Dynamical correlation functions of the transverse spin and energy density for the one-dimensional spin-½ Ising model with a transverse field

Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-½ Ising model with a transverse field $H=-\Gamma {\Sigma }_i{S}_i^x-J{\Sigma }_i{S}_i^z{S}_{i+1\mathrm{}}^z$. Explicit results are obtained for $〈\left\{S^x\right(q, \omega \left)S^x\mathrm{}\right(-q\left)\right\}〉$, $〈\left\{E\right(q, \omega ), E(-q\left)\right\}〉$, and their Fourier transforms in the limits $T=\infty$ and $T=0\mathrm{}$ and for the autocorrelation functions $〈\{S_i^x, S_i^x\mathrm{}(t\left)\right\}〉$ and $〈\{E_i, E_i\mathrm{}(t\left)\right\}〉$ in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.