Quantization of the Jackiw-Teitelboim model

TL;DR

This paper analyzes the Jackiw-Teitelboim model in its BF formulation to reveal how partial gauge fixings and quantization orders yield inequivalent quantum theories, even though the classical theory has a finite number of global degrees of freedom. By exploring both Dirac quantization and time-gauge/φ·n gauges, it demonstrates that a noncompact internal gauge group (SL(2,R)) induces superselection sectors and Virasoro-like structures, leading to distinct spectra for Dirac observables depending on the chosen quantization route. The work highlights the noncommutativity of reduction and quantization in a simple gravity model, clarifies the role of diffeomorphism-like generators, and exposes deep quantization ambiguities that bear on loop-quantization approaches in low dimensions. A notable aside is the null sector, which remains subtle and is addressed in a dedicated note, showing a winding-number observable that stays classical in the proposed treatment. Overall, the paper maps how different canonical and gauge-fixing choices influence the quantum theory, with implications for 2D gravity and noncompact gauge groups.

Abstract

We study the phase space structure of the Jackiw-Teitelboim model in its connection variables formulation where the gauge group of the field theory is given by local SL(2,R) (or SU(2) for the Euclidean model), i.e. the de Sitter group in two dimensions. In order to make the connection with two dimensional gravity explicit, a partial gauge fixing of the de Sitter symmetry can be introduced that reduces it to spacetime diffeomorphisms. This can be done in different ways. Having no local physical degrees of freedom, the reduced phase space of the model is finite dimensional. The simplicity of this gauge field theory allows for studying different avenues for quantization, which may use various (partial) gauge fixings. We show that reduction and quantization are noncommuting operations: the representation of basic variables as operators in a Hilbert space depend on the order chosen for the latter. Moreover, a representation that is natural in one case may not even be available in the other leading to inequivalent quantum theories.