I've been reading E. S. Fradkin and G. A. Vilkovisky, “Quantization of Relativistic Systems with Constraints: Equivalence of Canonical and Covariant Formalisms in Quantum Theory of Gravitational Field.” 1977. [Online]. Available: https://inspirehep.net/literature/120057 , and I am having some trouble understanding the way he solves the equations of motion for the generating functional. The paper does point to references that apparently explain this better but I haven't been able to find online versions of them and due to the pandemic I do not have access to my institutions library.
Let $\eta^A=(q^i,p_i)$ be the canonical variables of some system with Hamiltonian $H$ and subject to only second-class constraints $\Psi^a$. Let $J_A$ be external sources and define the interaction Hamiltonian $H_\text{int}=\eta^AJ_A$. Let $Z[J]$ be the generating functional for Green's functions $$\langle 0|T(\eta^{A_1}(t_1)\cdots\eta^{A_n}(t_n))|0\rangle=\frac{\delta Z[J]}{\delta iJ_{A_1}(t_1)\cdots\delta iJ_{A_n}(t_n)}|_{J=0}.$$ In other words, for every polynomial $F[\eta]$ we have $$\langle 0|TF[\eta]|0\rangle=F[\eta]|_{\eta=\frac{\delta}{\delta iJ}}Z[J]|_{J=0}.$$
Now, the first thing is that if we want an operator equation $F[\eta]=0$ to be satisfied by our theory, we want that all matrix elements of this form vanish $$0=\langle 0|T(F[\eta]\eta^{A_1}(t_1)\cdots\eta^{A_n}(t_n))|0\rangle=\frac{\delta }{\delta iJ_{A_1}(t_1)\cdots\delta iJ_{A_n}(t_n)}F[\eta]|_{\eta=\frac{\delta}{\delta iJ}}Z[J]|_{J=0}.$$ This determines the Taylor coefficients forming the equation of motion for the generating functional $$0=F[\eta]|_{\eta=\frac{\delta}{\delta iJ}}Z[J].$$ My first question is whether this logic is indeed correct or if I am missing some subtlety here.
Now, in our particular case the equations of motion are $$\left(\dot{\eta}-\{\eta,H+H_\text{int}\}_D\right)|_{\eta=\frac{\delta}{\delta iJ}}Z[J]=0,$$ $$\Psi^a|_{\eta=\frac{\delta}{\delta iJ}}Z[J]=0,$$ where $$\{A,B\}_D=\{A,B\}-\{A,\Psi^a\}Q_{ab}\{\Psi^b,B\},$$ is the Dirac bracket. In here $Q_{ab}$ is the inverse matrix to $\{\Psi^a,\Psi^b\}$. With our logic above, it is clear that the $J$'s appearing in the bracket are on the left of the $\delta/\delta iJ$.The paper proceeds to say that these equations can be equivalently written as $$\left(\dot{\eta}-\left\{\eta,H+H_\text{int}+\Psi^a\xi_a-\frac{1}{2i}\delta^{(1)}(0)\log\det\{\Psi^a,\Psi^b\}\right\}\right)|_{\eta=\frac{\delta}{\delta iJ}}Z[J]=0,$$ $$\Psi^a|_{\eta=\frac{\delta}{\delta iJ}}Z[J]=0.$$ I don't understand how this comes about. He mentions that the $\xi_a$ are determined by the consistency conditions. I imagine this means that (as in the classical case) they are given by demanding that the constraints are constant $$\xi_a=-Q_{ab}\{\Psi^b,H\}.$$ Is this correct? In that case however, I would obtain the equation above by expanding the Dirac bracket except for the $\det\{\Psi^a,\Psi^b\}$ part. Where does this come from?
Finally, my last question is how does one integrate this equation. The paper states the solution is $$Z[J]=\int\mathcal{D}q\mathcal{D}p\mathcal{D}\xi\prod_{t}\sqrt{\det\{\Psi^\bullet,\Psi^\bullet\}}\exp\left(i\int dt\,(p\dot{q}-H-H_\text{int}-\Psi\xi)\right).$$
