Hamilton's equation for generating functional 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
$$Z[J]=\int\mathcal{D}q\mathcal{D}p\mathcal{D}\xi\prod_{t}\sqrt{\det\{\Psi^\bullet,\Psi^\bullet\}}\exp\left(i\int dt\,(p\dot{q}-H-H_\text{int}-\Psi\xi)\right).$$
 A: *

*We start from the Hamiltonian action
$$ S ~=~\int \! dt~ L, \tag{1.15a}$$
with Hamiltonian Lagrangian
$$\begin{align}  L~=~&\vartheta_I(z) \dot{z}^{I} - H - \Psi^a\xi_a\cr
&- i\hbar \delta(0) \ln {\rm Pf}(Q^{\cdot\cdot})
- i\hbar \delta(0) \ln {\rm Pf}(\omega_{\cdot\cdot}) , \end{align} \tag{1.15b}$$
and with Hamiltonian
$$ H~=~H_0 + H_{\rm int}, \tag{2.5a}$$
$$z^I~=~(q^i,p_i), \qquad  H_{\rm int} ~=~-J_I z^I-j^a\xi_a,\tag{2.5b}$$
Here $\Psi^a$ are $2m$ second-class constraints,
$$ Q^{ab}~=~\{\Psi^a,\Psi^b\} \quad \text{invertible},\tag{1.17}$$
$\xi_a$ are Lagrange multipliers,
and $$\vartheta~=~\vartheta_I\mathrm{d}z^I\tag{A}$$ is a symplectic 1-form potential, where
$$ \omega_{IJ}~=~\partial_I\vartheta_J - \partial_J\vartheta_I\tag{B}$$
is the components of a symplectic 2-form
$$\omega~=~\frac{1}{2}\mathrm{d}z^I \omega_{IJ} \wedge \mathrm{d}z^J~=~\mathrm{d}\vartheta. \tag{C}$$
The appearance of $\delta(0)$ is e.g. explained in this Phys.SE post.


*The classical EOMs are
$$ \Psi^a~\approx~0, \tag{1.19}$$
$$ \dot{z}^I~\approx~\{z^I,H\}_D
~\approx~ \{z^I,H\}+\{z^I,\Psi^a\}\xi_a, \tag{1.18}$$
$$\xi_a~\approx~ -Q^{-1}_{ab}\{\Psi^b,H\}. \tag{1.20}$$


*The solution to the corresponding path integral
$$\begin{align} Z[J,j] ~=~&\int\! {\cal D}\frac{z}{\sqrt{\hbar}}~{\cal D}\xi~  \exp\left(\frac{i}{\hbar}S\right)\cr
~=~&\int\! {\cal D}\frac{z}{\sqrt{\hbar}}~
\exp\left(\frac{i}{\hbar}\int\!dt \left(\vartheta_I(z) \dot{z}^{I} - H \right)\right)\cr
&\times \delta(\Psi^{\cdot}){\rm Pf}(Q^{\cdot\cdot}) {\rm Pf}(\omega_{\cdot\cdot})
\end{align}\tag{2.8}$$
was first written down in Ref. 2. The Pfaffians are needed to ensure that the path integral (2.8) is invariant under reparametrizations of the $z^I$-coordinates and the second-class constraints $\Psi^a$.


*Ignoring boundary terms, we immediately derive that the solution (2.8) satisfies
$$\begin{align} 0~=~&\frac{\hbar}{i} \int\! {\cal D}\frac{z}{\sqrt{\hbar}}~{\cal D}\xi~ \frac{\delta e^{iS/\hbar} }{\delta z^I(t)}  \cr
~=~&\int\! {\cal D}\frac{z}{\sqrt{\hbar}}~{\cal D}\xi~ \frac{\delta S}{\delta z^I(t)} \exp\left(\frac{i}{\hbar}S\right) \cr
~=~&\left. \frac{\delta S}{\delta z^I(t)} \right|_{z=\frac{\hbar}{i}\frac{\delta}{\delta J},\xi=\frac{\hbar}{i}\frac{\delta}{\delta j}}Z[J,j],
\end{align} \tag{2.7a}$$
and
$$\begin{align} 0~=~&\frac{\hbar}{i} \int\! {\cal D}\frac{z}{\sqrt{\hbar}}~{\cal D}\xi~ \frac{\delta e^{iS/\hbar} }{\delta \xi_a(t)}  \cr
~=~&\int\! {\cal D}\frac{z}{\sqrt{\hbar}}~{\cal D}\xi~ \frac{\delta S}{\delta  \xi_a(t)} \exp\left(\frac{i}{\hbar}S\right) \cr
~=~&\left. \Psi^a(t) \right|_{z=\frac{\hbar}{i}\frac{\delta}{\delta J},\xi=\frac{\hbar}{i}\frac{\delta}{\delta j}}Z[J,j].
\end{align}\tag{2.7b} $$
In practice it is a bit more challenging to derive the solution (2.8) from the consistency conditions (2.7), cf. OP's question.
References:

*

*E.S. Fradkin & G.A. Vilkovisky, Quantization of Relativistic Systems with Constraints: Equivalence of Canonical and Covariant Formalisms in Quantum Theory of Gravitational Field, 1977.


*P. Senjanovic, Path Integral Quantization of Field Theories with Second Class Constraints, Annals Phys. 100 (1976) 227.
