D.1 Non-Interacting FERMIons

(D.1) |

where is the single-particle energy measured with respect to the FERMI energy and are the FERMIon annihilation and creation operators, respectively (Appendix A). The time-evolution of the annihilation operator in the HEISENBERG picture is (Appendix B)

(D.2) |

so the operator obeys the equation

(D.3) |

which has the solution

(D.4) |

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

(D.5) |

The non-interacting real-time GREEN's functions (Section 3.7.1) for FERMIons in momentum representation are now given by

where is the average occupation number of the state . The GREEN's functions depend only on time differences. One usually Fourier transforms the time difference coordinate, , to energy

where is a small positive number. Assuming that the particles are in thermal equilibrium one obtains , where is the FERMI-DIRAC distribution function (Appendix C.1). The result (D.7) shows that and provide information about the statistics, such as occupation or un-occupation of the states, and and provide information about the states regardless of their occupation. The spectral function for FERMIons is therefore defined as

(D.8) |

where the following relation is used

(D.9) |

where indicates the principal value. Under equilibrium the lesser and greater GREEN's functions can be rewritten as