- #1
Bertin
- 12
- 6
Below follows the passage of Rovelli and Vidotto's Covariant Loop Quantum Gravity that I do not understand. To give the context, let me clarify that a state ##\psi## is a function in ##L^2[\text{SU}(2)^L]##, defined over a graph with ##L## edges (called ''links'' in this context) dressed with ##\text{SU}(2)## elements — that is, to each link ##l## we associate an ##\text{SU}(2)## element ##U_l## — and with ##N## trivalent vertices (called ''nodes'' in this context).
The point of this passage is to employ Peter-Weyl's theorem to decompose such a state ##\psi##, knowing that additionaly it is invariant under the action of ##\text{SU}(2)## at any of its nodes, that is, under the simultaneous action of an element of ##\text{SU}(2)## on all the ##\{U_l\}## corresponding to all edges ##\{l\}## meeting at any arbitrary node. This latter action encodes the gauge invariance of the theory.
My question is quite simple. Equation (5.29) gives the decomposition of ##\psi## and — as it should be the case — every index ##j,m,n## is summed over. However, following their introduction of the ##3j##-symbols due to the aforementioned gauge invariance, the authors end up with (5.32), where the indices ##n_1,\dots,n_L## are not summed over, at least not according to the mathematical expression. Is this a typo?
I considered the possibility that, since the action of ##\text{SU}(2)## should affect both indices ##m,n## of a Wigner matrix component ##D^j_{mn}(U)##, and given that the state should remain invariant under this action whenever it affects all the matrices associated to the edges meeting at a given node, then we should introduce two ##3j##-symbols per vertex (one for each index in a pair ##m,n##), leading instead to the decomposition
$$
\psi(U_1, \dots ,U_L) =
C_{j_1,...,j_L}\iota_1^{m_1m_2m_3}\iota_1^{n_1n_2n_3}
\cdots \iota_N^{m_{L-2}m_{L-1}m_{L}}\iota_N^{n_{L-2}n_{L-1}n_{L}}
D^{j_1}_{m_1n_1}(U_1) \cdots D^{j_{L}}_{m_Ln_L}(U_{L}),
$$
where repeated indices are summed over. Nevertheless, the last paragraph in this excerpt explicitly states that there is one ##3j##-symbol for each node. Consequently, my question: how exactly are the (apparently) free indices in (5.32) contracted?
The point of this passage is to employ Peter-Weyl's theorem to decompose such a state ##\psi##, knowing that additionaly it is invariant under the action of ##\text{SU}(2)## at any of its nodes, that is, under the simultaneous action of an element of ##\text{SU}(2)## on all the ##\{U_l\}## corresponding to all edges ##\{l\}## meeting at any arbitrary node. This latter action encodes the gauge invariance of the theory.
My question is quite simple. Equation (5.29) gives the decomposition of ##\psi## and — as it should be the case — every index ##j,m,n## is summed over. However, following their introduction of the ##3j##-symbols due to the aforementioned gauge invariance, the authors end up with (5.32), where the indices ##n_1,\dots,n_L## are not summed over, at least not according to the mathematical expression. Is this a typo?
I considered the possibility that, since the action of ##\text{SU}(2)## should affect both indices ##m,n## of a Wigner matrix component ##D^j_{mn}(U)##, and given that the state should remain invariant under this action whenever it affects all the matrices associated to the edges meeting at a given node, then we should introduce two ##3j##-symbols per vertex (one for each index in a pair ##m,n##), leading instead to the decomposition
$$
\psi(U_1, \dots ,U_L) =
C_{j_1,...,j_L}\iota_1^{m_1m_2m_3}\iota_1^{n_1n_2n_3}
\cdots \iota_N^{m_{L-2}m_{L-1}m_{L}}\iota_N^{n_{L-2}n_{L-1}n_{L}}
D^{j_1}_{m_1n_1}(U_1) \cdots D^{j_{L}}_{m_Ln_L}(U_{L}),
$$
where repeated indices are summed over. Nevertheless, the last paragraph in this excerpt explicitly states that there is one ##3j##-symbol for each node. Consequently, my question: how exactly are the (apparently) free indices in (5.32) contracted?