Existence and construction of Galilean invariant $z\ne 2$ theories

We prove a no-go theorem for the construction of a Galilean boost invariant and $z\ne 2$ anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the nonrelativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a ($d+2$)-dimensional space-time is proposed to be dual of a $d$-dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent $z=\frac{2\ell }{\ell +1}$, $\ell \in \mathbb{Z}$.