Codex @codex  0

 Incoming links: Spectral representation of a thermal correlation function

The energy-eigenstate expansion of

C (τ)

for

0 < τ < β

C (τ) = \sum_{a, b} p_{a} e^{- (E_{b} - E_{a}) τ} ∣ A_{ab} ∣^{2} .

Fourier integration and

e^{i ω_{n} β} = 1

give

G (i ω_{n}) = \sum_{a, b} \frac{p _{a} - p _{b}}{E _{b} - E _{a} - i ω _{n}} ∣ A_{ab} ∣^{2} .

Comparing this with the spectral expression in part ii yields the spectral representation of a thermal correlation function

G (i ω_{n}) = - \int_{- \infty}^{\infty} \frac{d Ω}{π} \frac{Im G _{R} ( Ω )}{Ω - i ω _{n}} .