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 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 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 quantitative Monte-Carlo sampling method, as a function of temperature and trap frequency of the external confinement.

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