|
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.".〔(2007 International Conference on Combinatorial physics )〕 "Physical combinatorics might be defined naively as combinatorics guided by ideas or insights from physics".〔(Physical Combinatorics ), Masaki Kashiwara, Tetsuji Miwa, Springer, 2000, ISBN 0-8176-4175-0〕 Combinatorics has always played an important role in quantum field theory and statistical physics. However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer,〔A. Connes, D. Kreimer, (Renormalization in quantum field theory and the Riemann-Hilbert problem I ), Commun. Math. Phys. 210 (2000), 249-273〕 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,〔A. Connes, D. Kreimer, (Renormalization in quantum field theory and the Riemann-Hilbert problem II ), Commun. Math. Phys. 216 (2001), 215-241〕 the fact that the Slavnov–Taylor identities of gauge theories generate a Hopf ideal,〔W. D. van Suijlekom, (Renormalization of gauge fields: A Hopf algebra approach ), Commun. Math. Phys. 276 (2007), 773-798〕 the quantization of fields〔C. Brouder, B. Fauser, A. Frabetti, R. Oeckl, (Quantum field theory and Hopf algebra cohomology ), J. Phys. A: Math. Gen. 37 (2004), 5895-5927〕 and strings〔T. Asakawa, M. Mori, S. Watamura, (Hopf Algebra Symmetry and String Theory ), Prog. Theor. Phys. 120 (2008), 659-689〕 and a completely algebraic description of the combinatorics of quantum field theory.〔C. Brouder, (Quantum field theory meets Hopf algebra ), ''Mathematische Nachrichten'' 282 (2009), 1664-1690〕 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== *Mathematical physics *Statistical physics *Ising model *Percolation theory *Tutte polynomial *Partition function *Hopf algebra *Combinatorics and dynamical systems *Bit-string physics *Combinatorial hierarchy *Quantum mechanics 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Combinatorics and physics」の詳細全文を読む スポンサード リンク
|