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

TL;DR

This paper demonstrates that extending the Schrödinger Lie algebra to a maximal parabolic sub-algebra, combined with dualisation, can derive causality conditions for response functions in non-equilibrium statistical mechanics.

Contribution

It shows how algebraic extensions of the Schrödinger symmetry can be used to derive causality conditions in non-equilibrium systems.

Findings

Extension of Schrödinger algebra leads to causality in response functions.

Dualisation approach is effective in deriving physical response conditions.

Logarithmic extension of the algebra is compatible with causality derivation.

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.