With the convention $G_R(t)=-i\Theta(t)\langle[q(t),q(0)]\rangle$, the <retarded Green function> is
$$
G^R_{qq}(\omega,k)=
\frac{\omega_k/s}{(\omega+i0)^2-\omega_k^2}
=\frac{\rho_sk^2/s^2}{(\omega+i0)^2-\omega_k^2}.
$$
Defining the <spectral function> by $\varrho=-2\operatorname{Im}G_R$ gives
$$
\varrho(\omega,k)=\frac\pi s
\left[\delta(\omega-\omega_k)-\delta(\omega+\omega_k)\right].
$$
Other common spectral-weight conventions differ by an overall factor.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-337/2/e/solution