[Paper Review] Can causal dynamical triangulations probe factor-ordering issues?

Rachel Maitra|arXiv (Cornell University)|Oct 12, 2009

Algebraic structures and combinatorial models1 references3 citations

TL;DR

This paper investigates whether causal dynamical triangulations (CDT) can resolve factor-ordering ambiguities in quantum cosmology by comparing CDT simulations of 3-volume fluctuations with path integral computations in Friedmann-Robertson-Walker (FRW) minisuperspace models using different factor-ordering prescriptions. By varying the path integral measure via a 2-parameter family of operator orderings, the study aims to identify which quantum cosmological measure best matches the non-symmetry-reduced CDT results, potentially revealing the correct continuum limit measure for full quantum gravity.

ABSTRACT

The causal dynamical triangulations (CDT) program has for the first time allowed for path-integral computation of correlation functions in full general relativity without symmetry reductions and taking into account Lorentzian signature. One of the most exciting recent results in CDT is the strong agreement of these computations with (minisuperspace) path integral calculations in quantum cosmology. Herein I will describe my current project to compute minisuperspace (Friedman-Robertson-Walker) path integrals with a range of different measures corresponding to various factor orderings of the Friedman-Robertson-Walker Hamiltonian. The aim is to compare with CDT results and ask whether CDT can shed light on factor-ordering ambiguities in quantum cosmology models.

Motivation & Objective

• To determine if CDT simulations can distinguish between different factor-ordering choices in FRW quantum cosmology.
• To compare CDT-generated 3-volume covariance matrices with quadratic approximations of FRW path integrals under varying factor-ordering prescriptions.
• To identify the path integral measure in minisuperspace quantum cosmology that best approximates the non-symmetry-reduced CDT result.
• To explore whether the best-fit factor ordering remains stable across increasing cosmological model complexity, suggesting a universal feature of full quantum gravity.
• To assess whether CDT can serve as a non-perturbative probe of measure ambiguities in canonical quantum gravity.

Proposed method

• Construct a 2-parameter family of factor-ordering prescriptions for the FRW Hamiltonian, corresponding to different path integral measures in minisuperspace.
• Compute the quadratic approximation of the FRW path integral for each measure in the 2-parameter family, using WKB analysis and short-time propagators.
• Convert scale factor correlators into spatial 3-volume correlators to match CDT data, which is computed in terms of 3-volume fluctuations.
• Perform Monte Carlo simulations in CDT to compute the 3-volume covariance matrix for a fixed number of 4-simplices, $N_4$.
• Systematically increase $N_4$ and determine the best-fit factor-ordering parameters $(i,k)_{N_4}$ at each stage to extrapolate to the continuum limit.
• Use finite-size scaling to identify the limiting $(i,k)$ that best matches the CDT covariance matrix in the $N_4 \to \infty$ limit.

Experimental results

Research questions

• RQ1 Which factor-ordering prescription in FRW quantum cosmology yields a path integral measure that best matches the 3-volume fluctuations computed in CDT simulations?
• RQ2 Does the best-fit factor ordering for the scale factor remain consistent across increasing model complexity, such as from FRW to Bianchi IX cosmologies?
• RQ3 Can CDT distinguish between different path integral measures arising from operator ordering ambiguities in minisuperspace models?
• RQ4 Does the continuum limit of CDT correspond to a specific factor ordering in FRW quantum cosmology, and if so, which one?
• RQ5 Is the factor-ordering dependence in the path integral measure detectable in the covariance matrix of 3-volume fluctuations?

Key findings

• The path integral measure dependence on factor ordering enters explicitly through the prefactor $\left(a''a'\right)^{\frac{1}{4}+\frac{k-i}{2}}$ in the WKB approximation, which affects propagators and correlators.
• The WKB fluctuation factor $\varphi_0(a) \propto \left|1 - \frac{\Lambda}{3}a^2\right|^{-1/4} \cdot a^{\frac{k-i}{2}}$ shows explicit dependence on the asymmetry parameter $k-i$, linking factor ordering to observable correlation functions.
• The study identifies a 2-parameter family of measures $[\mathcal{D}a]_{(i,k)}$ that can be systematically compared to CDT data, enabling a quantitative fit.
• By computing the FRW path integral to quadratic order and comparing to CDT data across increasing $N_4$, the method aims to stabilize at a unique $(i,k)$ in the continuum limit.
• The best-fit $(i,k)$ in the $N_4 \to \infty$ limit would represent the effective measure of full quantum gravity with only the scale factor remaining.
• The approach suggests that CDT may resolve the long-standing ambiguity in quantum cosmology by selecting a physically preferred factor ordering through non-perturbative comparison.