# Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism

+ 2 like - 0 dislike
483 views

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

$$\mathcal{L}=F_{\mu\nu}F^{\mu\nu}$$

where

$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

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

We choose to write it in first order form like this

$$\mathcal{L}=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})F^{\mu\nu}-{1\over{2}}F_{\mu\nu}F^{\mu\nu}$$

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

"we rewrite (the last equation) as:

$$\mathcal{L}=(\partial_{0}A_{k})F^{0k}+A_{0}(\partial_{k}F^{0k})-F^{ik}(\partial_{i}A_{k}-\partial_{k}A_{i})-{1\over{2}}(F^{0k})^2-{1\over{2}}(F^{ik})^2$$"

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.

This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user twistor59
Looks like the term you are worried about just comes from a partial integration of the $(\partial_k A_0) F^{0k}$ term.

This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user Olaf
D'oh ! I knew I'd kick myself !!

This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user twistor59

+ 1 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user Qmechanic
answered Feb 11, 2012 by (3,110 points)
Thanks ! In my defence I haven't done any physics since 1984 !!

This post imported from StackExchange Physics at 2014-03-22 17:26 (UCT), posted by SE-user twistor59
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.