Integer-quaternion formulation of Lambek’s representation of fundamental particles and their interactions

Lambek’s unified classification of the elementary interaction-quanta of the “Standard model” is formulated in terms of the 24 units of the integerquaternion ring, i.e., the tetrahedral group Q24. An extension of Lambek’s scheme to the octahedral group Q48 may enable to take all three generations of leptons and quarks into account, as well as to provide a quantitative explanation for flavor-mixing. 1 Lambek’s quaternion assignment to fundamental particles and interactions In 2000, Joachim Lambek proposed a remarkable unified classification of elementary interaction quanta, i.e., particles, in which the four first-generation leptons and quarks (i.e., the electron and neutrino, and the up and down quarks) are treated on the same footing, together with their antiparticles and the gauge bosons responsible for their weak, electromagnetic, and strong interactions [1]. While the problem of classifying elementary particles is not new, and numerous studies have been published on the possibility of using the quaternion algebra to explain their properties, e.g., Refs. [2, 3, 4, 5, 6, 7], Lambek’s idea provides a unique classification scheme for the 24 fundamental quantum-number-changing quanta of the current “Standard model.”