[Paper Review] Poisson Structure Induced Field Theories and Models of 1+1 Dimensional Gravity

Thomas Strobl|ArXiv.org|Nov 28, 2000

Black Holes and Theoretical Physics14 references3 citations

TL;DR

This paper introduces a unified framework for 1+1 dimensional gravity models using Poisson structures in the target space, showing that diverse gravity actions—such as Jackiw-Teitelboim, Katanaev-Volovich, and string-inspired models—can be derived from a single first-order action involving maps from spacetime to a Poisson manifold. The key contribution is the identification of a universal Poisson structure that generates these theories, enabling a consistent quantization procedure and resolving the issue of time in quantum gravity through gauge-fixing and Dirac observables.

ABSTRACT

PhD thesis TU-Vienna, May 1994. Table of Contents: 1. Introduction, 2. Poisson Structure Induced Two Dimensional Field Theories, 3. Models of Gravity in 1+1 Dimensions

Motivation & Objective

• To unify various 1+1 dimensional gravity models—such as Jackiw-Teitelboim, Katanaev-Volovich, and string-inspired actions—under a single geometric framework.
• To identify the underlying Poisson structure in the target space that generates the dynamics of these gravity models.
• To develop a consistent quantization procedure for these models by formulating them as Poisson structure-induced field theories.
• To address the 'problem of time' in quantum gravity by implementing gauge conditions that break diffeomorphism invariance while preserving physical consistency.
• To demonstrate that quantum wave functions can be constructed via Fourier transforms and operator ordering, yielding time-dependent Hamiltonians and physical states.

Proposed method

• Formulate a general first-order action: ∫(Aᵢ ∧ dXⁱ + ½Pⁱʲ(X)Aᵢ ∧ Aⱼ), where Xⁱ maps spacetime to a Poisson manifold with Poisson tensor P.
• Show that linear Poisson structures yield Yang-Mills-like theories, while quadratic structures reproduce known 2D gravity actions including the Katanaev-Volovich model.
• Implement gauge conditions such as X₊ = 1, X₃ = τ, and ∂e₁⁻ = 0 to break diffeomorphism invariance and define a physical time evolution.
• Use Fourier transformation of constraints to derive quantum wave functions in the B-polarization, ensuring consistency with Dirac observables.
• Apply operator ordering derived from Fourier transforms to avoid quantum anomalies and ensure nontrivial kernel of constraints.
• Construct time evolution via effective Hamiltonians, including terms like −(γ²/2)∫trE²dx¹ − β²π²τ²/p₂², showing time dependence in the quantum theory.

Experimental results

Research questions

• RQ1 Can diverse 1+1 dimensional gravity models be unified under a single geometric framework based on Poisson structures?
• RQ2 How does the Poisson tensor in the target space determine the dynamics of different 2D gravity actions?
• RQ3 What is the role of gauge-fixing in resolving the problem of time in quantum gravity?
• RQ4 How can consistent quantum wave functions be constructed in the presence of constraints and diffeomorphism invariance?
• RQ5 What is the structure of the quantum Hamiltonian and how does it evolve in time in the absence of a fixed background?

Key findings

• The general action ∫(Aᵢ ∧ dXⁱ + ½Pⁱʲ(X)Aᵢ ∧ Aⱼ) unifies multiple 2D gravity models, including Jackiw-Teitelboim and Katanaev-Volovich, via different choices of the Poisson tensor P.
• The quantum wave functions derived via Fourier transform of constraints, such as Ψ ∝ exp[i/ℏ ∫(E₃∂φ + ...)], are consistent with the full set of quantum constraints and yield nontrivial physical states.
• The effective Hamiltonian acting on wave functions includes a time-dependent term proportional to β²π²τ²/p₂², indicating explicit time evolution in the quantum theory.
• Operator ordering derived from Fourier transforms preserves the constraint algebra and avoids quantum anomalies, unlike alternative orderings that lead to an empty kernel.
• The gauge condition ∂e₁⁻ = 0 can be implemented as an operator condition in the B-polarization, though ∮e₁⁻ = const remains inadmissible in this representation.
• The limit β → 0 recovers the standard Yang-Mills theory on a cylinder, confirming consistency of the method with known results in the absence of gravity.