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 $$Z[J]=\int\mathcal{D}q\mathcal{D}p\mathcal{D}\xi\prod_{t}\sqrt{\det\{\Psi^\bullet,\Psi^\bullet\}}\exp\left(i\int dt\,(p\dot{q}-H-H_\text{int}-\Psi\xi)\right).$$

  1. 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.

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

  3. 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 $\Psi^a$.

  4. 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.


  1. 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.

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


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