2
$\begingroup$

I've been learning Green's Functions (Zero-Temprature) from Fetter and Walecka's Quantum Theory of Many-Particle Systems, and find it hard to understand the proof of Goldstone's Theorem, in page 111, the equation is: $$[E-E_0]^{(n)}=⟨\Phi_0|\hat{H_1}\frac{1}{E_0-\hat{H_0}+i\epsilon n\hbar}\hat{H_1}\frac{1}{E_0-\hat{H_0}+i\epsilon (n-1)\hbar}...\hat{H_1}\frac{1}{E_0-\hat{H_0}+i\epsilon \hbar}\hat{H_1}|\Phi_0⟩_C$$ where C stands for connected. He noted that the limitation to connected diagrams ensures that $|\Phi_0⟩$ can not appear as an intermediate state and is nondegenerate. I'm confused with this statement.

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ Are you familiar with diagrammatic perturbation theory? You can prove the claim graphically to yourself if so. A disconnected diagram contributing to this correction would be the above expression with a $|\Phi_0><\Phi_0|$ inserted somewhere. This would be exactly a case where the state in question appears as an intermediate state. $\endgroup$
    – hulsey
    Commented May 10, 2021 at 17:08

0

Reset to default