Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Procli Diadochi sphaera: astronomiam discere incipientibvs incipientibus vtilissima utilissima

Procli Diadochi sphaera: astronomiam discere incipientibvs incipientibus vtilissima utilissim

Procli sphaera

Thoma Linacro Britanno interpret

Procli insignis philosophi compendiaria de Motu disputatio, posteriores quinque Aristotelis de auscultatione naturali libros, mira brevitate complectens

Mit Druckermarke BebelsMit Vorwort von Simon Grynaeus vom 19.8.1531Offizin dem Kolophon entnommen: "Basileae, per Io. Bebelium, et Mich. Ysingrinium, mense Augusto, anno M. D. XXXI."Originaltitel: Institutio physic

Tabulae institutionum astronomicarum : adiuncta est Sphaera Procli cum textu Graeco e regione in Latinam linguam converso

Georgii Henischi

Eis ton tou Platōnos Timaion hypomnēmatōn Proklou Bib. Ē ... = : In Platonis Timaeon Commentariorum Procli Libri quinque ... ; Kai eis tēn tou autou Politikēn chalepoterōn zētēmatōn hapantōn exēgēsis ... = Et in eiusdem Politicen Difficiliorum quaestionum omnium enarratio

Impressum gemäss Hieronymus/Griechischer GeistDer vorliegende Druck mit den Kommentaren des Proklos ist ursprünglich Teil von: Platon, Opera omnia. Basel : Johannes Walder, 1534 (VD16 P 3275