Combinatorics and physics
Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
Overview[edit]
- "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[edit]
- Mathematical physics
- Statistical physics
- Ising model
- Percolation theory
- Tutte polynomial
- Partition function
- Hopf algebra
- Combinatorics and dynamical systems
- Quantum mechanics
References[edit]
- ^ 2007 International Conference on Combinatorial physics
- ^ Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0-8176-4175-0
- ^ David Ruelle (1999). Statistical Mechanics, Rigorous Results. World Scientific. ISBN 978-981-02-3862-9.
- ^ A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I, Commun. Math. Phys. 210 (2000), 249-273
- ^ A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II, Commun. Math. Phys. 216 (2001), 215-241
- ^ W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276 (2007), 773-798
- ^ C. Brouder, B. Fauser, A. Frabetti, R. Oeckl, Quantum field theory and Hopf algebra cohomology, J. Phys. A: Math. Gen. 37 (2004), 5895-5927
- ^ T. Asakawa, M. Mori, S. Watamura, Hopf Algebra Symmetry and String Theory, Prog. Theor. Phys. 120 (2008), 659-689
- ^ C. Brouder, Quantum field theory meets Hopf algebra, Mathematische Nachrichten 282 (2009), 1664-1690
Further reading[edit]
- 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[edit]
- "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[edit]
- 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