A Causal Algebra for Liouville Exponentials

A causal Poisson bracket algebra for Liouville exponentials on a cylinder is derived using an exchange algebra for free fields describing the in and out asymptotics. The causal algebra involves an even number of space-time points with a minimum of four. A quantum realisation of the algebra is obtained which preserves causality and the local form of non-equal time brackets. Following Polyakov’s work on the relativistic string [1] the quantum Liouville theory became the subject of intense study. Indeed, the quantisation of Liouville theory is a deep problem in its own right. Early approaches [2, 3, 4, 5, 6] were based on the canonical quantisation of structures present in the classical Liouville theory. In particular, exact forms were obtained for operators corresponding to negative integer and half-integer powers of the Liouville exponential. For arbitrary powers a formal power series can be obtained [6, 7]. In the 1990’s attention was focussed on the construction of correlation functions. Dorn and Otto [8] and Zamolodchikov and Zamolodchikov [9] provided formulae for three-point functions and gave a self-consistent framework for computing n-point functions. However, an explicit construction of arbitrary Liouville exponential operators is lacking (see however [10, 11]). In [12] it was noted that the Liouville exponential obeys a causal algebra at the classical and quantum level. The Poisson bracket of Liouville exponentials at different