Combinatorics and physics

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."^[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 Riemann–Hilbert 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] 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

1. 2007 International Conference on Combinatorial physics
2. Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0-8176-4175-0
3. David Ruelle (1999). Statistical Mechanics, Rigorous Results. World Scientific. ISBN 978-981-02-3862-9.
4. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I, Commun. Math. Phys. 210 (2000), 249-273
5. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II, Commun. Math. Phys. 216 (2001), 215-241
6. W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276 (2007), 773-798
7. C. Brouder, B. Fauser, A. Frabetti, R. Oeckl, Quantum field theory and Hopf algebra cohomology, J. Phys. A: Math. Gen. 37 (2004), 5895-5927
8. T. Asakawa, M. Mori, S. Watamura, Hopf Algebra Symmetry and String Theory, Prog. Theor. Phys. 120 (2008), 659-689
9. C. Brouder, Quantum field theory meets Hopf algebra, Mathematische Nachrichten 282 (2009), 1664-1690

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, ISBN 981-02-2212-2
• Physical Combinatorics and Quasiparticles, Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte
• Fitzgerald, Hannah. "Physical Combinatorics of Non-Unitary Minimal Models" (PDF). CiteSeerX 10.1.1.46.4129. Archived from the original (PDF) on 4 March 2016. Retrieved 17 August 2014.
• 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, ISBN 0-8218-3551-3

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 981-02-4578-5
• Physics and combinatorics 2000: proceedings of the Nagoya 2000 International Workshop, Anatol N. Kirillov, Nadejda Liskova, World Scientific, 2001, ISBN 981-02-4642-0
• 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, ISBN 3-540-40312-4
• 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