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Homework Help: Statistical and spectral function in thermal state

  1. Dec 4, 2013 #1
    1. The problem statement, all variables and given/known data
    The statistical and spectral functions for bosonic operators [itex]\phi_a[/itex] are:
    [itex] G_{ab}(t,\vec{x})=\frac{1}{2}\langle \{\phi_a(t,\vec{x}),\phi_b(0,\vec{0})\}\rangle [/itex],
    [itex] \rho_{ab}(t,\vec{x})=\langle [\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]\rangle [/itex].
    The expectation values are in static thermal equilibrium in the grand canonical ensemble with Hamiltonian [itex]H'=H-\mu Q[/itex].

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

    2. Relevant equations

    3. The attempt at a solution
    [itex]G_{ab}(\omega,\vec{k})[/itex] and [itex]\rho_{ab}(\omega,\vec{k})[/itex] re defined as Fourier transforms of the above expressions. The expectation values are given as
    [itex] \langle A \rangle = \frac{tr\, e^{\beta H'}A}{tr\, e^{\beta H'}}[/itex]
    [itex] \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}[/itex]
    [itex] = \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}[/itex]
    [itex] = \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}[/itex]
    [itex] = \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}[/itex]

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