6 research outputs found
Renormalization Group Functions of the \phi^4 Theory in the Strong Coupling Limit: Analytical Results
The previous attempts of reconstructing the Gell-Mann-Low function \beta(g)
of the \phi^4 theory by summing perturbation series give the asymptotic
behavior \beta(g) = \beta_\infty g^\alpha in the limit g\to \infty, where
\alpha \approx 1 for the space dimensions d = 2,3,4. It can be hypothesized
that the asymptotic behavior is \beta(g) ~ g for all values of d. The
consideration of the zero-dimensional case supports this hypothesis and reveals
the mechanism of its appearance: it is associated with a zero of one of the
functional integrals. The generalization of the analysis confirms the
asymptotic behavior \beta(g)=\beta_\infty g in the general d-dimensional case.
The asymptotic behavior of other renormalization group functions is constant.
The connection with the zero-charge problem and triviality of the \phi^4 theory
is discussed.Comment: PDF, 17 page
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 ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . 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.