First-Order Gauge-Invariant Generalization of the Quantum Rigid Rotor

A first-order gauge-invariant formulation for the two-dimensional quantum rigid rotor is long known in the theoretical physics community as an isolated peculiar model. Parallel to that fact, the longstanding constraints abelianization problem, aiming at the conversion from second- to first-class systems for quantization purposes, has been approached a number of times in the literature with a handful of different forms and techniques and still continues to be a source of lively and interesting discussions. Connecting these two points, we develop a new systematic method for converting second-class systems to first-class ones, valid for a family of systems encompassing the quantum rigid rotor as a special case. In particular, the gauge invariance of the quantum rigid rotor is fully clarified and generalized in the context of arbitrary translations along the radial momentum direction. Our method differs substantially from previous ones as it does not rely neither on the introduction of new auxiliary variables nor on the a priori interpretation of the second-class constraints as coming from a gauge-fixing process.