Conformal Invariance and Field Theory in Two Dimensions

The relevance of the representation theory of the conformal group to quantum field theory is illustrated in two dimensions by the Thirring model. Space-time position operators and their complex extensions are defined as operator-valued functions of the generators. These complex position variables coincide with the Gel'fand-Naimark labels and can be interpreted as labels for nonorthogonal coherent states. A Hilbert-space metric then becomes necessary. It is given by the matrix elements of the metric operator $G$ and is nontrivial for the nonanalytic supplementary series and the analytic representations. In this case ${G}^{\ensuremath{-}1}$ gives the two-point function for the Thirring model. For nonanalytic representations only weak (infinitesimal) conformal invariance holds for interacting fields if causal and spectral properties are imposed, while those properties become compatible with strong (global) conformal invariance in the case of analytic representations which lead either to free fields or to interacting fields with a quantized value of the coupling constant.