×

zbMATH — the first resource for mathematics

Geometric analysis of \(\phi^ 4\) fields and Ising models. I, II. (English) Zbl 0533.58034
This is the first part of a detailed proof of a result announced earlier [the author, Phys. Rev. Lett. 47, 1-4 (1981); Mathematical Problems in Theoretical Physics, Proc. IAMP Conf. Berlin, 1981, Lect. Notes Phys. 153, 37–46 (1982)], concerning the free character of Euclidean \(\phi^ 4\) field theory in \(d>4\) dimensions. The analysis is nonperturbative and exploits the representation of field variables (or of spins in the isomorphic Ising models) as source-sink creation operators in a system of random currents. The onset of long-range order is attributed to percolation in an ensemble of sourceless currents and the interaction in the \(\phi^ 4\) field (or the critical behavior of the Ising models) is directly related to the intersection properties of long current clusters. Insight into the critical nature of \(d=4\) is derived from an analogy (due to Kurt Symanzik) with intersection properties of Brownian paths. It is also shown that in high dimensions \(d>4\) the critical behavior of Ising models is in exact agreement with mean field theory, but for \(d=2\) this is not true and hyperscaling is ”universal”.
Part I of the paper deals with Ising systems. Part II deals with the \(\phi^ 4\) field theory. An Appendix discusses the geometric aspects of the criticality of \(d=4\). Part III, which will be published separately, deals with random-surface phenomena, roughening transitions, stochastic geometry of \(\mathbb Z(2)\) lattice gauge theory and frustration in spin glasses.
Reviewer: M.E.Mayer

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
58J90 Applications of PDEs on manifolds
81T08 Constructive quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aizenman, M.: Phys. Rev. Lett.47, 1 (1981) [and a contribution in Mathematical problems in theoretical physics. (Proceedings, Berlin 1981). Berlin, Heidelberg, New York: Springer (to appear) · doi:10.1103/PhysRevLett.47.1
[2] Symanzik, K.: Euclidean quantum field theory. In: Local quantum theory. Jost, R. (ed.). New York: Academic Press 1969
[3] Fröhlich, J.: Nucl Phys.B200, [FS4] 281 (1982) · doi:10.1016/0550-3213(82)90088-8
[4] McBryan, O.A., Rosen, J.: Commun. Math. Phys.51, 97 (1976) · doi:10.1007/BF01609341
[5] Simon, B.: Commun. Math. Phys.77, 111 (1980) · doi:10.1007/BF01982711
[6] Griffiths, R., Hurst, C., Sherman, S.: J. Math. Phys.11, 790 (1970) · doi:10.1063/1.1665211
[7] Newman, C., Schulman, L.: J. Stat. Phys.26, 613 (1981) · Zbl 0509.60095 · doi:10.1007/BF01011437
[8] Sokal, A.D.: Phys. Lett.71A, 451 (1979)
[9] Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976) · doi:10.1007/BF01608557
[10] Kakutani, S.: Proc. Japan Acad.20, 648 (1944) · Zbl 0063.03106 · doi:10.3792/pia/1195572742
[11] Dvoretsky, A., Erdös, P., Kakutani, S.: Acta Sci. Math. (Szeged)12, 75 (1950)
[12] Lebowitz, J.: Commun. Math. Phys.35, 87 (1974) · doi:10.1007/BF01646608
[13] Glimm, J., Jaffe, A.: Ann. Inst. Henri Poincaré A22, 97 (1975)
[14] Glimm, J., Jaffe, A.: Commun. Math. Phys.51, 1 (1976) · doi:10.1007/BF01609048
[15] Sokal, A.D.: Ann. Inst. Henri Poincaré (to appear)
[16] Glimm, J., Jaffe, A.: Phys. Rev. D10, 536 (1974)
[17] Widom, R.: J. Chem. Phys.43, 3892 (1965) · doi:10.1063/1.1696617
[18] Kadanoff, L.P., et al.: Rev. Mod. Phys.39, 395 (1967) · doi:10.1103/RevModPhys.39.395
[19] Fisher, M.: Rep. Prog. Phys.30, 615 (1967), and references therein · doi:10.1088/0034-4885/30/2/306
[20] Schrader, R.: Phys. Rev. B,14, 172 (1976) · doi:10.1103/PhysRevB.14.172
[21] Lieb, E.H., Sokal, A.D.: In preparation
[22] Constructive quantum field theory. Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973
[23] Simon, B.: TheP(?)2 euclidean (quantum) field theory. Princeton, NS: Princeton University Press 1974
[24] Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0461.46051
[25] Simon, B., Griffiths, R.: Commun. Math. Phys.33, 145 (1973) · doi:10.1007/BF01645626
[26] Sokal, A.D.: Mean-field bounds and correlation inequalities. J. Stat. Phys. (to appear)
[27] Newman, C.: Commun. Math. Phys.41, 1 (1975) · doi:10.1007/BF01608542
[28] Newman, C.: Z. Wahrscheinlichkeitstheorie33, 75 (1975) · Zbl 0297.60053 · doi:10.1007/BF00538350
[29] Schrader, R.: Phys. Rev. B15, 2798 (1977)
[30] Messager, A., Miracle-Sole, S.: J. Stat. Phys.17, 245 (1977) · doi:10.1007/BF01040105
[31] Aizenman, M.: On brownian motion ind=4 dimensions (in preparation)
[32] Graham, R.: Correlation inequalities for the truncated two-point function of an Ising ferromagnet (J. Stat. Phys., to appear) and: An improvement of the GHS inequality for the Ising ferromagnet (J. Stat. Phys., to appear)
[33] Aizenman, M., Graham, R.: A bound on the renormalized coupling in the ? d 4 field theory ind=4 dimensions (in preparation)
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.