I have a doubt about the lagrangian of the Schrodinger equation. If we consider the wave function $\psi(\textbf{x},t)$ that satisfy the Schrodinger equation as a field, one way of construct the Lagrange Density $\mathcal{L}$ is try an ansatz
$$\mathcal{L} = i\hbar \psi^{*}\dot{\psi} - \frac{\hbar^{2}}{2m}(\nabla \psi^{*})\cdot (\nabla \psi) - V(\textbf{x},t)\psi^{*} \psi,$$
and show that this leads to the Schrodinger equation using the euler-Lagrange equation for $\mathcal{L}$:
$$\frac{\partial \mathcal{L}}{\partial \psi^{*}} - \partial_{\mu} \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \psi^{*})} = 0.$$
In the book "Field Quantization" of Greiner, it's said that we can consider the field $\psi(\textbf{x},t)$ and its complex conjugated $\psi^{*}(\textbf{x},t)$ as independents fields, so the Lagrange Density will be a function of $\psi$ and $\psi^{*}$ and their partial derivatives. What's the reason for this? The book dosen't give explanations.
