Compact quantum systems: Internal geometry of relativistic systems

Abstract

A generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Qi and corresponding momentum operators Pi, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.