l = 0, . . . , k o=1 n o + k(k − 1)/2: for s = k o=1 n o + k(k − 1)/2 with nilpotent operator Q (n) k on any |χ m (n) k , m = 0, . . . , s: Q 2 (n) k |χ m (n) k = 0. The corresponding formal BRST-like (as for integer HS fields [67]) gauge-invariant action ) first, contains second order derivative l 0 , second the operator K ′ realizing hermitian conjugation for additional parts to the constraints o ′ I . 4 The dependence on L 0 , η 0 from the BRST operator Q (n) k (3.33) and from the whole set of the vectors |χ l (n) k maybe removed by means of partial gauge-fixing procedure, based on the extraction of the zero-mode ghosts q 0 , η 0 from Q: (1 + δ ln )ϑ + lm η l+ n P mn + n,l<m (1 + δ mn )ϑ + lm η m n P +ln + 1 2 l<m,n≤m η + nm η n l λ lm

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As the results of gauge-fixing procedure with help of the equations of motion and the set of gauge transformations (3.49) all the components in powers of q 0 , η 0 vector are removed except for two fields for each level, |χ (s(0) 0 (n) k , |χ (s(1) 0 (n) k , for s = 0, . . . , k o=1 n o + k(k − 1)/2. Therefore, the representation holds: which leads to the first-order independent equations of motion for the rest vectors which follows from the Lagrangian action with help of supermatrix multiplication δ |χ for s = 0, 1, . . . , s max = k o=1 n o + k(k − 1)/2 with a finite number of reducibility stage to be equal to (s max − 1).
Now, we have all the tools to pass to construction of the constrained BRST-BFV approach for the same initial fermionic HS field.

Derivation of the constrained BRST operator, spin operator and BRST-extended algebraic constraints
Here, we consider the same main objects as for unconstrained BRST approach but with extracted set of second-class constraints o a , o + a together with g i 0 (related to ∆ ab (g 0 ) in (3.21)) from the HS symmetry superalgebra A f (Y (k), R 1,d−1 ), which should be imposed on H tot in compatible way with the constrained BRST and spin operator for the superalgebra of the rest first-class constraints o A with space-time derivatives only.

Reduction from unconstrained BRST operator
Starting from the representation (3.31)-(3.33) for BRST-BFV operator Q ′ (3.26) written for the converted constraints O I , let us extract from Q ′ (hence from Q, σ i (G)) the terms corresponding to the Minkowski space R 1,d−1 isometries subsuperalgebra (being the ideal in A f (Y (k), R 1,d−1 )) of the first-class constraints subsystem {o A } and the summands corresponding to superalgebra isomorphic to osp(1|2k) of the second-class constraints {O a , O + a } only: 5 5 In this section and later on, we understand the subscript "C" in σ i C (G), QC (oA), |χ l c in the sense that the corresponding objects belong to the constrained BRST-BFV approach unless stated otherwise.

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and respective Hermitian conjugated constraints O + a . The set of the constraints {T i , L lm , T rs ; T + i , L + lm , T + rs } augmented by the constrained spin operator σ i c (G) satisfies to the same algebraic relations as the superalgebra of {o a , o + a ; g i 0 } given by ; σ i C (g) acting on H ⊗ H gh|m holds. We will call the set of the constraints {O a , O + a } (B=B + =0) as the BRST extended constraints. From the supercommutator's relations it follows that algebraically independent set for a half of the BRST extended constraints, O a is composed from k elements: (4.8) Indeed, the rest (dependent) elements {O a } from {O a } = {Oǎ, O a } are generated as follows: The nilpotency of unconstrained BRST-BFV operator Q ′ for A f (Y (k), R 1,d−1 ) with following from it equations on unconstrained operators Q, σ i (g) (3.35), but in H tot , leads

