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- In the referenced paper, an explanation as to how to insert $\hat{\sigma}_{j}^{z}$ into a correlation function is given. I am seeking clarity on this technical point mentioned in the paper.

I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising model (with only an interaction term in the Hamiltonian). The correlation function ##\mathcal{G}## up to page 7 deals only with raising and lowering operators of the form ##\sigma^{\pm}_{j}## on sites ##j## (##\alpha_j = 0## if ##\hat{\sigma}_{j}^{\pm}## appears in the correlation function ##\mathcal{G}## and ##\alpha_j = 0## otherwise). To insert operators of the form ##\hat{\sigma}_{j}^{z}## into the correlation function ##\mathcal{G}##, the following is stated:

Can anyone see the reasoning behind the last sentence:The insertion of an operator ##\hat{\sigma}_{j}^{z}## inside a correlation function ##\mathcal{G}##, which we denote by writing ##\mathcal{G} \mapsto \mathcal{G}^{z}_{j}##, is relatively straightforward. If ##\alpha_j = 0##, then clearly the substitution ##\hat{\alpha}_{j}^{z} \mapsto \alpha_{j}^{z}(t)## does the trick. If ##\alpha_j = 1##, ##\hat{\alpha}_{j}^{z}## can be inserted by recognizing that the variable ##\phi_j## couples to ##\hat{\alpha}_{j}^{z}## as a source term, and thus the insertion of ##\hat{\alpha}_{j}^{z}(t)## is equivalent to applying ##i \frac{\partial}{\partial \phi_j}## to ##\mathcal{G}##.

Thanks for any assistance.If ##\alpha_j = 1##, ##\hat{\alpha}_{j}^{z}## can be inserted by recognizing that the variable ##\phi_j## couples to ##\hat{\alpha}_{j}^{z}## as a source term, and thus the insertion of ##\hat{\alpha}_{j}^{z}(t)## is equivalent to applying ##i \frac{\partial}{\partial \phi_j}## to ##\mathcal{G}##.