From Wikipedia, the free encyclopedia
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
References
^
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
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