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to the nontrivial equations for extracted operators (4.1)-(4.5): Finally, from the left-hand side of the last equations in (4.12) at the first degree in ghost operator C α we have respectively for O a (B=B + =0) = (T j , L lm , T rs ). Therefore, the operators Q C , O a (B=B + =0) , , gh |ψ c = (0, 0, 0), |ψ c ∈ ker Q C Im Q C (4.18) = |ψ | σ i (G), gh |ψ = (0, 0), |ψ ∈ ker Q Im Q . Equivalently, in terms of the respective Q-and Q C -complexes the equivalence above means that the found set of states, H σ i C Oa,o A may be presented according to (4.16), (4.17) as: Corollary 3 represents the basis for the equivalent description of the Lagrangian dynamics for free half-integer HS field on R 1,d−1 subject to Y (s 1 , . . . , s k ) both on a base of unconstrained BRST method and in terms of constrained gauge-invariant Lagrangian formulation with use of constrained BRST approach. Note, earlier the Lagrangian dynamics for free half-integer HS field was developed only within unconstrained BRST method. Because of, HS symmetry algebra A(Y (k), R 1,d−1 ) [67] for free integer HS field on R 1,d−1 subject to Y (s 1 , . . . , s k ), in fact, contains in the A f (Y (k), R 1,d−1 ) the Corollary 3 solves the same problem for free integer HS field.
Hence, in order to prove that the set of solutions for the irreducibility representations conditions (3.2)-(3.4) [or in the form (3.7), (3.9)] for half-integer HS fields of fixed spin is equivalent to the Lagrangian equations of motion for more wider set of HS fields to be subject to the reducible gauge transformations, it is enough to consider the constrained gauge-invariant Lagrangian formulation.

Self-consistent constrained BRST, spin operators, off-shell constraints from HS symmetry superalgebra
The constrained operator quantities Q C , σ i C (g) and algebraically independent BRST extended constraints {Oā} (4.8) may be derived without appealing to the unconstrained
Starting from the Poincare group irreducibility conditions (3.2)-(3.4) extracting the massless HS field of spin s = (n 1 +1/2, n 2 +1/2, . . . , n k +1/2), realized equivalently for Fock space vector (3.5) by the equations (3.7), (3.9), we consider only the subsuperalgebra of Minkowski space R 1,d−1 isometries (3.23) {o A } = {t 0 , l 0 , l i , l + i } as the dynamical constraints being used to construct Lagrangian formulation, whereas the rest (algebraic) primary constraints t i , t rs (3.13), (3.8) should be imposed as off-shell (i.e. holonomic) constraints on the respective solutions of the Lagrangian equations of motion, but in a consistent way. From the algebraic viewpoint, the set of operators t i , t rs appears by the derivations of the subalgebra {o A } : [A, o A } ∈ {o A } for any A ∈ {t i , t rs }. The constraints t i , t rs itself generates the superalgebra of total set of constraints {o a } = {t i , t rs , l lm } being by the restriction of O a (4.9) on the initial Fock space H. The algebraically independent set of {o a } according to (4.8), (4.9) is given by: The constrained BRST operator Q C for the system of the first-class constraints {o A } in the Hilbert space H C , H C = H ⊗ H o A gh has the form (4.4). The consistency condition requires that the set of off-shell constraints t i , t rs as well as the number particle operator being additively extended in powers of the ghosts variables C A , P A in H C to T i , T rs and σ i C (g): with preservation of the vanishing ghost number: gh T i , T lm , σ i C (g) = (0, 0, 0) should satisfy to the equations which we call by the generating equations for superalgebra of the constrained BRST, Q C spin operators σ i C (g) and extended in H C off-shell constraints T i , T rs . Explicit calculations show, that the solutions for unknown σ i C (g), T i , T rs exists for quadratic in powers of ghosts C A , P A approximation in the form: Comparison with the structure of the BRST extended constraints T i , T rs (4.6) and σ i C (G) (4.1) restricted to H C , permits to state on their coincidence (for h i = 0 in case Therefore, the superalgebra of the constrained operators {Q C , T i , T rs , σ i C (g)} derived in self-consistent way coincides with the superalgebra {Q C , T i , T rs , σ i C (g) h i =0 } of the operators derived from unconstrained BRST formulation.

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As to the difference of the constrained spin operators: self-consistent, σ i C (g), and σ i C (g) then because of the coincidence the set of proper eigen-vectors, |χ l c (n) k ∈ H C , (gh(|χ l c ) = −l) (e.g. due to its supercommutativity: [ σ i C (g), σ i C (g)} = 0) the corresponding equations: with stabilized h i C ≡ h C , ∀i = 1, . . . , k, in fact, determine identical spin value distribution: for |χ c having the representation following from (3.40) for B + = 0, (C a , P a ) = 0 and for

Constrained gauge-invariant Lagrangian formulations
In this section. we consider the construction of constrained gauge-invariant Lagrangian formulations for mixed-symmetric HS fields in case of massless field with fixed generalized half-integer spin, then in case of one with fixed generalized integer spin, and in case of massive fields.

