Barut-Zanghi (BZ) Lagrangian derivation

I'm trying to derive the BZ Lagrangian (density) from the Dirac Hamiltonian density and some questions popped up.
BZ Lagrangian is $$\mathcal{L} = \frac{i}{2}(\dot{\bar{\psi}}\psi - \bar{\psi}\dot{\psi}) + p_\mu(\dot{x}^\mu - \bar{\psi}\gamma^\mu\psi)\tag{1}$$ where the four-potential $$A_\mu$$ is set to zero for simplicity.
The first question is: how can I derive it? As it can be seen, the first right member is the time derivative inside the Dirac Lagrangian (which corresponds to the conjugate momenta) and if I set $$p_\mu\gamma^\mu = m$$, then second member in brackets is the second term of Dirac Lagrangian.
I though that it can be derived from the Legendre transform starting with the Dirac Hamiltonian and the conjugate momenta (that I know) and inserting the other conjugate momenta linked to the dynamical variables $$(x_\mu, p_\mu)$$, i.e. $$p_\mu\dot{x}^\mu$$ .
The problem is that there's the divergence term of the Dirac Hamiltonian that doesn't appear. Any advice?
The second question is the following: I saw that the Lagrangian density depends on fields and their derivatives: in this case we have $$\psi$$,$$\bar{\psi}$$ as fields and their derivative there aren't. Although, in this case we also have a dependence from the dynamical variables $$(x_\mu, p_\mu)$$. How can I justify this?

1. The Barut-Zanghi (BZ) Lagrangian \begin{align} L ~=~& \frac{i}{2}(\dot{\bar{\psi}}\psi - \bar{\psi}\dot{\psi}) + p_{\mu}\dot{x}^{\mu} \cr &-(p_{\mu}-qA_{\mu}) \bar{\psi}\gamma^{\mu}\psi, \cr \bar{\psi} ~=~&\psi^{\dagger}\gamma^0 ,\end{align}\tag{1} is a point-mechanical (rather than a field-theoretic) model, i.e. all the variables depend only on time. In contrast the spinor in the Dirac equation is a field.
2. The BZ Hamiltonian $$H~=~ (p_{\mu}-qA_{\mu}) \bar{\psi}\gamma^{\mu}\psi\tag{2}$$ is minus the terms without time derivatives in the BZ Lagrangian (1), cf. e.g. the Faddeev-Jackiw method.