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 $p_0 = -E_p$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.