Group Theoretical Quantization and the Example of a Phase Space S^1 x R^+

Abstract

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^1 x R^+ (interpreted as restrictions of T^*R and T^*S^1 to positive coordinate and momentum, respectively). In this framework (and under an additional, natural condition) an SO_+(1,2)-action on S^1 x R^+ results as the unique admissible group action. 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^1 x R^+ 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.

41 pages, LaTeX2e

References in corpus (4)

• Group Theoretical Quantization of a Phase Space S^1 x R^+ and the Mass Spectrum of Schwarzschild Black Holes in D Space-Time Dimensions
• Refined Algebraic Quantization in the oscillator representation of SL(2,R)
• Symplectic Cuts and Projection Quantization
• An SL(2,R) Model of Constrained Systems: Algebraic Constraint Quantization

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• Group Theoretical Quantization of a Phase Space S^1 x R^+ and the Mass Spectrum of Schwarzschild Black Holes in D Space-Time Dimensions
• Quantum field theory and its symmetry reduction
• Refined Algebraic Quantization in the oscillator representation of SL(2,R)
• A New Look at the Quantum Mechanics of the Harmonic Oscillator
• Group averaging in the (p,q) oscillator representation of SL(2,R)
• Quantization and spacetime topology
• Quantization of the Zigzag Model
• Refined algebraic quantisation with the triangular subgroup of SL(2,R)
• Symplectic Cuts and Projection Quantization
• Quantum cutting and a Szegö limit theorem