I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure).

The topic is the Faddeev-Jackiw treatment of Lagrangians which are singular (Hessian vanishes) - similar to what Dirac does, but without the need to differentiate between first and second class constraints. Just looking at classical stuff here, no quantization.

Starting with the Maxwell Lagrangian




we see that it's second order in time derivatives acting on A.

We choose to write it in first order form like this


where we're treating $F^{\mu\nu}$ now as an auxilliary, independent variable. Having defined this, Faddeev says

"we rewrite (the last equation) as:


My question is how does he arrive at this from the previous equation ? I don't see how just expanding the indices into time and space values ever gets me to $A_{0}(\partial_{k}F^{0k})$

I can see how there's something special about $A_{0}$, since when I write out the EOM for the first order Lagrangian, $A_{0}$ drops out, which indeed it should do because we'll end up with it being a Lagrange multiplier. I just can't see how you end up with that term, with $A_{0}$ multiplying $\partial_{k}F^{0k}$.

It's clearly correct since $A_{0}divE$ is just the Gauss law constraint.

  • $\begingroup$ Looks like the term you are worried about just comes from a partial integration of the $(\partial_k A_0) F^{0k}$ term. $\endgroup$
    – Olaf
    Feb 11, 2012 at 14:37
  • $\begingroup$ D'oh ! I knew I'd kick myself !! $\endgroup$
    – twistor59
    Feb 11, 2012 at 14:44

1 Answer 1


Faddeev has implicitly dropped a total 4-divergence term $d_{\mu}(A_0 F^{0\mu})$ in the Lagrangian density ${\cal L}$. This does not affect the equations of motion, i.e., Maxwell's equations.

  • $\begingroup$ Thanks ! In my defence I haven't done any physics since 1984 !! $\endgroup$
    – twistor59
    Feb 11, 2012 at 14:47
  • $\begingroup$ @twistor59. You are welcome. Thank you for the Faddeev link! $\endgroup$
    – Qmechanic
    Feb 11, 2012 at 14:53

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