Constrained Lagrangian formulation for half-integer HS fields
From the nilpotent BRST operator Q C , spin operator σ i C (σ i C ≡ σ i C (g)), algebraically independent BRST extended constraints T i , T rs , i, r, s = 1, . . . , k, r < s we have the JHEP09(2018)104 spectral problem, analogous to one for unconstrained case (3.41)-(3.44), but for |χ l c ∈ H l C :  The corresponding BRST-like constrained gauge-invariant action from which follows the equations of motion Q C |χ 0 c (n) k = 0, contains second order operator l 0 , but less terms in comparison with its unconstrained analog S (2) (n) k (3.50). Again, repeating the procedure of the removing the dependence on l 0 , η 0 , q 0 from the BRST operator Q C (4.4) and from the whole set of the vectors |χ l c (n) k as it was done in the section 3.1 by means of partial gauge-fixing we come to the: Statement 4. The first-order constrained gauge-invariant Lagrangian formulation for half-integer HS field, Ψ (µ 1 )n 1 ,...,(µ k )n k (x) with generalized spin (s) k = (n) k + ( 1 2 , . . . , 1 2 ), is determined by the action, δ |χ = 0, m = 0, 1) with off-shell algebraically independent BRST-extended constraints imposed on the whole set of field and gauge parameters: The proof is based on the extraction of the zero-mode ghosts q 0 , η 0 from Q C and |χ l c (n) k : ≡ 0) which lead after gauge-fixing procedure with help of the equations of motion and the set of gauge transformations (5.6) to removing of all the components in powers of q 0 , η 0 vector except for two fields for each level, l = 0, . . . , k. As the result, the representation |χ l (n) k = |χ which in terms of q 0 -independent vectors is written as: . (5.16) and To get (5.17) from (5.15) we have used the last equations in the first row of (5.16) and operator identity, The solution for the gamma-traceless equations (5.16) leads to η + i -independence of |χ l(1) 0|c (n) k and, at most, linear in η + i -dependence of |χ l(1) 0|c (n) k , due to algebraic consequences from the second and third equations in (5.16): P i P j |χ l(0) 0|c = 0 and (P i ) 2 = 0.

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Consider, now the half of the gauge transformations and equations of motion (for l = −1): (n) k , l = −1, 0, . . . , k − 1, (5.18) and apply to them from the left the operator P i . Because of, η + i -independence of |χ where, e = 0, 1; l = 0, . . . , k − 1; 1 2 appears by unit 2 × 2 matrix. For massive case this theorem is proved for unconstrained BRST Lagrangian formulation in [68] by explicit resolution of BRST Q-complex within first order Lagrangian formulation. Here we reached the equivalence in question for massless case without above JHEP09(2018)104 procedure but with using the results of study the structure of the physical states for topological dynamical system subject to the first and second-class constraints on a base of BRST-BFV approach.
There are some consequences from suggested construction. First, the constrained BRST Lagrangian formulation due to nilpotency of Q C on H C without spin operator imposing may be used to determine Lagrangian formulation for so called fermionic HS fields with continuous spin (i.e. with not restricted by the spin constraints set of spin-tensor fields), suggested for totally-symmetric integer HS fields in d = 4 in [93]. Second, without offshell constraints (5.10) imposing, but with generalized spin condition (given by the middle set in (5.1)-(5.3)) we will have so-called generalized triplet formulation for half-integer HS fields on Minkovski space-time with many auxiliary spin-tensors with different generalized spin values. Third, including of the part BRST-extended constraints, corresponding to the gamma-trace constraints: T i |χ l(m) c (n) k = 0, (with obvious resolution the mixed-symmetric constraints T rs for whole set of field and gauge parameters |χ l(m) c (n) k ) into Lagrangian dynamics with help of additional gauge transformations and Lagrangian multipliers permits to construct so-called generalized quartet formulation for half-integer HS field with spin (s) k = (n) k + (1/2) k , suggested for totally-symmetric integer and half-integer HS fields in [71,72].

