Phase space of Jackiw-Teitelboim gravity with positive cosmological constant (2409.12943v2)

Abstract: In this paper we construct the classical phase space of Jackiw-Teitelboim gravity with positive cosmological constant on spatial slices with circle topology. This turns out to be somewhat more intricate than in the case of negative cosmological constant; this phase space has many singular points and is not even Hausdorff. Nonetheless, it admits a group-theoretic description which is quite amenable to quantization.

• The paper introduces a refined analysis of the classical phase space in JT gravity, emphasizing singular solutions and structural obstructions.
• It employs group-theoretical methods on cylindrical slices to classify timelike, spacelike, and null solutions in a non-Hausdorff context.
• The work proposes quantization strategies by constructing a robust symplectic form that accommodates gauge symmetries and singular configurations.

An Examination of the Phase Space of Jackiw-Teitelboim Gravity with Positive Cosmological Constant

The paper "Phase space of Jackiw-Teitelboim gravity with positive cosmological constant" by Alonso-Monsalve, Harlow, and Jefferson explores a refined approach to understanding classical solutions within Jackiw-Teitelboim (JT) gravity when considering a positive cosmological constant. The focus is on defining the classical phase space, identifying significant obstructions, and proposing quantization avenues that can address the unique challenges presented by this setup.

Overview

Jackiw-Teitelboim gravity is a model in 1+1 dimensions that provides a simplified context for exploring aspects of quantum gravity. In scenarios involving a positive cosmological constant, the paper argues that the analysis becomes complex due to a variety of singular solutions and unconventional topological properties of the phase space. This configuration starkly contrasts with negative cosmological constant scenarios, such as those associated with Anti-de Sitter spacetime.

Phase Space Analysis

The authors construct the phase space of the JT model by focusing on cylindrical spatial slices. They accommodate the complications arising from the singular nature of the phase space, which is non-Hausdorff in certain aspects, by leveraging group-theoretical methods. By doing so, they describe solutions using conjugacy classes of the universal cover of the Lorentz group $\widetilde{SO^+}(1,2)$, and introduce identifications through the adjoint and inverse actions. The framework allows the authors to classify solutions into timelike, spacelike, and null categories, with particular attention to topological equivalences.

Symplectic Structure

An essential part of their exposition involves the construction of a symplectic form on the phase space using the covariant phase space formalism. This symplectic structure must account for the intricacies introduced by gauge symmetries. The authors demonstrate how the complexity of solutions, particularly at singular points, requires special treatment to ensure the symplectic form remains well-defined and invertible.

Implications and Quantization

Blocking and gluing processes inherent to defining the JT model's phase space hold direct implications for both a deeper understanding of quantum gravity and cosmology more broadly. Particularly, understanding these gluings sheds light on non-local aspects of the theory such as diffeomorphism invariance and size modes which affect how wave functions behave across different universe states.

The paper anticipates that canonical quantization approaches to their constructed phase space, defined either directly through a cotangent bundle or by constraining group actions, would aid in realizing new quantum mechanical models. This quantization can facilitate further exploration of quantum cosmology and valid solutions in JT gravity that include embedding in larger, potentially realistic scenarios.

Concluding Remarks

While this paper does not attempt to solve the fundamental issues in quantum cosmology, it builds a robust framework that enriches the understanding of JT gravity under positive cosmological conditions. It opens pathways for further inquiry into the quantization of gravity, with ramifications for non-perturbative quantum mechanics and its application to larger cosmological models. Future work will likely expand on these findings by considering more complex manifolds and higher-dimensional analogs, potentially bringing us closer to a comprehensive quantum theory of gravity.