Generalized Matrix Mechanics

Until at the end of 19th century, it was usually believed that any experimental results could be explained with classical mechanics (CM). The black body radiation phenomena crushed this belief, and the concept of energy quanta was introduced by Planck in 1900 to overcome the difficulty. Afterwards quantum mechanics (QM) has been applied to far broader areas of physics with indisputable success. When one witnesses the triumphs, it is natural to ask the following questions. 1. Why does QM describe a microscopic world so successfully? 2. Does QM hold true without limit? 3. If there are limitations, how is QM modified beyond it? Unfortunately we have no definite answers to them, although there are some conjectures. We expect that a generalization of CM and/or QM gives us a hint to answer the above questions. Hence it would be a meaningful task to construct a new mechanics based on CM and/or QM. Nambu proposed a generalization of Hamiltonian dynamics by the extention of phase space based on the Liouville theorem and made a suggestion on its quantization. 1) The structure of this mechanics has been studied in the framework of the constrained system 2) and in a geometric and algebraic formulation. 3) There are several works towards quantization of Nambu mechanics. 3), 4), 5), 6), 7), 8) This approach is quite interesting, but it is not a unique way to explore a new mechanics. There is a possibility to examine a generalization of QM directly, and we take a trial on this possibility. In this paper, we propose a generalization of Heisenbergs’ matrix mechanics based on many-index objects (we refer it as M-matrix).∗∗) It is shown that there exists a solution describing a harmonic oscillator and many-index objects lead to a generalization of spin algebra. A conjecture on the operator formalism is also given.