Nonassociative Quantum Mechanics and Strong Correspondence Principle

Vladimir Dzhunushaliev

Джунушалиев Владимир Джумакадырович

It is shown that some operators in quantum mechanics have hidden structures that are unobservable in principle. These structures are based on a supersymmetric decomposition of the momentum operator, and a nonassociative decomposition of the spin operator.

Gary R. Gruber

Yu.I. Zaparovannyi

Jens Köplinger, Джунушалиев Владимир Джумакадырович

This paper follows recent steps towards a nonassociative quantum theory and points out the mathematical structure behind the proposed modifications to conventional quantum theory. An N=1 supersymmetry model and a strong force glueball ansatz is highlighted. Using nonassociative complex octonion algebra, it is shown how the Lorentz Lie algebra can be understood as a four dimensional generalization of the algebra of spin-1/2 operators in physics. Probability is speculated to become an emergent phenomenon from some nonassociative geometry in which to better understand the fluxes involved. A prototype nonassociative quantum theory in one dimension is brought forward to illustrate how normed division algebras may aid in modeling isospin properties that are similar to observed field and particle symmetries in nature. This prototype is built from a principle of self-duality between types of active and passive transformations and supplied with a modified Born rule that models observation, similar to conventional quantum mechanics. Solutions on the complex numbers, quaternions and octonions are discussed. The Hopf coquasigroup structure of the octonionic eigenvalue relation is shown and advertised as a tool for future investigation into the complete solution set of the model.

Джунушалиев Владимир Джумакадырович

It is shown that some operators in quantum mechanics have hidden structures that\nare unobservable in principle. These structures are based on a supersymmetric\ndecomposition of the momentum operator, and a nonassociative decomposition of\nthe spin operator.