Constrained Lagrangian formulation for integer HS fields
Another direct consequence from the procedure of constrained Lagrangian formulation for half-integer HS fields concerns, the equivalence among the unconstrained BRST Lagrangian formulation developed in [66,67] and constrained BRST Lagrangian formulations for the integer HS irreducible representations of the Poincare group in Minkowski spacetime with fixed generalized spin (s) k = (s 1 , s 2 , . . . , s k ), s i ≥ s j , i > j. and constrained BRST Lagrangian formulation, in fact, suggested here in full details (considered without spin operator for totally-symmetric case in [76,77] and for mixed-symmetric case in [78]).
Indeed, the HS symmetry algebra A(Y (k), R 1,d−1 ) for integer HS fields in R 1,d−1 [67] (described by tensor fields, Φ (µ 1 )s 1 ,...,(µ k )s k (x) subject to the d'Alamber, traceless and divergentless equations:   and, thus, determining the gauge theory of [k(k + 1) − 1]-stage of reducibility. Here, the standard Grassman-even scalar product: χ B |φ B , for Lorentz-scalar vectors is used, instead of Grassman-odd scalar product and Lorentz-spinor vectors for the fermionic HS fields. The operator K B is reduced from K and explicitly given in [67] (see for K B′ denoted in [67] as K ′ eq. (3.14), as well as for the BRST operator Q B eq. (5.2)). The form of the field (l = 0) and gauge parameters (l = 1, . . . , k(k + 1)) |χ l B (s) k , gh(|χ l B ) = −l is determined accordingly to (3.40) for n 0 c = n b0 = n ae = n bg = 0 with component vector |Φ(a + i ) ... ) being by the proper eigen-states for generalized spin operator σ i so that the vector |χ 0 B (n) k contains the physical field |Φ . In the corresponding spin (3.47) and ghost number (3.48) distributions for |χ l B (s) k one should put n 0 c = n b0 = n ae = n bg = 0 and n i = s i .
Note, the Statements 1, 2, 3 are valid for the integer HS field case. The constrained BRST Lagrangian formulation for integer HS field in Minkowski space with generalized spin (s) k may be derived equivalently both in the self-consistent way and from the unconstrained BRST Lagrangian formulation above as it was done respectively in the subsection 4.2 and subsection 4.1. The final result for constrained BRST Lagrangian formulation for integer HS field under consideration can be determined by the Statement 5. The constrained gauge-invariant Lagrangian formulation for integer HS field, Φ (µ 1 )s 1 ,...,(µ k )s k (x) with generalized spin (s) k , is determined by the action and sequence of reducible gauge transformation,

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The set of |χ l c|B (s) k is determined by the decomposition (4.29) for n b0 ≡ 0 and appears by the set of proper eigen-states for the constrained spin operator σ i c|B ≡ σ i c with proper eigen-values: which determine the same spin and ghost number grading (4.28), (4.30) [but for n b0 ≡ 0] as for the constrained half-integer Lagrangian formulation. The constrained and unconstrained Lagrangian formulations for integer HS field in Minkowski space-time with generalized spin (s) k are equivalent, but former one contains less auxiliary HS fields as compared to latter formulation. Concluding, the subsection, note the same comments in the end of the previous subsection, concerning bosonic HS fields with continuous spin, generalized triplet formulation and generalized quartet formulation are valid as well.
Let us now shortly consider the Lagrangian formulations for the massive HS fields in Minkowski space-time subject to Young diagram with k rows.

On constrained Lagrangian formulations for massive fields
The unconstrained BRST Lagrangian formulations for massive half-integer and massive integer HS fields in d-dimensional Minkowski space-time with generalized respective spins, (n) k + ( 1 2 ) k , (s) k were elaborated in [67,68] (for k = 1 and example, see [94,95]). It was done on a base of the derivation of the HS symmetry superalgebra A f m (Y (k), R 1,d−1 ) for massive half-integer HS fields in R 1,d−1 and HS symmetry algebra A m (Y (k), R 1,d−1 ) for massive integer HS fields from respective HS symmetry superalgebra A f (Y (k), R 1,d ) and algebra A(Y (k), R 1,d ) for massless HS fields in R 1,d with help of the dimensional reduction (see subsections 3.3 in [67,68]).

