Abstract
In this talk methods for a rigorous control of the renormalization group (RG) flow of field theories are discussed. The RG equations involve the flow of an infinite number of local partition functions. By the method of exact beta-function the RG equations are reduced to flow equations of a finite number of coupling constants. Generating functions of Greens functions are expressed by polymer activities. Polymer activities are useful for solving the large volume and large field problem in field theory. The RG flow of the polymer activities is studied by the introduction of polymer algebras. The definition of products and recursive functions replaces cluster expansion techniques. Norms of these products and recursive functions are basic tools and simplify a RG analysis for field theories. The methods will be discussed at examples of the Φ 4-model, the O(N) σ-model and hierarchical scalar field theory (infrared fixed points).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
T. Balaban. Renormalization Group Approach to Lattice Gauge Field Theories. I. Generation of Effective Actions in a Small Field Approximation and a Coupling Constant Renormalization in Four Dimensions, Commun. Math. Phys. 109, 249 (1987).
T. Balaban. Renormalization Group Approach to Lattice Gauge Field Theories. II. Cluster Expansions, Commun. Math. Phys. 116, 1 (1988).
T. L. Bell, K. Wilson. Finite-lattice approximations to renormalization groups, Phys. Rev. B, Vol. 11, 3431 (1974).
D. Brydges. A short course on cluster expansions, In Critical Phenomena, Random Systems, Gauge Theories 1984.
D. Brydges, H. Yau. Grad φ Perturbations of Massless Gaussian Fields, Commun. Math. Phys. 129, 351 (1990).
P. Collet, J. P. Eckmann. The ε-expansion for the Hierarchical Model, Commun. Math. Phys. 55, 67 (1977).
P. Collet, J. P. Eckmann. A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics, Lecture Notes in Physics, Vol. 74, Berlin, Heidelberg, New York, Springer 1978.
J. Dimock, T. Hurd. A Renormalization Group Analysis of the Kosterlitz-Thouless Phase, Commun. Math. Phys. 137, 263–287 (1991).
J. Dimock, T. Hurd. A Renormalization Group Analysis of correlation functions for the dipole gas, J. Stat. Phys. 66, 1277–1318 (1992).
J. Dimock, T. Hurd. A Renormalization Group Analysis of the Kosterlitz-Thouless Phase, Commun. Math. Phys. 156, 547–580 (1993).
G. Felder. Renormalization Group in the Local Approximation, Commun. Math. Phys. 111, 101–121 (1987).
J. Feldman, J. Magnen, V. Rivasseau, R. Sénéor. Construction and Borel Summability of Infrared φ 44 by a Phase Space Expansion, Commun. Math. Phys. 109, 137 (1987).
K. Gawędzki, A. Kupiainen. A Rigorous Block Spin Approach to Massless Lattice Theories, Commun. Math. Phys. 77, 31 (1980).
J. Gawędzki, A. Kupiainen. Block Spin Renormalization Group for Dipole Gas and (∇Φ)4, Ann. Phys. 147, 198 (1983).
K. Gawędzki, A. Kupiainen. Massless Lattice φ 4 4 4 Theory: Rigorous Control of a Renormalizable Asymptotically Free Model, Commun. Math. Phys. 99, 197–252 (1985).
J. Glimm, A. Jaffe. The Positivity of the ϕ 43 Hamiltonian, Fortschr. Phys. 21, 327 (1973).
J. Glimm, A. Jaffe. Quantum Physics, Springer Verlag, Heidelberg 1987, Second Edition.
C. Gruber, H. Kunz. General Properties of Polymer Systems, Commun. Math. Phys. 22, 133 (1971).
H. Koch, P. Wittwer. A Non-Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories, Commun. Math. Phys. 106, 495–532 (1986).
H. Koch, P. Wittwer. On the Renormalization Group Transformation for Scalar Hierarchical Models, Commun. Math. Phys. 138, 537 (1991).
G. Mack, A. Pordt. Convergent Perturbation Expansions for Euclidean Quantum Field Theory, Commun. Math. Phys. 97, 267 (1985).
G. Mack, A. Pordt. Convergent Weak Coupling Expansions for Lattice Field Theories that Look Like Perturbation Series, Rev. Math. Phys. 1, 47 (1989).
J. Magnen, R. Sénéor. Phase Space Cell Expansion and Bore] Summability of the Euclidean Φ 3 4-Theory, Commun. Math. Phys. 56, 237 (1977).
K. Pinn, A. Pordt, C. Wieczerkowski. Algebraic Computation of Hierarchical Renormalization Group Fixed Points and their ε-Expansions, Preprint Munster MS-TPI-94-02.
A. Pordt, C. Wieczerkowski. Nonassociative Algebaas and Nonperturbative Field Theory for Hierarchical Models, Preprint Miinster MS-TPI-9404.
A. Pordt. A Tree Formula for Use in Cluster Expansions with General Multibody Interactions, Z. Phys. C 31 (1986).
A. Pordt. Renormalization Theory for Hierarchical Models, Helv. Phys. Acta, Vol. 66, 105 (1993).
V. Rivasseau. From Perturbatioe to Constructive Renormalization, Princeton University Press 1991.
D. Ruelle. Statistical Mechanics, W. A. Benjamin Inc., 1969.
K. G. Wilson. Renormalization Group and Critical Phenomena. II. Phase Space Cell Analysis of Critical Behavior, Phys. Rev. B4, 3184 (1971).
K. G. Wilson. The Renormalization Group and Critical Phenomena, Rev. Mod. Phys. 55, 583 (1983).
K. G. Wilson, J. Kogut. The Renormalization Group and the ge-expansion, Phys. Rep. 12, 76 (1974).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag
About this paper
Cite this paper
Pordt, A. (1995). On renormalization group flows and polymer algebras. In: Rivasseau, V. (eds) Constructive Physics Results in Field Theory, Statistical Mechanics and Condensed Matter Physics. Lecture Notes in Physics, vol 446. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59190-7_22
Download citation
DOI: https://doi.org/10.1007/3-540-59190-7_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59190-0
Online ISBN: 978-3-540-49222-1
eBook Packages: Springer Book Archive