D.2 Non-Interacting Bosons

(D.11) |

where is the energy of mode with the polarization , , and are the Bosons annihilation and creation operators. The time-evolution of the annihilation operator in the HEISENBERG picture is

(D.12) |

so the operator obeys the equation

(D.13) |

which has the solution

(D.14) |

The creation operator for Bosons is the just the HERMITian conjugate of , i.e.

(D.15) |

The non-interacting real-time GREEN's functions for Bosons in momentum representation are now given by

where , , , and is the occupation number of the state , where under thermal equilibrium one obtains , with denoting the Bose-EINSTEIN distribution function (Appendix C.2). The GREEN's functions depend only on time differences. One usually Fourier transforms the time difference coordinate, , to energy