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Secondly, the additional parts o ′ I (B, B + ) coincide with those of the massless case, whereas the converted set of constraints has the form O m I =ǒ I + o ′ I , no longer with a central chargẽ γm, for massive HS fields withǒ I = o I +ô I (b i , b + i ) (for additional bosonic 2k-oscillators [b i , b + j ] = δ ij acting in the Fock space H m ), determined by adding the terms induced by dimensional reduction: The set of O m I satisfies the same HS symmetry superalgebra A f (Y (k), R 1,d−1 ) as for massless half-integer Poincare group irreducible representations, but for even-valued d.
Note, the b + l -independent vectors |χ (n ′ ) l have the decomposition in powers of oscillators presented by (3.40).
The unconstrained Lagrangian formulation for massive half-integer HS field of spin (n) k + ( 1 2 ) k is determined for d = 2N , N ∈ N almost the same relations as for massless case (3.57), (3.58) with use of (3.51)-(3.53) for Q(O m ): where the operator K m|(n) k , which realizes the hermitian conjugation in H m c|tot : K m = 1 ⊗ K ′ ⊗ 1 m ⊗ 1 gh [with 1 m being by unit operator in H m ] as well as ∆Q = ∆Q(O m ) are obtained after substitution for of the proper eigen-values for real constants h i m from the spectrum problem for generalized spin equations, for e = 0, 1; s = 0, . . . , s max : as follows, K m|(n) k = K m h i m →h i m (n) . From the spin and ghost number distributions (3.47), (3.48) for massless HS field the only first one is modified as: (1 + δ ij )(n ij + n f ij + n pij ) + n f i + n pi + r<i (p ri + n f ri + n λri ) − r>i (p ir + n f ir + n λir ) , i = 1, . . . , k.

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In turn, omitting the details of the derivation of the constrained BRST Lagrangian formulation for massive half-integer HS field, Ψ (µ 1 )n 1 ,,...,(µ k )n k , of spin (n) k + ( 1 2 ) k , the final expressions for the gauge-invariant action, sequence of reducible (of the same stage reducibility as for massless case) gauge transformations and independent off-shell BRSTextended constraints are given by the expressions almost coinciding with (5.8)-(5.10), but in terms of the constrained operators and vectors on H m C , H m C = H C ⊗ H m : Here, the constrained field and gauge parameters are by the proper eigen-functions for the constrained massive spin operator, σ i m|C determined as in (4.1): which together with constrained BRST operator, Q m C : Q m C = Q C (ǒ A ) (4.4), constraints T m i , T m rs and algebraically dependent constraints L m ij : L m ij = L ij +l ij , forms the same closed superalgebra with respect to [ , }-multiplication as theirs analogs from massless case, namely, nilpotent Q m C supercommutes with any from T m i , T m rs , L m ij , σ i m|C , which satisfy to the relations in the table 1 for t i , l ij , t rs , g i 0 . Note, it is easy to establish again that both approaches: first, reduced from unconstrained BRST Lagrangian formulation and, second, obtained in self-consistent way; to derive the constrained BRST Lagrangian formulation for massive HS fields developed in the subsections 4.1, 4.2 are equivalent. The equivalence among the unconstrained and constrained BRST Lagrangian formulations for the same massive half-integer HS field of spin (n) k + ( 1 2 ) k follows, in fact, from the validity of the Statement 4 and Corollary 3 because of the constraints subsystem,ǒ A = {ť 0 ,ľ 0 ,ľ i ,ľ + i } appears by the first-class ones in The problem of derivation of the (un)constrained Lagrangian formulation for massive half-integer HS fields in odd-valued dimensions, d = 2N + 1, may be solved within BRST-BFV approach by means of calculation of whole pairings for ghost C, P and auxiliary B, B + oscillators in (5.35) and (5.39) with use of the partial gauge-fixing procedure up to the representation of the respective Lagrangian formulations where the only initial γ µmatrices will survive, withoutγ object due to the property described in the footnote 2. The last Lagrangian formulations obtained for d = 2N can be extrapolated to be by
To be complete, we remind the form of unconstrained BRST Lagrangian formulation for massive integer HS field of spin (s) k in R 1,d−1 [67] and present new results for respective constrained BRST Lagrangian formulation. For the former case we ignore, first, spinor- ] structure for the operators and spinor structure for the states, and extracting from the massive HS symmetry superalgebra A f m (Y (k), R 1,d−1 ) the (2k + 1) Grassman-odd elements:ť 0 ,ť i ,ť + i , to get massive HS symmetry algebra A m (Y (k), R 1,d−1 ) because of having instead of the wave equation in (5.21) the Klein-Gordon equation and, thus, determining the gauge theory of [k(k + 1) − 1]-stage of reducibility. The form of the field (l = 0) and gauge parameters (l = 1, . . . , k(k + 1)) |χ l m|B (s) k , with the same Grassman and ghost number grading as ones for massless case is determined accordingly to (5.46) According to the Statement 5, the constrained gauge-invariant Lagrangian formulation for massive integer HS field, Φ (µ 1 )s 1 ,...,(µ k )s k (x) with generalized spin (s) k , is determined by the action and sequence of reducible gauge transformation,

