# Statistical and spectral function in thermal state

1. Dec 4, 2013

### physicus

1. The problem statement, all variables and given/known data
The statistical and spectral functions for bosonic operators $\phi_a$ are:
$G_{ab}(t,\vec{x})=\frac{1}{2}\langle \{\phi_a(t,\vec{x}),\phi_b(0,\vec{0})\}\rangle$,
$\rho_{ab}(t,\vec{x})=\langle [\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]\rangle$.
The expectation values are in static thermal equilibrium in the grand canonical ensemble with Hamiltonian $H'=H-\mu Q$.

Show that
$G_{ab}(\omega,\vec{k})=\frac{1}{2}\frac{1+e^{-\beta \omega}}{1-e^{-\beta\omega}}\rho_{ab}(\omega,\vec{k})$
using the definitions of $G_{ab}(\omega,\vec{k})$ and $\rho_{ab}(\omega,\vec{k})$ in the basis of H' and inserting a complete set of states.

2. Relevant equations

3. The attempt at a solution
$G_{ab}(\omega,\vec{k})$ and $\rho_{ab}(\omega,\vec{k})$ re defined as Fourier transforms of the above expressions. The expectation values are given as
$\langle A \rangle = \frac{tr\, e^{\beta H'}A}{tr\, e^{\beta H'}}$
so
$\rho_{ab}(\omega,\vec{k})=\int d^dx dt\, \langle [\phi_a(t,\vec{x}),\phi_b(0,\vec{0})] \rangle e^{-i\vec{k}\vec{x}+i\omega t}$
$= \frac{1}{tr\, e^{\beta H'}}\int d^dx dt \sum_\alpha \langle\alpha|e^{\beta H'}[\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]|\alpha\rangle e^{-i\vec{k}\vec{x}+i\omega t}$
$= \frac{1}{tr\, e^{\beta H'}}\int d^dx dt \sum_{\alpha,\gamma} \langle\alpha|e^{\beta \omega_\gamma}|\gamma\rangle\langle\gamma|[\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]|\alpha\rangle e^{-i\vec{k}\vec{x}+i\omega t}$
$= \frac{1}{tr\, e^{\beta H'}}\int d^dx dt \sum_\alpha e^{\beta \omega_\alpha}\langle\alpha|[\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]|\alpha\rangle e^{-i\vec{k}\vec{x}+i\omega t}$

However, I don't know how the operators $\phi_a(t,\vec{x})$ act on the eigenstates of H' which I called $|\alpha\rangle$ here. So I don't really know how to proceed.