Extension of Faddeev-Jackiw first-order Lagrangian formalism to fields

Question

In this paper, Toms discusses the method that Faddeev and Jackiw proposed for quantization of constrained theories. In section III.B, he applies this method to a field theory, but I have several doubts about how this transition is done.

A first-order Lagrangian for a mechanical system can be written as

\begin{equation}\tag{2.1} L=A_\alpha(\xi)\dot{\xi}^\alpha+L_v(\xi), \end{equation}

whose Euler-Lagrange equations are

\begin{equation}\tag{2.2} F_{\alpha\beta}\dot{\xi}^\beta=-\frac{\partial L_v}{\partial\xi^\alpha}, \end{equation}

with

\begin{equation}\tag{2.3} F_{\alpha\beta}=\frac{\partial A_\beta}{\partial\xi^\alpha}-\frac{\partial A_\alpha}{\partial\xi^\beta} \end{equation}

being the symplectic two-form. So far so good.

Later, on section III.B, he says that the traduction of expresion (2.3) to field theory is

\begin{equation}\tag{3.41} F_{\alpha\beta}=\frac{\delta A_\beta}{\delta\xi^\alpha}-\frac{\delta A_\alpha}{\delta\xi^\beta}, \end{equation}

but I can't see how this araises from Euler-Lagrange equations, i.e.

\begin{equation}\tag{EL} \frac{\delta S}{\delta\phi^\alpha}=\frac{\partial\mathcal{L}}{\partial\phi^\alpha}-\partial_\mu\Bigg(\frac{\partial\mathcal{L}}{\partial({\partial_\mu\phi^\alpha})}\Bigg)=0, \end{equation}

with

\begin{equation} S[\phi]=\int d^4{x} \mathcal{L}(\phi,\partial\phi). \end{equation}

On a field theory, I suppose that a first-orden lagrangian density would be written as

\begin{equation}\tag{LD} \mathcal{L}=A_\alpha(\phi,\nabla\phi)\dot{\phi}^\alpha+\mathcal{L}_v(\phi,\nabla\phi), \end{equation}

where $\phi^\alpha$ are the fields and $\nabla\phi$ indicates dependence in spatial derivatives. However, if I insert that lagrangian density into (EL) I can't get an expression equivalent to (2.2). Moreover, the "$A$"s in expresion (3.41) must be functionals. Which is the relation between those "$A$"s and the ones in (LD)?

Answer 1

OP is essentially asking the following.

How does the point-mechanical Faddeev-Jackiw formulas from chapter II generalizes to the field-theoretic formulas in section III.B?

Well, let's see. The field-theoretic first-order Lagrangian is $$ L~=~\int\!d^3{\bf x}~A_{\alpha}({\bf x}) \dot{\xi}^{\alpha}({\bf x})~-~H.\tag{2.1'}$$ An infinitesimal variation reads $$\begin{align} \delta L ~\sim~& \int\!d^3{\bf x}\left( \delta A_{\alpha}({\bf x}) \dot{\xi}^{\alpha}({\bf x}) -\frac{dA_{\alpha}({\bf x})}{dt} \delta\xi^{\alpha}({\bf x})\right) ~-~\delta H \cr ~=~&\int\!d^3{\bf x} \int\!d^3{\bf x}^{\prime} \left( \delta\xi^{\alpha}({\bf x}) \frac{\delta A_{\beta}({\bf x}^{\prime})}{\delta \xi^{\alpha}({\bf x})} \dot{\xi}^{\beta}({\bf x}^{\prime}) ~-~ \dot{\xi}^{\beta}({\bf x}^{\prime}) \frac{\delta A_{\alpha}({\bf x})}{\delta \xi^{\beta}({\bf x}^{\prime})} \delta\xi^{\alpha}({\bf x}) \right) \cr &~-~ \delta H\cr ~=~&\int\!d^3{\bf x} \int\!d^3{\bf x}^{\prime} \delta\xi^{\alpha}({\bf x}) F_{\alpha\beta}({\bf x},{\bf x}^{\prime})\dot{\xi}^{\beta}({\bf x}^{\prime}) ~-~ \int\!d^3{\bf x}~ \delta \xi^{\alpha}({\bf x}) \frac{\delta H}{\delta \xi^{\alpha}({\bf x})}. \end{align}$$ Here the $\sim$ symbol means equality modulo total time derivative terms. Also we have defined the components of symplectic 2-form $$ F_{\alpha\beta}({\bf x},{\bf x}^{\prime})~=~\frac{\delta A_{\beta}({\bf x}^{\prime})}{\delta\xi^{\alpha}({\bf x})}-\frac{\delta A_{\alpha}({\bf x})}{\delta\xi^{\beta}({\bf x}^{\prime})}.\tag{2.3'/3.41'} $$ (Judging from eq. (3.42) it becomes clear that eq. (3.41) is bi-local.)

The Euler-Lagrange equations are Hamilton's equations $$ \int\!d^3{\bf x}^{\prime} F_{\alpha\beta}({\bf x},{\bf x}^{\prime})\dot{\xi}^{\beta}({\bf x}^{\prime}) ~=~ \frac{\delta H}{\delta \xi^{\alpha}({\bf x})}.\tag{2.2'} $$

References:

1. D.J. Toms, Faddeev-Jackiw quantization and the path integral, arXiv:1508.07432.