[Paper Review] Graviton propagator asymptotics and the classical limit of ELPR/FK spin foam models

Aleksandar Miković, Marko Vojinović|arXiv (Cornell University)|Mar 8, 2011

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper investigates the classical limit of ELPR/FK spin foam models by analyzing the large-distance asymptotics of the graviton propagator via the Hartle-Hawking wavefunction. Using the stationary phase method, it shows that the standard vertex amplitude yields an incorrect $ S = O(1) $ scaling, leading to a $ r^4 $-law behavior instead of the desired $ r^{-2} $, but proposes a redefinition of the vertex amplitude to achieve the correct asymptotics.

ABSTRACT

We study the classical limit of the ELPR/FK spin foam models by analyzing the large-distance asymptotics of the corresponding graviton propagators. This is done by examining the large-spin asymptotics of the Hartle-Hawking wavefunction which is peaked around a classical flat spatial geometry. By using the stationary phase method we determine the wavefunction asymptotics. The obtained asymptotics does not give the desired large-distance asymptotics for the corresponding graviton propagator. However, we show that the ELPR/FK vertex amplitude can be redefined such that the corresponding Hartle-Hawking wavefunction gives the desired asymptotics for the graviton propagator.

Motivation & Objective

• To determine whether the ELPR/FK spin foam model yields the correct large-distance asymptotics for the graviton propagator in the classical limit.
• To assess whether the Hartle-Hawking wavefunction, derived from the spin foam state sum, exhibits the Gaussian asymptotic form required for a semiclassical limit.
• To identify the conditions under which the wavefunction asymptotics matches the form $ \Psi \approx \mathcal{A} \exp\left[-\frac{1}{2j_0} \sum_{a,b} \alpha_{ab}(j_a - j_0)(j_b - j_0) \right] $, which is necessary for the correct propagator behavior.
• To determine whether the vertex amplitude can be redefined such that the resulting wavefunction yields the desired $ r^{-2} $ large-distance behavior of the graviton propagator.

Proposed method

• The Hartle-Hawking wavefunction is constructed as the spin foam state sum with a boundary spin network, satisfying the Hamiltonian constraint.
• The large-spin asymptotics of the wavefunction are analyzed using the stationary phase method, focusing on critical points of the action derived from the spin foam amplitude.
• The Hessian matrix of the logarithm of the amplitude is computed to determine the width of the Gaussian asymptotics, with the Schur complement used to isolate the boundary spin dependence.
• The $ j_0 $-dependence of the effective matrix $ S $, which controls the propagator asymptotics, is analyzed using Theorem 1, which establishes the scaling of the Hessian determinant.
• The analysis compares the Lorentzian ELPR/FK model with the Euclidean case studied previously, highlighting differences in the role of insertion functions and boundary conditions.
• A redefinition of the vertex amplitude is proposed to modify the $ S $-matrix scaling from $ O(1) $ to $ O(1/j_0) $, ensuring the correct $ r^{-2} $ large-distance behavior.

Experimental results

Research questions

• RQ1 Does the standard ELPR/FK vertex amplitude lead to a graviton propagator with the correct large-distance asymptotics in the classical limit?
• RQ2 What is the scaling behavior of the effective matrix $ S $ in the Gaussian asymptotics of the Hartle-Hawking wavefunction, and does it match the required $ O(1/j_0) $ form?
• RQ3 Can the vertex amplitude be redefined such that the resulting wavefunction yields the correct $ r^{-2} $ large-distance behavior for the graviton propagator?
• RQ4 Why does the standard model yield $ S = O(1) $, and is this a generic feature of known spin foam models?
• RQ5 How does the asymptotic structure of the Lorentzian ELPR/FK model compare to the Euclidean case, particularly in terms of wavefunction Gaussianity and propagator behavior?

Key findings

• The standard ELPR/FK vertex amplitude leads to $ S = O(1) $, which results in a graviton propagator that behaves as $ r^4 $ at large distances, inconsistent with the classical $ r^{-2} $ behavior.
• The $ S = O(1) $ scaling is a generic feature of known spin foam models, arising from the asymptotic behavior of the vertex amplitude.
• The wavefunction asymptotics are Gaussian only if the Hessian matrix of the amplitude's logarithm leads to a Schur complement $ S $ of order $ O(1/j_0) $, which is not satisfied in the standard model.
• The analysis shows that the wavefunction does not satisfy the required Gaussian form $ \Psi \approx \mathcal{A} \exp\left[-\frac{1}{2j_0} \sum \alpha_{ab}(j_a - j_0)(j_b - j_0) \right] $ under standard vertex amplitudes.
• A redefinition of the vertex amplitude is proposed to achieve $ S = O(1/j_0) $, which would yield the correct $ r^{-2} $ large-distance asymptotics for the graviton propagator.
• The comparison with the Euclidean case confirms that the $ S = O(1) $ scaling is a common obstacle, and that the proposed redefinition is necessary to achieve the correct semiclassical limit.