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It has long been known that two-point functions of conformal field theory(CFT) are nothing but the integral kernels of intertwining operators for twoequivalent representations of conformal algebra. Such intertwining operatorsare known to fulfill some operator identities---the intertwining relations---inthe representation space of conformal algebra. Meanwhile, it has been knownthat the S-matrix operator in scattering theory is nothing but the intertwiningoperator between the Hilbert spaces of in- and out-particles. Inspired by thisalgebraic resemblance, in this paper we develop a simple Lie-algebraic approachto momentum-space two-point functions of thermal CFT living on the hyperbolicspacetime $\mathbb{H}^{1}\times\mathbb{H}^{d-1}$ by exploiting the idea ofKerimov's intertwining operator approach to exact S-matrix. We show that inthermal CFT on $\mathbb{H}^{1}\times\mathbb{H}^{d-1}$ the intertwiningrelations reduce to certain linear recurrence relations for two-point functionsin the complex momentum space. By solving these recurrence relations, we obtainthe momentum-space representations of advanced and retarded two-point functionsas well as positive- and negative-frequency two-point Wightman functions for ascalar primary operator in arbitrary spacetime dimension $d\geq3$.

Intertwining Operator in Thermal CFT$_{d}$