Symmetry of the Renormalization Group in $p$-Adic Models

Bosonic and fermionic fields are considered on a ball in a $d$-dimensional $p$-adic space. These fields are defined by a Hamiltonian whose Gaussian part is invariant with respect to the Wilson renormalization group (RG) $R(\alpha)$ with parameter $\alpha$ and the non-Gaussian part is a formal series of finite-particle Hamiltonians. Let $F$ be a functional map applied only to the non-Gaussian part of $H$. A new symmetry of the renormalization group is defined by the commutator relation $R(\alpha )FH=FR(2d-\alpha )H$. As a consequence of this symmetry, the non-Gaussian branch of the stable points of the RG with $\alpha =d/2$ bifurcates from the fixed point corresponding to a constant (zero) random field.