A Non-Associative Quantum Mechanics

E. Kapuścik

A critical examination of the original Heisenberg approach to quantum mechanics is presented. It is shown that, using the same kind of arguments as Heisenberg, it is possible to construct new non-associative algebras of quantum-mechanical observables. Two examples of such algebras are explicitly constructed.

Susumu Ôkubo

Elihu Lubkin, Thelma Lubkin

Susumu Ôkubo

Within an algebraic framework of non-associative quantum mechanics, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory as follows. The underlying non-associative operator algebra A is assumed to be flexible and Lie-admissible so that it is compatible with the Heisenberg equation of motion and quantization. A Hilbert space can be introduced into the theory as a faithful representation of the associated Lie algebra A/sup -/ of the Lie-admissible algebra A. However, if A/sup -/ is the standard Heisenberg algebra, and if the representation is irreducible, then it is shown that A must be associative, reproducing the standard associative quantum mechanics. Some discussions to circumvent this No-Go theorem are discussed. Especiallyyu if we are interested in the formulation of infinite component wave equations, then this difficulty can be avoided.

G. Cvijanovich

Natália Bebiano, João da Providência

In this article, we review the quantum mechanical setting associated with a non-self-adjoint Hamiltonian with a real spectrum. The spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non-self-adjoint operators with implications in quantum mechanics are discussed.