large N limit




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In theoretical physics, one often considers gauge theory models whose symmetries are groups of large matrices, notably in the special unitary group SU(N)SU(N). The limit of such theories for NN\to \infty (“large number of colours-limit”) has often remarkable properties and string-like features.

This limit is sometimes appearing in its own right, but sometimes it is just considered as an approximation for a system with fixed finite NN. One of the features is that in the large N limit is that non-planar Feynman diagrams lose their importance and that the correlation functions satisfy certain decoupling/factorization rule. The behaviour is studied in terms of expansion in 1/N1/N whose square has a simialr role to Planck constant in semiclassical approximation limit of quantum mechanics.


On the large N limit of QCD at fixed 't Hooft coupling in terms of planar Feynman diagrams:

See also:

  • A. A. Migdal, Loop equations and 1/N expansion, Physics Reports, 102 (4), 199-290 (1983) doi

  • Laurence G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982) pdf

  • Sidney Coleman, 1/N, in Aspects of Symmetry, Cambridge University Press 1985.

  • A. V. Manohar, Large N QCD, L:es Houches Lecture 2004, pdf

  • A. Jevicki, Instantons and the 1/N1/N expansion in nonlinear σ\sigma models, Phys. Rev. D 20, 3331–3335 (1979) pdf

  • Juan Maldacena, The large NN limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2:231-252, 1998 hep-th/9711200; Wilson loops in large N field theories, hep-th/9803002.

  • E. Brézin, S.R. Wadia, eds. The Large N Expansion in Quantum Field Theory and Statistical Physics, a book collection of reprinted historical articles, gBooks

  • Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks

  • M. Bershadsky, Z. Kakushadze, C. Vafa, String expansion as large N expansion of gauge theories, Nucl.Phys. B523 (1998) 59-72hep-th/9803076, doi

  • G.T. Horowitz, H. Ooguri, Spectrum of large N gauge theory from supergravity, hep-th/9802116

  • Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, arxiv/1309.7333

  • wikipedia 1/N expansion

category: physics

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