Yang–Baxter equation

"It has been known for some time that the Yang-Baxter equations can be solved using s. More recently it was discovered … that the YBE for the N state could be solved using special curves of (N – 1)2."

"About 40 years ago, in the study of quantum s … , in particular in the framework of the … , new algebraic structures arose, the generalizations of which were later called quantum groups … The Yang-Baxter equations became a unifying basis of all these investigations. The most important nontrivial examples of quantum groups are quantizations (or deformations) of ordinary classical s and algebras (more precisely, one considers the deformations of the algebra of functions of a Lie group and the universal enveloping of a Lie algebra). The quantization is accompanied by the introduction of an additional parameter q (the deformation parameter), which plays a role analogous to the role of in quantum mechanics. In the limit q → 1, the quantum groups and algebras go over into the classical ones."

"At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the , commuting in statistical mechanics, and factorizable in field theory."

"The origins of es is one of the biggest mysteries in modern physics since they are beyond the realm of the Standard Model. As massive particles, neutrinos undergo throughout their propagation. In this paper we show that when a neutrino oscillates from a flavor state α to a flavor state β, it follows three possible paths consistent with the Quantum Yang- Baxter Equations. These trajectories define the transition probabilities of the oscillations. Moreover, we define a probability matrix for flavor transitions consistent with the Quantum Yang-Baxter Equations, and estimate the values of the three neutrino mass eigenvalues within the framework of the triangular formulation."