Abstract
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.
Highlights
BRST-BFV method to construct constrained Lagrangian formulations forreducible half-integer higher-spin Poincare group representations in Minkowski space is suggested
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 = (n1 + 1/2, n2 + 1/2, . . . , nk + 1/2), in Minkowski space-time R1,d−1 of any dimension in the “metric-like” formulation
This fact guarantees a common set of eigenstates in H ⊗ HgohA, which depends on less ghost coordinates and momenta than the ones in the unconstrained Lagrangian formulation (3.57), (3.58) [68] for the same spin-tensor field, and ensures the consistency of dynamics in the constrained formulation with holonomic off-shell constraints
Summary
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 on the interacting levels. The physical states in HΓ for the dynamical system with first - TA(Γ) and second-class Θα constraints, permitting for the latter the division on two sets with only first-class constraints, Θα(Γ) → Θ′α(Γ) = θα, θα subject to operator analog of the relations (2.43), maybe equivalently presented by nilpotent BRST-BFV operator Qc = CαΦα(Γc) + CATc|A(Γc) + ”more′′, Qc ≡ Ωc| min(Γc, Γgh|m) (2.37), constructed with respect to the system of converted first Tc|A(Γc) and second-class constraints Φα(Γc) in the Hilbert space H(Qc) = HΓ ⊗ Hζ ⊗ Hgh|m ⊗ H2|gh|m in the form: H1p,h2ys = |ψ | TA(Γ), θα(Γ) |ψ = (0, 0), |ψ ∈ HΓ (2.56). The non-trivial solution of the first equations in (3.35) , (3.36) requires choosing the representation in Htot and conversion the second-class constraints subsystem {oa, o+a } into first-class one
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