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The set of |χ l m|c|B (s) k is determined by (4.29) and (5.34) for n b0 ≡ 0 and appears by the set of proper eigen-states for the massive constrained spin operator σ i m|C|B ≡ σ i m|C with proper eigen-values: Again, as well as for the massive half-integer HS fields from the dimensional reduction procedure it follows that both approaches: first, reduced from unconstrained BRST Lagrangian formulation and, second, obtained in self-consistent way; to get the constrained BRST Lagrangian formulation for massive HS fields developed in the subsections 4.1, 4.2 are equivalent. The equivalence among the unconstrained and constrained BRST Lagrangian formulations for the same massive integer HS field of spin (s) k follows, in fact, from the validity of the Statement 4 and Corollary 3 adapted to integer spin case because of the constraints subsystem,ǒ m A = {ľ 0 ,ľ i ,ľ + i } appears by the first-class ones in H m .
6 Example: spin (n + 1 2 ) totally-symmetric field Here, we, firstly, realize the general prescriptions of our constrained BRST-BFV Lagrangian formulations in the known case of totally-symmetric fermionic fields in the metric-like formulation.
The constrained Lagrangian formulation for the massive totally-symmetric spin-tensor may be obtained according to the recipe (5.39)-(5.41) of subsection 5.3. Note that the constrained Lagrangian formulation for such HS fields in constant-curvature spaces was considered using the metric-like formalism in ref. [96] beyond the BRST-BFV approach.

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with the only independent gauge parameter |ξ m|0 n−1 . In turn, the (ť 1 ) 3 -constraint (6.47) at a fixed degree in (b + ) k for k = 0, . . . , n is rewritten in an unfolded b + -independent form, with k = 0, . . . , n − 3 and the spin index omitted in |Ψ m|k n−k , From the gauge transformations for the field vectors |Ψ k n−k at a fixed degree in (b + ) k for k = 0, . . . , n, with allowance for (6.49), one can remove the field |Ψ n 0 by using the gauge parameter |ξ m|n−1 0 , so that a constraint appears for the independent gauge parameter |ξ m|0 n−1 : In the spin-tensor notation, the off-shell constraints and gauge transformations read formulations can be derived in spin-tensor forms for the massive spin-tensor field following the recipes for the above massless case with any d = 2N . In the massless case determined by the vanishing of all Ψ (ν) n−k m|k , k ≥ 1 with m = 0 in (6.56)-(6.60), we derive a Fang-Fronsdal Lagrangian formulation containing no dependence (!) on either odd or even values of d. Secondly, in view of the spin-tensor representation for the constrained Lagrangian and its gauge symmetries in the massive case not depending on the Grassmann-odd gammamatricesγ µ , we may use Lagrangian formulation (6.55)-(6.57), (6.59), (6.60) for d = 2N +1 dimensions as well, thus providing its applicability for any d ≥ 4. 8 Let us consider a massive field of spin s = 5/2, first examined in [97] for d = 4. In this case, the off-shell constraints for the independent gauge parameters and the spin-tensor fields Ψ mµν parameterizing the configuration space, look trivial for the fields, which is implied by (6.56), and have the representation (6.55) for ξ, albeit for arbitrary d (for d = 2N by construction and for d = 2N + 1 by extrapolation of the Lagrangian formulation (6.55)-(6.57), (6.59) from even to odd dimensions), Using the gauge transformations (6.57), we find where the constraint (6.62) has been resolved, with the gauge parameter ξ m|1 expressed in terms of the γ-trace of ξ µ m|0 . Using (6.65), we resolve the Stueckelberg gauge symmetries, thereby removing the field Ψ m|2 with a γ-traceless residual gauge parameter ξ µ m|0 , γ µ ξ µ m|0 = 0. We then decompose the field Ψ µ m|1 as follows: whence the spin-tensor Ψ µ m|1|⊥ is gauged away entirely by using the remaining degrees of freedom in ξ µ m|0 . Thus the theory of a massive field of spin 5/2 is described, in accordance with [96], by the non-gauge unconstrained fields Ψ µν m|0 , Ψ m|1|L whose dynamics is governed

