# 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. Aug 6, 2016 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. Aug 6, 2016 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. Aug 6, 2016 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. Aug 6, 2016 at 21:17
• Furthermore, it treats the free Maxwell field, which doesn't give me problems. Aug 17, 2016 at 21:32

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