Hierarchical Dyson model and \(p\)-adic conformal invariance
Let \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers. We call the group, generated by the inversion, scaling transformations and all isometries of \(P (\mathbb{Q}_p) \equiv \mathbb{Q}_p \cup \{\infty\}\) the conformal group \(K\). Let \(\varphi\) be a generated random field over a space of test functions \({\mathcal L}\) (we consider functions from \(P (\mathbb{Q}_p)\) to \(\mathbb{R})\). In the space of test functions we introduce the group transformation \(C_a\): \(g(x) \to g'(x) = \lambda (x)^{a/2} g(f(x))\), \(f \in K\). Here \(g \in {\mathcal L}\), \(0 < a < 2\), and \(\lambda (x)\) is the Radon-Nikodym derivative of the measure \(df(x)\) with respect to the Haar measure \(dx\). We say that the field \(\varphi\) is conformal invariant with parameter \(a\) if it is a fixed point of any transformation of the dual group \(C_a'\). The main result of this paper is a rigorous construction of the scale invariant non-Gaussian random field, that represents a continuum limit of the hierarchical Dyson model, and a proof of the conformal invariance of the field. This field is the \(\varphi^4\)-model of \(p\)-adic quantum field theory [see \textit{E. Yu. Lerner} and \textit{M. D. Missarov}, Theor. Math. Phys. 78, No. 2, 177-184 (1989); translation from Teor. Mat. Fiz. 78, No. 2, 248-257 (1989)]. The problem is reformulated in terms of the hierarchical model defined on the Bruhat-Tits tree. The renormalization-group transformation for random fields on the Bruhat-Tits tree is defined. The condition of conformal invariance goes over into a condition of a new symmetry of the hierarchical model.
continuum limit of the hierarchical Dyson model
scientific article; zbMATH DE number 610023