BRST-BFV and BRST-BV Lagrangians for Bosonic Fields with Continuous Spin on $R^{1,d-1}$

Gauge-invariant Lagrangian descriptions for a free bosonic scalar field of continuous spin in a $d$-dimensional Minkowski space-time using a metric-like formulation are constructed on the basis of a constrained BRST--BFV approach we propose. The resulting formulations contain different sets of auxiliary fields, depending on the manner of a partial gauge-fixing and a resolution of some of the equations of motion for a BRST-unfolded first-stage reducible gauge theory. To reach an equivalence of the resulting BRST-unfolded constrained Lagrangian equations of motion with the initial irreducible Poincare group conditions of a Bargmann--Wigner type, it is demonstrated that one should replace the field in these conditions by a class of gauge-equivalent configurations. Triplet-like, doublet-like and non-gauge constrained Lagrangian descriptions, as well as a quartet-like unconstrained Lagrangian formulation, are derived using Fronsdal fields, in particular. BRST--BV actions, in the minimal sector of the respective field and antifield configurations, are constructed in an explicit way.