Theory of Single-Particle Time-Correlation Functions

We present a general theory for the calculation of the single-particle time-correlation function $\alpha \mathrm{}\left(t\right)=〈[a,a^{†}\mathrm{}(t\left)\right]〉$ which is the canonical average of the commutator between the particle annihilation $a$ and creation $a^{†}\mathrm{}\left(t\right)\mathrm{}$ at time $t$. The theory is based on the projection-operator method. The complex spectral function $\hat{\alpha }\left(\omega \right)$ is expressed in terms of the natural frequency of oscillation $\Omega$ and the width function $\hat{\gamma }\left(\omega \right)$. From the analytical property of $\hat{\alpha }\left(\omega \right)$ in the complex $\omega$ plane for the weak-coupling limit, the long-time behavior of the correlation function $\alpha \mathrm{}\left(t\right)\mathrm{}$ and the relaxation functions is obtained. For a harmonic oscillator immersed in a heat bath, the perturbation calculations for $\Omega$ and $\hat{\gamma }\left(\omega \right)$ are given in the power of the coupling constant. By means of this series, the spectral function $\hat{\alpha }\left(\omega \right)$ for a single normal mode of an anharmonic system is explicitly calculated as a function of the frequency $\omega$ and the temperature $T$. As a possible application of the results the electrical conductivity due to a localized mode is discussed.