The Semiclassical limit of $SU(3)$ Gauge Field Coherent States: Peakedness and Overlap Functions

TL;DR

This work extends Thiemann’s gauge-field coherent-state framework from SU(2) to SU(3) within loop quantum gravity by constructing SU(3) coherent states on graphs using the heat-kernel complexifier. The authors derive the edge states as a convergent sum over SU(3) irreps, perform a detailed asymptotic analysis in the semiclassical limit t→0, and recast the leading behavior in terms magnitude ⟨v⟩ and determinant η of the complexified group element gh^{-1}. They provide numerical verification of the peakedness of both the coherent states and their overlaps via MCMC and compute the leading order of the overlap amplitude in the combined limit t→0 and g→g', establishing a Gaussian-like kernel essential for a coherent-state path integral. The results furnish a foundational tool for semiclassical analyses of SU(3) gauge fields coupled to gravity, enabling effective dynamics and quantum-gravity corrections to non-Abelian gauge theories in a background-independent setting.

Abstract

By using the heat kernel method, we construct diffeomorphism-covariant coherent states for the $SU(3)$ gauge group. We numerically demonstrate that these states exhibit the required semiclassical properties in the semiclassical limit: the peakedness property of the probability distribution and the peakedness property of the overlap function. We also provide the leading order term of the overlap amplitude in the combined limit where $t \rightarrow 0$ and $g\rightarrow g'$. This work provides the essential tool for deriving effective dynamics for $SU(3)$ gauge fields coupled to gravity via a coherent state path integral.

Figures (2)

• Figure 1: MCMC sampling results of the peak position of the coherent state for a randomly chosen $H$ and $u$. The diagonal subplots show the marginal distribution of each $\delta v^a$ for $a = 1,2,\cdots, 8$. The lower triangle subplots show the covariance between any pair of $\delta v^a$. In each of these subplots, the 2-dimensional histogram of the sampled points is shown by the grey scale, the blue cross resepents the mean value of the samples, and the green ellipse represents the contour of the standard deviation at one sigma, two sigma, three sigma, and four sigma levels respectively. The results show that the mean values of all $\delta v^a$ parameters are close to zero, and the covariance between any pair is smaller than standard value for two sigma level. Thus, the results confirms that the probability density $p^t_g(h)$ is sharply peaked at $h=u$ in the semiclassical limit.
• Figure 2: MCMC sampling results of the peak position of the overlap function. The diagonal subplots show the marginal distribution of each $\delta v^a$ and $\delta w^a$ for $a = 1,2,\cdots, 8$. The lower triangle subplots show the covariance between any pair of $\delta v^a$ and $\delta w^a$. In each of these subplots, the 2-dimensional histogram of the sampled points is shown by the grey scale, the blue cross locates the mean value of the samples, and the green ellipse represents the contour of the standard deviation at one sigma, two sigma, three sigma, and four sigma levels respectively. The results show that the mean values of all $\delta v^a$ and $\delta w^a$ parameters are close to zero, and the covariance between any pair is smaller than standard value for two sigma level. Thus, the results confirms that the overlap function is sharply peaked at $g=g'$ in the semiclassical limit.