Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

Abstract

Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in d spatial dimensions. For d = 1 spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for d = 1 and d = 2 and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber–Ising chain with spatially long-ranged initial conditions is described.