Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
Overview "Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory."2007 International Conference on Combinatorial physics "Physical combinatorics might be defined naively as combinatorics guided by ideas or insights from physics"Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000,
Combinatorics has always played an important role in quantum field theory and statistical physics. However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer,A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I, Commun. Math. Phys. 210 (2000), 249-273 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 Riemann–Hilbert problem,A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II, Commun. Math. Phys. 216 (2001), 215-241 the fact that the Slavnov–Taylor identities of gauge theories generate a Hopf ideal,W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276 (2007), 773-798 the quantization of fieldsC. Brouder, B. Fauser, A. Frabetti, R. Oeckl, Quantum field theory and Hopf algebra cohomology, J. Phys. A: Math. Gen. 37 (2004), 5895-5927 and strings,T. Asakawa, M. Mori, S. Watamura, Hopf Algebra Symmetry and String Theory, Prog. Theor. Phys. 120 (2008), 659-689 and a completely algebraic description of the combinatorics of quantum field theory.C. Brouder, Quantum field theory meets Hopf algebra, Mathematische Nachrichten 282 (2009), 1664-1690 An important example of applying combinatorics to physics is the enumeration of alternating sign matrix in the solution of ice-type models. The corresponding ice-type model is the six vertex model with domain wall boundary conditions.
See also Mathematical physics Statistical physics Ising model Percolation theory Tutte polynomial Partition function Hopf algebra Combinatorics and dynamical systems Quantum mechanics
References
Further reading Some Open Problems in Combinatorial Physics, G. Duchamp, H. Cheballah One-parameter 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 Bit-String Physics: a Novel "Theory of Everything", H. Pierre Noyes Combinatorial Physics, Ted Bastin, Clive W. Kilmister, World Scientific, 1995, Physical Combinatorics and Quasiparticles, Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte
Paths, Crystals and Fermionic Formulae, G.Hatayama, A.Kuniba, M.Okado, T.Takagi, Z.Tsuboi On powers of Stirling matrices, István Mező "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 Non-commutative 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, 83-112 (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 19-21, 2001, DIMACS Center, Jaroslav Nešetřil, Peter Winkler, AMS Bookstore, 2001,
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, Physics and combinatorics 2000: proceedings of the Nagoya 2000 International Workshop, Anatol N. Kirillov, Nadejda Liskova, World Scientific, 2001, Asymptotic combinatorics with applications to mathematical physics: a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 9-20, 2001, Anatoliĭ, Moiseevich Vershik, Springer, 2002, 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 19–23, 2007