Doubt on the consistency condition of the secondary constraint of electrodynamics

On the Dirac's method for the electromagnetism, demanding consistency on the secondary constraint $X$ (which should be identically achieved since there are no further constraints),

$$X_{\mathbf{x}}[\pi, \lambda] \equiv \int d^3x \,\lambda(\mathbf{x}) (\partial_i\pi^i-j^0)(\mathbf{x}) \approx 0$$

I get

$$\begin{split} 0\overset{!}{\approx }\dot{X}\approx & \{X(\mathbf{x}),H_{P}(\mathbf{y})\} =-\int d^3z \, \frac{\delta H_P(\mathbf{y})}{\delta A^{\mu}(\mathbf{z})}\frac{\delta X(\mathbf{x})}{\delta \pi^{\mu}(\mathbf{z})} \\ \approx&\int d^3z \, \frac{\delta H_P(\mathbf{y})}{\delta A^{i}(\mathbf{z})}\partial_i [\lambda(\mathbf{x}) \delta(\mathbf{x}-\mathbf{z})] \\ \therefore & \; \; 0\approx \;\partial_i \frac{\delta H_P}{\delta A^{i}}=\partial_i \partial_j F^{ji}-\partial_i j^i = -\partial_i j^i \end{split}$$

With $H_P$ being the primary hamiltonian for the electrodynamics with sources, with density

$$\mathscr{H}_P=\frac{1}{4}F^{i j} F_{i j} -\frac{1}{2}\pi^{i}\pi_{i} -A_0 \partial_i \pi^{i} + j^{\mu}A_{\mu} + \lambda_1 \pi^0$$

Now, the first term of $\dot{X}$ is, of course, identically zero, which guarantees the consistency of the secondary constraint on electrodynamics without sources. The problem is that I'm left with the term $\partial_i j^i$ that has no reason to be zero (differently from $\partial_{\mu} j^{\mu}=0$, of course).

It's been a while since I'm stuck here and I'd be extremely grateful if someone could at least point me at the right direction.

Edit: I started off with the lagrangian density

$$\mathscr{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}-j^{\mu}A_{\mu}$$

• You could refer 'Lectures on Quantum field theory's by Ashok Das. It has a chapter on constrained systems and deals with electromagnetic field presenting all the necessary calculations. – BoundaryGraviton Aug 6 '16 at 19:55
• @BoundaryGraviton, thanks for the suggestion! Which part of the book? I just went through the Maxwell field chapter, but it treats only the quantization, basically. – GaloisFan Aug 6 '16 at 20:12
• As I have already said in the previous comment, there is a separate chapter on constrained systems. I think it is titled Dirac's constrained systems. It deals with covariant quantization of the e.m. field and quantization of Dirac filed both of which are done using Dirac brackets. – BoundaryGraviton Aug 6 '16 at 21:00
• @BoundaryGraviton, you're right, sorry. Either way, he just states that the consistency of the secondary constraint is identicaly true (as I said) and the chapter in general doesn't really clear up my doubts. – GaloisFan Aug 6 '16 at 21:17
• Furthermore, it treats the free Maxwell field, which doesn't give me problems. – GaloisFan Aug 17 '16 at 21:32

$$\dot{F}(x)=\{F,H\}+\frac{\partial F}{\partial t}$$
$$0 \overset{!}{\approx} \partial_i \partial_jF^{ij}-\partial_ij^{i}-\partial_0j^0=-\partial_\mu j^\mu \approx 0$$