On some properties of renormalization group transformation in the Euclidean models

Exact solution of the hierarchical fermionic model generates list of conjectures for the renormalization group (RG) properties in the Euclidean models. Here we discuss two of them. Let us consider fields in the unit ball of the d-dimensional Euclidean space given by the Hamiltonians of the type H(a) + H. Here H(a) is the Gaussian part of the Hamiltonian which is invariant under Wilsons renormalization group transformation with parameter a. Let R(a) denote the renormalization group transformation and let F denote the functional Fourier transformation in the space of non-Gaussian Hamiltonians H. We prove that FR(a)=R(2d-a)F in terms of formal functional integrals and discuss some consequences of this commutation relation. If we consider lattice 2N-component bosonic and fermionic fields given with Hamiltonians of the type H(a) + H, where Gaussian Hamiltonians H(a) are invariant under block-spin RG transformation with parameter a and the non-Gaussian parts H are sums of the O(2N)-invariant polynomials with given sets of coupling constants. Then RG transformation R(a;N) in the coupling constants space of fermionic model formally is equal to the RG transformation R(a;-N) in the coupling constants space of (-2N)-component bosonic model.