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.
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.