Gauge invariant Lagrangian construction for massive higher spin fermionic fields

TL;DR

The paper develops a BRST-BFV, operator-based framework to construct gauge-invariant Lagrangians for massive half-integer higher-spin fermions in arbitrary dimensions. It encodes the irreducible spin-$s$ conditions as a closed set of constraints acting in an auxiliary Fock space, introduces a new representation of the constraint algebra with extra oscillators and a modified scalar product to ensure Hermitian consistency, and builds a BRST operator that yields a reducible gauge theory in which all constraints emerge from the equations of motion. The method is validated through explicit spin-1/2 and spin-3/2 examples, recovering the Dirac and Rarita–Schwinger Lagrangians (the latter after gauge fixing and elimination of auxiliaries). The approach avoids imposing off-shell constraints a priori and provides a systematic route to gauge-invariant Lagrangians for arbitrary half-integer spins, with potential extensions to AdS spaces and mixed-symmetry fields.

Abstract

We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincare group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case hep-th/0410215, the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.