Group theoretical quantization and the example of a phase space S¹×ℝ⁺

The group theoretical quantization scheme is reconsidered by means of elementary systems. Already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions. A systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed, two-dimensional prototypes of which are T*ℝ⁺ and S=S¹×ℝ⁺ (interpreted as restrictions of T*ℝ and T*S¹ to positive coordinate and momentum, respectively). In this framework (and under an additional, natural condition) an SO^↑ (1,2)-action on S results as the unique admissible group action. Furthermore, for symplectic manifolds which are (specific) parts of phase spaces with known quantum theory a simple "projection method" of quantization is formulated. For T*ℝ⁺ and S equivalent results to those of more established (but more involved) quantization schemes are obtained. The approach may be of interest, e.g., in attempts to quantize gravity theories where demanding nondegenerate metrics of a fixed signature imposes similar constraints.