4
$\begingroup$

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

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

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – BoundaryGraviton Aug 6 '16 at 19:55
  • $\begingroup$ @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. $\endgroup$ – GaloisFan Aug 6 '16 at 20:12
  • $\begingroup$ 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. $\endgroup$ – BoundaryGraviton Aug 6 '16 at 21:00
  • $\begingroup$ @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. $\endgroup$ – GaloisFan Aug 6 '16 at 21:17
  • $\begingroup$ Furthermore, it treats the free Maxwell field, which doesn't give me problems. $\endgroup$ – GaloisFan Aug 17 '16 at 21:32
0
$\begingroup$

I found my mistake a few weeks after posting the question and only now decided to post the solution here:

The silly error was just the lack of the second term in

$$ \dot{F}(x)=\{F,H\}+\frac{\partial F}{\partial t} $$

Which would then give me something like

$$ 0 \overset{!}{\approx} \partial_i \partial_jF^{ij}-\partial_ij^{i}-\partial_0j^0=-\partial_\mu j^\mu \approx 0 $$

So that, indeed, the two constraints found up to here exhaust the constraints of the theory.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.