Semiclassical limit of 4-dimensional spin foam models

We study the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states. We show that, in the semiclassical limit, the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed if the face spins do not correspond to a discrete geometry. When they do arise from a geometry, the amplitudes reduce to the exponential of $\mathrm{i}$ times the Regge action. Remarkably, the dependence on the Immirzi parameter disappears in this limit.

Figure 1

(a) Face $f$ of dual complex ${\Delta }^{*}$. (b) Subdivision of face $f$ into wedges. The arrows indicate starting point and orientation for wedge holonomies.

Figure 2

Variation of the group variable ${\mathbf{h}}_{fe}$ on the edge $fe$ in the interior of the face $f$.

Figure 3

Variation of the group variable ${\mathbf{g}}_{ev}$ on the segment $ev$ between faces.

Figure 4

The wedge holonomies $G_{e_{i}f}(f)$ have their starting and end points at the center of the face.

Figure 5

Choice of labelling at neighboring vertices $v$ and $v^{\prime }$.