  A theoretical particle-number conserving quantum field theory based on the
concept of imaginary time is presented and applied to the scenario of a
coherent atomic laser field at ultra-cold temperatures. The proposed
theoretical model describes the analytical derivation of the frequency comb
spectrum for an atomic laser realized from modeling a coherent atomic beam of
condensate and non-condensate quantum field components released from a trapped
Bose-Einstein condensate at a given repetition phase and frequency. The
condensate part of the atomic vapor is assumed to be subjected to thermal noise
induced by the temperature of the surrounding thermal atomic cloud. This new
quantum approach uses time periodicity and an orthogonal decomposition of the
quantum field in a complex-valued quantum field representation to derive and
model the quantum field's forward- and backward-propagating components as a
standing wave field in the same unique time and temperature domain without
quantitative singularities at finite temperatures. The complex-valued atom
laser field, the resulting frequency comb, and the repetition frequency
distribution with the varying shape of envelopes are numerically monitored
within a Monte-Carlo sampling method, as a function of temperature and trap
frequency of the external confinement.
