  The free Schr\"odinger 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 Schr\"odinger
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.
