Combinatorics and physics

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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] 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.

See also[edit]

References[edit]

  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[edit]

Combinatorics and statistical physics[edit]

Conference proceedings[edit]