On Scale Versus Conformal Symmetry in Turbulence

We consider the statistical description of steady state fully developed incompressible fluid turbulence at the inertial range of scales in any number of spatial dimensions. We show that turbulence statistics is scale but not conformally covariant, with the only possible exception being the direct enstrophy cascade in two space dimensions. We argue that the same conclusions hold for compressible non-relativistic turbulence as well as for relativistic turbulence. We discuss the modification of our conclusions in the presence of vacuum expectation values of negative dimension operators. We consider the issue of non-locality of the stress-energy tensor of inertial range turbulence field theory.