Causality from dynamical symmetry: an example from local scale-invariance

Abstract

Physical ageing phenomena far from equilibrium naturally lead to dynamical scaling. It has been proposed to consider the consequences of an extension to a larger Lie algebra of local scale-transformation. The best-tested applications of this are explicitly computed co-variant two-point functions which have been compared to non-equilibrium response functions in a large variety of statistical mechanics models. It is shown that the extension of the Schr\"odinger Lie algebra $\mathfrak{sch}(1)$ to a maximal parabolic sub-algebra, when combined with a dualisation approach, is sufficient to derive the causality condition required for the interpretation of a two-point function as a physical response function. The proof is presented for the recent logarithmic extension of the differential operator representation of the Schr\"odinger algebra.