×

zbMATH — the first resource for mathematics

The renormalization group according to Balaban. III: Convergence. (English) Zbl 1302.81139
Summary: This is an expository account of Balaban’s approach to the renormalization group. The method is illustrated with a treatment of the ultraviolet problem for the scalar \(\phi^4\) model on a toroidal lattice in dimension \(d=3\). In this third paper we demonstrate convergence of the expansion and complete the proof of a stability bound.
For Part I see [the author, Rev. Math. Phys. 25, No. 7, Article ID 1330010, 64 p. (2013; Zbl 1275.81068)] and for Part II see [the author, J. Math. Phys. 54, No. 9, 092301, 85 p. (2013; Zbl 1284.81217)].

MSC:
81T17 Renormalization group methods applied to problems in quantum field theory
81T10 Model quantum field theories
81T25 Quantum field theory on lattices
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Balaban, T., (Higgs)_{2,3} quantum fields in a finite volume I, Commun. Math. Phys., 85, 603-636, (1982)
[2] Balaban, T., (Higgs)_{2,3} quantum fields in a finite volume II, Commun. Math. Phys., 86, 555-594, (1982)
[3] Balaban, T., (Higgs)_{2,3} quantum fields in a finite volume III, Commun. Math. Phys., 88, 411-445, (1983)
[4] Balaban, T., Regularity and decay of lattice green’s functions, Commun. Math. Phys., 89, 571-597, (1983) · Zbl 0555.35113
[5] Balaban, T., A low temperature expansion for classical N-vector models I, Commun. Math. Phys., 167, 103-154, (1995) · Zbl 0811.60100
[6] Balaban, T., Variational problems for classical N-vector models, Commun. Math. Phys., 175, 607-642, (1996) · Zbl 0838.60092
[7] Balaban, T., Localization expansions I, Commun. Math. Phys., 182, 33-82, (1996) · Zbl 0878.60072
[8] Balaban, T., A low temperature expansion for classical N-vector models II, Commun. Math. Phys., 182, 675-721, (1996) · Zbl 0869.60095
[9] Balaban, T., A low temperature expansion for classical N-vector models III, Commun. Math. Phys., 196, 485-521, (1998) · Zbl 0924.60096
[10] Balaban, T., Renormalization and localization expansions II, Commun. Math. Phys., 198, 1-45, (1998) · Zbl 0932.81020
[11] Balaban, T., Large field renormalization operation for classical N-vector models, Commun. Math. Phys., 198, 493-534, (1998) · Zbl 0940.81034
[12] Balaban, T.; O’Carroll, M., Low temperature properties for correlation functions in classcial \(N\)-vector spin models, Commun. Math. Phys., 199, 493-520, (1999) · Zbl 0932.81024
[13] Brydges, D.; Dimock, J.; Hurd, T.R., The short distance behavior of \({ϕ^4_3}\), Commun. Math. Phys., 172, 143-186, (1995) · Zbl 0858.60095
[14] Dimock, J.: The renormalization group according to Balaban - I. small fields. Rev. Math. Phys. 25, 1330010, 1-64 (2013) · Zbl 1275.81068
[15] Dimock, J.: The renormalization group according to Balaban - II. large fields. J. Math. Phys. 54, 092301, 1-85 (2013). · Zbl 1284.81217
[16] Feldman, J.; Osterwalder, K., The wightman axioms and the mass gap for weakly coupled \({ϕ^4_3}\) quantum field theories, Ann. Phys., 97, 80-135, (1976)
[17] Glimm, J.; Jaffe, A., Positivity of the \({ϕ^4_3}\) Hamiltonian, Fortschritte der Physik, 21, 327-376, (1973)
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.