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JT Gravity in de Sitter Space and the Problem of Time

Kanhu Kishore Nanda, Sunil Kumar Sake, Sandip P. Trivedi

TL;DR

The work develops a canonical quantisation of JT gravity in de Sitter space and confronts the problem of time by treating the dilaton as a physical clock. It constructs gauge-invariant states governed by a Klein-Gordon-like constraint, and introduces a conserved, finite KG norm along with a gauge-invariant probability density for the spatial length, $\,p(l,\phi)$. A central finding is that, although many states can satisfy the norm and classical-limit requirements, these requirements strongly restrict the state space, yet leave an infinite set of viable states, including diverse Gaussian and delta-function coefficient profiles. The results illuminate how classical cosmological behavior and decoherence between expanding/contracting branches can emerge in a simple quantum-gravitational setting, and they clarify the special role and limitations of the Hartle-Hawking state within this framework.

Abstract

We discuss the canonical quantisation of JT gravity in de Sitter space, following earlier work by Henneaux, with particular attention to the problem of time. Choosing the dilaton as the physical clock, we define a norm and operator expectation values for states and explore the classical limit. We find that requiring a conserved and finite norm and well-defined expectation values for operators imposes significant restrictions on states, as does the requirement of a classical limit. However, these requirements can all be met, with the dilaton providing a satisfactory physical clock. We construct several examples and analyse them in detail. We find that in fact an infinite number of states exist which meet the various conditions mentioned above.

JT Gravity in de Sitter Space and the Problem of Time

TL;DR

The work develops a canonical quantisation of JT gravity in de Sitter space and confronts the problem of time by treating the dilaton as a physical clock. It constructs gauge-invariant states governed by a Klein-Gordon-like constraint, and introduces a conserved, finite KG norm along with a gauge-invariant probability density for the spatial length, . A central finding is that, although many states can satisfy the norm and classical-limit requirements, these requirements strongly restrict the state space, yet leave an infinite set of viable states, including diverse Gaussian and delta-function coefficient profiles. The results illuminate how classical cosmological behavior and decoherence between expanding/contracting branches can emerge in a simple quantum-gravitational setting, and they clarify the special role and limitations of the Hartle-Hawking state within this framework.

Abstract

We discuss the canonical quantisation of JT gravity in de Sitter space, following earlier work by Henneaux, with particular attention to the problem of time. Choosing the dilaton as the physical clock, we define a norm and operator expectation values for states and explore the classical limit. We find that requiring a conserved and finite norm and well-defined expectation values for operators imposes significant restrictions on states, as does the requirement of a classical limit. However, these requirements can all be met, with the dilaton providing a satisfactory physical clock. We construct several examples and analyse them in detail. We find that in fact an infinite number of states exist which meet the various conditions mentioned above.
Paper Structure (42 sections, 382 equations, 5 figures)

This paper contains 42 sections, 382 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Canonical Quantization of JT gravity in dS space
  3. Classical Solutions
  4. Classical Phase space and A Symplectic Form
  5. Canonical Quantization
  6. Minisuperspace Quantization
  7. General Solutions
  8. Physical Properties and The Problem of Time
  9. Norm and Probability density
  10. Conservation of Norm
  11. Finiteness of Norm
  12. Positivity of Probability and Expectation Values of Operators
  13. Decoherence Between Branches
  14. Examples
  15. The Hartle-Hawking Wave Function and A Check
  16. ...and 27 more sections

Figures (5)

  • Figure 1: Penrose diagrams for (a) Global dS (b) Black hole in dS
  • Figure 2: Milne wedge and the horizons
  • Figure 3: Integration contour for M
  • Figure 4: We have taken $M_0=1$ in both the figures.(a)$\phi=0$, (b) $\phi=10$.
  • Figure 5: Contribution to the reduced density matrix of a single universe from the density matrix of the two-universe sector.