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

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Abstract

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.

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Notes

  1. For simplicity, we are considering mechanical systems. The description of fields can in principle be achieved by allowing the discrete indexes to take continuous values through a limiting process.

  2. We are assuming that the given function T(qk) satisfies

    $$ \frac{\partial T(q_{k})}{\partial q_{i}}\neq 0 $$

    for each \(i=1,\dots ,n\).

  3. We assume the order of the partial derivatives always commute, i.e., Tij = Tji.

  4. That prescription was actually originally discussed by Dirac in [3].

  5. This choice explicitly shows that the present generalization is not restricted to the form (60).

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Authors and Affiliations

  1. Departamento de Ciências Exatas e Naturais, Universidade Estadual do Sudoeste da Bahia, Rodovia BR 415, km 03, s/n - Itapetinga, Bahia, 45700-000, Brazil

    Suzicleide Lopes de Oliveira, Camila Messias Barbosa Santos & Ronaldo Thibes

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  1. Suzicleide Lopes de Oliveira
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  2. Camila Messias Barbosa Santos
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  3. Ronaldo Thibes
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Correspondence to Ronaldo Thibes.

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Lopes de Oliveira, S., Santos, C.M.B. & Thibes, R. First-Order Gauge-Invariant Generalization of the Quantum Rigid Rotor. Braz J Phys 50, 480–488 (2020). https://doi.org/10.1007/s13538-020-00749-8

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