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

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

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

whose Euler-Lagrange equations are

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

with

$$$$\tag{2.3} F_{\alpha\beta}=\frac{\partial A_\beta}{\partial\xi^\alpha}-\frac{\partial A_\alpha}{\partial\xi^\beta}$$$$

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

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

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

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

with

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

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

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

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)?

## 1 Answer

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.
• Thank you very much.
– AFG
Feb 20, 2021 at 11:55