Momentum in complex scalar field

Question

Consider a complex scalar field $\psi(x)$ with Lagrangian density

$$ \mathcal{L} = \partial_\mu\psi^* \partial^\mu\psi - M^2\psi^*\psi. $$

Expand the complex field operator as a sum

$$ \psi = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_\mathbf{p}}}(b_\mathbf{p}e^{i\mathbf{p}\cdot\mathbf{x}} + c_\mathbf{p}^\dagger e^{-i\mathbf{p}\cdot\mathbf{x}}). $$

From the Lagranigan density we obtain the momentum $\pi = \dot\psi^*$. From the above information, how do we derive the expression for $\pi$:

$$ \pi = \int \frac{d^3p}{(2\pi)^3} i \sqrt{\frac{E_\mathbf{p}}{2}} (b_\mathbf{p}^\dagger e^{-i\mathbf{p}\cdot\mathbf{x}}- c_\mathbf{p} e^{i\mathbf{p}\cdot\mathbf{x}})~? $$

Answer 1

You complex conjugate the expression for $\psi$ and then you derive as $\partial_t$, where $t = x^0$ and $p_0 = -E_p$.