an:03979405
Zbl 0606.22018
Barut, A. O.; Bracken, A. J.
Compact quantum systems: Internal geometry of relativistic systems
EN
J. Math. Phys. 26, 2515-2519 (1985).
00149969
1985
j
22E70 22E60 17A70 81T60
kinematical algebra; wave equations; canonical commutation relations; finite quantum systems
A generalization is presented of the kinematical algebra \({\mathfrak so}(5)\), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac's electron. The algebra \({\mathfrak so}(n+2)\) 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 \(Q_ i\) and corresponding momentum operators \(p_ i\), occurring within the \({\mathfrak so}(n+2)\) algebra, go over into the usual canonical commutation relations.
The \({\mathfrak so}(n+2)\) algebra is contrasted with the \({\mathfrak sl}(1,n)\) superalgebra introduced recently by Palev in a similar context: because \({\mathfrak so}(n+2)\) 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.