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

Abstract

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, Ô α  = (Ô a , Ô a + ), from a total super-algebra of constraints, second, in self-consistent way by means of finding BRST-extended initial off-shell algebraic constraints, Ô a . In both cases, the latter constraints supercommute on the constraint surface with constrained BRST operator Q C and spin operators σ C i . The closedness of the superalgebra {Q C , Ô a , σ C i } guarantees that the final gauge-invariant Lagrangian formulation is compatible with the off-shell algebraic constraints Ô a imposed on the field and gauge parameter vectors of the Hilbert space not depending from the ghosts and conversion auxiliary oscillators related to Ô a , 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.