Schrödinger invariance and spacetime symmetries

Abstract

The free Schrödinger equation with mass M can be turned into a non-massive Klein–Gordon equation via Fourier transformation with respect to M . The kinematic symmetry algebra sch d of the free d-dimensional Schrödinger equation with M fixed appears therefore naturally as a parabolic subalgebra of the complexified conformal algebra conf d+2 in d+2 dimensions. The explicit classification of the parabolic subalgebras of conf 3 yields physically interesting dynamic symmetry algebras. This allows us to propose a new dynamic symmetry group relevant for the description of ageing far from thermal equilibrium, with a dynamical exponent z=2. The Ward identities resulting from the invariance under conf d+2 and its parabolic subalgebras are derived and the corresponding free-field energy–momentum tensor is constructed. We also derive the scaling form and the causality conditions for the two- and three-point functions and their relationship with response functions in the context of Martin–Siggia–Rose theory.