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by an action which follows from (6.60) for n = 2 in the Fang-Fronsdal-like notation For an arbitrary massive spin-tensor field of spin n+1/2, we can derive from (6.59) or (6.60) a Lagrangian non-gauge description (equivalent to the gauge-invariant one) whose configuration space contains only the constrained fields (due to (6.56)) Ψ (µ)n m|0 , Ψ The derivations of the constrained Lagrangain formulations for massless and massive spintensor fields of spin s = n + 1/2 according to general prescriptions of the suggested constrained BRST-BFV approach presents the basic results of the subsection.

Conclusion
In this paper, we have developed a constrained BRST-BFV method to construct gaugeinvariant Lagrangian formulations for free massless and massive half-integer spin-tensor fields with an arbitrary fixed generalized spin s = (n 1 + 1/2, n 2 + 1/2, . . . , n k + 1/2), in Minkowski space-time R 1,d−1 of any dimension in the "metric-like" formulation. This result is presented in Statement 4 and explicitly includes, for the field Ψ (µ 1 )n 1 ,...,(µ k )n k , a first-order Lagrangian action S C|(n) k (5.8) invariant with respect to reducible gauge transformations (5.9) (which determine a gauge theory of (k − 1)-th stage of reducibility), and off-shell independent BRST-extended constraints T i , T rs (4.23), (4.25) imposed on the whole set of field (incorporating the initial spin-tensor Ψ (µ 1 )n 1 ,...,(µ k )n k ) and gauge parameter vectors (5.10) from the resultant Hilbert space H ⊗ H o A gh , whose vectors have the representation (4.29). The crucial point is that the superalgebra formed by a constrained BRST operator (only for the first-class constraint system o A with a subsuperalgebra of Minkowski space R 1,d−1 isometries (3.23) in the HS symmetry superalgebra A f (Y (k), R 1,d−1 )), a generalized spin operator, and BRST extended second-class constraints {Q C , σ i C (g), T i , T rs , L ij } is closed with respect to the [ , }-multiplication in H ⊗ H o A gh and forms, with the exception of Q C and with an addition of T + i , T + rs , L + ij , an osp(1|2k) orthosymplectic superalgebra. This fact guarantees a common set of eigenstates in H ⊗ H o A gh , which depends on less ghost coordinates and momenta (hence, for a smaller set of auxiliary fields and gauge parameters) JHEP09(2018)104 than the ones in the unconstrained Lagrangian formulation (3.57), (3.58) [68] for the same spin-tensor field, and therefore also ensures the consistency of dynamics in the constrained formulation with holonomic off-shell constraints. It is shown in the Theorem (5.19), (5.20), on the basis of general results in operator quantization for dynamical systems with firstand second-class constraints, that the Lagrangian dynamics for the same element of an irreducible half-integer HS representation of Poincare group in R 1,d−1 subject to Y (s 1 , . . . , s k ) in the constrained and unconstrained BRST-BFV approaches are equivalent, i.e., for both dynamics equivalent to irreducibility conditions with a given spin (3.2)-(3.4). The equivalence of the constrained and unconstrained BRST-BFV methods in question is two-fold. First, it is based on derivation starting from the unconstrained HS symmetry superalgebra A f (Y (k), R 1,d−1 ) and respective BRST operators Q ′ (3.26) and Q (3.33), as one disregards the additional (due to conversion) oscillators B a , B a+ given by (4.1)-(4.7), and resulting in Statement 3 and Corollary 3 in (4.18), (4.19), or, equivalently, in terms of the Q-and Q C -complexes in (4.20), (4.21). Second, in a self-consistent form, the nilpotent constrained BRST operator Q C , the spin operators σ i C (g), and the off-shell BRST-extended constraints T i , T rs are derived explicitly in section 4.2 from the irreducibility conditions (3.2)-(3.4), on a basis of solving the generating equations (4.24). The constrained Lagrangian formulation is then obtained from the second-order one (5.5)-(5.7) by partial gauge-fixing, thereby eliminating the zero-mode ghost operators q 0 , η 0 , p 0 , P 0 for the Dirac and D'Alembert operators from the whole set of gauge parameters and field vectors, in a way compatible with off-shell BRST-extended constraints having the form (5.16).
