# Combinatorics and physics

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 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]. The important example of editing combinatorics and physics is relation between enumeration of Alternating sign matrix and Ice-type model. Corresponding ice-type model is six vertex model with domain wall boundary conditions.

## References

1. ^ 2007 International Conference on Combinatorial physics
2. ^ Physical Combinatorics, Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0817641750
3. ^ David Ruelle (1999). Statistical Mechanics, Rigorous Results. World Scientific. ISBN 978-9810238629.
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, Math. Nachr. 282 (2009), 1664-1690

• "On cluster expansions in graph theory and physics", N BIGGS - The Quarterly Journal of Mathematics, 1978 - Oxford Univ Press

### Conference proceedings

• Proc. of Combinatorics and Physics, Los Alamos, August 1998
• Proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March 19-23, 2007

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