Dynamical derivation of vacuum operator-product expansion in Euclidean conformal quantum field theory

Abstract

An expansion of the type ${〈\ensuremath{\phi}({x}_{1})\ensuremath{\cdots}\ensuremath{\phi}({x}_{n})〉}_{0}={〈\ensuremath{\phi}({x}_{1})\ensuremath{\phi}({x}_{2})〉}_{0}{〈\ensuremath{\phi}({x}_{3})\ensuremath{\cdots}\ensuremath{\phi}({x}_{n})〉}_{0}+\ensuremath{\Sigma}\stackrel{}{{\ensuremath{\chi}}_{l}}{C}^{2}({\ensuremath{\chi}}_{l})\ensuremath{\int}(\mathrm{dp}){Q}^{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{l}}({x}_{1},{x}_{2};\ensuremath{-}p){w}_{{\ensuremath{\chi}}_{l}} (p){Q}^{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{l}}(p;{x}_{1},\dots{},{x}_{4})$ is derived, where ${\ensuremath{\chi}}_{l}=[l,{c}_{l}]$ are labels for infinite-dimensional symmetric tensor representations of the Euclidean conformal group ${\mathrm{O}}^{\ensuremath{\uparrow}}(2h+1,1)$,${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{l}=[l,\ensuremath{-}{c}_{l}]$, the constants $C ({\ensuremath{\chi}}_{l})$ are real, and ${Q}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}$ and ${w}_{\ensuremath{\chi}}$ have the properties of vacuum expectation values of field products. The starting point is an infinite set of coupled nonlinear integral equations for Euclidean Green's functions in $2h$ space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green's functions of the corresponding Gell-Mann-Low limit theory are expanded in conformal partial waves. The dynamical equations imply the existence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of the expansion we use some differential relations between partially equivalent exceptional representations of ${\mathrm{O}}^{\ensuremath{\uparrow}}(2h+1,1)$, established in an earlier paper. This work completes the group-theoretical derivation of the vacuum operator-product expansion undertaken by Mack in 1973.