Thermal coherent states, a broader class of mixed coherent states, and generalized thermo-field dynamics

We introduce coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at nonzero temperature (T ≠ 0), as a "random" (or "thermal" or "noisy") basis in Hilbert space. A resolution of the identity for these states is proven and is used to generalize the usual pure (T = 0) coherent state formalism to the mixed (T ≠ 0) case. This new formalism for thermal coherent states is then further generalized to a broader class of so-called negative-binomial mixed states. It is known that the negative-binomial distribution is itself intimately related to the discrete series of SU(1,1) representations. We consider the pure SU(1,1) coherent states in the two-mode harmonic oscillator space, and show how our negative-binomial mixed states arise from taking the partial trace with respect to one of these two modes. This observation is then used to show how the formalism of thermo-field dynamics may be generalized to a correspondingly much broader negative-binomial-field dynamics, which we expect to have many uses.