Convergence of Migdal-Kadanoff iterations: A simple and general proof. (English) Zbl 0664.43004

Migdal-Kadanoff recursion relations for lattice gauge theories realize exact renormalization group transformations on hierarchical lattices embedded in \({\mathbb{Z}}^ d\). The authors prove a convergence of Migdal- Kadanoff iterations using the following constructions. Let \({\mathcal G}\) be a compact connected Lie group and \({\mathcal F}\) be the class of real-valued functions on \({\mathcal G}\). Defining for arbitrary \(r\in {\mathbb{N}}\setminus \{1\}\) and \(q\in {\mathbb{N}}\) the mapping \({\mathcal T}: {\mathcal F}\to {\mathcal F}\) by \({\mathcal T}(g)=K(g^{*r})^ q\) which involves a multiple convolution product of r factors, and \(K\in {\mathbb{R}}_+\) is determined by the normalization condition at the group unit, one can demonstrate that these cover Migdal’s recursion relation for gauge models with compact connected gauge groups. Interchanging the convolution and product in the definition of \({\mathcal T}\), we are lead to Kadanoff’s recursion relation. Estimating the rate of convergence, the authors derive lower bounds depending on the coupling strength of the initial action, for the string tension of hierarchical gauge models and for the mass gap in the hierarchical \({\mathcal O}({\mathcal N})\) Heisenberg models in their respective critical four and two dimensions.
Reviewer: E.Kryachko


43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A75 Harmonic analysis on specific compact groups
81T17 Renormalization group methods applied to problems in quantum field theory
22E70 Applications of Lie groups to the sciences; explicit representations
Full Text: DOI


[1] Migdal, A. A., JETP 69, 810, 1457 (1975).
[2] Kadanoff, L., Ann. Phys. (N. Y.) 100, 359 (1976).
[3] Bleher, P. M. and Zalys, E., Commun. Math. Phys. 67, 17 (1979).
[4] Collet, P. and Eckmann, J.-P., Commun. Math. Phys. 93, 379 (1984). · Zbl 0553.60098
[5] Ito, K. R., Commun. Math. Phys. 95, 247 (1984).
[6] Ito, K. R., Phys. Rev. Lett. 54, 2383; 55, 558 (1985).
[7] Ito, K. R., Commun. Math. Phys. 110, 237 (1987).
[8] Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis II, Springer, Berlin, 1970. · Zbl 0213.40103
[9] Bröcker, T. and tom Dieck, T., Representations of Compact Lie Groups, Springer, New York, 1985. · Zbl 0581.22009
[10] Müller, V. F. and Schiemann, J., Commun. Math. Phys. 97, 605 (1985). · Zbl 1223.81150
[11] Guth, A., Phys. Rev. D21, 2291 (1980). Fröhlich, J. and Spencer, T., Commun. Math. Phys. 81, 527 (1981); 83, 411, (1982).
[12] José, J., Kadanoff, L., Kirkpatrick, S., and Nelson, D., Phys. Rev. B 16, 1217 (1977).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.