# Momentum in complex scalar field

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 $$\pi$$:

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

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