Coherent $3j$-symbol representation for the loop quantum gravity intertwiner space

We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.

Figure 1

The norm of the $3j$ symbol (3.21) as a function of the polar angles of the vector ${\overrightarrow{n}}_3$. The spins $j_1$, $j_2$ and $j_3$ are all equal to a common value $j$, and the first two vectors are fixed to ${\overrightarrow{n}}_1=(0,0,1)$ and ${\overrightarrow{n}}_2=(\frac{\sqrt{3}}{2},0,-\frac{1}{2})$. In the upper diagram $j=20$, and in the lower diagram $j=100$. In both cases, the norm is peaked on the angles $(\theta ,\varphi )=(\frac{2}{3}\pi ,\pi )$, corresponding to the closed configuration ${\overrightarrow{n}}_3=(-\frac{\sqrt{3}}{2},0,-\frac{1}{2})=-{\overrightarrow{n}}_1-{\overrightarrow{n}}_2$, but the peak is sharper when the spin $j$ is larger.

Figure 2

The norm of (3.21) as a function of the polar angles of ${\overrightarrow{n}}_3$ when $j_1=j_2=50$ and $j_3=86\approx 50\sqrt{2}$. The maximum occurs at the angles $(\theta ,\varphi )=(\frac{1}{2}\pi ,\frac{5}{4}\pi )$, corresponding to the vector ${\overrightarrow{n}}_3=(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)$. Therefore the norm is peaked on the approximately closed configuration ${\overrightarrow{n}}_1+{\overrightarrow{n}}_2+\sqrt{2}{\overrightarrow{n}}_3=0$.

Figure 3

A plot of the function $|F({\theta }_{\eta },{\varphi }_{\eta })|/j^{2}l$, when the spins of the three-valent intertwiner $\iota$ are given by $j_1=j_2=j$ and $j_3\approx \sqrt{2}j$. The function $F({\theta }_{\eta },{\varphi }_{\eta })$ determines the preferred orientation of the vectors ${\overrightarrow{n}}_{{\eta }_1}={\overrightarrow{n}}_{{\eta }_2}\equiv {\overrightarrow{n}}_{\eta }$, created by the action of the operator $C_{12}$ on the intertwiner $\iota$. The maxima of the function $|F({\theta }_{\eta },{\varphi }_{\eta })|$ are located at $({\theta }_{\eta },{\varphi }_{\eta })=(\frac{1}{2}\pi ,\frac{1}{2}\pi )$ and $({\theta }_{\eta },{\varphi }_{\eta })=(\frac{1}{2}\pi ,\frac{3}{2}\pi )$. Both points correspond to the vector ${\overrightarrow{n}}_{\eta }$ orthogonal to the plane of the vectors ${\overrightarrow{n}}_1$, ${\overrightarrow{n}}_2$ and ${\overrightarrow{n}}_3$.