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.

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Foundations of Mechanics, Second Edition

Preface to the Second Edition. Since the first edition of this book appeared in 1967, there has been a great deal of activity in the field of symplectic geometry and Hamiltonian systems. In addition to the recent textbooks of Arnold, Arnold-Avez, Godbillon, Guillemin-Sternberg, Siegel-Moser, and Souriau, there have been many research articles published. Two good collections are "Symposia Mathematica," vol. XIV, and "Géométrie Symplectique el Physique Mathématique," CNRS, Colloque Internationaux, no. 237. There are also important survey articles, such as Weinstein [1977b]. The text and bibliography contain many of the important new references we are aware of. We have continued to find the classic works, especially Whittaker [1959], invaluable. The basic audience for the book remains the same: mathematicians, physicists, and engineers interested in geometrical methods in mechanics, assuming a background in calculus, linear algebra, some classical analysis, and point set topology. We include most of the basic results in manifold theory, as well as some key facts from point set topology and Lie group theory. Other things used without proof are clearly noted. We have updated the material on symmetry groups and qualitative theory, added new sections on the rigid body, topology and mechanics, and quantization, and other topics, and have made numerous corrections and additions. In fact, some of the results in this edition are new. We have made two major changes in notation: we now use f^* for pull-back (the first edition used f[sub]*), in accordance with standard usage, and have adopted the "Bourbaki" convention for wedge product. The latter eliminates many annoying factors of 2. A. N. Kolmogorov's address at the 1954 International Congress of Mathematicians marked an important historical point in the development of the theory, and is reproduced as an appendix. The work of Kolmogorov, Arnold, and Moser and its application to Laplace's question of stability of the solar system remains one of the goals of the exposition. For complete details of all tbe theorems needed in this direction, outside references will have to be consulted, such as Siegel-Moser [1971] and Moser [1973a]. We are pleased to acknowledge valuable assistance from Paul Chernoff, Wlodek Tulczyjew, Morris Hirsh, Alan Weinstein, and our invaluable assistant authors, Richard Cushman and Tudor Ratiu, who all contributed some of their original material for incorporation into the text. Also, we are grateful to Ethan Akin, Kentaro Mikami, Judy Arms, Harold Naparst, Michael Buchner, Ed Nelson, Robert Cahn, Sheldon Newhouse, Emil Chorosoff, George Oster, André Deprit, Jean-Paul Penot, Bob Devaney, Joel Robbin, Hans Duistermaat, Clark Robinson, John Guckenheimer, David Rod, Martin Gutzwiller, William Satzer, Richard Hansen, Dieter Schmidt, Morris Kirsch, Mike Shub, Michael Hoffman, Steve Smale, Andrei Iacob, Rich Spencer, Robert Jantzen, Mike Spivak, Therese Langer, Dan Sunday, Ken Meyer, Floris Takens, [and] Randy Wohl for contributions, remarks, and corrections which we have included in this edition. Further, we express our gratitude to Chris Shaw, who made exceptional efforts to transfom our sketches into the graphics which illustrate the text, to Peter Coha for his assistance in organizing the Museum and Bibliography, and to Ruthie Cephas, Jody Hilbun, Marnie McElhiney, Ruth (Bionic Fingers) Suzuki, and Ikuko Workman for their superb typing job. Theoretical mechanics is an ever-expanding subject. We will appreciate comments from readers regarding new results and shortcomings in this edition. RALPH ABRAHAM, JERROLD E. MARSDEN</p