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

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 $t\to 0$ and $g\to g^{\mathrm{\prime }}$. This work provides the essential tool for deriving effective dynamics for $SU(3)$ gauge fields coupled to gravity via a coherent state path integral.