Emergence of scalar matter from spin foam model

A spin foam model of three dimensional gravity nonminimally coupled with a scalar field is studied. By discretization of the scalar field, the model is worked out precisely in a purely combinational way. It is shown that the quantum physics of the scalar matter is totally encoded into the modified dynamics of $SU(2)$ spin-network states which describe the quantum geometry of space. It turns out that the physics of the scalar matter coupled with gravity manifested in the low energy scale can be viewed as the phenomena that emerged from this microscopic construction. This gives rise to a radical observation on the issue of the unification of geometry and matter.

Figure 1

(a) The $N_f$ virtual faces. (b) Three bunches of faces that join at each edge.

Figure 2

The edge integral of Eq. (12).

Figure 3

The couplings at a vertex: the double lines represent the bunches of representations.

Figure 4

Casting ${\widehat{T}}_{\gamma ,N,j,\iota }$ to $T_{\gamma ,N,j,\iota }$.

Figure 5

Evolutions of the nodes.