 Combinatorics and physics

Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
"Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory."^{[1]}.
"Physical combinatorics might be defined naively as combinatorics guided by ideas or insights from physics"^{[2]}.
Combinatorics has always played an important role in quantum field theory and statistical physics.^{[3]} However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer^{[4]}, showing that the renormalization of Feynman diagrams can be described by a Hopf algebra.
Combinatorial physics can be characterized by the use of algebraic concepts to interpret and solve physical problems involving combinatorics. It gives rise to a particularly harmonious collaboration between mathematicians and physicists.
Among the significant physical results of combinatorial physics we may mention the reinterpretation of renormalization as a RiemannHilbert problem^{[5]}, the fact that the Slavnov–Taylor identities of gauge theories generate a Hopf ideal^{[6]}, the quantization of fields^{[7]} and strings^{[8]} and a completely algebraic description of the combinatorics of quantum field theory^{[9]}. The important example of editing combinatorics and physics is relation between enumeration of Alternating sign matrix and Icetype model. Corresponding icetype model is six vertex model with domain wall boundary conditions.
Contents
See also
 Mathematical physics
 Statistical physics
 Ising model
 Percolation theory
 Tutte polynomial
 Partition function
 Hopf algebra
 Combinatorics and dynamical systems
 Bitstring physics
 Combinatorial hierarchy
 Quantum mechanics
References
 ^ 2007 International Conference on Combinatorial physics
 ^ Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0817641750
 ^ David Ruelle (1999). Statistical Mechanics, Rigorous Results. World Scientific. ISBN 9789810238629.
 ^ A. Connes, D. Kreimer, Renormalization in quantum field theory and the RiemannHilbert problem I, Commun. Math. Phys. 210 (2000), 249273
 ^ A. Connes, D. Kreimer, Renormalization in quantum field theory and the RiemannHilbert problem II, Commun. Math. Phys. 216 (2001), 215241
 ^ W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276 (2007), 773798
 ^ C. Brouder, B. Fauser, A. Frabetti, R. Oeckl, Quantum field theory and Hopf algebra cohomology, J. Phys. A: Math. Gen. 37 (2004), 58955927
 ^ T. Asakawa, M. Mori, S. Watamura, Hopf Algebra Symmetry and String Theory, Prog. Theor. Phys. 120 (2008), 659689
 ^ C. Brouder, Quantum field theory meets Hopf algebra, Math. Nachr. 282 (2009), 16641690
Further reading
 Some Open Problems in Combinatorial Physics, G. Duchamp, H. Cheballah
 Oneparameter groups and combinatorial physics, G. Duchamp, K.A. Penson, A.I. Solomon, A.Horzela, P.Blasiak
 Combinatorial Physics, Normal Order and Model Feynman Graphs, A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela, K.A. Penson
 Hopf Algebras in General and in Combinatorial Physics: a practical introduction, G. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, A.I. Solomon
 Discrete and Combinatorial Physics
 BitString Physics: a Novel "Theory of Everything", H. Pierre Noyes
 Combinatorial Physics, Ted Bastin, Clive W. Kilmister, World Scientific, 1995, ISBN 9810222122
 Physical Combinatorics and Quasiparticles, Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte
 Physical Combinatorics of NonUnitary Minimal Models, Hannah Fitzgerald
 Paths, Crystals and Fermionic Formulae, G.Hatayama, A.Kuniba, M.Okado, T.Takagi, Z.Tsuboi
 On powers of Stirling matrices, Istvan Mezo
 "On cluster expansions in graph theory and physics", N BIGGS  The Quarterly Journal of Mathematics, 1978  Oxford Univ Press
 Enumeration Of Rational Curves Via Torus Actions, Maxim Kontsevich, 1995
 Noncommutative Calculus and Discrete Physics, Louis H. Kauffman, February 1, 2008
 Sequential cavity method for computing free energy and surface pressure, David Gamarnik, Dmitriy Katz, July 9, 2008
Combinatorics and statistical physics
 "Graph Theory and Statistical Physics", J.W. Essam, Discrete Mathematics, 1, 83112 (1971).
 Combinatorics In Statistical Physics
 Hard Constraints and the Bethe Lattice: Adventures at the Interface of Combinatorics and Statistical Physics, Graham Brightwell, Peter Winkler
 Graphs, Morphisms, and Statistical Physics: DIMACS Workshop Graphs, Morphisms and Statistical Physics, March 1921, 2001, DIMACS Center, Jaroslav Nešetřil, Peter Winkler, AMS Bookstore, 2001, ISBN 0821835513
Conference proceedings
 Proc. of Combinatorics and Physics, Los Alamos, August 1998
 Physics and Combinatorics 1999: Proceedings of the Nagoya 1999 International Workshop, Anatol N. Kirillov, Akihiro Tsuchiya, Hiroshi Umemura, World Scientific, 2001, ISBN 9810245785
 Physics and combinatorics 2000: proceedings of the Nagoya 2000 International Workshop, Anatol N. Kirillov, Nadejda Liskova, World Scientific, 2001, ISBN 9810246420
 Asymptotic combinatorics with applications to mathematical physics: a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 920, 2001, Anatoliĭ, Moiseevich Vershik, Springer, 2002, ISBN 3540403124
 Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics, 10–15 July 2005, Dunk Island, Queensland, Australia
 Proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March 1923, 2007
Categories: Physics stubs
 Mathematics stubs
 Mathematical physics
 Quantum mechanics
 Combinatorics
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