Graduate Quantum Ising model correlation function query

[Danny Boy]

In this paper, on quantum Ising model dynamics, they consider the Hamiltonian
$$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$
and the correlation function
$$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot \hat{\sigma}^{a_1}_{j_1}(t_1^*)\hat{\sigma}^{b_m}_{k_m}(t_m) \cdot\cdot\cdot \hat{\sigma}^{b_1}_{k_1}(t_1)) \rangle$$
where $a,b= \pm$ and the time dependence of the Heisenberg picture
$$\hat{\sigma}_{j}^{a}(t) = e^{it\mathcal{H}}\hat{\sigma}_{j}^{a}e^{-it\mathcal{H}}$$
where the time ordering operator $\mathcal{T_C}$ orders operators along a closed path $\mathcal{C}$.

Question: Can anyone see the reasoning behind the following statement on page 6:
If an operator $\hat{\sigma}^{a=\pm}_{j}$ occurs in $\mathcal{G}$ one or more times, the operator $\hat{\sigma}^{z}_{j}$ (appearing in the time evolution operator) is forced to take on a well defined value $\sigma_{j}^{z}(t)$ at all points in time.

Thanks for any assistance.

 

[Danny Boy]

I think the idea is after $t_1$ (denoting the first occurrence of the operator $\hat{\sigma}^{a=\pm}_{j}$) since this results in the state on site $j$ being an eigenstate of $\hat{\sigma}_{j}^{z}$, hence it commutes with the Ising Hamiltonian above. It is still not clear how it it well defined before time $t_1$?