Invariant manifolds of \(p\)-adic renormalization groups (Q1803220)

Properties of \(p\)-adic models of quantum field theory and statistical mechanics are studied. The \(p\)-adic renormalization transformation is generalized to the case of noninteger dimension and the invariant manifolds of renormalization groups in terms of \((4-d)\)-expansion and dimensional renormalization are discussed. A generalized Hamiltonian is defined as a sequence of functions with a complex number \(\nu\) (dimension). In order to construct nontrivial invariant manifolds of a renormalization group one considers the class of discretized Hamiltonians. It is shown that the non-Gaussian branch of the fixed points of the renormalization group bifurcates from the Gaussian one at \(\varepsilon=0\), where \(\varepsilon= 4-\nu\). It is concluded that, unlike the real case, there is no anomalous dimension in this theory, because \(p\)-adic self energy amplitudes have no divergences.