Quantization of the Jackiw-Teitelboim model

TL;DR

This paper explores the phase space and quantization of the Jackiw-Teitelboim model, revealing that the order of reduction and quantization affects the resulting quantum theories due to their noncommuting nature.

Contribution

It demonstrates that reduction and quantization operations do not commute in the Jackiw-Teitelboim model, leading to inequivalent quantum representations depending on the order.

Findings

Reduction and quantization are noncommuting operations.

Different gauge fixings lead to different quantum representations.

Some natural representations in one approach are unavailable in others.

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.