As a byproduct, we have derived the constrained BRST-BFV Lagrangian formulation (5.25)-(5.27), for the integer mixed-symmetric HS field Φ (µ 1 )s 1 ,...,(µ k )s k in R 1,d−1 subject to the irreducibility conditions (5.21), from the unconstrained formulation, albeit with BRST-extended traceless off-shell constraints, L ij , instead of gamma-traceless ones, T i . The stages of reducibility for both integer and half-integer massless HS fields coincide and can be used as a starting point for constrained BRST-BFV Lagrangian formulations to accommodate SUSY models of HS fields.
It should be noted that the constrained BRST-BFV approach, as well as the unconstrained one, implies automatically a gauge-invariant Lagrangian description, reflecting the general fact of BV-BFV duality [98][99][100][101], which reproduces a Lagrangian action for the initial non-Lagrangian equations (reflecting the fact that the (spin)-tensor belongs to an irreducible representation space of the Poincare group) by means of a Hamiltonian object.
It is shown in section 5.3 that the case of massive half-integer and integer HS fields with a corresponding arbitrary Young tableaux Y (s 1 , . . . , s k ) allows one to obtain constrained gauge-invariant Lagrangian formulations, for fermionic (5.39)-(5.41), initially for even dimensions d, and bosonic (5.47)-(5.49) fields for any d, derived from the respective unconstrained formulations (5.35), (5.36) and (5.22), (5.23). This was achieved by dimensional reduction procedure for the respective massless HS symmetry (super)algebra, A (f ) (Y (k), R 1,d ) in R 1,d , to the massive one, A (f ) m (Y (k), R 1,d−1 ) in R 1,d−1 , which means conversion for the sets of differential constraints, being this time second-class constraints. In both cases, the constrained BRST operator for differential constraints, the generalized spin operators, and the BRST-extended off-shell constraints are modified by k pairs of Concluding, we present some ways of extending the results obtained in this paper. First, the development of a Lagrangian construction for tensor and spin-tensor fields with an arbitrary index symmetry in AdS space. Second, the derivation of constrained BRST-BFV Lagrangian formulations for irreducible representations of the SUSY Poincare supergroup along the lines of [102]. Third, the development of the constrained and unconstrained BRST-BV method to construct respective minimal field-antifield BV actions for half-integer HS fields in terms of Fock-space vectors. Fourth, the construction of a quantum action for HS fields within an N = 1 BRST approach, 9 where the space-time variables x µ are to be considered on equal footing with the total Fock space variables. Fifth, a consistent deformation of the (un)constrained BRST-BFV and BRST-BV approaches applied to bosonic and fermionic mixed-symmetric HS fields will make it possible to construct an interacting theory with mixed-symmetry fermionic HS fields, including the case of curved (AdS) backgrounds. We intend to carry out a study of these problems in our forthcoming works.

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The presentation for the generating functional of Green's functions (A.26), (A.27) in terms of BRST-invariant second-class constraints for the dynamical system in question and Statement 6 present the main results of the appendix. In case of topological system, i.e. for H 0 = 0, the last result is immediately corresponds to the case of constrained BRST-BFV approach for the Lagrangian formulations for HS fields in the section 5 without extracted from x µ time coordinate and with special presentation for the invertible supermatrix ∆ αβ (Γ,Γ gh|m ) (A.23) in terms of constrained generalized spin operator σ i C ≡ σ i C (g) (4.23), (4.25).
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