Recurrence relations for spin foam vertices

Abstract

We study recurrence relations for various Wigner 3nj-symbols and the non-topological 10j-symbol. For the 6j-symbol and the 15j-symbols which correspond to basic amplitudes of 3d and 4d topological spin foam models, recurrence relations are obtained from the invariance under Pachner moves and can be interpreted as quantizations of the constraints of the underlying classical field theories. We also derive recurrences from the action of holonomy operators on spin network functionals, making a more precise link between the topological Pachner moves and the classical constraints. Interestingly, our recurrence relations apply to any SU(2) invariant symbol, depending on the cycles of the corresponding spin network graph. Another method is used for non-topological objects such as the 10j-symbol and pseudo-isoceles 6j-symbols. The recurrence relations are also interpreted in terms of elementary geometric properties. Finally, we discuss the extension of the recurrences to take into account boundary states which leads to equations similar to Ward identities for correlation functions in the Barrett-Crane model.

Figures (8)

• Figure 1: The identity (\ref{['recurrencebe']}). On the left hand side, the tetrahedron $ABCD$ with edges $(g,h,j,k,a,b)$ is glued via the triangle $(g,h,j)$ to the flattened tetrahedron $BCDD'$ with an edge spin being $1/2$. On the right hand side, the three tetrahedra $ACDD'$, $ABDD'$ and $ABCD'$ share the edge $AD'$ with spin $k\pm\frac{1}{2}$. The tetrahedra $ACDD'$ and $ABDD'$ are "flattened" because of the spin $\frac{1}{2}$ along $DD'$. Thus, from the point of view of the initial tetrahedron $ABCD$, the move can be seen as a "small" displacement of the point $D$ to $D'$, resulting in some small length shifts.
• Figure 2: The functional $\varphi_{\{j_l\}}(g_1..g_6)$ can be represented by a tetrahedron whose links are labelled by the spins $j_i$ and the group elements $g_i$ ($i=1,\cdots,6$). The action of the holonomy operator $\chi_{j}(g_4g_5g_6)$ translates, upon recoupling, into the functional depicted above, where intersections of links stand for $3jm$-symbols, and the spins $k_1,k_2,k_3$ are summed over. The dashed lines are the links carrying the spin $j$.
• Figure 3: The top figure displays a reducible symbol: the cycle $(j_1 j_2 j_3)$ can be factorized because there is an unique intertwiner from $l_1\otimes l_2\otimes l_3$ to $\mathbb{C}$. One can then use the recurrence relations of the 6j-symbol to shift the spins $(j_i)$. For a cycle made of four links, one needs to sum over the intertwiners $z$ between $l_4\otimes l_2\rightarrow l_1\otimes l_3$. One can then use recurrence relations which shift some $(j_i)$ provided they do not depend on $z$, but only on the spins which join the cycle.
• Figure 4: The left picture describes the configuration with two 4-simplices, while the right picture displays the structure of four 4-simplices which are glued to each other along different tetrahedra. The arrows give the orientations of the dual edges and the boxes stand for integration of the group elements carried by these dual edges. Once the right hand side is regularised, the two situations have the same BF amplitude. To regularise it, we have dropped the flatness condition for the face which passes through $l$, $q$ and $p$. As a consequence, one obtains only one 15j-symbol on the right hand side, since only three lines pass through $l$, $q$ and $p$: three 4-simplex amplitudes are reduced to $\{12j\}$-symbols.
• Figure 5: The two sides of the 2-4 move. On the left hand side, the 4-simplices $(abcde)$ and $(aa'cde)$ are glued along the tetrahedron $(acde)$. On the right hand side the four 4-simplices $(a'bcde)$, $(a'bacd)$, $(a'bade)$ and $(a'bace)$. This configuration has four 'bulk' triangles (given by $(a'bc)$, $(a'bd)$, $(a'be)$ and $(aa'b)$), which are shared by three 4-simplices, and six 'bulk' tetrahedra (made of the points $a'$ and $b$ together with any choice of two others), which are shared by two 4-simplices. When the spins of the triangles $(aa'e)$ and $(aa'c)$ are taken to be zero, we may consider $a'$ as being very close to $a$. Then, the areas of the initial 4-simplex $(abcde)$ on the left and those of $(a'bcde)$ on the right only differ by some slight shifts for the three triangles sharing the edge $(ad)$.