Renormalization-group approach in the theory of turbulence: The dimensions of composite operators

In the framework of the renormalization-group approach in the theory of turbulence proposed by De Dominieis and Martin [1], the problem of renormalization and determination of the critical dimensions of composite operators is discussed. The renormalization of the system of operators of canonical dimension $4$, which includes the operator $F=\varphi\Delta\varphi$, where $\varphi$ is the velocity field, is considered. It is shown that the critical dimension $\Delta_F$ associated with this operator is exactly equal to the Kolmogorov dimension: $\Delta_F=0$. The Appendix gives brief proofs of, first, a theorem on the equivalence of an arbitrary stochastic problem and quantum field theory and, second, a theorem that determines the restriction of the Green's functions of a stochastic problem to a simultaneity surface.